Dynamics of Machinery I - 1 File Download

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Dynamics of Machinery I

Mircea Radeş Universitatea Politehnica Bucureşti

2007

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Preface

This textbook is based on the first part of the Dynamics of Machinery lecture course given since 1993 to students of the English Stream in the Department of Engineering Sciences (D.E.S.), now F.I.L.S., at the University Politehnica of Bucharest. It grew in time from a postgraduate course taught in Romanian between 1985 and 1990 at the Strength of Materials Chair.

Dynamics of Machinery, as a stand alone subject, was first introduced in the curricula of mechanical engineering at D.E.S. in 1993. To sustain it, we published Dynamics of Machinery in 1995, followed by Dinamica sistemelor rotor-lagăre in 1996 and Rotating Machinery in 2003.

As seen from the Table of Contents, this book is application oriented and limited to what can be taught in an one-semester (28 hours) lecture course. It also contains many exercises to support the tutorial, where the students are guided to write simple finite element computer programs in Matlab, and to assist them in solving problems as homework.

The course aims to: (a) increase the knowledge of machinery vibrations; (b) further the understanding of dynamic phenomena in machines; (c) provide the necessary physical basis for the development of engineering solutions to machinery problems; and (d) make the students familiar with machine condition monitoring techniques and fault diagnosis.

As a course taught for non-native speakers, it has been considered useful to reproduce, as language patterns, some sentences from English texts.

Finite element modeling of rotor-bearing systems and hydrodynamic bearings are treated in the second part. Analysis of rolling element bearings, machine condition monitoring and fault diagnosis, balancing of rotors as well as elements of the dynamic analysis of reciprocating machines are presented in the third part. No reference is made to the vibration of discs, impellers and blades.

August 2007 Mircea Radeş

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Prefaţă

Lucrarea se bazează pe prima parte a cursului de Dinamica maşinilor predat din 1993 studenţilor Filierei Engleze a Facultăţii de Inginerie în Limbi Străine (F.I.L.S.) la Universitatea Politehnica Bucureşti. Conţinutul cursului s-a lărgit în timp, pornind de la un curs postuniversitar organizat între 1985 şi 1990 în cadrul Catedrei de Rezistenţa materialelor.

Dinamica maşinilor a fost introdusă în planul de învăţământ al F.I.L.S. în 1993. Pentru a susţine cursul, am publicat Dynamics of Machinery la U. P. B. în 1995, urmată de Dinamica sistemelor rotor-lagăre în 1996 şi Rotating Machinery în 2005, ultima conţinând materialul ilustrativ utilizat în cadrul cursului.

După cum reiese din Tabla de materii, cursul este orientat spre aplicaţii inginereşti, fiind limitat la ceea ce se poate preda în 28 ore. Materialul prezentat conţine multe exerciţii rezolvate care susţin seminarul, în cadrul căruia studenţii sunt îndrumaţi să scrie programe simple cu elemente finite în Matlab, fiind utile şi la rezolvarea temelor de casă.

Cursul are un loc bine definit în planul de învăţământ, urmărind: a) descrierea fenomenelor dinamice specifice maşinilor; b) modelarea sistemelor rotor-lagăre şi analiza acestora cu metoda elementelor finite; c) înarmarea studenţilor cu baza fizică necesară în rezolvarea problemelor de vibraţii ale maşinilor; şi d) familiarizarea cu metodele de supraveghere a stării maşinilor şi diagnosticare a defectelor.

Fiind un curs predat unor studenţi a căror limbă maternă nu este limba engleză, au fost reproduse unele expresii şi fraze din lucrări scrise de vorbitori nativi ai acestei limbi.

În partea a doua se prezintă modelarea cu elemente finite a sistemelor rotor-lagăre şi lagărele hidrodinamice. În partea a treia se tratează lagărele cu rulmenţi, echilibrarea rotorilor, măsurarea vibraţiilor pentru supravegherea funcţionării maşinilor şi diagnosticarea defectelor, precum şi elemente de dinamica maşinilor cu mecanism bielă-manivelă. Nu se tratează vibraţiile paletelor, discurilor paletate şi ale roţilor centrifugale.

August 2007 Mircea Radeş

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Contents

Preface i

Contents iii

1. Rotor-bearing systems 1 1.1 Evolution of rotating machinery 1 1.2 Rotor-bearing dynamics 22 1.3 Rotor precession 24 1.4 Modeling the rotor 26 1.5 Evolution of rotor design philosophy 29 1.6 Historical perspective 32

2. Simple rotors in rigid bearings 39 2.1 Simple rotor models 39

2.2 Symmetric undamped rotors 40

2.2.1 Equations of motion 41

2.2.2 Steady state response 43

2.3 Damped symmetric rotors 46

2.3.1 Effect of viscous external damping 47

2.3.2 Effect of viscous internal damping 54

2.3.3 Combined external and internal damping 62

2.3.4 Gravity loading 65

2.3.5 Effect of shaft bow 66

2.3.6 Rotor precession in rigid bearings 67

2.4 Undamped asymmetric rotors 68

2.4.1 Reference frames 69

2.4.2 Inertia torques on a spinning disc 69

2.4.3 Equations of motion for elastically supported discs 72

2.4.4 Natural frequencies of precession 75

2.4.5 Response to harmonic excitation 81

2.4.6 Campbell diagrams 87

2.4.7 Effect of gyroscopic torque on critical speeds 97

2.4.8 Remarks on the precession of asymmetric rotors 98

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MECHANICAL VIBRATIONS iv

3. Simple rotors in flexible bearings 101 3.1 Symmetric rotors in flexible bearings 101

3.1.1 Effect of bearing flexibility 102

3.1.2 Effect of external damping 109

3.1.3 Effect of external and internal damping 117

3.1 4 Effect of bearing damping 119

3.1.5 Combined effect of bearing damping and shaft mass 131 3.2 Symmetric rotors in fluid film bearings 136

3.2.1 Unbalance response 136

3.2.2 Stability of precession motion 142 3.3 Asymmetric rotors in flexible bearings 145

3.3.1 Equations of motion 145

3.3.2 Natural frequencies of precession 148

3.3.3 Unbalance response 152

3.3.4 Effect of bearing damping 156

3.3.5 Mixed modes of precession 158

3.4 Simulation examples 168

4. Rotor dynamic analysis 207 4.1 Undamped critical speeds 207

4.1.1 Effect of support flexibility 207

4.1.2 Critical speed map 209

4.1.3 Influence of stator inertia 217 4.2 Damped critical speeds 219

4.2.1 Linear bearing models 219

4.2.2 Equations of damped motion 220

4.2.3 Eigenvalue problem of damped rotor systems 220

4.2.4 Campbell diagrams 222

4.2.5 Orbits and precession mode shapes 223 4.3 Peak response critical speeds 224 4.4 Stability analysis 227 4.5 Simulation examples 231 4.6 Planar modes of precession 273 Index 283

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1. ROTOR-BEARING SYSTEMS

The first part of the Dynamics of Machinery is devoted to rotor-bearing systems, including the effects of seals and bearing supports. The flexibilities of discs and blades are neglected, so that the Rotor Bearing Dynamics does not include the vibration analysis of impellers and bladed-disc assemblies.

1.1. Evolution of rotating machinery

Interest in the vibration of rotating machinery has been due primarily to the fact that more than 80 percent of the problems involve vibration. In the continuing effort to develop more power per kilogram of metal in a machine, designs have approached the physical limits of materials and vibration problems have increased. These, together with the extremely high cost associated with forced outages, for machines with continuous operating regime, have determined the development of research activity and design procedures in two fields of primary practical interest: the Dynamics of Rotor-Bearing Systems and the Vibrations of Bladed Disc Assemblies.

1.1.1 Steam turbines

Of significance for the technical advancement in this field is the development of steam turbines in Europe [1]. From the first single stage impulse turbine built in 1883 by the Swedish engineer Gustaf de Laval (with a speed of 30000 rpm reduced to 3000 rpm by gearing), and the first multistage reaction turbine built in 1884 by Charles Parsons (having a speed of 18000 rpm and an output of 10 HP), to the turbines of today nuclear power stations, the evolution has been spectacular.

Early in 1901 the Brown Boveri Company built a steam turbine of 250 kW at 3000 rpm, coupled directly to an a.c. generator. From 1907 onwards, a double impulse Curtis wheel (invented in 1896) was mounted before the reaction

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DYNAMICS OF MACHINERY 2

stage, which was replaced by single-row versions on two to three impulse wheels. In 1914, a turbine of 25 MW at 1000 rpm was the largest single-cylinder steam turbine in the world. The first systematic studies of Rotor Dynamics started in 1916, carried out by professor Aurel Stodola at the Swiss Federal Institute of Tehnology in Zürich.

After 1920, the high price of coal imposed the increase of steam turbine efficiency. Among other means, this was achieved by the reduction in the diameter and the increase in the number of stages, hence by the increase of the shaft length, a major incentive for developing the Dynamics of Rotor-Bearing Systems.

The maximum unit output of a turbine is largely dependent on the available last-stage blade length. The permissible blade length to diameter ratio has an influence on the machine efficiency. Shafts should be as slender as possible, to ensure small rotor diameter and large blade length. Otherwise, increased shaft weight gives rise to an increase in the average specific bearing loading.

Increasing the cross-section of a machine is limited by the mechanical stresses and the size of pieces that can be transported. This is compensated by the increase of the active length, eventually with a tandem arrangement, having a long shaft line, in which the mechanical power is produced in several turbine cylinders.

The first super-pressure three-cylinder (high, intermediate and low pressure) turbine was built by BBC in 1929, and had an output of 36 MW at 3000 rpm. The steam flowed through high pressure and intermediary pressure rotors in opposite directions, to balance the thrust. Rotors, which previously were composed of keyed and shrunk-on wheels on a continuous shaft, started to be welded from solid discs, allowing larger rotor diameters and increased ratings. The increased efficiency of steam turbines lowered the amount of coal required for producing 1 kWh of electrical energy from 0.75 kg during the war to 0.45 kg in 1927. The output of the largest turbines in Europe had reached 50 to 60 MW by the mid twenties, when, for large units, turbines of 1500 rpm were coupled to four-pole generators. A 165 MW two-shaft turboset was built in 1926-1928, with the high-pressure shaft rotating at 1800 rpm, and the low-pressure shaft at 1200 rpm.

In 1948, the largest steam turboset of single-shaft design (Fig. 1.1) had four cylinders, a length of 27 m (without the station service generator), an output of 110 MW and speed of 3000 rpm [2]. In 1950, turbosets of 125 MW were built in Europe and of 230 MW in the U.S.A., then, in 1956 - with ratings of 175 MW, and in 1964 - with ratings of 550 MW and two shafts.

In 1972, the first 1300 MW cross-compound turboset was built at 3600 rpm, provided with two shaft lines for two 722 MVA generators. Figure 1.2 shows a longitudinal section of the high-pressure turbine of a 1300 MW unit at 1800 rpm.

Current designs have generators of 1635 MVA at 1500 rpm, and of 1447 MVA at 3000 rpm. At present time, turbosets of 1700-2000 MW at 1500 or 1800 rpm, and of 1500-1700 MW at 3000 or 3600 rpm are currently built.

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1. ROTOR-BEARING SYSTEMS 3

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DYNAMICS OF MACHINERY 4

Generally, the shaft line has a length of 8 to m20 in turbosets of 1 to 50 MW, between 25 and m30 in those of 100 to 150 MW, and exceeds m75 in turbosets beyond 1000 MW.

Fig. 1.2 (from [3])

The increase of the rotor length has been accompanied by the increase of the number of stages (or discs on a shaft), and the number of bearings and couplings between shafts in a line. Adding the increase of seal complexity and the problems raised by the non-uniform thermal expansion at start-up, all doubled by strength of materials problems raised by the increase in size, one can easily understand the complexity of the dynamic calculations of the rotors of such machines.

Figure 1.3 shows a typical axial section in an industrial back-pressure turbine of an early design [4]. The steam is expanded in the turbine from the live-steam pressure to the exhaust pressure in two principal parts.

In the first part, the steam is accelerated in the nozzle segments 1, thus gaining kinetic energy, which is utilized in the blades of the impulse wheel 2. The disc of the impulse stage is integral with the shaft. Usually, the nozzles are machined into several segments fixed into the cylinder by a cover ring. The blades of the impulse wheel are milled from chromium steel bars. The roots are fixed into the slot in the impulse wheel with spacers gripping the upset feet of the blades. In some designs, the flat outer ends are welded together in groups, thus forming an interrupted shroud.

The second or reaction part consists of stationary and moving rows of blades 3 fixed with suitably shaped spacers into slots in the casing and rotor.

The glands 4 prevent the steam flowing out of the casing along the shaft. Labyrinth seals allow a very small amount of steam to escape into specially

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1. ROTOR-BEARING SYSTEMS 5

provided channels. Due to the turbulence of the steam, the pressure drop is sufficiently high to allow the gland to be made relatively short. The labyrinth strips are caulked into grooves in the rotor shaft whereas the corresponding grooves are machined into a separate bushing of the casing. The risk of damaging the rotor by distortion caused by friction in the seals is avoided, as the heat transfer from the tips of the thin labyrinth strips to the shaft is very small.

Fig. 1.3 (from [4])

The balancing piston 5 is positioned between the impulse wheel and the gland at the steam inlet end. The chamber between is interconnected with the exhaust. Generally, the balancing ring is integral with the shaft. In older designs it was shrunk-on but this design can give rise to instability due to rotating dry friction. This arrangement counteracts the axial forces imposed on the rotor by the steam flow.

The bearing 6 at the steam inlet end is a combined thrust and journal bearing, to reduce the rotor length. The thrust part of it acts in both axial directions on the thrust collars 7 to absorb any excess forces of the balancing piston. Usually tilting bronze pads are fitted on flexible steel rings according to the Mitchell principle.

The journal bearing of the combined bearing and that at the opposite end 8 are lined with white metal cast into separate shells. Tilting pad bearings are used in some designs.

The rotor 9 is machined from high-quality steel forging. After the blades are fitted, the rotor is balanced and subjected to a 20 percent overspeed test for a few minutes. A high-alloy chromium steel is used for high pressures and temperatures. Figure 1.4 shows presently used steam turbine rotor designs [5].

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DYNAMICS OF MACHINERY 6

Turbines running at high speeds require reduction gearing to drive alternators with 2 or 4 poles, running at 3000 or 1500 rpm (for 50 Hz).

As a rule, the pinion and gear wheel shafts are connected to the driving and driven machines by means of couplings. They must be able to compensate for small errors in alignment and thermal expansion in the machine without affecting the reduction gearing. The coupling hubs are integral with the forged shafts.

Fig. 1.4 (from [5])

The first steam turbine built in Romania in 1953 at Reşiţa, was a 3 MW at 3000 rpm turbine. In 1967, the first two-cylinder 50 MW turbine was built. Twenty years later, the 330 MW four-cylinder condensing turbine was manufactured at I.M.G. Bucureşti, under a Rateau-Schneider license. Rotors have a monoblock construction, having the discs in common with the shaft. At present, General Turbo S.A. manufactures 700 MW turbines.

1.1.2 Gas turbines

The development of gas turbines is more recent. From the first gas turbine for airplanes, designed by Whittle in 1937, and the first stationary turbine built by Brown Boveri in 1939, turbines of 80 MW at 3000 rpm and 72 MW at 3600 rpm are found in power plants, while 16 MW turbines are working with blast-furnace gases. The progress is mainly due to blade cooling and limitation of the effects of corrosion and erosion. State-of-the-art gas turbines built by ABB have 265 MW at 3000 rpm and 183 MW at 3600 rpm.

The simplest type of open circuit stationary gas turbine installation comprises a compressor, a combustion chamber, and a gas turbine. In the

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1. ROTOR-BEARING SYSTEMS 7

arrangement from Fig. 1.5, the compressor and turbine rotors form a single shaft line, while the generator 7 is coupled via a clutch 6. The starter 9 is used to launch the generator when operating as a compensator. The starter 5 is used to launch the turbine while the generator turns. Part of the compressed air is used for the fuel combustion. The remainder (approx. 70%) is used for cooling the shell of the combustion chamber and some components of the turbine, and is mixed with the hot gases.

Fig. 1.5 (from [6])

The volume of the expanded gas in the turbine is much larger than the volume of the compressed air in the compressor, due to the heating in the combustion chamber. The difference between the work produced by the turbine and the work absorbed by compressor and friction losses is the work supplied to the electrical generator. It is a function of the compressor and turbine thermodynamic efficiencies and the turbine inlet temperature.

Fig. 1.6 (from Power, Jan 1980, p.27)

A design with concentric shafts, resembling the aircraft gas turbines, is shown in Fig. 1.6.

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DYNAMICS OF MACHINERY 8

Figure 1.7 shows the Rolls-Royce RB.211 turbofan rotors. The three-stage low pressure (LP) turbine drives the single-stage LP fan which has no inlet guide vanes. The single-stage intermediate pressure (IP) turbine drives the seven-stage IP compressor. The single-stage air-cooled high pressure (HP) turbine drives the six-stage HP compressor.

Fig. 1.7 (adapted from [7])

The eight main bearings are located in four rigid panels (not shown). The three thrust ball bearings are grouped in a stiff intermediate casing. Oil squeeze-film damping is provided between each roller bearing and housing to reduce engine vibration. The short HP system needs only two bearings located away from the combustion zone for longer life.

The single-stage LP fan has 33 blades with mid-span clappers and fir-tree roots. The seven-stage IP axial compressor has drum construction. It consists of seven discs electron beam welded into two drums of five and two stages bolted together between stages 5 and 6. The blade retention is by dovetail roots and lockplates. The six-stage HP compressor consists of two electron beam welded drums bolted through the stage 3 disc with blades retained by dovetail roots and lockplates.

The three-shaft concept has two basic advantages: simplicity and rigidity. Each compressor runs at its optimum speed, thus permitting a higher pressure ratio per stage. This results in fewer stages and fewer parts, to attain the pressure ratio, than in the case of alternative designs. The short, large diameter shafts give good vibration characteristics and a very smooth engine. The short carcase and the positioning of the engine mounting points give a very rigid structure. This allows the rotors to run with smaller tip clearances and thus improved efficiency.

Gas turbines manufactured in Romania are: 1) the Viper 632-41, Rolls-Royce license, 8-stage axial compressor and 2-stage turbine at 13,800 rpm; 2) the Alouette III B, Turbomeca license, 422 kW, 33,480 rpm; and 3) the Turmo IV CA, Turbomeca license, 1115 kW.

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1. ROTOR-BEARING SYSTEMS 9

1.1.3 Axial compressors

Although patents for axial compressors were taken out as long ago as 1884, it is only in the early 1950's that they become the most versatile form for gas-turbine work. In the aircraft field, where high performance is at a premium, the axial compressor is now used exclusively. It is only for some industrial applications that other compressor types offer serious competition.

Fig. 1.8 (from [8])

The axial-flow compressor resembles the axial-flow steam or gas turbine in general appearance. Usually multistage, one observes rows of blades on a single shaft with blade length varying monotonically as the shaft is traversed. The difference is, of course, that the blades are shorter at the outlet end of the compressor, whereas the turbine receives gas or vapour on short blades and exhausts it from long blades.

In Fig. 1.8 the numbers have the following designations: 1 and 13 - bearings, 2 - seals, 3 - prewhirler, 4 - intake duct, 5 - rotor blades, 6 - stator blades, 7 - straightener stator blades, 8 - discharge duct, 9 - diffuser, 10 - coupling, 11 - gas turbine shaft, 12 - drum-type rotor, 14 - stator casing.

In practically all existing axial compressor designs, the rotor is supported by one bearing at the gas inlet end and by a second bearing at the gas delivery end. In aircraft practice, ball and roller bearings are universally used, on account of their

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compactness, small lubricating oil requirements, and insensitivity to momentarily cessations of oil flow as may occur during acrobatic flying.

1.1.4 Centrifugal compressors

Although centrifugal compressors are slightly less efficient than axial-flow compressors, they are easier to manufacture and are thus preferred in applications where simplicity, ruggedness, and cheapness are primary requirements. Additionally, a single stage of a centrifugal compressor can produce a pressure ratio of 5 times that of a single stage of an axial-flow compressor. Thus, centrifugal compressors find application in power station plants, petrochemical industry, gas injection and liquefaction, ground-vehicle turbochargers, locomotives, ships, auxiliary power units, etc.

Fig. 1.9 (from [9])

A typical high-pressure compressor design is shown schematically in Fig. 1.9. Apart from shaft, impellers, bearings and coupling, modeled as for other machines, items of major concern in rotor dynamic analyses are the gas labyrinths, the oil ring seals and the aerodynamic cross coupling at impellers. Furthermore, squeeze film dampers are used to stabilize compressors with problems.

Multistage centrifugal compressors have relatively slender shafts. Usually, impellers are mounted on almost half of the rotor length, the other part being necessary for the centre seal, the balance drum, the oil seals, the radial bearings and the thrust bearing. The shaft diameter is kept small to increase the impeller eye. In comparison with the drum rotor of axial compressors, the shaft of centrifugal compressors is more flexible, having relatively low natural frequencies which favour instabilities.

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1. ROTOR-BEARING SYSTEMS 11

Vibrations of a centrifugal compressor are controlled by: bearings, shaft geometry, gas seals and oil bushings, fluid forces on impellers, and other factors. Squeeze film dampers are used in centrifugal compressors to eliminate instabilities or to alter the speed at which they occur.

In the case of centrifugal compressors, undamped critical speed maps are of little interest. For typical compressor precession modes which are heavily damped, second mode in particular, the damped natural frequency can be as much as 2 to 9 times lower than the expected peak response speed.

Shop testing, carried out after compressor is constructed but before it is commissioned, can reveal problems prior to start-up. Bode plots, obtained during run-up measurements, are used to check that the critical speeds are not within the operating speed range. Separation margins of the critical speeds from the intended operating speed range are defined in API Standard 617; resonances must be 20 percent above the maximum continuous speed and/or 15 percent below the operating speeds [10]. Compliance with present specifications requires calculation of deflections at each seal along the rotor, as a percentage of the total clearance.

Modern multistage compressors are typically designed to operate through and above several critical speeds so as to maximize the work done by a given size machine. For example, a 425 mm diameter impeller for an industrial centrifugal compressor can be designed for a work load well in excess of 2000 HP by running at speeds approaching 9000 rpm. Up to eight stages are used to obtain the required pressure rise. Process compressors and units used for natural gas injection can have discharge pressures of the order of 650 bar and can drive gases with high density. The result of this combination of supercritical speed, high pressure and high work load has been an increasing tendency for such machines to exhibit problems of nonsynchronous rotor whirling. This is why stability analysis is of prime interest.

While many rotating machines operate below the first critical speed (point A in Fig. 1.10), turbomachinery operate above the first critical speed (point B). Until mid seventies any further shift of the resonance - and hence any increase in the maximum number of stages per casing - was precluded by the bearing stability limit. This was then raised by means of stronger bearing designs until operation above the second critical speed became possible (point C).

High pressure compressors operating on fixed lobe bearings could generate a violent shaft whip condition just above twice the first natural frequency. By going to tilting-pad bearings, that threshold speed can be raised to well over two times the first natural frequency. Attempts to raise speed further came up against another stability limit: rotor instability due to gap excitation. Using vortex brakes before labyrinths this boundary has been pushed back and the way is open in principle to still higher speed ratios (point D).

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Exhaust-gas turbocharging is used to increase the mean effective pressure (m.e.p.) of diesel engines. It has applications in stationary plants for electricity generation, in ships' auxiliary and propulsion machinery and in railway traction.

Fig. 1.10 (from [11])

One of the oldest applications was in marine engines. In 1923, BBC and the Vulkan shipyard manufactured turbochargers for the 10-cylinder four-stroke engines from the vessels 'Preussen' and 'Hansestadt Danzig'. The engines, which were designed for an uncharged performance of 1700 HP each at 235 rpm provided, when charged, a cruising power of 2400 HP at 275 rpm and a temporary overload of 4025 HP at 320 rpm (for a m.e.p. = 8.4). Turbocharging of two-stroke marine engines began after 1950.

For the relatively short turbocharger rotors, which are almost always equipped with single-stage compressor and turbine wheels, two bearings are sufficient. One of these is a combined radial-axial bearing, the other a pure radial bearing. Two bearing layouts have proved successful on the market: 1) bearings at the shaft ends (external bearings), used predominantly in large machines, and 2) bearings between the compressor and turbine wheel (internal bearings) used mainly for small turbochargers. In both arrangements the axial bearing is located near the compressor wheel, to keep the axial clearance in that region small.

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1. ROTOR-BEARING SYSTEMS 13

In the variant with external bearings (Fig. 1.11, a), the large distance between the bearings reduces the radial bearing forces and requires smaller clearances at the compressor wheel and turbine wheel. The frictional losses in the bearings are smaller, particularly at part load. The shaft ends can be kept small in diameter and are simple to equip with a lubricating oil pump and centrifuge, thus rendering rolling-contact bearings and self-lubrication possible.

Fig. 1.11

Internal bearings (Fig. 1.11, b) offer advantages in fitting a turbocharger with axial air and gas inlets to the engine. Small turbochargers do not, however, have an axial-flow turbine, but a radial-flow turbine with axial gas outlet. For specific applications internal bearings have advantages, which relate mainly to the wider variety of ways of fitting the turbocharger to the engine.

In automotive applications, a floating bush bearing is used due to size and cost considerations. This type of bearing has a thin bush rotating freely between the journal and the fixed bush, forming two hydrodynamic oil films [12]. This turbocharger shows peculiar behaviour yet to be explained theoretically: 1) it has stable operation at very high shaft speeds, though at lower speeds it can exhibit instability in either a conical mode or an in-phase bending mode; and 2) some designs have a third flexible critical speed, very difficult to balance out; with a high amplification factor, leading to rubbing and bearing distress.

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1.1.5 Fans and blowers

Fans can be either radial-flow or axial-flow machines. The ratio discharge pressure vs. suction pressure is defined as the pressure ratio. Fans are designed for pressure ratios lower than or equal to 1.1. Centrifugal fans absorb powers between 0.05 kW and 1 MW, have flow rates up to 3·105 m3/h and discharge pressures up to 1000 mm H2O (~104 N/m2). Blowers are single-stage uncooled compressors with pressure ratios between 1.1 and 4, and discharge pressures up to 3.5·105 N/m2. Compressors have pressure ratios larger than 4, so they usually require interstage cooling.

Fig. 1.12 (from [13])

Fig. 1.13 (from [13])

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1. ROTOR-BEARING SYSTEMS 15

The design from Fig. 1.12 is a medium-pressure blower, with labyrinth seals, and overhung design.

The arrangement from Fig. 1.13 is with double suction and single exhaust. The symmetrical rotor has a disc at the middle.

Centrifugal fans used for forced- or induced-draft and primary-air service generally have large diameter rotors, operating from 500 to 900 rpm in pillow-block bearings, supported on structural steel or concrete foundations.

As a rule, the major problem with fans is unbalance caused by 1) uneven buildup or loss of deposited material; and 2) misalignment. Both are characterized by changes in vibration at or near the rotational frequency.

1.1.6 Centrifugal pumps

Centrifugal pumps are used in services involving boiler feed, water injection, reactor charge, etc. Instability problems encountered in the space shuttle hydrogen fuel turbopumps and safety requirements of nuclear main coolant pumps have prompted research interest in annular seals.

It is now recognized that turbulent flow annular seals in multi-stage pumps and in straddle-mounted single-stage pumps have a dramatic effect on the dynamics of the machine. Stiffness and damping properties provided by seals represent the dominant forces exerted on pump shafts, excluding the fluid forces of flow through the impellers, particularly at part-flow operating conditions. For these systems, the hydrodynamics of oil-lubricated journal bearings is dominated by seal properties.

Typical multi-stage centrifugal pumps have more inter-stage fluid annuli than they have journal bearings. The fluid annuli are distributed between the journal bearings where precession amplitudes are highest and can therefore be 'exercised' more as dampers than can be the bearings.

In typical applications, shaft resonant critical speeds are rarely observed at centrifugal pumps because of the high damping capability afforded by seals. Problems encountered with boiler feed pumps have been produced by excessive wear in seals, yielding a decrease in the dynamic forces exerted by the seals.

Centrifugal pumps have comparatively slender shafts and relatively flexible cantilevered bearing housings (Fig. 1.14).

Fine clearance annular seals are used in pumps primarily to prevent leakage between regions of different pressure within the pump. The rotordynamic behaviour of pumps is critically dependent on forces developed by annular seals, between the impeller shroud and the stator, between the impeller back disc and the stator, and between the impeller and diffuser.

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Fig. 1.14 (from [14])

1.1.7 Hydraulic turbines

Hydraulic turbines have traditionally been used to convert hydraulic energy into electricity. The first effective radial inward flow reaction turbine was developed around 1850 by Francis, in Lowell, Massachusetts. Around 1880 Pelton invented the split bucket with a central edge for impulse turbines. The modern Pelton turbine with a double elliptic bucket, a notch for the jet and a needle control for the nozzle was first used around 1900.

The axial flow turbine, with adjustable runner blades, was developed by Kaplan in Austria, between 1910-1924. The horizontal bulb turbines have a relatively straighter flow path through the intake and draft tube, with lower friction losses. In the Straflo (straight flow) design, the turbine and generator form an integral unit without a driving shaft.

With hydraulic turbines, despite the low rotating speeds (200-1800 rpm), problems occur owing to the vertical position of most machines, due to transients and cavitation. Rotors are very robust and stiff, problems being raised by bearings and the supporting structure.

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1. ROTOR-BEARING SYSTEMS 17

Fig. 1.15

The hydro power plant at Grand Coulée (U.S.A.) has a 960.000 hp Francis turbine driving a synchronous generator of 718 MVA at 85.7 rpm. The rotor has a diameter in excess of 9 m and a weight exceeding 400 tons, the main shaft having 3.3 m diameter and more than 12 m length.

The world’s largest hydroelectric plant Itaipu, on the Rio Paraná, which forms the border between Brazil and Paraguay, near the city of Foz do Iguaçu, consists of 18 generating sets of 824/737 MVA, driven by Francis turbines, with a total rating of 12,600 MW. Turbines have rotors of 300 tons and 8 m diameter, the main shaft has 150 tons and 2.5 m diameter, while the synchronous generator has 2000 tons and 16 m diameter, running at respectively 90.9 rpm for 50 Hz generators, and 92.3 rpm for 60 Hz generators (Fig. 1.15).

The hydro power plant at Ilha Solteira, Brazil, has sets of 160 MW at 85.8 rpm. The rotor shaft has 6.33 m length, 1.4 m outer diameter and 0.4 m inner diameter. The generator has 495 tons and the Francis turbine has 145 tons. The first critical speed is about 222 rpm.

The hydroelectric power plant at Corbeni-Argeş has four Francis turbines with nominal speed 428.6 rpm, gross head 250 m, nominal water flow 20 m3/s and individual rated power 50 MW.

An axial cross-section of a vertical axis Kaplan turbine is presented in Fig. 1.16 where 1 – runner with adjustable blades, 2 – draft tube, 3 – guide vanes, 4 – lower guide bearing, 5 – stay vanes and ring support, 6 – concrete spiral casing, 7 – control ring with servo-motor for the stay vanes, 8 – thrust bearing, 9 – upper guide

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DYNAMICS OF MACHINERY 18

bearing, 10 – servo-motor for adjustment of runner blades, 11 – runner blades control rod inside the turbine shaft, and 12 – generator.

Fig. 1.16 (from [15])

The Porţile de Fier I hydroelectric power plant has eight Kaplan turbines of 194 MW, head 27 m, nominal water flow 840 m3/s, speed 71.43 rpm, 6 blades and rotor diameter 9.5 m.

The Porţile de Fier II hydroelectric power plant has eight double-regulated bulb units type KOT 28-7.45, with the bulb upstream and the turbine overhung downstream. The unit has three guide bearings and a thrust bearing, 16 stator blades and 4 rotor blades, and the following parameters: head 7.45 m, nominal water flow 432 m3/s, rated power 27 MW, rotor diameter 7.5 m.

1.1.8 Turbo-generators

The turbo-alternator was developed by C. E. L. Brown and first marketed by Brown Boveri in 1901. With a cylindrical rotor having embedded windings, it has proven to be the only possible design for high speeds, as when driven direct by a steam turbine. Such alternators are available for ratings between 500 kVA and

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1. ROTOR-BEARING SYSTEMS 19

20,000 kVA and higher, but are not normally used below 2500 kW, because salient-pole machines with end-shield bearings are more economical. Beyond 2500 kW, an alternator running at 3000 (or 3600) rpm permits a more economical gear to be used than a 1500 (or 1800) rpm alternator for the same turbine [16].

The marked increase in the unit ratings of turbo-generators has not, for the most part, been accompanied by a corresponding increase in the size of machines because of the increase in the specific electric ratings. For example, between 1940-1975, the maximum power of electric generators increased from 100 to 1600 MVA, whereas in 1940 a 3000 rpm turbo-generator weighed 2 kg per kW of output, and its 1975 counterpart weighed only 0.5 kg/kW.

Alternator rotors have been also designed to be progressively longer and more flexible. The forging of a 120 MW rotor had approximately 30 tons and 8 m distance between bearing centres, while a 500 MW rotor had 70 tons and 12 m. Modern rotors have two or three critical speeds below their operating speed of 3000 rpm.

Fig. 1.17 (from [16])

The rotor of small units is a solid cylindrical forging of high-quality steel with slots milled in it to accommodate the field winding. For larger units, several hollow cylinders are fitted over a central draw-bolt threaded at both ends, to which the two shaft extensions are fastened by shrinking. The specially formed winding is a single layer of copper strip insulated with glass-fibre which is pressed and baked into the slots. To secure the end sections, end-bells forged from solid-drawn non-magnetic steel with ventilation holes or slots are used.

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DYNAMICS OF MACHINERY 20

Rotors of electrical machines are different from rotors with bladed discs or impellers, being more massive, but occasionally rising problems due to asymmetrical stiffness properties.

Figure 1.17 is a cutaway perspective drawing of a 400 MVA, 3000 rpm generator with water-cooled stator winding and forced hydrogen direct cooling in the rotor. Due to the high flux density and current loadings, generators of over 500 MW employing these cooling methods must have their stator cores mounted in a flexible suspension. This is necessary in order to isolate the foundations from the enormous magnetic vibration forces arising between rotor and stator.

Two-pole generator rotors have axial slots machined to match more closely the principal stiffnesses. They are intended to reduce the parametric vibrations induced by the variation of the cross-section second moment of area about the horizontal axis, during rotation.

The second order (or 'twice per revolution') forced vibration which arises from the dual flexural rigidity is virtually inescapable in a two-pole machine; where the motion is excited by the weight of the rotor. This is a source of considerable difficulty, largely because it can be cured only at the design stage and cannot be 'balanced'. Certain 'trimming' modifications can be made but these present problems of their own. In fact it would be very difficult to design accurately an alternator rotor so as to have axial symmetry in a dynamical sense. The rotor is, in effect, a large rotating electromagnet, having a north pole and a south pole on opposite sides of the rotor and having slots cut in it, in which copper conductors are embedded to provide the magnetic field.

Fig. 1.18 (from [17])

The cross-section of a 120 MW alternator rotor after slotting is shown in Fig. 1.18, a. It is clear from the figure that the flexural rigidity of the shaft is

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1. ROTOR-BEARING SYSTEMS 21

unlikely to be the same for bending about the horizontal and the vertical neutral axes, even after copper conductors and steel wedges have been placed in the slots.

In attempts to equalize these rigidities, one of two schemes is usually adopted. In the first, the pole faces are slotted as shown in Fig. 1.18, b. In order to maintain the magnetic flux density, the slots in the pole faces are filled with steel bars that are wedged in. The second technique is to build a rotor in the manner of Fig. 1.18, a and then to cut lateral slots across the poles at intervals along the length of the rotor.

Figure 1.19 shows the different cross-sections in a turbo-generator rotor: A-A rectangular slots for field winding and smaller slots in the pole area, and B-B cross-cuts to ensure uniform flexibility with respect to the vertical and horizontal cross-section principal axes.

Fig. 1.19 (from [18])

Alternator rotors are supported in plain bearings. These hydrodynamic bearings present unequal dynamical stiffnesses in the vertical and horizontal directions. Asymmetry of the bearings introduces a split of critical speeds but cannot by itself cause second order vibration.

For small machines, e.g. electrical motors, having relatively low rotational speeds and rolling-ball bearings, the balancing and the dynamic calculation of the rotor does not generally raise problems. On the contrary, large machines, having long and flexible rotors, sliding bearings, seals, pedestals and relatively flexible casings, with high speeds, have determined the continuous development and improvement of dynamic calculations and vibration measurement.

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1.2 Rotor-bearing dynamics

Rotor-Bearing Dynamics has got its own status, apart from Mechanical Vibrations and Structural Dynamics, becoming an interdisciplinary research field, as soon as the importance of the effects of bearings and seals on the rotor dynamic response has been recognized.

The scope of Rotor-Bearing Dynamics is the study of the interaction between rotor, stator and the working fluid, for the design, construction and operation of smooth-running machines in which allowable vibration and dynamic stress levels are not overpassed, within the whole operating range.

Smooth machine operation is characterized by small, stable rotor precession orbits, and by the absence of any instability throughout the machine operating range.

In order to understand the dynamic response of a rotating machine it is necessary to have, early in the design stage, information on the following aspects of its behavior:

1. Lateral critical speeds of the rotor-bearing-pedestal-foundation system; effects of the stiffness and damping of bearings, seals, supporting structure and foundation on the location of critical speeds within the machine operating range.

2. Unbalance response: orbits of the rotor precession as a response to different unbalance distributions, throughout the whole operating range of the machine, and vibration amplitudes due to rotor unbalance.

3. Rotor speed at onset of instability: the threshold speed for unstable whirling due to the rotor/bearing and/or working fluid interaction, as well as the consequences of its crossing.

4. Time transient response analysis, to a blade loss, mainly for gas turbine engines operating at supercritical speeds, or when passing through a critical speed.

5. System torsional critical speeds, especially at geared rotors, eventually the transient response of the shaft line to electric disturbances applied to the generator.

Practical measures regarding the balancing and the monitoring of the dynamic state of rotors are added to these:

6. Balancing of rotors: calculation and attachment (removal) of correction masses such that the centrifugal forces on the rotor due to these additional masses and the inherent unbalance forces are in equilibrium.

7. Machinery monitoring: measurement of the parameters characterizing the dynamic state of machines and trending their time evolution, in order to detect any damage, to anticipate serious faults, determining the outage.

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1. ROTOR-BEARING SYSTEMS 23

The capability of predicting the performances of a rotor-bearing system is dependent firstly on the information about bearing properties, fluid-rotor interaction and the unbalance distribution along the rotor. In this respect, in recent years, important progress has been achieved in determining the dynamic coefficients of bearings and seals, and in the identification of the spatial distribution of unbalance for flexible rotors. The direct result is the development of computer programs helping in modeling most of the dynamic phenomena occurring during the operation of rotating machinery.

Generally, the following dynamic characteristics of rotating machinery are of interest:

a. Rotor lateral critical speeds in the operating range.

b. Unbalance response amplitudes at critical speeds.

c. Threshold speed of instabilities produced by bearings, seals or other fluid-structure interactions.

d. Bearing transmitted forces.

e. The overshoot ratio, of maximum transient response relative to the steady-state response.

f. System torsional critical speeds.

g. Gear dynamic loads.

h. Vibration amplitudes in casing and supporting structure.

The following can be added to this list:

i. Natural frequencies of bladed discs, impellers, wheels.

j. Frequencies and mode shapes of blades and blade buckets.

k. Blade flutter frequencies.

l. Rotating stall and surge thresholds.

m. Noise radiated by rotating machinery.

In the following, only the first three issues are treated. Problems not treated in this book are:

a. Shafts with dissimilar principal moments of inertia;

b. Cracked rotors;

c. Reverse precession due to dry-friction contact between rotor and stator;

d. Partial rubbing conditions;

e. Transient critical-speed transition.

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1.3 Rotor precession

The most important sources of machinery vibration are the residual rotor unbalance and rotor instability.

Most rotors have at least two bearings. With horizontal rotors, the rotor weight is distributed between all the bearings. The rotation axis is coincident with the static elastic line under the own weight. If the weight effect is neglected, the rotation axis coincides with the line connecting the bearing centres.

Any rotational asymmetry due to manufacturing, or produced during operation, makes the line connecting the centroids of rotor cross-sections not to coincide with the rotation axis. Hence, as the rotor is brought up in speed, the centrifugal forces due to dissymmetry cause it to deflect. For example, a 50 tons rotor, with its mass centre off-set by 25 μm from the axis of rotation, experiences a force of approximately 13 tons force, when rotating at 3000 rpm. The rotating centrifugal forces are transferred to the bearings and their supports, and produce unwanted vibrations.

While the bearings and the casing vibrate, the rotor has a precession motion. For isotropic bearings, at constant speed, the deflected shape of the rotor remains unchanged during the motion, any cross-section traces out a circular whirling orbit. The motion appears as a vibration only when the whirl amplitude is measured in any fixed direction.

Despite the analogy often used in describing vibration and precession, their practical implications are different. The remedy for resonance – internal damping – is totally inefficient in the case of critical speeds, since the shape of the deflected rotor does not change (or changes very slightly) during the precession motion at constant speed. Moreover, at a critical speed, if the deflections are not limited, a rotor bends rather than damages by fatigue, phenomenon produced by the lateral vibrations. Instead, journal bearings, small clearance liquid seals, or viscous sleeves are the major source of damping in most cases. Without this damping or a similar source, it would be very difficult to pass through a critical speed. That is why bearings and seals play a major role in the dynamics of the rotor systems.

If identical orbits are traced out with successive rotor rotations, the motion is said to be stable precession. If the orbit increases in size with successive rotations, the motion is an unstable whirl. It may subsequently grow until the orbit becomes bounded either by system internal forces, or by some external constraint, e.g. bearing rub, guard ring, shut-down, etc.

Some typical orbits are shown in Fig. 1.20. The circular orbit (Fig. 1.20, a) represents the synchronous whirling of a rotor in isotropic radial supports. The absence of loops within the orbit denotes synchronous whirl.

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1. ROTOR-BEARING SYSTEMS 25

An elliptical orbit (Fig. 1.20, b) may arise from orthotropic supports, i.e. from dissimilar bearing or pedestal stiffnesses in the horizontal and vertical directions. Inclination of ellipse axes occurs due to cross-coupled stiffnesses and damping properties.

a b c

d e f

Fig. 1.20 (from [19])

If the precession is non-synchronous, i.e. the rotor whirls at a frequency other than the rotational frequency, the orbit will contain a loop as in Fig. 1.20, c, characteristic for the half-frequency whirl due to the instability of motion in hydrodynamic bearings ("oil whirl"). An internal loop indicates that the precession is in the direction of rotation.

Other non-synchronous excitations may occur at several times rotational frequency, giving rise to multi-lobe whirl orbits depicted in Fig. 1.20, d, as in the case of multi-pole electrical generators.

Instabilities such as the half-frequency whirl are frequently bounded. The whirl is initiated by crossing the onset of instability speed, and then it develops in a growing transient, whose radius increases until a new equilibrium orbit is reached (Fig. 1.20, e).

Another type of transient condition is shown in Fig. 1.20, f. The rotor is initially operating in a small stable unbalance whirl condition. The rotor system then receives a transverse shock, and the journal displaces abruptly in a radial

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direction within the bearing clearance, but without contacting the bearing surface. Following the impact, the rotor motion is a damped decaying spiral transient, as it returns to its original small unbalance whirl condition.

Many other types of whirl orbits have been observed, such as those associated with system non-linearities and nonsymmetric clearance effects.

1.4 Modeling the rotor

For the mechanical design of rotor, bearings and supporting structure, one has to take into account that they work as a whole, responding together to the dynamic loading, and interacting. The rotor is part of a dynamic system, its behavior being determined by the location and stiffness of bearings, seals, pedestals and foundation, as well as their damping properties. The casing and foundation masses also play an important role.

The rotor is the main part in any piece of rotating machinery. Its function is to generate or transmit power. It consists of a shaft on which such components as turbine wheels, impeller wheels, gears, or the rotor of an electric machine may be mounted. The rotor is never completely rigid and in many applications it is actually quite flexible. However, in practice, rigid rotors are considered to be those running below 1/3 of the first bending critical speed. Elastic rotors operate near or beyond the first bending critical speed, so that the centrifugal forces due to the residual unbalance cause it to deflect.

In most machines, rotors have shafts with axisymmetric cross-section. If, in some parts, the cross-section is not symmetrical, then the bending stiffness with respect to a fixed axis is variable during the rotation giving rise to non-synchronous motions and instabilities (e.g. two-pole generators and cracked rotors). The rotor shaft can be modeled as a Timoshenko-type beam, accounting for the shear and rotational inertia, including also the effect of gyroscopic couples. The discs – usually rigid – are included by lumped parameters: the mass, and the polar and diametral mass moments of inertia. More advanced calculations consider the disc flexibility. Rotors of individual machines are joined by couplings (locked spline, double-hinged, sliding spline, flex plate).

Bearings are selected as a function of static load and speed, taking into account the dynamic loading, available space, energy losses, simplity of design solution, as well as durability and reliability requirements.

In early studies, bearings were considered as rigid supports (Fig. 1.21, a). Later, their radial stiffness and damping has been taken into account (Fig. 1.21, b). In rolling bearings and air bearings, the damping is usually neglected. The stiffness and damping characteristics of journal bearings are functions of running speed and loading. At rolling bearings, the stiffness is considered independent of speed and

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1. ROTOR-BEARING SYSTEMS 27

loading. Generally, only the bearing translational radial stiffness is taken into account, the angular stiffness being relatively small (one tenth).

With journal bearings, under steady-state hydrodynamic conditions, the total pressure force equals the static load on the bearing. If the centre of the rotating journal is in motion, as for instance during synchronous precession, additional pressures are set up in the lubricant film, which act as dynamic forces on the journal in addition to the static forces. The dynamic force depends on both the relative displacement and the velocity of the journal centre motion but, in contrast to conventional elastic forces, the dynamic force does not have the same direction as the imposed motion, being phase shifted in space and time.

Fig. 1.21

Resolving the dynamic force into two components along fixed coordinate axes in the bearing, say Oy and Oz, and likewise resolving the journal centre motion into y and z displacements, the dynamic force components may be expressed by:

.⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡+

⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡=

⎭⎬⎫

⎩⎨⎧

zy

cccc

zy

kkkk

ff

zzzy

yzyy

zzzy

yzyy

z

y&

& (1.1)

The above equations are exact only for very small amplitudes, but in practice they prove to be valid even for amplitudes as large as a third of the bearing clearance.

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DYNAMICS OF MACHINERY 28

The four stiffness coefficients zzzyyzyy k,k,k,k and the four damping coefficients , zzzyyzyy cc,c,c are calculated from lubrication theory by linearizing the non-

linear bearing forces. They are properties of the particular bearing, being functions of the bearing configuration and the lubricant properties. More important, they belong to a given steady-state journal center position, changing with the speed of rotor.

The inequality of cross-coupling stiffnesses zyyz kk ≠ is the source of a certain type of self-excited precession known as oil whirl, fractional frequency whirl, or half-frequency whirl. Because of the speed dependence of the eight bearing coefficients, the effective damping is negative at low speeds and may become positive at higher speeds.

Active magnetic bearings are applied in industrial centrifugal compressors, turbo expanders and centrifugal pumps. The principle is an electromagnetic shaft suspension, without physical contact between rotor and stator. Sensors located near the electromagnets observe the rotor position. This rotor position signal feeds into an electronic controller which feeds, in a closed loop, the power amplifiers of the electromagnets.

Short annular seals with gas or fluid are usually considered isotropic. The diagonal terms of their stiffness and damping matrices are equal, while the off-diagonal terms are equal, but with reversed sign. The two force components by which the seals act upon the rotor can be written as

. + ⎭⎬⎫

⎩⎨⎧

⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡−

+⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡−

=⎭⎬⎫

⎩⎨⎧

zy

Mzy

CccC

zy

KkkK

ff

z

y&&

&&

&

& (1.2)

The inertial term is negligible at gas seals, where direct stiffnesses can be very small, even negative. For long annular clearance seals, like those used to break down large pressure differences in multi-stage pumps, angular dynamic coefficients are introduced, because the rotor is acted upon by couples, and forces give rise to tilting shaft motions, and moments produce linear displacements.

Radial seals in centrifugal pumps are either balance disks or the radial gap of mechanical seals. Impellers generate motion-dependent forces and moments in the flow fields between impeller tip and casing (volute or diffuser) and in the leakage flow fields developed between impeller shrouds and casing.

Squeeze-film dampers are used in gas turbines as a means of reducing vibrations and transmitted forces due to unbalance. A squeeze film is an annulus of oil supplied between the outer race of a rolling-element bearing (or the bush of a sleeve bearing) and its housing. It can be considered as a parallel element of a vibration isolator, or as a series element in a bearing housing.

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Flexible pedestals are considered in the dynamic response of machines especially for blowers and fans, centrifugal pumps and turbosets with flexible casings and cantilevered bearings. The calculation model (Fig. 1.21, c) includes the stiffness and damping of bearing supports and, in some cases, also their equivalent mass.

The foundation, the sole plate and the soil are seldom included in the calculation model (Fig. 1.21, d), their influence on the rotor response being generally smaller. However, in some cases, especially for large fans on concrete pedestals, the elasticity of the subgrade is taken into account.

Generally, for bearings and pedestals, even with frequency independent characteristics, the horizontal stiffness is lower than the vertical stiffness. This anisotropy doubles the number of critical speeds. In some cases, due to the high damping level, the separation of the two criticals in a pair due to orthotropy does not show up in the unbalance response of rotors. It is also possible for some whirl modes, especially the backward ones, to be overcritically damped, thus appearing neither in the natural frequency diagrams nor in those of the unbalance response

1.5 Evolution of rotor design philosophy

Calculation methods and the interpretation of the results of the dynamic analysis of rotors had a spectacular evolution.

Until the late 1950's, calculations were made numerically or using the graphic method developed by Mohr. It was common practice to assume rigid supports and to treat one span at a time in the model. Analysis was limited to the determination of undamped critical speeds and the objective was to avoid having a running speed at a critical speed, in other words, each span was 'tuned' to avoid certain frequencies. The purpose of an undamped analysis was to provide a close, initial estimate of the critical speeds.

Many specifications explicitly require that operating speeds differ from critical speeds by safe margins. In the API Standard 610, the critical speed is required to be at least 20% greater or 15% less than any operating speed [20]. Compliance with such specifications requires that critical speeds be calculated as part of design and selection procedures of rotating machines. In some cases well established procedures are used. In other cases, e.g. machines that handle liquids, like centrifugal pumps, specific calculation procedures are used, the ‘dry’ running critical speeds being different from the ‘wet’ running criticals.

It was recognized that large discrepancies existed between calculations and tests, and efforts were made to improve the analyses. The rigid supports were replaced by elastic springs with stiffness equal to that of the oil film in the bearings. Later, the effect of pedestals was added. It was recognized that entire

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rotor-bearing systems, rather than single spans, should be analyzed, taking advantage of the advent of high-speed digital computers. After whole systems were studied, it became clear that a change in philosophy was required, involving a switch from tuning to response.

In some cases, the large discrepancies between calculations and tests were diminished introducing the effect of damping and determining the damped critical speeds. Also, calculating the unbalance response, it came out that the speeds at which the radius of synchronous whirling orbits is a maximum – referred to as peak response critical speeds – are different from both the undamped and damped critical speeds, approaching the latter. It was recognized that not all potentially critical speeds are indeed critical, the large damping in bearing smoothing the unbalance response curves so that the passage through a critical speed may take place without an increase in the response amplitude.

At present, numerical simulations are used in the predictive design stage, and rotor designs are accepted or rejected on the basis of the unbalance response at the journals as a function of running speed. In contrast to lateral vibration, torsional natural frequencies are tuned to avoid coincidence with running speed and known exciting frequencies.

Fig. 1.22

Figure 1.22 shows the response of a rotor journal versus the ratio of natural frequency to running speed. Values calculated for rigid supports are denoted by R1 and R2, while values for bearings treated as elastic springs are indicated by E1, E2 and E3. The continuous line shows the unbalance response calculated considering both the stiffness and the damping in bearings. D1, D2, D3

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are the actual critical speeds, measured by the peaks in the unbalance response characteristic. It can be seen that critical speeds based on rigid support calculations can be seriously in error, that criticals calculated assuming elastic supports can be more accurate and that neither calculation can be used to determine a response level because damping has been neglected.

Figure 1.23 shows an analysis of a complete system consisting of high pressure turbine (HPT), intermediate pressure turbine (IPT), low pressure turbine (LPT), generator (G) and exciter (E).

Fig. 1.23

Assuming elastic supports at the bearings, 22 critical speeds were calculated between zero and the running speed. With such a large spectrum of natural frequencies, the desired separation between the operating speed and the critical speeds has limited application, so that there is a need to change the basic philosophy of design and commissioning of a rotating machinery.

In the field of compressors and turbines for industry or power plants, the actual trend to increase the size and the operating speeds has lead to a new generation of machines for which, inevitably, one or two critical speeds are within the range of operating speeds.

As machinery become larger, the elasticity of the lubricant film in bearings and the flexibility of supports play a more important role with respect to

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the rotor stiffness, determining a decrease of the critical speeds and their interference with the range of operating speeds. As the bearing dynamic characteristics are not exactly known, critical speeds cannot be determined accurately, so that the traditional criterion aiming at operation at or near to a critical cannot offer the necessary safe limit

In practice it has been found out that one can operate perfectly safe and reliable at well damped critical speeds if the vibration levels do not exceed the allowable levels and if the rotor has not a pronounced sensitivity to mass unbalance. It means that the unbalance response of the rotor can give the most useful information about the soundness of a design solution. Carrying out this calculation for different unbalance distributions, judiciously chosen so as to enhance the deflection at different unbalance critical speeds, it can be established how critical each of the critical is and what measures have to be taken so that the vibration amplitudes remain within normal limits, even in the presence of unbalances that occur during the normal operation (erosion, deposits, component failures, thermal strains, etc.).

1.6 Historical perspective

The first analysis of critical speeds of a uniform elastic shaft has been made in 1869 by Rankine [21], who devised the term ‘critical speed’. The phenomenon was incorrectly thought to be an unstable condition, the rotor being unable to run beyond that speed. In this case art preceded science, for in 1895 some commercial centrifuges and steam turbines were already running supercritically. Gustaf de Laval first demonstrated experimentally that a (single stage steam) turbine could operate above the rotor’s lowest bending resonance speed and supercritical operation could be smoother than subcritical. In many European papers, the rotor model consisting of a central disk, mounted on a massless flexible shaft supported at its ends, is referred to as the Laval rotor. Although the first correct solution for an undamped model has been given by Föppl [22], who was the first to demonstrate analytically that that a rotor could operate supercritically, the confusion persisted until the publication in 1919 of Jeffcott’s paper [23] using a model with damping. This simple model is called the Jeffcott rotor in recent papers.

In 1894 Dunkerley [24] published results of his studies on critical speeds of shafts with many discs, and gave his well-known method with its experimental verification. In 1916 Stodola [25] published an analysis of the bearing influence on the flexible shaft whirling. He also introduced the gyroscopic couples on disks.

Hysteretic whirl was first investigated by Newkirk [26] in 1924 during studies about a series of failures of blast-furnace compressors. It was observed that at speeds above the first critical speed, these units would enter into a violent

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whirling in which the rotor centerline precessed at a rate equal to the first critical speed. If the unit rotational speed was increased above its initial whirl speed, the whirl amplitude would increase, leading to eventual rotor failure. During the course of the investigation, Kimball in 1924 suggested that internal shaft friction could be responsible for the shaft whirling. In 1925, Kimball and Lovell performed extensive tests on the internal friction of various materials. At the end Newkirk concluded that the internal friction created by shrink fits of the impellers and spacers is a more active cause of the whirl instability than the material hysteresis in the rotating shaft.

In 1925, Newkirk and Taylor [27] observed oil film whirl and resonant whipping. The true upper limit for safe operating speeds has been thus revealed, namely the threshold speed of rotor-bearing instability. This typically occurs at speeds between two and three times the lowest resonant frequency, wherefrom the name of half-frequency and sub-synchronous whirl. The phenomenon was explained only in 1952 by Poritsky [28], who showed that the destabilizing influence comes from the hydrodynamic journal bearing which loses its ability to damp the lowest rotor-bearing bending resonant mode.

In 1933 Smith [29] published a review of the basic rotor dynamics problems, discussing qualitatively the effect of gyroscopic coupling, and simultaneous asymmetries of the bearing and shaft flexibilities. Between 1932-1935, Robertson [30] presented a series of papers on the subjects of bearing whirl, rotor transient whirl, and hysteretic whirl.

In 1946, Prohl [31] published a transfer matrix procedure for determining the critical speeds of a multi-disc single shaft rotor, allowing for the inclusion of gyroscopic effects, but restricted to isotropic elastic supports. Between 1955-1965, Hagg and Sankey [32], Sternlicht [33], Lund [34] and others have developed the theory of hydrodynamic bearings, Yamamoto [35] studied the rolling bearings and Sternlicht [36], Pan and Cheng investigated the rotor instability in gas bearings.

In 1948 Green [37] studied the gyroscopic effect of a rigid disc on the whirling of a flexible overhang rotor, being credited with the initial generalization of Jeffcott’s model to account for rigid-body dynamics. In 1957 Downham [38] has experimentally confirmed the existence of backward whirling.

Between 1963-1967, Lund [39] and Glienicke [40] presented values of the linearized stiffness and damping coefficients for a series of hydrodynamic bearings, first presented by Sternlicht in 1959 [33]. Lund [41, 42] expanded the transfer matrix method of Myklestad and Prohl for calculating damped unbalance response and damped natural frequencies of a flexible rotor with asymmetric supports. Ruhl [43] and Nordmann [44] have first used the finite element method for the dynamic analysis of rotor-bearing systems in their doctoral theses, but the first papers using this method were published by Ruhl and Booker [45] in 1972, and Gasch [46] in 1973. Reduction of the finite element model has been used

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DYNAMICS OF MACHINERY 34

starting in 1980 by Rouch and Kao [47], and Jäcker [48], the latter introducing also the effect of foundation on the rotor response.

The study of the effect of annular fluid seals was initiated by Lomakin [49] in 1958, and then developed by Black [50] and Childs [51]. The effect of gas seals has been studied by Benckert and Wachter [52], and Iwatsubo [53]. The study of instabilities due to unequal gaps between rotor and stator as a result of the rotor eccentricity was initiated by Thomas [54] and Alford [55].

In Romania, the first book with elements of machinery dynamics was published in 1958 by Gh. Buzdugan and L. Hamburger [56]. The lubrication theory has been developed by N. Tipei [57] and V. N. Constantinescu [58-60]. Books on sliding bearings were published by Tipei et al [61] and Constantinescu et al [62]. The first PhD thesis on Rotordynamics was presented in 1971 by M. Rădoi [63], using a computer program developed at INCREST [64], based on Lund’s transfer matrix method [65].

References

1. Hohn, A. and Spechtenhauser, A., Present state and possible applications of turbosets for industrial and medium-sized power plants, Brown Boveri Review, Vol.63, No.6, pp 321-332, June 1976.

2. Hard, F., 75 years of Brown Boveri steam turbines, Brown Boveri Review, Vol.63, No.2, pp 85-93, 1976.

3. Somm, E., Developing Brown Boveri Steam Turbines to Achieve Still Higher Unit Outputs, Brown Boveri Review, Vol.63, No.2, pp 94-105, 1976.

4. * * * Back-Pressure Turbosets for Industrial Use, Brown Boveri Publication 3090 E, 1967.

5. Bertilsson, J. E., and Berg, U., Steam Turbine Rotor Reliability, EPRI Workshop on Rotor Forgings for Turbines and Generators, Palo Alto, California, Sept 13-17, 1980.

6. * * * Turbine à gaz de 6000 kW de l'Electricité de France (E.D.F.) à St-Dizier, Revue Brown Boveri, Vol.47, No.1/2, pp 37-42, 1960.

7. * * * RB.211 Technology & Description, Rolls-Royce Publ. TS2100, Issue 18, Nov.1977.

8. Kostyuk, A. G., and Frolov, V. V., Steam and Gas Turbines (in Russian), Energoatomizdat, Moskow, 1985.

9. Wachel, J. C., Rotordynamic Instability Field Problems, NASA CP 2250, pp 1-19, 1982.

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1. ROTOR-BEARING SYSTEMS 35

10. API Standard 617, Centrifugal Compressors for Petroleum, Chemical and Gas Service Industries, American Petroleum Institute, Washington, 1995.

11. Meiners, K., Compressors in Energy Technology, Sulzer Technical Review, Vol.62, No.4, pp 143-148, 1980.

12. Shaw, M. C., and Macks E. F., Analysis and Lubrication of Bearings, McGraw Hill, New York, 1949.

13. Eck, B., Ventilatoren, Springer, Berlin, 1957.

14. Pfleiderer, C., and Petermann, H., Strömungsmaschinen, 6.Aufl., Springer, Berlin, 1990.

15. Siekmann, H., Wasserturbinen, Dubbel. Taschenbuch für den Maschinenbau, 17. Aufl., Springer, Berlin, pp R30-R36, 1990.

16. Krick, N., and Noser, R., The Growth of Turbo-Generators, Brown Boveri Review, Vol.63, No.2, pp 148-155, 1976.

17. Bishop, R. E. D., and Parkinson, A. G., Second Order Vibration of Flexible Shafts, Phil. Trans. Royal Society, Series A, Vol.259, A.1095, pp 1-31, 1965.

18. * * * Caractéristiques de construction des alternateurs de grande puissance, Revue ABB, No.1, 11 pag. 1989.

19. Rieger, N. F., and Crofoot, J. F., Vibrations of Rotating Machinery. Part I: Rotor-Bearing Dynamics, The Vibration Institute, Illinois, Nov 1977.

20. API Standard 610, Centrifugal Pumps for General Refinery Services, American Petroleum Institute, Washington, 1979.

21. Rankine, W. J. M., On the centrifugal force of rotating shafts, The Engineer, Vol.27, p.249, Apr.1869.

22. Föppl, A., Das Problem der Laval'schen Turbinewelle, Civilingenieur, Vol.41, pp.332-342, 1895.

23. Jeffcott, N., Lateral vibration of loaded shafts in the neighbourhood of a whirling speed – The effect of want of balance, Philosophical Magazine, Series 6, Vol.37, pp.304-314, 1919.

24. Dunkerley, S., On the whirling and vibration of shafts, Trans. Roy. Soc. (London), Vol.185, Series A, pp.279-360, 1894.

25. Stodola, A., Neuere Beobachtungen uber die Kritischen Umlaufzahlen von Wellen, Schweizer.Bauzeitung, Vol.68, pp.210-214, 1916.

26. Newkirk, B. L., Shaft whipping, General Electric Review, Vol.27, pp.169-178, 1924.

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DYNAMICS OF MACHINERY 36

27. Newkirk, B. L. and Taylor H. D., Oil film whirl – An investigation of disturbances on oil films in journal bearings, General Electric Review, Vol.28, 1925.

28. Poritsky, H., Contribution to the theory of oil whip, Trans. ASME, Vol.75, pp.1153-1161, 1953.

29. Smith, D. M., The motion of a rotor carried by a flexible shaft in flexible bearings, Proc. Roy. Soc. London, Series A, Vol.142, pp.92-118, 1933.

30. Robertson, D., The vibration of revolving shafts, Phil. Mag. Series 7, Vol.13, pp.862, 1932; The whirling of shafts, The Engineer, Vol.158, pp.216-217, 228-231, 1934; Transient whirling of a rotor, Phil. Mag., Series 7, Vol.20, pp.793, 1935.

31. Prohl, M. A., A general method for calculating critical speeds of flexible rotors, Trans ASME, Vol.67, J. Appl. Mech., Vol.12, No.3, pp.A142-A148, Sept.1945.

32. Hagg, A. C. and Sankey, G. O., Some dynamic properties of oil-film journal bearings with reference to the unbalance vibration of rotors, Trans. ASME, J. Appl. Mech., Vol.23, pp.302-306, 1956.

33. Sternlicht, B., Elastic and damping properties of cylindrical journal bearings, Trans. ASME, J. Basic Eng., Series D, Vol.81, pp.101-108, 1959.

34. Lund, J. W., The stability of an elastic rotor in journal bearings with flexible damped supports, Trans. ASME, J. Basic Eng., Vol.87, 1965.

35. Yamamoto, T., On the critical speed of a shaft supported in ball bearing, Trans. Soc. Mech. Engrs. (Japan), Vol.20, No.99, pp.750-760, 1954.

36. Sternlicht, B., Gas-lubricated cylindrical journal bearings of the finite length, Trans. ASME, J. Appl. Mech., Paper 61-APM-17, 1961.

37. Green, R. B., Gyroscopic effects on the critical speeds of flexible rotors, Trans. ASME, J. Appl. Mech., Vol.70, pp.369-376, 1948.

38. Downham, E., Theory of shaft whirling. A fundamental approach to shaft whirling, The Engineer, pp.519-522, 552-555, 660-665, 1957.

39. Lund, J. W., Rotor Bearing Dynamics Design Technology. Part III, Design Handbook for Fluid-Film Type Bearings, M.T.I. Report AFSCR 65-TR-45, 1965.

40. Glienicke, J., Feder- und Dämpfungskonstanten von Gleitlagern für Turbomaschinen und deren Einfluss auf das Schwingungsverhalten eines einfachen Rotors, Dissertation, T. H. Karlsruhe, 1966.

41. Lund, J. W., Stability and damped critical speeds of a flexible rotor in fluid-film bearings, Trans. ASME, J. Engng. Ind., Series B, Vol.96, No.2, pp.509-517, May 1974.

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1. ROTOR-BEARING SYSTEMS 37

42. Lund, J. W. and Sternlicht, B., Rotor-bearing dynamics with emphasis on attenuation, Trans. ASME, J. Basic Engng., Vol.84, No.4, pp.491, 1962.

43. Ruhl, R. L., Dynamics of distributed parameter turborotor systems: Transfer matrix and finite element techniques, Ph. D. Thesis, Cornell Univ., Ithaca, N. Y., Jan.1970.

44. Nordmann, R., Ein Näherungsverfahren zur Berechnung der Eigenwerte und Eigenformen von Turborotoren mit Gleitlagern, Spalterregung, ausserer und innerer Dämpfung, Dissertation, T. H. Darmstadt, 1974.

45. Ruhl, R. L. and Booker J. F., A finite element model for distributed parameter turborotor systems, Trans. ASME, Series B, J. Eng. Industry, Vol.94, No.1, pp.126-132, Febr.1972.

46. Gasch, R., Unwucht-erzwungene Schwingungen und Stabilität von Turbinenläufern, Konstruktion, Vol.25, Heft 5, pp.161-168, 1973.

47. Rouch, K. and Kao, J., Dynamic reduction in rotor dynamics by the finite element method, J. Mechanical Design, Vol.102, pp.360-368, 1980.

48. Jäcker, M., Vibration analysis of large rotor-bearing-foundation systems using a model condensation for the reduction of unknowns, Proc. Second Int. Conf. "Vibration in Rotating Machinery", Cambridge, U.K., Paper C280, pp.195-202, 1980.

49. Lomakin, A., Calculation of critical number of revolutions and the conditions necessary for dynamic stability of rotors in high-pressure hydraulic machines when taking into account forces originating in sealings, Power and Mechanical Engineering, April 1958 (in Russian).

50. Black, H., Effects of hydraulic forces on annular pressure seals on the vibrations of centrifugal pump rotors, Journal of Mechanical Engineering Science, Vol.11, No.2, pp.206-213, 1969.

51. Childs, D. and Kim C.-H., Analysis and testing of rotordynamic coefficients of turbulent annular seals, J. of Tribology, Vol.107, pp.296-306, 1985.

52. Benckert, H. and Wachter, J., Studies on vibrations stimulated by lateral forces in sealing gaps, AGARD Proc. No.237 Conf. Seal Technology in Gas-Turbine Engines, London, pp.9.1-9.11, 1978.

53. Iwatsubo, T., Evaluation of instability forces of labyrinth seals in turbines or compressors, Rotordynamic Instability Problems in High-Performance Turbomachinery, NASA CP No.2133, pp.139-167, 1980.

54. Thomas, H., Instabile Eigenschwingungen von Turbinenläufern angefacht durch die Spaltstromungen Stopfbuschen und Beschauflungen, Bull. de l'AIM, vol.71, pp.1039-1063, 1958.

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DYNAMICS OF MACHINERY 38

55. Alford, J., Protecting turbomachinery from self-excited rotor whirl, Trans. ASME, J. Engng. Power, pp.333-344, 1965.

56. Buzdugan, Gh. and Hamburger, L., Teoria vibraţiilor, Editura tehnică, Bucureşti, 1958.

57. Tipei, N., Hidro-aerodinamica lubrificaţiei, Editura Academiei, Bucureşti, 1957.

58. Constantinescu, V. N., Lubrificaţia cu gaze, Editura Academiei, Bucureşti, 1963.

59. Constantinescu, V. N., Aplicaţii industriale ale lagărelor cu aer, Editura Academiei, Bucureşti, 1968.

60. Constantinescu, V. N., Teoria lubrificaţiei în regim turbulent, Editura Academiei, Bucureşti, 1965.

61. Tipei, N., Constantinescu, V. N., Nica, Al., and Biţă, O., Lagăre cu alunecare, Editura Academiei, Bucureşti, 1961.

62. Constantinescu, V. N., Nica, Al., Pascovici, M. D., Ceptureanu, Gh., and Nedelcu, Şt., Lagăre cu alunecare, Editura tehnică, Bucureşti, 1980.

63. Rădoi, M., Contribuţii la studiul dinamicei şi stabilităţii rotorilor, cu considerarea influenţei reazemelor, Teză de doctorat, Inst. Politehnic Timişoara, 1971.

64. Biţă, O., Program pentru calculul răspunsului dinamic al unui rotor, INCREST, Bucureşti, 1973.

65. Lund, J. W., Rotor-Bearings Dynamic Design Technology Part III: Design Handbook for Fluid-Film Bearings, Mechanical Technology Inc. Report AFAPL-Tr-65-45, 1965.

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2. SIMPLE ROTORS IN RIGID BEARINGS

Simple single-disc rotors with massless shafts supported in rigid bearings are considered in this chapter. The effect of damping and gyroscopic couples on the rotor precession is examined in detail.

2.1 Simple rotor models

The simplest flexible rotor consists of a rigid disc, fixed on a flexible shaft of axi-symmetric cross section, supported at the ends in identical bearings. The symmetric rotor, with a massless shaft supported in rigid bearings (Fig. 2.1) is known as the Laval-Jeffcott model [1, 2]. Generally, only the first precession mode is studied for which, because of the symmetry, the disc rotary inertia can be neglected. The model serves to the introduction of the concepts of critical speed and synchronous precession.

Fig. 2.1 Fig. 2.2

The Stodola-Green model [3-5] consists of a flexible shaft with an over-hung disc, not necessarily thin (Fig. 2.2). The model is used to examine the influence of the disc rotary inertia and gyroscopic torques on the rotor precession, the concepts of forward and backward precession, as well as the effect of the unbalance due to the skew mounting of the disc on the shaft.

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DYNAMICS OF MACHINERY 40

In the following, the rotor shaft is considered to be rigidly supported. This is possible when the shaft stiffness is much lower than (less than 10% of) the combined stiffness of bearings and pedestals. The model simplification allows the stepwise introduction of the influence of mass unbalance, external and internal damping, and gyroscopic coupling, neglecting the bearing flexibility and damping.

2.2 Symmetric undamped rotors

Consider a rotor which consists of a flexible shaft of circular cross-section, supported at the ends in rigid bearings, and carrying a thin rigid disc in the symmetry plane, at mid distance between bearings (Fig. 2.3, a).

Fig. 2.3

Let the point G be the disc mass centre and point C - the disc geometric centre, where the geometric axis of the shaft intersects the disc plane. The disc has mass m and polar moment of inertia GJ . The bearing line intersects the disc at point O. Denote e=GC the offset of the disc mass centre G with respect to the point C.

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2. SIMPLE ROTORS IN RIGID BEARINGS 41

The shaft stiffness coefficient is denoted by k (the ratio of a force applied to the shaft middle and the static deflection produced at the same point). For the symmetric rotor, it is 348 l/EIk = where l is the span between bearings, I is the shaft cross-section second moment of area, and E is the shaft material Young's modulus.

In this section, the shaft mass, the damping forces and the static deflection (of the horizontal shaft) under the disc weight are neglected. It is assumed that the non-rotating shaft is rectilinear and the rotor motion is studied with respect to this static equilibrium position. Later on (Section 2.3.4) the effect of gravity on the horizontal rotor will be studied.

An inertial coordinate system with the origin in O is considered. The Ox axis coincides with the bearing line (axis of the non-rotating shaft). The horizontal axis Oz and the vertical axis Oy are in the disc median plane (Fig. 2.3, b).

The disc motion in its own plane can be described by the variation in time of either the coordinates Cy and Cz of the geometric centre C, or the coordinates

Gy and Gz of the mass centre G.

Under the action of an external torque ( )tM , the disc turns and, at a given time t, the line CG makes an angle θ (positive anti-clockwise) with the axis Oy.

2.2.1 Equations of motion

The disc equations of motion can be written using d'Alembert's principle. The disc is isolated and subjected to the elastic restoring force due to the shaft flexibility, the external torque, the inertia force and the inertia torque (Fig. 2.3, c) that must be in dynamic equilibrium [6].

The resulting equations of motion are

).t(MθezkθeykθJ

,zkzm,ykym

CCG

CG

CG

=+

=+=+

cos -sin

0 0

&&

&&

&&

(2.1)

The coordinates of the points C and G are related through

.θezz,θeyy

CG

CG

sin cos

+=+=

(2.2)

Substituting (2.2) into (2.1) we obtain

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DYNAMICS OF MACHINERY 42

).t(M)θzθy(ekθJ

,θθemθθemzkzm

,θθemθθemykym

CCG

CC

CC

cossin

cos sin

sin cos 2

2

=−+

−=+

+=+

&&

&&&&&

&&&&&

(2.3)

Denoting ,imJ GG2 = where Gi is the disc gyration radius with respect to

the spinning axis, the third equation (2.3) can be written

.ie

iz

iy

mk

J)t(M

GG

C

G

C

G cossin ⎟⎟

⎞⎜⎜⎝

⎛−−= θθθ&&

Because Gie << and GCC iz,y << , the second term in the right hand side can be neglected with respect to the first one. For steady-state motion, when the active and resistant torques balance each other, 0)( ,tM = hence .0≅θ&& As a result, the running speed is constant, .,Ωθ const==& and

.t 0 θΩθ += (2.4)

The first two equations (2.3) become

.θtemzkzm

,θtemykym

CC

CC

) (sin

) ( cos

02

02

+=+

+=+

ΩΩ

ΩΩ

&&

&& (2.5)

Equations (2.5) are uncoupled. For each equation, the total solution is the sum of the solution of the homogeneous equation and the particular solution corresponding to the right-hand side.

The solutions of the homogeneous equations (for 0=e , hence for a perfectly balanced disc) have the form

,tZtz

,tYty

znCC

ynCC

)(sin )(

)( cos )(

θω

θω

+=

+= (2.6)

where

mk

n =ω (2.7)

is the natural frequency of the rotor free precession.

The particular solutions, corresponding to the harmonic excitation due to the mass unbalance, describe the steady-state motion and have the form

.tzz,tyy

CC

CC

) (sin ) (cos

0

0

θΩθΩ

+=+=

(2.8)

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2. SIMPLE ROTORS IN RIGID BEARINGS 43

Because, in reality, the free motion decays due to the inherent damping and dies out after a short time interval, in the following only the forced steady-state motion will be studied.

2.2.2 Steady state response

Substitution of (2.8) into (2.5) gives the magnitudes of the displacements of point C along the axes Oy and Oz, respectively:

.e/

/emkemzy

n

n

nCC

)(1)( 2

2

22

2

2

2

ωΩωΩ

ΩωΩ

ΩΩ

−=

−=

−== (2.9)

Due to the disc mass unbalance, the point C moves along a circle of radius

,/

/ezyrn

nCCC

)(1)(

2

222

ωΩωΩ

−=+= (2.10)

with the angular speed Ω .

The above calculations can be written in a more compact form using complex numbers.

Fig. 2.4

In the plane yOz, Oy is taken as the real axis and Oz as the imaginary axis (Fig. 2.4). The following notations are used

,zyr,zyr

GGG

CCC

i i

+=+=

(2.11)

where .1 i −=

Multiplying the second equation (2.2) by i, by addition to the first one and using the notation (2.4), we find

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DYNAMICS OF MACHINERY 44

[ ]) (sin i) (cos i i 00 θΩθΩ +⋅++++=+ ttezyzy CCGG

or .err t

CG) ( i 0e θΩ ++= (2.12)

Analogously, equations (2.5) are written under the form

[ ]) (sini) (cos )i()i( 002 θΩθΩΩ +⋅++=+++ ttemzykzym CCCC &&&&

or .emrkrm t

CC) ( i2 0e θΩΩ +=+&& (2.13)

The steady-state solution of equation (2.13) is

.eΩ/ωΩ/ωr θΩt

n

nC

)( i2

20e

)(1)( +

−= (2.14)

Inserting (2.14) into (2.12) we obtain the displacement of the disc mass centre

.eΩ/ω

r θΩt

nG

)( i2

0e )(1

1 +

−= (2.15)

The point G moves along a circle of radius GO=Gr , with the angular speed Ω .

Considering the line GC as a vector in the complex plane, we can write

.e θΩt )( i 0e CG += (2.16)

Denoting

,rr,rr tGG

tCC

) ( i)+ ( i 00 e e θΩθΩ +== (2.17)

because from equations (2.14) and (2.15) it comes out that Cr and Gr are real quantities, the vectors GO and CO are collinear with GC , hence the points O, C and G are collinear.

At Ω = const. the relative position of these points is fixed. The shaft deflected axis is located in the plane defined by the axis Ox and the line GCO , having a constant bend. The shaft rotates around the Ox axis in this deflected shape so that, at any section, the bending stresses are time invariable. As the motion of point C around the bearings line is executed with the same angular speed Ω as the rotation of point G around the shaft axis (point C), the motion is referred to as a synchronous precession.

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2. SIMPLE ROTORS IN RIGID BEARINGS 45

Figure 2.5 shows the speed dependence of the radius Cr of the orbit of point C (solid line) and the radius Gr of the orbit of point G (broken line).

Fig. 2.5

For nΩ ω= , the radii Cr and Gr become infinite, the shaft bend grows indefinitely, as do the bending stresses. This state has been considered critical and the corresponding speed

mkn n

cr ππω 30 30

== (2.18)

has been referred to as the rotor undamped critical speed. The critical angular speed ncr ωΩ = corresponds to the natural circular frequency of the rotor undamped lateral vibrations.

In the undercritical range, for nΩ ω<1 , ,COGO 1111 > point 1G is outside the segment 11CO , moving as in Fig. 2.6, a along a circle of radius larger than that of point 1C . In the overcritical range, for nΩ ω>2 , ,COGO 2222 < the point 2G is located between the points 2O and 2C , moving as in Fig. 2.6, b along a circle of radius smaller than that of point .C2

At very large angular speeds, for ,Ω nω>>3 the point 3G coincides with

3O , hence the disc mass centre tends to the bearing line (Fig. 2.6, c). The shaft deflection becomes practically equal to the offset e. It is said that the rotor is self-balanced. This is the optimal operating regime in the overcritical range, since the dynamic forces in bearings have the minimum value ek .

The results of the above analysis are of theoretical interest. In practice, large (but not infinite) rotor shaft deflections could be anticipated at the rotor critical speed, when the rotor speed coincides with its flexural natural frequency.

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DYNAMICS OF MACHINERY 46

The findings are limited to the synchronous motion, condition in which equal precession and rotation rates are assumed for the rotor. The orbits of the rotor points are circles only if the shaft is circumferentially symmetric.

Fig. 2.6

2.3 Damped symmetric rotors

The rotor motion takes place in the presence of friction forces arising due to the rotor interaction with its stationary environment and due to the relative motion of its particles and components during bending.

In the following, distinction will be made between external and internal friction forces. The external friction forces, producing the "external damping", limit the precession radius at the critical speed and stabilize the motion. The internal friction forces act at joints, between components mounted with shrink fits, or arise from the internal friction in the shaft material. They produce the "internal damping", attenuating the magnitude of the precession motion at the critical speed, and being able to produce, at higher speeds, unstable motions, as a result of the character of tangential follower forces.

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2. SIMPLE ROTORS IN RIGID BEARINGS 47

2.3.1 Effect of viscous external damping

It is considered that, due to the rotor motion relative to the stationary environment, the disc from figure 2.3 is acted upon by a viscous damping force proportional to the absolute tangential velocity of the disc centre. Let )( Ce yc &− and

)( Cezc &− be the components of this force along the axes of the inertial coordinate system yOz, where ec is the coefficient of external viscous damping.

2.3.1.1 Equations of motion

The equations of motion of the symmetrical rotor (2.5) become

.temzkzczm

,temykycym

CCeC

CCeC

) (sin

) (cos

02

02

θΩΩ

θΩΩ

+=++

+=++

&&&

&&& (2.19)

Denoting CCC zyr i+= and adding the first equation (2.19) to the second one, multiplied by 1i −= , we obtain

.Ωemrkrcrm θtΩCCeC

) ( i2 0e +=++ &&& (2.20)

2.3.1.2 Free damped precession

Substituting solutions of the form tC Rtr λe )( = into the equation (2.20)

with zero right-hand side, we find the characteristic equation

02 =++ kcm eλλ ,

whose solutions are

⎟⎠⎞⎜

⎝⎛ −±−=−⎟

⎠⎞

⎜⎝⎛±−= 2

2

21 1i 22 een

ee, m

km

cm

c ζζωλ (2.21)

where mkn =ω is the natural frequency of the undamped system, referred to as the undamped natural frequency, and

m

cmk

c

n

eee ω

ζ22

== (2.22)

is the external damping ratio.

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DYNAMICS OF MACHINERY 48

The roots (2.21) can also be written

ede, ωαλ i21 ±= (2.23)

where ene ζωα = is a negative attenuation (decay) factor and 21 ened ζωω −= is the damped natural frequency, i.e. the frequency of the damped free motion of the perfectly balanced rotor.

The general solution is

ttttC

edeede RRtr ωαωα i2

i1 ee ee )( −+= , (2.24)

where the integration constants 1R and 2R are determined from the initial

conditions for the displacement and velocity, )(rC 0 and ).(rC 0&

In order to determine the orbit of point C, the free undamped motion will be considered first, when .c eee 0=== αζ The solution (2.24) becomes

.RRtr ttC

nn ωω i2

i1 ee )( −+= (2.25)

In the complex plane, this represents the sum of two vectors of length

1R and 2R , respectively, rotating in opposite directions with angular velocity

.nω The tip of the resultant vector moves along an ellipse (Fig. 2.7, a). The major

semiaxis 21 RRa += is directed along the bisector of the angle between the

two vectors. The minor semiaxis is .RRb 21 −=

a b

Fig. 2.7

In the case of the damped motion, the solution (2.24) represents the sum of two vectors rotating in opposite directions with angular speed ned ωω ≅

) 1for ( <<eζ . For 0<eα , the factor teαe produces a decrease in magnitude, hence a motion along a converging spiral (Fig. 2.7, b).

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2. SIMPLE ROTORS IN RIGID BEARINGS 49

2.3.1.3 Steady-state precession due to unbalance

Substitution of the steady-state solution ) ( i 0e )( θΩ += t

CC r~tr (2.26)

into equation (2.20) gives 22 ) i ( ΩΩΩ emr~kcm Ce =++− .

After transformations, the expression of the displacement of point C is obtained as:

[ ][ ]

.rr//

///e

///er~

IR CCnen

nenn

nen

nC

i 2 ])(1[

)(2i )(1)(=

)(2 i)(1)(

222

22

2

2

+=+−

−−

=+−

=

ωΩζωΩ

ωΩζωΩωΩ

ωΩζωΩωΩ

(2.27)

The solution (2.26) is written under the form

) ( i 0e CtCC r~r θθΩ ++= (2.28)

where

[ ]

,Ω/ωΩ/ωθ

,Ω/ωΩ/ω

Ω/ωer~

n

neC

nen

nC

2

222

2

)(1)(2 tan

)(2])(1[

)(

−−=

+−=

ζ

ζ (2.29)

so that the vector Cr=CO is not collinear with .e θtΩ ) ( i 0e GC +=

Fig. 2.8

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DYNAMICS OF MACHINERY 50

Figure 2.8 shows the relative position of these vectors in the case of

rotation at undercritical speeds .Ω n )( ω< Because ,C 0tan <θ the angle Cθ is

negative and the vector CO lags the line GC . The real part 'CO)( =ℜ Cre

represents the component in phase with GC and the imaginary component

'CC)( =ℑ Crm represents the component in quadrature with (perpendicular to)

GC .

The radius of the disc mass centre orbit is

)( i 0e θΩtCG err ++=

hence

IR GGnen

neCG rre

///er~r~ i

)(2i )(1)(2i1

2 +=+−

+=+=

ωΩζωΩωΩζ (2.30)

so that ) ( i 0e Gt

GG r~r θθΩ ++= (2.31) where

[ ] [ ],

//

/er~

nen

neG

222

22

)(2)(1

)(41

ωΩζωΩ

ωΩζ

+−

+= (2.32)

222

3

)(4)(1)(2 tan

nen

neG //

/ωΩζωΩ

ωΩζθ+−

−=

In figure 2.8, for 0 tan << Gn θ,ωΩ and the vector GO lags the line GC with an angle Gθ .

2.3.1.4 Unbalance response diagrams

Figures 2.9, a and b illustrate the speed-dependence of the modulus and phase angle of the complex quantities Cr~ and Gr~ , for a value .conste =ζ The maximum displacements in the two orbits occur at speeds which are different from the critical speed of the undamped system. However, because practically ,e 1<<ζ it can be considered that the maximum displacement is )/(e eζ2 and takes place at

.mkn ==ωΩ

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2. SIMPLE ROTORS IN RIGID BEARINGS 51

The peak response speeds are slightly different from the undamped and the damped critical speeds, and are different for points C and G.

a b

Fig. 2.9

Eliminating )( n/ ωΩ between the expressions of the real and imaginary components of Cr~ , we obtain the locus of the tip of the vector Cr~ in the complex plane, referred to as the polar plot, Nyquist plot or the hodograph of the respective vector.

Such a curve, of equation

( ) ( ) 04

22

222222 =−+++

IIRRIR Ce

CCCCC rerrrerrζ

(2.33)

is plotted in Fig. 2.10 with solid line, for a value .conste =ζ

At Ω =0, point C coincides with the origin O. With increasing speed, point C moves along the curve clockwise. At nωΩ = point C lies on the negative imaginary semiaxis, and for ,∞=Ω it lies on the negative real semiaxis, at a distance e from the origin.

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DYNAMICS OF MACHINERY 52

In Fig. 2.10, the polar diagram of the displacement Gr~ is drawn with thin line. The relative positions of the vectors CO and GC at three different rotational speeds are also plotted.

Fig. 2.10

At ,nωΩ < vectors CO and GC are drawn in the lower part of the figure. In this relative position, they turn at uniform angular velocity Ω around the point O, anticlockwise, the points C and G moving in circles.

At nωΩ = the vector GC is perpendicular to CO (the position on the left of Fig. 2.10). At ,nωΩ > the vector GO becomes smaller than CO (the upper position) and at ∞=Ω the point G coincides with O, the rotor is "self-balanced".

2.3.1.5 Synchronous precession

The relative orientations of points C and G are sketched in Fig. 2.11 for three speeds. Note the difference compared to Fig. 2.6, drawn for undamped rotors.

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2. SIMPLE ROTORS IN RIGID BEARINGS 53

As the rotor rotates, the eccentric mass tends to pull the rotor toward the bearings on the side the point G is located, called the ‘heavy spot’. The point where the precession radius is a maximum is called the ‘high spot’. The opposite point is the ‘low spot’.

Fig. 2.11

When the damping is neglected (Fig. 2.6), the angle 0=Cθ at nωΩ < ,

and 0180=Cθ at nωΩ > . There is a sudden change from 0 to 0180 at nωΩ = .

The heavy point G coincides with the ‘high spot’ at speeds nωΩ < , and then suddenly coincides with the ‘low spot’ above the critical speed.

In the presence of damping, there is a continuous change of angle Cθ with speed, as given by (2.29) and shown in Fig. 2.9, a, with a higher rate around the critical speed. The heavy spot S no more coincides with the high spot H. At very low speeds, the high spot is almost in phase with the unbalanced (heavy) mass, and the “heavy side flies out”. As speed increases, the high spot begins to lag

the heavy spot. At nωΩ = , the phase lag is 090 , and for at nωΩ > it tends to 0180 , and “the heavy side flies in” (Fig. 2.11).

As point C travels in a circle with angular speed Ω , line GC rotates at the same angular speed around C, so that, in the steady state precession, the line segments CO and GC have no relative motion with respect to each other. They rotate about O as though they were a rigid body.

This implies that, in the synchronous precession, the shaft supported in rigid bearings does not bend back and forth during the motion, but simply revolves in a bowed position, with constant orbit radii. At constant rotor speed, stresses are constant in a given point. This means that internal damping in the shaft material

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DYNAMICS OF MACHINERY 54

will be ineffective against the forced precession. The solution is to introduce damping in bearings and/or bearing supports.

Hence, the external damping reduces the synchronous rotor response and determines a gradual change, with the speed increase, of the relative position of lines CO and GC , that are no more collinear. By increasing the speed, a rotor can be "passed through the critical speed" and can operate beyond the critical speed.

There are at least three ways to reduce the amplitude of synchronous precession: a) balancing the rotor; b) changing either the operating speed or the critical speed; and c) adding damping.

2.3.2 Effect of viscous internal damping

Internal damping in rotors is produced by either the material hysteretic damping or by the Coulomb damping due to rubbing at the interface of shrink-fitted parts. To emphasize the difference between external and internal damping, they are often referred to as, respectively, stationary and rotating damping.

2.3.2.1 Rotating damping

The force due to internal rotor damping is defined as

r& ii cF −= , (2.34)

where ic is the coefficient of internal viscous damping, and r& is the time rate of change of the shaft deflection at the point of disc attachment.

For a non-rotating shaft, this force is proportional to the absolute velocity of the respective point, hence the internal viscous damping plays the same role as the external damping.

For the rotating shaft, the rate of change of shaft deflection, equal to the velocity of the relative displacement of its points, is different from the absolute velocity. Equation (2.34) holds only in a rotating coordinate system fixed to the rotor, hence the name of rotating damping.

Figure 2.12 [6] shows a simple model which illustrates the action of internal viscous damping. If the rotor has a synchronous precession, .constr = , and the damping force (2.34) is zero. Internal damping forces are produced only by alternating bending stresses and strains, when the rotor orbits are non-circular.

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2. SIMPLE ROTORS IN RIGID BEARINGS 55

Fig. 2.12 (from [6])

An interesting description of the nature of internal friction can be found in an early paper by A. L. Kimball [7] and is adapted in the following.

“When a horizontal shaft is supported at its two ends and sags upward in the middle due to a centrifugal force, its lower fibers evidently are in compression and its upper ones are in tension.

If the shaft is revolving, every fiber must have its length alternately decreased and increased once every revolution, due to the alternating compression and tension to which the fiber is subjected. The amount of this alternating increase and decrease of length of the fibers depends upon how much the shaft bends and how far the fiber is from the centre of the shaft. At the centre of the shaft the change of length of the fibers is zero as they lie on the axis of the shaft.

In all metals, a frictional resistance to this change of length exists in a greater or less degree. When a fiber is shortening, a frictional compression is set up. When it is lengthening a frictional tension arises. These frictional stresses are very different from the elastic stresses. The elastic stresses are proportional to the amount that the length of the fibers is changed, and have their maximum and minimum values at the bottom and top of the shaft. The frictional stresses arise during the change of length of the fibers; a frictional tension results when the length of the fibers is increasing and a frictional compression results when the length of the fibers is decreasing.

Fig. 2.13 shows a cross-section of the shaft near its middle point. All of the fibers in the lower half are in elastic compression, EC , and those in the upper half are in elastic tension, ET . When the shaft is rotating anticlockwise, all the fibers in the right hand half are increasing in length, resulting in a frictional tension

FT and the fibers in the left half of the shaft are decreasing in length, producing a frictional compression FC .

The elastic stresses produce an inward restoring force indicated by elF in the figure, whose direction is from the elastic tension toward the elastic compression side of the shaft. So also the frictional stresses produce a corresponding transverse reaction iF . The magnitude of iF is far smaller than that

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DYNAMICS OF MACHINERY 56

of elF , however. If the shaft is placed on ends, a whirling motion of the shaft is likely to build up provided the rotational speed exceeds the natural whirling speed.

Fig. 2.13

In the preceding discussion, the frictional reaction iF has been shown to arise in the fibers during compression and elongation. This is not the only sort of friction which may cause the reaction iF , however.

Any frictional resistance which arises within a revolving deflected rotor, while one-half of the cross section is stretching and the other half is shortening, also must produce a frictional reaction component iF . For example, the rotor may be a shaft with rings shrunk on it. In this case, friction may take place between the surface of the shaft and the inner surface of the rings, due to a working of the shaft in the rings as it revolves. The surface fibers of the deflected shaft go through a cycle of elastic lengthening and shortening for every complete revolution of the shaft. This produces a friction against the inside surface of the rings which may be as great as to cause the shaft to take a slight permanent set when deflected a small amount”.

2.3.2.2 Motion in the rotating coordinate system

Consider the coordinate system ξηζO fixed to the disc, rotating at the running speed Ω (Fig. 2.14). The rotating coordinate frame is selected so that the axis ηO makes an angle θ with respect to GC (the unbalance). The axis ζO is perpendicular to ηO and in the disc plane (see Fig. 2.3). For the symmetric rotor, the axis ξO coincides with the axis Ox of the stationary coordinate system. At a given time t, the axis ηO makes an angle tΩ with the axis Oy .

The coordinates of any arbitrary point P, in the two coordinate systems, are related by the following equations

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2. SIMPLE ROTORS IN RIGID BEARINGS 57

cossin

sin cos =

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡ −

⎭⎬⎫

⎩⎨⎧

ζη

tΩtΩtΩtΩ

zy

. (2.35)

Introducing the complex variables [6]

i ,zyr += (2.36)

,ζηρ i+= (2.37)

it is found that

e i ,ρr tΩ= (2.38)

tΩrρ ie −= . (2.39)

Note that OP=r in the stationary Oxyz system, while OP=ρ in the rotor-fixed ξηζO system.

Fig. 2.14

For undamped rotors, the equation of motion of the disc centre, in stationary coordinates, has the form (2.13):

.emrkrm )tCC

0 ( i2e θΩΩ +=+&& (2.40)

In order to change to rotating coordinates, we use equation (2.38) tΩ

CCrieρ= (2.41)

and, by successive differentiation with respect to time, we obtain

,Ωr tΩCCC

ie ) i( ρρ +=& (2.42)

.ρΩρΩρr tΩCCCC

i2 e )2i( −+= &&&&& (2.43)

On substituting (2.41) and (2.43) into (2.40) we obtain

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DYNAMICS OF MACHINERY 58

.emmkmm CCC0i22 e )( 2i θΩρΩρΩρ =−++ &&& (2.44)

In order to take into account the internal damping, a term proportional to the relative velocity, Ci ρc & , is added in equation (2.44). The equation of motion with internal damping is

0i22 e )( 2i θΩρΩρΩρρ emmkmcm CCCiC =−+++ &&&& (2.45) or

0i222 e )( 2i θΩρΩωρΩρ emc

CnCi

C =−+⎟⎠⎞

⎜⎝⎛ ++ &&& , (2.46)

where the notation (2.7) has been used.

Expressing Cρ in complex form as CCC ζηρ i+= , separating the real and the imaginary parts, equation (2.46) is splitted into a set of two coupled equations

.θΩeζΩω ηΩζ

mcζ

,θΩeηΩωζΩηmcη

CnCCi

C

CnCCi

C

0222

0222

sin)(2

cos)( 2

=−+−+

=−+−+

&&&&

&&&&

(2.47)

Because the right-hand sides contain constant terms, the particular solutions of equations (2.47) will be also constants, namely

,ΩωθΩe, ζ

ΩωθΩeη

nC

nC 22

02

220

2 sincos −

=−

= (2.48)

hence

.e

nC 22

2

ΩωΩρ−

= (2.48, a)

It means that, in the case of rotation with constant angular speed, the disc centre C has a fixed position with respect to the rotating coordinate system.

In the steady-state regime, when the shaft deflected shape remains unchanged, the internal damping has no influence on the magnitude of the rotor precession.

2.3.2.3 Motion in the stationary coordinate system

Because the internal damping force has a fixed position with respect to the rotating coordinate system, and rotates with the same angular speed, its expression in the stationary coordinate system is

.ρcF tΩCii

ie)( &−= (2.49)

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2. SIMPLE ROTORS IN RIGID BEARINGS 59

Equation (2.39) yields

,rr tCCC

ie)i( ΩΩρ −−= && (2.50)

hence

.rΩr CCtΩ

C i e i −= &&ρ (2.51)

Substituting (2.51) into (2.49) we obtain

,yΩzczΩyczyΩzycrΩrcF

CCiCCi

CCCCiCCii

)( i )( ]) i( i i[ )i(

−++==+−+=−=−

&&

&&& (2.52)

so that the components of the internal damping force along the axes of the stationary coordinate system are

)( )( CCiiCCii yΩzcF,zΩycFzy

−−=+−= && , (2.53)

or, in matrix form,

⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡−

−⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡−=

⎭⎬⎫

⎩⎨⎧

C

C

i

i

C

C

i

i

i

i

zy

cc

zy

cc

FF

z

y

0 0

00

ΩΩ

&

&. (2.54)

Considering that the force (2.52) acts upon the disc, the above terms are added with opposite sign in the left-hand side of equations (2.5) to obtain the equations of the motion with internal damping

⎭⎬⎫

⎩⎨⎧

++

=⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡−

+⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡+

⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡) (sin ) ( cos

00

00

0

02

θΩθΩ

ΩΩ

Ωtt

emzy

kcck

zy

cc

zy

mm

C

C

i

i

C

C

i

i

C

C

&

&

&&

&&.

(2.55)

Internal damping yields skew-symmetric (cross-coupled) terms in the stiffness matrix. They produce destabilizing tangential forces.

Equations (2.55) can be written in complex notation as

)( i2 0e) i( θΩtCiCiC ΩemrcΩkrcrm +=−++ &&&

or

.ΩermcΩr

mcr θΩt

Ci

nCi

C)( i22 0e i +=⎟

⎠⎞

⎜⎝⎛ −++ ω&&& (2.56)

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DYNAMICS OF MACHINERY 60

2.3.2.4 Rotor stability

The study of the motion of the perfectly balanced rotor is carried out substituting 0=e in equation (2.56) and looking for a solution of the form

tCC Rr λe=

for the homogeneous equation.

Denoting

n

ii m

ζ2

= , nωλΛ = ,

nωΩη = , (2.57)

we obtain the characteristic equation

( ) 0 2 i122 =−++ ηζΛζΛ ii (2.58) with the roots

ηζζζΛ iii, 2 i1221 +−±−= , (2.59)

where iζ is the internal damping ratio.

Because 2iζ is small with respect to the other terms, it can be replaced by

22ηζ i on condition that η is not much greater than 1. This gives

( )( ) . i 1

i 1

2

1

−+−=

+−−=

ηζΛ

ηζΛ

i

i , (2.60)

The solution of the homogeneous equation has the form

.RRtr tC

tCC

22

11

ee )( λλ +=

For nωΩ > , the real part of the root 1Λ is positive and its associated motion is divergent. At the passage through the 'resonance', when nωΩ ≥ , the motion becomes unstable, the shaft deflection increasing suddenly.

Equations (2.60) show that for 1<η the motion associated with 1Λ is forward, while the motion associated with 2Λ is backward. For 1>η the motion associated with 1Λ is divergent. Increasing the rotor speed Ω , the real part of 1Λ decreases and the real part of 2Λ increases. The stability of the forward component decreases but that of the backward component increases.

This is a general result. Forces which tend to destabilize forward precession modes of a rotor, generally stabilize the backward precession modes.

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2. SIMPLE ROTORS IN RIGID BEARINGS 61

2.3.2.5 Rotor whirling due to internal friction

The previous analysis has shown that in the presence of internal damping the rotor becomes unstable for angular speeds greater than ns ωΩ = - the onset speed of instability and the displacement Cr grows unbounded in time.

A simple physical explanation of this phenomenon can be given considering that the point C moves in a circle of radius CR with angular speed nω , hence

.Rtr tCC

n ie )( ω= In this case, the internal damping force is

CniCCii rΩcrΩrcF )( i ) i( −−=−−= ω& . (2.61)

This force is proportional to the shaft deflection Cr but rotated 090 behind, i.e. it is a tangential force. At nωΩ < , 0<iF , so it is a genuine damping force against the rotor revolving motion.

When the rotor traverses the critical speed, the sign of the damping force changes. At nωΩ > , 0>iF and a "negative damping" force occurs. The work of this force is positive, energy is introduced into the system, and the disc displacement grows unbounded. The force acts tangentially in the direction of motion, giving rise to a diverging spiral orbit, hence to instability.

The instability due to internal rotor friction was studied during the 1920’s by B. L. Newkirk [8] in connection with a series of failures of blast furnace compressors designed to operate above the first critical speed. The following are adapted from a paper by Gunter and Trumpler [9].

“It was observed that at speeds above the first critical speed, these compressors would enter into a violent whirling in which the rotor centreline precessed at a rate equal to the first critical speed. If the unit rotational speed was increased above its initial whirl speed, the whirl amplitude would increase, leading to eventual rotor failure” [8]. It was “concluded that the internal friction created by shrink fits of the impellers and spacers was the predominant cause of the observed whirl instability. In tests on an experimental test rotor, when all shrink fits were removed, no whirl instability would develop. A special test rotor was constructed with rings on hubs shrunk on the shaft [7]. Measurements showed that the frictional effect of shrink fits is a more active cause of shaft whirling than the internal friction within the shaft itself and long clamping fits always lead to trouble with supercritical speed rotors” [10].

“For the case of a hub or a sleeve which is fastened to a shaft, which is afterwards deflected, either the surface fibres of the shaft must slip inside the sleeve as they alternately elongate and contract, or the sleeve itself must bend along with the shaft. Usually both actions occur simultaneously to an extent which

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DYNAMICS OF MACHINERY 62

depends upon the tightness of the shrink fit and the relative stiffness of the two parts”.

Robertson (1935) reported that even short, highly stressed shrink fits are not entirely devoid of problems [11]. He stated that even small, tight shrink fits may develop whirl instability provided the rotor is given a sufficiently large initial disturbance or displacement to initiate relative internal slippage in the fit. If long shrink fits such as compressor wheels and impeller spacers must be employed, it is important that these pieces be undercut along the central region of the inner bore so that the contact area is restricted to the ends of the shrink fit.

A similar effect can be produced by any friction which opposes a change of the deflection of the shaft, such as the friction which exists at the connections of flexible couplings, and even in “rigid” couplings. This group of friction forces was referred to as “hysteretic forces” and the corresponding instability – “hysteretic whirl”.

Extensive testing using an experimental test rotor uncovered the following features of this phenomenon [8]: 1) the onset speed of whirling or whirl amplitude was unaffected by refinement in rotor balance; 2) whirling always occurred above the first critical speed, never below it; 3) the whirl threshold speed could vary widely between machines of similar construction; 4) the precession (or whirl) speed was constant regardless of the unit rotational speed; 5) whirling was encountered only with built-up rotors; 6) increasing the foundation flexibility would increase the whirl threshold speed; 7) distortion or misalignment of the bearing housing would increase stability; 8) introducing damping into the foundation would increase the whirl threshold speed; 9) a small disturbance was sometimes required to initiate the whirl motion in a well balanced rotor. If the foundation flexibility is increased, the rotor stability will be improved only if damping is incorporated into the system” [12].

In order to avoid the instability induced by internal friction, the balancing pistons of steam turbine rotors are no more shrunk on but machined integral with the shaft, built-up rotors with shrunk-on discs are used only in the low pressure turbines, and spline teeth couplings between the rotors of the turbine sections are replaced by other designs without Coulomb friction.

2.3.3 Combined external and internal viscous damping

Considering both external and internal damping, the equations of motion of the disc centre with respect to the fixed coordinate system become [13]

.tΩΩemzkyΩczcczm

,tΩΩemykzΩcyccym

CCiCieC

CCiCieC

)(sin)(

)(cos)(

02

02

θ

θ

+=+−++

+=++++

&&&

&&& (2.62)

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2. SIMPLE ROTORS IN RIGID BEARINGS 63

Denoting

mk

n =2ω , n

ii m

ζ2

= , n

ee m

ζ2

= , ie ζζζ += , (2.63)

equations (2.62) can be written in matrix form as

⎭⎬⎫

⎩⎨⎧

++

=⎭⎬⎫

⎩⎨⎧

⎥⎥⎦

⎢⎢⎣

−+

⎭⎬⎫

⎩⎨⎧

+⎭⎬⎫

⎩⎨⎧

) (sin ) ( cos

2 22

0

022

2

θΩθΩ

ΩωΩωζ

Ωωζωωζtt

ezy

zy

zy

C

C

nni

nin

C

Cn

C

C

&

&

&&

&&.

(2.64)

Introducing the following dimensionless quantities

nωλΛ = ,

nωΩη = , (2.65)

the study of the motion of the perfectly balanced rotor (e=0) leads to the characteristic equation

( ) ( ) 0 4141224 22234 =++++++ ηζΛζΛζΛζΛ i . (2.66)

If equation (2.66) is written under the form

,BBBB 0012

23

34 =++++ ΛΛΛΛ

then, according to the Routh-Hurwitz stability criterion [6], the roots do not have positive real parts (the system is stable) on condition that

,B,B,B,B 0 0 0 0 3210 >>>>

03021 ≥− BBBB ,

.BBBBBB 0230

21321 ≥−−

This yields the stability condition

022

≥−⎟⎟⎠

⎞⎜⎜⎝

⎛η

ζζ

i,

or

.ni

e 01 ≥−+ωΩ

ζζ (2.67)

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DYNAMICS OF MACHINERY 64

Hence, for angular speeds ⎟⎟⎠

⎞⎜⎜⎝

⎛+>

i

enωΩ

ζζ1 the rotor motion is unstable,

and the displacement of the point C with respect to O increases unbounded if there is no damping.

A perfectly balanced rotor, rotating with the angular speed Ω , cannot operate beyond the onset speed of instability

.i

ens ⎟⎟

⎞⎜⎜⎝

⎛+=ζζωΩ 1 (2.68)

In the presence of external damping, ,ns ωΩ > hence the external damping extends the range of stable operation conditions (Fig. 2.15).

As in all self-excited motions, the rotor tends to whirl at its natural frequency nω . The tangential force due to internal friction (2.61) is balanced by the force due to external damping Cne rc ω so that

CneCsni rcrΩc ωω =− )( ,

wherefrom the onset speed of instability (2.68) is easily obtained.

Below the onset speed of instability, the rotor motion is stable and synchronous. Above this speed, the rotor motion has a subsynchronous component which diverges exponentially with time. The associated precession motion is forward. Though the above derivations hold for zero eccentricity, it has been found that the occurrence of rotor instability is rather independent of the state of rotor balance.

Fig. 2.15

The case of internal damping produced by hubs or sleeves clamped to the shaft, shrink fits or spline teeth couplings, is treated in a similar way [3]. The onset speed of instability always exceeds the rotor first critical speed.

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2. SIMPLE ROTORS IN RIGID BEARINGS 65

2.3.4 Gravity loading

For horizontal rotors, the own weight changes the location of the centre of the disc orbit.

If we add the rotor weight gm (g is the acceleration of gravity) to the right-hand side of equations (2.56) with external damping, we obtain the equation of motion [6]

.gmemrckrccrm tCiCieC +=−+++ + ) ( i2 0e) i()( θΩΩΩ&&& (2.69)

The total solution is obtained summing the solution of the homogeneous equation, the particular solution due to unbalance and the particular solution

gCr

due to the gravity. The latter has the form

.Ωg

cΩkgmr

nini

Cg

ωζω 2 i1

1 i 2

−=

−= (2.70)

In the absence of internal damping, for ,i 0=ζ we obtain

kgmgrr

nCC stg

=== 2ω

which corresponds to the shaft static deflection under the disc weight.

Fig. 2.16

In the presence of viscous internal damping, the point C moves in a circle whose centre location depends on the running speed. When the running speed Ω increases, the point O' moves along a semicircle of radius 2/r

stC (Fig. 2.16).

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DYNAMICS OF MACHINERY 66

Within the range nωΩ ≤≤0 the displacement of the point O' is however very small, due to the generally very low value of the damping ratio. At nΩ ω= , the vector radius makes an angle iζ2 with the vertical line OO'.

2.3.5 Effect of shaft bow

Similar precession motions can occur due to an initial bend in the shaft, sometimes referred to as "elastic unbalance" [14].

The solution is identical to that of equation (2.27) except the factor 2Ωe is replaced by 2

naω , where a denotes the initial bend of the shaft at the point of disc attachment.

When the shaft is stationary, the point C is displaced a distance a from the bearing line, but the mass centre G is no longer offset from C. This initial bend must not be confused with any sag due to the disc weight.

If the shaft is turned slowly, then the bend rotates with the rotor, whereas a sag induced by the gravity remains approximately vertically downwards. The rotor behaves as if the previous mass unbalance force 2Ωem is replaced by a force

akam n =2ω .

The difference between the two forms of motion necessitates a modification of the concept of a "balanced" rotor.

Thus, if the rotor suffers from mass unbalance, the resulting precession can be balanced for all rotor speeds by attaching a small mass 1m to the circumference of the disc at the appropriate location (diametrally opposite to CG), such that 22

1 ΩΩ emRm = , where R is the radius of the disc. The magnitude of the required balancing mass is R/emm =1 and this mass will completely cancel the vibration at all rotor speeds.

If a similar mass 1m is attached to the initially bent shaft at a radius R,

however, the net force is of magnitude 21

2 Ωω Rmam n − . Thus the exciting force cannot be removed for all speeds Ω . The best that can be done is to select

R/amm =1 , so that the rotor does not whirl at its critical speed.

The only way to balance such a rotor for all speeds would be to straighten the initial bend, but this is not possible in practice.

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2. SIMPLE ROTORS IN RIGID BEARINGS 67

2.3.6 Rotor precession in rigid bearings

Although there is an obvious analogy between the analytical results of Section 2.3.1, on the one hand, and the familiar expressions for the linear vibration of a simple single-degree-of-freedom system, on the other, the forced motion of the rotor is not a true vibration.

A flexible shaft with a concentrated mass bows out in a simple bend. The magnitude of the deflection and its direction relative to the radial plane containing the unbalance are determined by the speed of rotation and by the external damping. The shaft does not experience any alternating stresses while precessing in rigid bearings because its points move along circular orbits. Hence there is no concern regarding the fatigue.

The centrifugal forces due to the unbalance cause the rotor to deflect. The bent rotor whirls around its neutral axis at the running speed. This is a synchronous precession which is actually not a vibration of the rotor in the normal sense of the word. The deflected shape of the rotor remains unchanged during the precession in rigid bearings. Only when the whirl amplitude is measured in any fixed direction, the motion appears as a vibration. The rotor does appear to vibrate only when the projection of the forced motion on any fixed radial plane is examined. Moreover, the bearings experience an oscillatory force in any such plane.

The remedy for resonance - the internal damping, does not contribute to limit the amplitude of motion, since the shape of the deflected rotor does not change during the whirl motion. As explained later, the major sources of damping for rotors are the journal bearings, small-clearance liquid seals or viscous sleeves.

For the Laval-Jeffcott rotor, the bend is greatest when the frequency corresponding to the rotational speed is equal (or nearly equal) to the natural frequency of transverse vibration that the rotor would have if it did not rotate and were simply executing forced undamped flexural vibrations. This will be referred to as a natural frequency of precession. Because the precession is synchronous, i.e. the angular speed of precession is equal to the angular speed of rotation, this frequency corresponds to the critical speed. The large shaft bow at the critical speed can produce stresses in the plastic range that can be limited by radial stops.

The external damping is a ‘stationary’ damping which limits the magnitude of the precession radius. The internal damping is a ‘rotating’ damping which can produce unstable whirling above the critical speed. At a certain speed, the internal friction force changes the direction and becomes a destabilizing tangential force acting in the sense of whirling, against the external damping force.

Note that the above analysis neglects the effects of the disc mass moments of inertia and the disc pitching motion when mounted off-centre. These will be treated in the following.

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DYNAMICS OF MACHINERY 68

2.4 Undamped asymmetric rotors

In this section, simple rotors are considered with shafts supporting a rigid disc attached either in-board off-centre, as in Fig. 2.17, or overhung, as in Fig. 2.2. The disc rotates with the angular speed Ω .

If the rotor speed and the disc mass moments of inertia are relatively small, than the disc can be modeled by a concentrated mass, and the problem can be reduced to the study of the lateral vibrations of a beam carrying a point mass.

In practical cases, the angular precession of the disc axis (tangent to the shaft axis) adds to the orbital motion of the centre of the shaft cross-section. This gives rise to inertia torques that influence the parameters of the rotor whirling motion.

Fig. 2.17

The disc rotary inertia due to the disc transverse mass moment of inertia resists any local angular acceleration due to the change of slope of the rotor. This contributes to the overall inertia of the rotor and tends to lower the system critical speeds. The gyroscopic couple resists any change in the angular momentum of the disc. For forward precession, this acts in opposition to rotatory inertia and introduces a so-called ‘gyroscopic stiffening’ effect, proportional to the disc polar mass moment of inertia and rotation speed.

The gyroscopic coupling yields pairs of forward and backward precession modes whose natural frequencies are, respectively, larger and lower than the associated zero-speed natural frequencies. Because the natural frequencies depend on the rotor speed, distinction should be made between rotor natural frequencies and critical speeds. Base excitation and harmonic forces with fixed direction in space excite both forward and backward critical speeds. The occurrence of backward precession is not desirable in practice, producing alternating bending stresses, which can shorten the fatigue life of the rotor. Rotor unbalance cannot excite backward modes.

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2. SIMPLE ROTORS IN RIGID BEARINGS 69

2.4.1 Reference frames

It is convenient to utilize several different reference frames: a) a stationary reference frame Oxyz; b) a rotating rectangular coordinate frame Gx'y'z' whose axes are collinear with the Oxyz axes, and c) a reference frame 111 zyGx which is translating and rotating with respect to Oxyz but is not fixed to the moving disc.

The bearing line crosses the non-rotating disc at point O. The moving axes Gx'y'z' have the origin at the disc mass centre G . They rotate around the point O, but remain collinear with the axes of the stationary frame. The axis 1Gx coincides with the rotor spin axis, while 11 Gz,Gy do not rotate about 1Gx . Axes

111 zyGx can be considered to be the principal axes of inertia of the disc.

Fig. 2.18

It is assumed that, due to the shaft deflection, the disc spinning axis 1Gx makes an angle Gϕ with the plane yOx (hence with y'Gx') and an angle Gψ with the plane zOx (hence with z'Gx') as shown in Fig. 2.18.

2.4.2 Inertia torques on a spinning rigid disc

The principal mass moments of inertia of the rotor disc with respect to the coordinate frame 111 zyxG are denoted :

TzyPx JJJ,JJ ===111

, (2.71)

where PJ is the polar mass moment of inertia and TJ is the diametral (transverse) mass moment of inertia.

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DYNAMICS OF MACHINERY 70

In order to determine the expressions of the torques acting on the disc (from the shaft), the angular momentum principle with respect to the point G is used.

The projections of the angular momentum vector along the axes of the 111 zyxG frame, taken to coincide with the principal axes of inertia of the perfectly

balanced disc, can be written

.ΩJ, KψJ, KJK PxGTzGTy ===111

&&ϕ (2.72)

The components zyx , K, KK of the angular momentum along the axes of the stationary frame zyxO are equal to the components along the corresponding axes of the frame zyxG ′′′ (Fig. 2.19).

Fig. 2.19

From Figs. 2.19, a and b it comes out that

.ψKψKK

,KKK

GxGyy'

GxGzz'

sincos

sincos

11

11

+=

−= ϕϕ

For small angles, ,, ψψψ ≅≅ sin 1cos so that

.ψKKKK

,KKKK

Gxyy'y

Gxzz'z

11

11

+==

−== ϕ (2.73)

On inserting expressions (2.72) into (2.73), we obtain

.ΩJψJK

,ψΩJJK

GPGTz

GPGTy

ϕ

ϕ

−=

+=

&

& (2.74)

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2. SIMPLE ROTORS IN RIGID BEARINGS 71

Fig. 2.20

Using the angular momentum principle, the components along the axes Oy and Oz, respectively, of the torque applied to the disc, as a result of the shaft deflection, can be written

GPGTzG

GPGTyG

ΩJψ JKM

,ψ ΩJ JKM

z

y

ϕ

ϕ

−==

+==

&&&

&&&&

(2.75, a)

or, in matrix form,

.0

0 +

00

=⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡−⎭

⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎭⎬⎫

⎩⎨⎧

G

G

P

P

G

G

T

T

G

G

JJ

JJ

MM

z

y

ψϕ

Ωψϕ

&

&

&&

&& (2.75, b)

When the disc is part of a rotor, the system equations of motion can be obtained from d'Alembert's principle if the right-hand sides of (2.75, b) are introduced with opposite signs as inertia torques acting on the disc (Fig. 2.20):

.00

=0

0

00

⎭⎬⎫

⎩⎨⎧

⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡−

−⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡−

⎭⎬⎫

⎩⎨⎧

G

G

P

P

G

G

T

T

G

G

JJ

JJ

MM

z

y

ψϕ

Ωψϕ

&

&

&&

&& (2.75, c)

↓ ↓ ↓ applied angular gyroscopic diametral acceleration torques torques inertia torques

The last term in the left-hand side of equation (2.75, c) describes the gyroscopic torques acting on the disc. They couple the equations of motion. The

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DYNAMICS OF MACHINERY 72

torque yM about the Oy axis is proportional to the angular velocity ψ& about the Oz axis and vice versa.

From Fig. 2.20 it can be seen that a torque yM , which produces a

rotation ϕ , gives rise to a gyroscopic torque ϕ&×ΩJP directed along the Oz axis, which "tends to rotate the spin axis Ox toward the Oy axis".

A torque zM , which produces a rotation ψ , gives rise to a gyroscopic torque ψΩJP &× directed along the negative Oy axis, hence "tends to rotate the spin axis Ox toward the Oz axis".

The general rule, also given by the vector product from the expression of the gyroscopic torque, can be stated as follows: "the spin vector Ω tries to move into the torque vector".

2.4.3 Equations of motion for elastically supported discs

The shaft is acted upon by torques of the same magnitude but opposite direction as those applied to the disc

.MM

,MM

zz

yy

GC

GC

−=

−= (2.76)

Moreover, the shaft is acted upon by the disc inertia forces, whose projections on the fixed frame axes have the following expressions

,zmFF

,ymFF

GGC

GGC

zz

yy

&&

&&

−=−=

−=−= (2.77)

where GG z,y are the coordinates of the disc mass centre.

Figure 2.21 illustrates the forces and torques acting on the shaft at the disc attachment point, as well as the corresponding deformations (positive, by the right hand rule, in the positive direction of the coordinate axes).

The equations of motion can be written using the (flexibility) influence coefficient method. The following notation is used: 11δ - deflection (at a point) produced by a unit force (applied at the same point); 21δ - rotation (of a cross-section, or the slope of the elastic line) produced by a unit force (applied at the same point); 22δ - rotation (of a cross-section) produced by a unit couple (applied at the same cross-section); 12δ - deflection (at a point) produced by a unit couple

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2. SIMPLE ROTORS IN RIGID BEARINGS 73

(applied at the same cross-section). According to Maxwell's reciprocity theorem, .2112 δδ =

Fig. 2.21

The displacements of the disc centre can be written

2221

1211

δδψ

δδ

zy

zy

CCC

CCC

MF

,MFy

+=

+= (2.78)

and

.)M(F

,)M(Fz

yz

yz

CCC

CCC

2221

1211

δδϕ

δδ

−+=−

−+= (2.79)

Substituting the expressions of forces (2.77) and torques (2.76), (2.75a) into equations (2.78) and (2.79), the following equations of motion are obtained

,)JJ(zm,)JJ(zmz,)JJ(ym,)JJ(ymy

GPGTGC

GPGTGC

GPGTGC

GPGTGC

2221

1211

2221

1211

δψΩϕδϕδψΩϕδδϕΩψδψδϕΩψδ

&&&&&

&&&&&

&&&&&

&&&&&

++−=−++−=−−−=

−−−=

or

.JJzm,JJym

,zJJzm,yJJym

CGPGTG

CGPGTG

CGPGTG

CGPGTG

0 0

0 0

222221

222221

121211

121211

=−−−=+−+

=+−−

=+−+

ϕψδΩϕδδψϕδΩψδδ

ψδΩϕδδϕδΩψδδ

&&&&&

&&&&&

&&&&&

&&&&&

(2.80)

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DYNAMICS OF MACHINERY 74

In matrix form

.zψy

zψy

J

zψy

Jm

Jm

C

C

C

C

G

G

G

G

P

P

G

G

G

G

T

T

⎪⎪⎭

⎪⎪⎬

⎪⎪⎩

⎪⎪⎨

=

⎪⎪⎭

⎪⎪⎬

⎪⎪⎩

⎪⎪⎨

+

⎪⎪⎭

⎪⎪⎬

⎪⎪⎩

⎪⎪⎨

−⎥⎥⎥⎥

⎢⎢⎢⎢

−⎥⎥⎥⎥

⎢⎢⎢⎢

+

+

⎪⎪⎭

⎪⎪⎬

⎪⎪⎩

⎪⎪⎨

−⎥⎥⎥⎥

⎢⎢⎢⎢

⎥⎥⎥⎥

⎢⎢⎢⎢

0000

000

000

00

00

2221

1211

2221

1211

2221

1211

2221

1211

ϕϕδδδδ

δδδδ

ϕδδδδ

δδδδ

&

&

&

&

&&

&&

&&

&&

(2.80, a)

In order to reduce the dimensionality of the governing equations, the following complex variables are introduced

.ψ,ψ,rzyrzy

CCCGGG

CCC,GGG

αϕαϕ =−=−

=+=+

i i i i (2.81)

The second equation (2.80) is multiplied by 1=i − and added to the first equation. The fourth equation (2.80) is multiplied by i and is added to the third equation. This produces the following set of two coupled equations

.JJrm

,rJJrm

CGPGTG

CGPGTG

0 i 0 i

222221

121211

=+−+=+−+

ααδΩαδδαδΩαδδ&&&&&

&&&&& (2.82)

Generally, if the disc has a running speed Ω and an offset e, then the mass G and the geometric centre C of the shaft cross-section do not coincide. The radii of the orbits of these points are related by

.err otCG

) ( ie θΩ ++= (2.83)

If the disc is attached at an angle α to the shaft, the slope of the shaft axis Cα and the inclination of the disc spinning axis (perpendicular to its plane)

Gα are related by

.tCG

) ( ie αθΩααα ++= (2.84)

Eliminating Gr and Gα between equations (2.82)-(2.84), the differential equations of the motion of the disc geometric centre C are obtained as:

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2. SIMPLE ROTORS IN RIGID BEARINGS 75

,JJem

JJrm,JJem

rJJrm

tPT

tCCPCTC

tPT

tCCPCTC

o

o

) ( i222

)( i221

222221

) ( i212

)( i211

121211

e )(e

ie )(e

i

α

α

θΩθΩ

θΩθΩ

ΩαδΩδ

ααδΩαδδΩαδΩδ

αδΩαδδ

++

++

−+=

=+−+−+=

=+−+

&&&&&

&&&&&

(2.85)

or, in matrix form,

e

e)(e

i000

0

0

ii

i

2221

12112

2221

1211

2221

1211

Ωt

PTC

C

C

C

PC

C

T

JJemΩ

r

rΩJ

rJ

m

o

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−⎥⎦

⎤⎢⎣

⎡=

⎭⎬⎫

⎩⎨⎧

+

+⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡−⎥

⎤⎢⎣

⎡+

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

αθ

θ

αδδδδ

α

αδδδδ

αδδδδ

&

&

&&

&&

(2.86)

Introducing the stiffness matrix as the inverse of the flexibility matrix

⎥⎦

⎤⎢⎣

⎡=⎥

⎤⎢⎣

⎡−

2221

12111

2221

1211 kkkk

δδδδ

,

the equation (2.86) has the simpler form

.e

e )(e

i0

00

00

ii

i2

2221

1211

Ω t

PT

C

C

C

C

PC

C

T

JJemΩ

rkkkkr

ΩJr

Jm

o

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−=

=⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡+

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡−

+⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

αθ

θ

α

ααα &

&

&&

&&

(2.87)

In the following, the rotor precession natural frequencies, the response to mass unbalance and the response to a harmonic force fixed in space will be studied [15]. The index C will be dropped to simplify the notation.

2.4.4 Natural modes of precession

To study the free precession of the asymmetric rotor, consider equations (2.85) with zero right-hand side

.JJrm

,rJJrm

PT

PT

0i0i

222221

121211

=+−+=+−+ααδΩαδδ

αδΩαδδ&&&&&

&&&&& (2.88)

The system is undamped, so that the solutions are of the form

.,Rr tt i i e e ωω Αα == (2.89)

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DYNAMICS OF MACHINERY 76

Substituting the solutions (2.89) into (2.88), we obtain the following homogeneous algebraic equations

.JJRm

,JJRm

PT

PT

0) 1()(

0) ()1(

22222

212

12 122

112

=+−+−

=−−−

ΑδΩωδωδω

ΑδΩωδωδω (2.90)

The condition to have non-trivial solutions is

0 1

1

22222

212

122

12112

=+−−−−

δωδωδωδωδωδω

PT

TP

JΩJmJJΩm (2.91)

and represents the characteristic equation, also termed the frequency equation.

It is a quartic in the variable ω , and has four roots (two positive and two negative), which correspond to the four natural frequencies iω ( )41,..,i = of an elastically-supported disc. They are functions of the rotational angular speed Ω .

Fig. 2.22

In order to draw the graph of this function it is useful to re-write equation (2.91) under the form

[ ] .mJ

JmJm

P

TT

222

1222112

2122211

42211

2

)(

)()(1

δδδδωω

δδδωδδωΩ−−

−++−= (2.92)

Substituting values of ω , equal to the natural frequencies, gives the corresponding values of Ω .

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2. SIMPLE ROTORS IN RIGID BEARINGS 77

Figure 2.22 presents the dependence )(Ωωω = for a rotor whose disc has the mass moments of inertia .JJ TP > The four curves in the diagram correspond to the four roots iω of equation (2.91). They are anti-symmetrical, two by two, with respect to the coordinate axes. The labeling is arbitrary.

The intersection points with the ω axis locate the natural frequencies at zero running speed ( )0=Ω . It can be noticed that the rotor rotation determines a variation of the natural frequencies with respect to those of the (non-rotating) rotor in transverse vibrations.

The ordinates of the horizontal asymptotes correspond to the natural frequencies of a rotor with zero disc angular precession. When ∞→Ω , the axial angular momentum becomes so large that the disc cannot be tipped out of its own plane and α remains zero during precession.

Fig. 2.23

For a complete description of the phenomenon, it is sufficient to use the curve branches located in the positive semiplane of ω (Fig. 2.23). The running speed Ω is considered to vary between (–∞) and (+∞) and the natural frequencies ω to vary between 0 and (+∞).

It must be noticed that equations (2.88) admit also solutions of the form

,,Rr tt i i e e ωω Αα −− ==

which implies the substitution of ω by ( )ω− in equation (2.87), or substitution of Ω by ( )Ω− .

When ω and Ω have the same sign (in this case - positive), the rotation of the deflected shaft around the bearing line has the same direction as the disc spinning motion. The motion is a forward precession, i.e. the precession motion is in the direction of rotation.

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DYNAMICS OF MACHINERY 78

When ω and Ω have opposite signs (in this case, 0>ω , 0<Ω ), the lines CO and GC rotate in opposite directions; the motion is a backward precession.

The directions of the rotor precession and rotor rotation are opposite. Points on the curves ( )Ωωω = , in the quadrant with 0>ω , 0>Ω , correspond to the forward precession; points in the quadrant 0>ω , 0<Ω correspond to backward precession (Fig. 2.23).

Fig. 2.24

Figure 2.24 illustrates the dependence ( )Ωωω = for a rotor whose disc has the mass moments of inertia TP JJ < (so-called ‘stick’ case).

For a disc

2

2RmJ P = , ( )223

12HRmJT += , (2.93)

where R is the disc radius, H is the disc length and m is the disc mass.

For thin discs, RH << and TP JJ 2= . For RH 3= , the mass moments of inertia are equal, TP JJ = .

In numerical simulations, it is useful to use dimensionless quantities [16]. The characteristic equation (2.91) can be written

0 11

1

11

222

0

22

2

11

21

11

12

0

22

2

=

⎟⎟⎠

⎞⎜⎜⎝

⎛−−−

⎟⎟⎠

⎞⎜⎜⎝

⎛−−−

δδ

ηωΩηη

δδ

δδ

ηωΩηη

l

ll

l

ll

l

P

TP

P

TP

JJ

mJ

JJ

mJ

(2.91, a)

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2. SIMPLE ROTORS IN RIGID BEARINGS 79

where

ωη = , (2.94)

l is rotor’s length (or span), and

11

01δ

ωm

= (2.95)

is the natural frequency when the disc is replaced by a concentrated mass.

Equation (2.91, a) becomes

010

12

23

03

44 =++−− η

ωΩηη

ωΩη aaaa , (2.96)

where

11

222

21 δ

δ l

lmJa P= ,

P

TJJaa 12 1+= ,

( )2

11

22122211

23 δ

δδδ l

l

−=

mJa P , 34 a

JJa

P

T= .

(2.97)

Substituting the natural frequencies iω ( )41,..,i = in the first equation (2.90) we obtain the amplitude ratios which define the mode shapes

i

iPi

T

ii R

JJ

mA

212

112

1

ωδωΩ

δω

⎟⎟⎠

⎞⎜⎜⎝

⎛−

−= . (2.98)

For calculations, equation (2.98) is written under the form

ll

l

i

iiP

TP

ii

R

JJ

mJ

A

11

12

0

22

2

1

δδ

ηωΩη

η

⎟⎟⎠

⎞⎜⎜⎝

⎛−

−= , (2.99)

where

ωη i

i = . (2.100)

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DYNAMICS OF MACHINERY 80

Example 2.1

Consider as an example a simply supported uniform shaft carrying an off-centre thin disc (Fig. 2.25). Determine the precession mode shapes taking

l40.a = , 21602 lm.JJ TP == , 080 ωΩ .= .

The flexibility matrix is

[ ] ( )( ) ⎥

⎥⎦

⎢⎢⎣

⎡=

⎥⎥⎦

⎢⎢⎣

+−−=

2800480048005760

331 2

33

22

2 ....

IEbaabbaabbaba

IE l

lll

l

ll

lδ .

Fig. 2.25

The frequency equation (2.96) has the form

03866711667461 234 =++−− ηηηη ...

with roots

820601 .−=η , 013712 .=η , 321713 .−=η , 728624 .=η .

The corresponding precession mode shapes are defined by (2.99)

l11 46642 R.A = , l22 69550 R.A = , l33 90132 R.A −= , l44 393931 R.A −= .

The four natural frequencies are

01 82060 ωω .−= , 02 01371 ωω .= , 03 32171 ωω .−= , 04 72862 ωω .= .

The mode shapes are shown in Fig. 2.26, considering l10.R i = .

For a stationary shaft ( )0=Ω , the characteristic equation is

0316674 24 =+− ηη . , with roots

962100201 .==− ηη , 800310403 .==− ηη .

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2. SIMPLE ROTORS IN RIGID BEARINGS 81

Fig. 2.26

The corresponding mode shapes are defined by

l010201 20521 R.AA == , l030403 371910 R.AA −== .

2.4.5 Response to harmonic excitation

Generally, the rotor forced precession motions are produced by external periodic perturbations. Unbalance, misalignment, blade passing and gear mesh have the excitation frequency a function of the running speed. Coincidence of an excitation frequency with a rotor natural frequency gives rise to a state similar to the resonance, wrongly referred to as ‘critical speed’ or simply ‘critical’, phenomenon characterized by very large values of the radius of the precession motion.

2.4.5.1 Unbalance response

The precession motions due to the unbalance have circular frequencies equal to the angular speed of the rotor. When a natural angular frequency iω , computed at a given rotational angular speed Ω , becomes equal to the rotor angular speed, a critical state takes place produced by the mass unbalance. The

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DYNAMICS OF MACHINERY 82

corresponding angular speed is called a critical angular speed. Measured in "rotations/minute" it is referred to as the peak response critical speed.

For ,0=α i.e. for an excitation by mass unbalance only, and taking 00 =θ , the disc steady-state response has the form

e iΩ tˆrr

⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧

αα, (2.101)

which describes a synchronous precession of angular speed Ω .

Equations (2.87) become

⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧

⎥⎥⎦

⎢⎢⎣

−−−

01

22

2221

122

11 Ωemˆr

Ω)JJ(kkkΩmk

PT α. (2.102)

The critical angular speeds are solutions of the equation

0 22221

122

11 =−−

−Ω)JJ(kk

kΩmk

PT , (2.103, a)

which is obtained by cancelling the denominator in the solutions of the precession produced by unbalance. They satisfy equation .ΩΩ crcr )( ω= When ,ΩΩ cr= the angular velocity of the disc precession motion is equal to the shaft rotational speed.

Substituting Ωω = into (2.91), the equation of the critical speeds can also be written

[ ] .ΩJJm

ΩJJm

PT

PT

01)(

)()(2

2211

42122211

=+−+−

−−−

δδ

δδδ (2.103, b)

It comes out that, in Figs. 2.23 and 2.24, the intersection points of the synchronous excitation line ω =Ω with the curves ( )Ωω i correspond to the critical speeds of (synchronous) forward precession.

In forward precession, the rotor with TP JJ > has only one critical speed (Fig. 2.23) and the rotor with TP JJ < has two critical speeds (Fig. 2.24).

The unbalance can excite a response at critical speeds only in the modes with forward precession.

In order to determine the synchronous response to mass unbalance excitation, consider equations (2.85)

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2. SIMPLE ROTORS IN RIGID BEARINGS 83

.emJJrm

,emrJJrmt

PT

tPT

Ω

Ω

ΩδααδΩαδδ

ΩδαδΩαδδi2

21222221

i211121211

e i

e i

=+−+

=+−+

&&&&&

&&&&& (2.104)

Substituting the solutions (2.101) into (2.104), we obtain the following set of algebraic equations

[ ] .emˆJJrm

,emˆJJrm

PT

PT

212

222

212

112

122

112

) (1)(

) ()1(

δΩαδΩδΩ

δΩαδΩδΩ

=−−+−

=−−− (2.105)

Equations (2.105) can be written in terms of dimensionless quantities as

.eˆJJ

mJr

,eˆJJ

mJr

P

TP

P

TP

l

ll

ll

l

ll

l

11 11

1 )1(

11

212

11

222

22

11

122

2

11

122

22

δδηα

δδ

ηδδ

η

ηαδδ

ηη

=⎥⎥⎦

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−−+−

=⎟⎟⎠

⎞⎜⎜⎝

⎛−−−

(2.106)

Using notations (2.94), (2.95) and (2.97), the solution for r can be written

( )[ ]

( ) ( ) 1

12

124

34

2234

+−−−

−−=

ηη

ηη

aaaa

aa

er . (2.107)

Fig. 2.27

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DYNAMICS OF MACHINERY 84

The plot of the amplitude ratio er versus frequency ratio 0ωΩη = gives the unbalance response curve, with peak(s) at the forward critical speed(s).

For the rotor from Example 2.1, the unbalance response curve is shown in Fig. 2.27.

As expected, there is a single peak at about 001371 ω. , corresponding to the first forward mode.

2.4.5.2 Response to a harmonic force fixed in space

A harmonic force having a fixed direction in space

tFtF ω cos )( 0= (2.108)

can be written under the form

( ).FtF tt i i0 ee2

)( ωω −+= . (2.109)

The first component produces a response of the form

t

f

fˆrr ωαα

ie ⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧

, (2.110)

which describes a forward precession of angular speed ω .

When the circular frequency becomes equal to the rotor angular speed, Ωω = , the first component can produce resonance in the modes with forward

precession.

The second component produces a response of the form

t

b

bˆrr ω

ααie −

⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧

, (2.111)

which describes a backward precession of angular speed (−ω ).

For Ωω −= , equations (2.87) become

01

2

02

2221

122

11

⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧

⎥⎥⎦

⎢⎢⎣

+−− F

ˆr

Ω)JJ(kkkmΩk

b

b

PT α. (2.112)

The backward (or asynchronous) critical speeds [4] are solutions of the equation

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2. SIMPLE ROTORS IN RIGID BEARINGS 85

0 22221

122

11 =+−

−Ω)JJ(kk

kΩmk

PT. (2.113)

The backward critical speeds can be excited only by forces rotating in a direction opposite to the rotor rotation. Such a force can be either a component of a force fixed in space, or the result of a kinematic excitation of the bearings or bearing supports.

In Figs. 2.23 and 2.24, the intersection points of the frequency curves with the asynchronous excitation line Ωω −= , locate the backward critical speeds.

In order to determine the frequency response to a harmonic unidirectional force, the unbalance excitation is replaced in the right hand side of equations (2.104) by the harmonic force (2.109), resulting in

( )( ).FJJrm

,FrJJrm

ttPT

ttPT

i i021222221

i i011121211

ee2

i

ee2

i

ωω

ωω

δααδΩαδδ

δαδΩαδδ

+=+−+

+=+−+

&&&&&

&&&&&

(2.114)

Because the forward and backward solutions are decoupled, they can be considered separately [15].

For the forward excitation component and solutions (2.110), the equations of motion are

⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧

⎥⎥⎦

⎢⎢⎣

+−−−−

21

110

22222

212

122

12112

2 1 1

δδ

αδωδωδωδωδωδω F

ˆr

JΩJmJJΩm

f

f

PT

TP . (2.115, a)

For the backward excitation component and solutions (2.111), the equations of motion are

⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧

⎥⎥⎦

⎢⎢⎣

−−−−−−

21

110

22222

212

122

12112

2 1 1

δδ

αδωδωδωδωδωδω F

ˆr

JΩJmJJΩm

b

b

PT

TP . (2.115, b)

The disc precession radius has a forward component

( )[ ]22110 1

2δΩωω

Δδ

PTf

f JJFr −−= (2.116, a)

and a backward component

( )[ ]22110 1

2δΩωω

Δδ

PTb

b JJFr +−= , (2.116, b)

where (2.91)

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DYNAMICS OF MACHINERY 86

2222

221

212

21211

2

1 1 δωδωδω

δωδωδωΔPT

TPf JΩJm

JJΩm+−−−−

= (2.117)

and

2222

221

212

21211

2

1 1 δωδωδωδωδωδωΔ

PT

TPb JΩJm

JJΩm−−−−−−

= . (2.118)

The disc precession orbit is an ellipse with the major and minor semiaxes

( )bf rra +=21 , ( )bf rrb −=

21 . (2.119)

Using notations (2.94), (2.95) and (2.97), the above solutions can be written

1

1

20

12

23

03

44

24

03

110

++−−

−+=

ηωΩηη

ωΩη

ηηωΩ

δ

aaaa

aaFrf , (2.120)

and

1

1

20

12

23

03

44

24

03

110

+−−+

−−=

ηωΩηη

ωΩη

ηηωΩ

δ

aaaa

aaFrb . (2.121)

Fig. 2.28

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2. SIMPLE ROTORS IN RIGID BEARINGS 87

For the rotor from Example 2.1, the major semiaxis of the disc unbalance response, ( )2110δFa , is plotted against excitation frequency in Fig. 2.28, with solid line, for 080 ωΩ .= and with broken line, for 040 ωΩ .= . For 080 ωΩ .= , the abscissae of the four peaks correspond to the four natural frequencies calculated in Example 2.1.

2.4.6 Campbell diagrams

The diagram of the natural frequencies of precession )(Ωωω ii = is usually plotted, in a condensed form, only in the first quadrant of the frame ΩO ω , as a Campbell diagram.

If the mirror images of the curves from the second quadrant of Figs. 2.23 and 2.24, corresponding to the backward precession, are drawn in the first quadrant, the Campbell diagrams from Fig. 2.29 are obtained. The notations are F for forward precession and B for backward precession. Note that the index of the natural frequencies has been changed.

a b

Fig. 2.29

The intersections with the line Ω=ω are the points whose abscissae define the critical angular speeds

icrΩ .

In practice, the excitation frequencies can be multiples of the running-speed frequency. The intersections of the curves representing the natural frequencies of forward precession with the lines of slope nΩ , where n = 1,2,3,4,....., help locating the possible critical angular speeds (Fig. 2.30). The

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DYNAMICS OF MACHINERY 88

magnitude of the response at the respective speed depends on both the magnitude of the forcing harmonic and the system damping.

Fig. 2.30

When the critical speeds are determined using the Campbell diagram, the remark made in Section 2.4.4 has to be taken into account: the unbalance can produce large deflections only at or near a forward critical speed.

Fig. 2.31

For the rotor from Example 2.1, the Campbell diagram is shown in Fig. 2.31. The synchronous excitation line is drawn with broken line.

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2. SIMPLE ROTORS IN RIGID BEARINGS 89

Example 2.2

Determine the mode shapes and plot the Campbell diagram for the cantilevered rotor shown in Fig. 2.32. Take 21602 lm.JJ TP == , 080 ωΩ .= .

Fig. 2.32

The flexibility matrix is

[ ]⎥⎥⎦

⎢⎢⎣

⎡=

6332

6

2

l

lll

IEδ .

The frequency equation (2.91, a) is

( )( ) 0

61 .2401151

61 .120 1 2

2

=−−−

−−−

..

.

ηηη

ηηη

l

l

or 0138402410960060 234 =++−− ηηηη ....

with roots

786801 .−=η , 049112 .=η , 874233 .−=η , 211954 .=η .

The corresponding mode shapes are defined by

l11 69051 R.A = , l22 44971 R.A = , l33 50475 R.A −= , l44 582111 R.A −= .

The four natural frequencies are

01 78680 ωω .−= , 02 04911 ωω .= , 03 87423 ωω .−= , 04 21195 ωω .= .

The mode shapes are shown in Fig. 2.33, considering l10.R i = .

For a stationary shaft ( )0=Ω , the characteristic equation is

050623 24 =+− ηη , with roots

916900201 .==− ηη , 452640403 .==− ηη .

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DYNAMICS OF MACHINERY 90

Fig. 2.33

Fig. 2.34

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2. SIMPLE ROTORS IN RIGID BEARINGS 91

The corresponding mode shapes are defined by

l010201 57971 R.AA == , l030403 91307 R.AA −== .

The Campbell diagram is illustrated in Fig. 2.34.

Example 2.3

Determine the precession mode shapes and plot the Campbell diagram for the simply supported rotor with overhang disc shown in Fig. 2.35. Take

21602 lm.JJ TP == , l20.c = , 080 ωΩ .= .

Fig. 2.35

The flexibility matrix is

[ ]( ) ( )

( ) ⎥⎥⎦

⎢⎢⎣

⎡=

⎥⎥⎥

⎢⎢⎢

++

++=

612602600480

33322

322

31 2

2

....

IEccc

cccc

IE l

lll

ll

llδ .

The characteristic equation (2.96) is

0126674666735111031940 234 =++−− ηηηη ....

with roots

200801 .−=η , 327312 .=η , 198533 .−=η , 672134 .=η .

The corresponding precession mode shapes are defined by

l11 12416 R.A = , l22 85524 R.A = , l33 38791 R.A −= , l44 78633 R.A −= .

The four natural frequencies are

01 20080 ωω .−= , 02 32731 ωω .= , 03 19853 ωω .−= , 04 67213 ωω .= .

The mode shapes are shown in Fig. 2.36, considering l10.R i = .

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DYNAMICS OF MACHINERY 92

Fig. 2.36

For a stationary shaft ( )0=Ω , the characteristic equation is

013043478311 24 =+− .. ηη , with roots

528700201 .==− ηη , 346430403 .==− ηη .

Fig. 2.37

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2. SIMPLE ROTORS IN RIGID BEARINGS 93

The corresponding mode shapes are defined by

l010201 94785 R.AA == , l030403 10162 R.AA −== .

The Campbell diagram is given in Fig. 2.37.

Example 2.4

Determine the precession mode shapes and plot the Campbell diagram for the symmetric rotor with a disc at the middle shown in Fig. 2.38. Take

21602 lm.JJ TP == , 080 ωΩ .= .

Fig. 2.38

This is a Laval-Jeffcott rotor with includes the effects of the disc mass moments of inertia. The translational and rotational motions of the disc are elastically decoupled, since the disc is located at the centre of the shaft.

The translational equations of the free motion are

⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡+

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡00

00

00

zy

kk

zy

mm

T

T

&&

&&,

where 348 lIEkT = is the translational stiffness.

The rotational equations of the free motion are

⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡+

⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡−

+⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡00

00

00

00

ψϕ

ψϕ

Ωψϕ

R

R

P

P

T

T

kk

JJ

JJ

&

&

&&

&&,

where lIEkR 12= is the rotational stiffness.

When the rotor is not rotating ( )0=Ω , there are two independent modes of lateral vibration of equal natural frequency mkTT =ω . One of the modes is a straight-line vibration in the z direction, and the other is a straight-line vibration in the y direction (Fig. 2.39).

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DYNAMICS OF MACHINERY 94

Fig. 2.39

If these two modes are given the same amplitude, they can be superposed with proper phasing to form a circular precession mode which is either forward or backward with respect to the rotor spin. For this model, these frequencies are independent of the rotor spin speed, T, ωω ±=21 .

For the angular motion, substituting solutions tt , i i e e ωω ΨψΦϕ ==

into the equations of motion, we obtain the following homogeneous algebraic equations

.JkJ

,JJk

RP

PR

0)( i

0 i)(

T2

T2

=−+−

=+−

ΨωΦΩω

ΨΩωΦω

The characteristic equation is

( ) 0)( 22T

2 =−− PR JJk Ωωω .

Denoting [17]

T

RR J

k=2ω ,

T

PJJ

=γ , ( )222

41 ΩγωωΩ += R ,

we obtain the natural frequencies

Ωγωω Ω 21

43 m=, ,

or

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2. SIMPLE ROTORS IN RIGID BEARINGS 95

T

R

T

P

T

P, J

kJJ

JJ

+⎟⎟⎠

⎞⎜⎜⎝

⎛±=

2

43 22ΩΩω .

Figure 2.40 shows the dependence of the two natural frequencies on the rotor speed. The synchronous excitation line as well as the asymptotes Ωγω =

and Ωγω21

= are also drawn in the figure.

Fig. 2.40

The synchronous line intersects the two natural frequency lines at

γωΩ += 13 R , γωΩ −= 14 R .

The mode shapes are given by the amplitude ratio

i

i

T2

±=−

=P

RJ

JkΩωω

ΦΨ .

For 3ωω = , ( ) i3 −=ΦΨ , the precession is backward. For 4ωω = , ( ) i4 +=ΦΨ , the precession is forward. The deflected shaft is planar. The rotation axis of the disc describes a cone with circular cross-section.

For a comparison with the previous examples, a solution based on notations (2.94), (2.95) and (2.97) is given in the following.

The flexibility matrix is

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DYNAMICS OF MACHINERY 96

[ ]⎥⎥⎦

⎢⎢⎣

⎡=

2500006250

3

2

..

IEllδ .

The characteristic equation (2.96) is

( ) ( ) 0115120320 22 =−−− ηηη ..

with roots

11 −=η , 12 =η , 140413 .−=η , 740424 .=η .

For a stationary shaft ( )0=Ω , the characteristic equation is

( ) ( ) 011253 22 =−− ηη . , with roots

10201 ±=,η , 767810403 ., ±=η .

Fig. 2.42

The Campbell diagram is shown in Fig. 2.42. The frequency ratio, 0ωω i , is plotted versus the speed ratio, 0ωΩ , where Tωω =0 . The overlaid lines of the translatory modes, T, ωω ±=21 , correspond to the Laval-Jeffcott rotor model. The only one critical speed excited by rotating unbalance is located at the intersection of the synchronous excitation line (dotted) with the line 2ω at Tcr ωΩ = .

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2. SIMPLE ROTORS IN RIGID BEARINGS 97

2.4.7 Effect of the gyroscopic torque on critical speeds

According to equations (2.76) and (2.75, a), in the case of forward precession, the disc acts on the shaft with a torque of components

.ΩJJM

,ΩJJM

GPGTC

GPGTC

z

y

) (

) (

ϕψ

ψϕ

&&&

&&&

−−=

+−= (2.122)

Generally, if the precession angular velocity is ω and the orbit is circular, as well as if the disc is perpendicular to the shaft axis, then

.t,t

CCG

CCG

cos sin

ωαψψωαϕϕ

==−==

(2.123)

The components of the torque applied to the shaft are

,JΩJM

,JΩJM

CTPC

CTPC

z

y

ψωω

ϕωω

2

2

⎟⎠⎞

⎜⎝⎛ −−=

⎟⎠⎞

⎜⎝⎛ −−=

(2.124)

and tend to decrease (for positive parenthesis) the slopes, hence to stiffen the shaft.

For synchronous forward precession, Ω =ω , and expressions (2.124) become

.ΩJJM

,ΩJJM

CPTC

CPTC

z

y

ψ

ϕ

2

2

)(

)(

−=

−= (2.125, a)

The gyroscopic torque produces an apparent decrease of TJ (or even the reverse effect) raising the critical speed.

For backward asynchronous precession, with ,Ω ω−= we obtain

,ΩJJM

,ΩJJM

CPTC

CPTC

z

y

ψ

ϕ

2

2

)(

)(

+=

+= (2.125, b)

so that there is an apparent increase of TJ , which lowers the respective critical speed.

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DYNAMICS OF MACHINERY 98

This explains the shape of the curves from figures 2.23 and 2.24, which indicate an increase of the forward precession natural frequencies and a decrease of the backward precession natural frequencies with the increase of the running speed.

2.4.8 Remarks on the precession of asymmetric rotors

The new phenomena introduced by the asymmetric rotor are the following:

a) The transverse disc inertia (rotary inertia) doubles the number of critical speeds. There is an 'inertial' effect, i.e. the reduction of the lowest natural frequency of the rotor in comparison to the natural frequency of the rotor with the disc modeled as a concentrated mass. The additional eigenvalue is associated with the additional rotational degree of freedom, viz. the disc rotation around its diameter. In the first mode of precession, the translatory and the angular motions are in phase. In the second mode, the translatory and the angular motions are out of phase. The rotary inertia effect acts also at zero running speed.

b) Due to the gyroscopic effects, the rotor natural frequencies depend on the running speed. Generally, gyroscopic torques double the number of natural frequencies. They occur in pairs corresponding to forward and backward precession. The gyroscopic effect does not act at zero running speed.

The number of critical speeds can be different from the number of natural frequencies of precession. Forward critical speeds can be encountered only in the case of corotating excitation. They are synchronous critical speeds. Backward critical speeds can be encountered only in the case of counter-rotating excitation. Backward critical speeds are referred to as asynchronous critical speeds. A constant direction harmonic force can produce both forward and backward precession.

The number of critical speeds depends on the disc inertia ratio TP J/J . For a thin disc, TP JJ > , there is only one forward synchronous critical speed. For a thick disc, TP JJ < , there are two critical speeds in forward synchronous precession. When TP JJ = , the system cannot pass through the second critical. Some drum washing machines are designed to have nearly equal axial and transverse moments of inertia. There are always two asynchronous critical speeds irrespective of the disc inertia ratio.

c) Discs mounted inclined on the shaft produce the so-called 'skew-unbalance' which is a source of synchronous rotor excitation analogous to the mass unbalance.

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2. SIMPLE ROTORS IN RIGID BEARINGS 99

Since ( ) α2sin21

TPxy JJJ −= , the effect of disc skewness can be

considered as a 'product of inertia' unbalance.

d) When the disc is attached at the middle of the shaft, it has a planar motion in the cylindrical modes of precession, there is no gyroscopic effect, the forward and backward natural frequencies coincide and are independent of the running speed. The corresponding curves in the Campbell diagram are overlapped straight lines.

e) For rotors in rigid bearings, all points move in circular orbits. The rotors have circular precession modes with planar deflected shapes.

f) Internal rotor damping from shrink-fit rubbing or material hysterezis can produce rotor instability. Below the onset speed of instability the rotor's motion is stable and synchronous. Above this speed, destabilizing forces which are normal to the radial displacement, and in the direction of shaft rotation, produce large whirl amplitudes which may result in damaged or destroyed equipment. Whirl amplitudes grow until they achieve a steady-state limit cycle.

References

1. Föppl, A., Das Problem der Lavalschen Turbinenwelle, Der Civilingenieur, Vol.4, pp.335-342, 1895.

2. Jeffcott, N., Lateral vibration of loaded shafts in the neighbourhood of a whirling speed - The effect of want of balance, Philosophical Magazine, Series 6, Vol.37, pp.304-314, 1919.

3. Childs, D., Turbomachinery Rotordynamics. Phenomena, Modeling and Analysis, Wiley, New York, 1993.

4. Stodola, A., Neuere Beobachtungen uber die Kritischen Umlaufzahlen von Wellen, Schweizer.Bauzeitung, Vol.68, pp.210-214, 1916.

5. Green, R., Gyroscopic effects on the critical speeds of flexible rotors, J. Appl. Mech, Vol.15, pp 369-376, 1948.

6. Gasch, R. and Pfützner, H., Rotordynamik, Springer, Berlin, 1975.

7. Kimball, A. L., Jr., Measurement of internal friction in a revolving deflected shaft, General Electric Review, Vol.28, No.8, pp.554-558, Aug 1925.

8. Newkirk, B. L., Shaft whipping, General Electric Review, Vol.27, p.169, 1924.

9. Gunter, E. J., Jr., and Trumpler, P. R., The influence of internal friction on the stability of high speed rotor with anisotropic supports, ASME Journal of Engineering for Industry, Series B, Vol.87, pp.1105-1113, Nov 1969.

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DYNAMICS OF MACHINERY 100

10. Kimball, A. L., Internal friction as a cause of shaft whirling, Philosophical Magazine, Vol.49, pp.724-727, 1925.

11. Robertson, D., Transient whirling of a rotor, Philosophical Magazine, Series 7, Vol.20, p.793, 1935.

12. Gunter, E. J., Jr., The influence of internal friction on the stability of high speed rotors, ASME Journal of Engineering for Industry, Series B, Vol.89, pp.683-688, Nov 1967.

13. Dimentberg, F., Flexural Vibrations of Rotating Shafts, Butterworths, London, 1961.

14. Bishop, R. E. D., and Parkinson, A. G., Vibration and balancing of flexible shafts, Appl. Mech. Reviews, Vol.21, No.5, 1968, pp.439-451.

15. Gasch, R., Nordmann, R., and Pfützner, H., Rotordynamik, Springer, Berlin, 2002.

16. Krämer, E., Dynamics of Rotors and Foundations, Springer, Berlin, 1993.

17. Ewins, D. J., Modal Testing: Theory, Practice and Applications, 2nd ed., Research Studies Press, Baldock, 2000.

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3. SIMPLE ROTORS IN FLEXIBLE BEARINGS

This chapter considers the effect of bearing flexibility and damping on the precession of flexible rotors. Only single-disc rotors supported in flexible bearings will be examined. Both isotropic and orthotropic constant parameter bearings, as well as hydrodynamic bearings with speed-dependent spring and damping coefficients, are considered.

3.1 Symmetric rotors in flexible bearings

This section considers single-disc rotor models, with a radially and longitudinally symmetric shaft, supported in identical flexible bearings and/or bearing supports. The disc rotary inertia is neglected and only the planar translatory disc precession is analyzed.

a b Fig. 3.1

It is supposed that the bearing flexibilities and damping are uncoupled as in rolling bearings on flexible supports, tilting pad journal bearings and squeeze-film supports with retaining springs designed to operate in the linear range.

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DYNAMICS OF MACHINERY 102

3.1.1 Effect of bearing flexibility

Consider a symmetric rotor supported in two identical anisotropic flexible bearings (Fig. 3.1, a). Such bearings have two principal directions of stiffness along which the radial stiffness has extreme values. For orthotropic bearings only the principal stiffnesses are considered.

The axes Oy and Oz are along the bearing principal directions of stiffness. Let 1k and 2k be the principal stiffnesses.

Let BB z,y be the components of the displacement of the journal centre along the axes of the stationary reference frame Oxyz (Fig. 3.1, b). The other notations are as for the rotors in rigid bearings (Fig. 2.3).

a b

Fig. 3.2

3.1.1.1 Equations of motion

Using d'Alembert's principle, the dynamic equilibrium of forces and torques acting on the disc (Fig. 3.2, a) is written as [1]:

( )( )( ) ( ) ( ) ,tMezzkeyykJ

,zzkzm,yykym

BCBCG

BCG

BCG

=−−−+

=−+=−+

θθθ cossin

00

&&

&&

&&

(3.1)

and the equilibrium of forces acting on the shaft (Fig. 3.2, b):

( )( ).zzkzk

,yykyk

BCB

BCB

−=−=

2

1

22

(3.2)

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3. SIMPLE ROTORS IN FLEXIBLE BEARINGS 103

The coordinates of points C and G are related by

.ezz,eyy

CG

CG

θθ

sincos

+=+=

(3.3)

In steady-state conditions, if ( ) 0=tM , then θ&& ≅0, the angular speed θ& =Ω =const. and the angular position 0 θΩθ += t . By a convenient selection of the time origin,

tΩθ = . (3.4)

Eliminating coordinates BB z,y and GG z,y between equations (3.1)-(3.3), and taking into account (3.4), the equations of motion of point C can be written

,temzkzm

,temykym

CzC

CyC

sin

cos 2

2

ΩΩ

ΩΩ

=+

=+

&&

&& (3.5)

where

kkkkk,

kkkkk zy +

=+

=2

2

1

122

22 . (3.6)

Equations (3.5) differ from equations (2.5), established for rotors in rigid bearings, only by the equivalent stiffnesses (3.6), which are different along Oy and Oz.

Because of the system symmetry, bearings are represented by springs connected in parallel, and the flexible shaft is connected in series with the bearings. The equivalent stiffnesses are computed from

.kkk

,kkk zy 21 2

111 2111

+=+=

Substituting k by yk in the first equation (2.5), and by zk in the second equation (2.5), we obtain equations (3.5).

The complete solutions of equations (3.5) are

,tetZtz

,tetYty

zzzCC

yyyCC

sin )(sin )(

cos )(cos )(

22

2

22

2

ΩΩω

Ωθω

ΩΩω

Ωθω

−++=

−++=

(3.7)

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DYNAMICS OF MACHINERY 104

where

mk,

mk z

zy

y == ωω (3.8)

are the natural frequencies of the lateral vibrations along Oy, and Oz, respectively.

Generally zy ωω ≠ and if 12 kk < then

mknyz =<< ωωω .

The natural frequencies of the rotor in flexible bearings are lower than the natural frequency of the rotor in rigid bearings.

3.1.1.2 Unbalance response

The steady-state motion is described by the particular solutions of equations (3.5)

.tetz)t(z

,tety)t(y

zCC

yCC

sin sin

cos cos

22

2

22

2

ΩΩω

ΩΩ

ΩΩω

ΩΩ

−==

−==

(3.9)

Figure 3.3 shows the magnitudes of the two motion components of point C as a function of the angular speed Ω . When zωΩ = and yωΩ = , the amplitude grows unbounded. The Laval-Jeffcott rotor in orthotropic bearings has two critical speeds.

Because ,ˆˆ CC zy ≠ equations (3.9) describe an ellipse. Eliminating the time between the two equations yields

122

=⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎞⎜⎜⎝

C

C

C

Czz

yy . (3.10)

The orbit of point C is an ellipse whose axes are collinear with the bearing principal stiffness axes.

Point C completes the ellipse in a time interval Ωπ2=T , equal to the disc rotation period, hence its motion is a synchronous precession.

At speeds zωΩ < and yωΩ > the point C moves along the ellipse in the same direction as the disc running speed; the precession is forward. At speeds

yz ωΩω << the point C moves along the ellipse in the opposite direction; the precession is backward.

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3. SIMPLE ROTORS IN FLEXIBLE BEARINGS 105

At *ΩΩ = and ,** ∞→=ΩΩ the orbit is circular. At speeds *ΩΩ < the ellipse major semiaxis is collinear with Oz and at speeds *ΩΩ > it

is collinear with Oy.

The bearing orthotropy doubles the number of critical speeds and produces the synchronous precession with elliptical orbits.

Fig. 3.3

Although the precession is synchronous, the angular velocity of the point C along the ellipse is variable, and Ω is the angular velocity in the circular motions that generate the ellipse.

As the rotor moves along the elliptical orbit, it speeds up or slows down to conserve energy and angular momentum. The precession speed is not the angular speed of the rotor along the ellipse. It is equal to the constant angular speed of the forward and backward uniform circular motions that, compounded, generate the elliptical motion.

Using complex representation, the radius vector of the disc centre precession orbit can be written as

tztyzyr CCCCC ΩΩ sin i cos i +=+= , (3.11) or

( ) ( )ttCttCC

zyr i i i i eei2

i ee2

ΩΩΩΩ −− −++= ,

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DYNAMICS OF MACHINERY 106

,rrzyzyr tb

tf

tCCtCCC

i i i i eee2

e2

ΩΩΩΩ −− +=−

++

= (3.11, a)

( ) ( ) trrtrrr bfbfC ΩΩ sin i cos −++= , (3.11, b) where

.er

,er

zy

zyb

zy

zyf

)()(2

)()(

22

2222

222

2222

2222

ΩωΩω

ωωΩ

ΩωΩω

ΩωωΩ

−−

−−=

−−

−+=

(3.12)

The first term in equation (3.11, a) represents (in the complex plane) a vector of length fr which rotates in the same direction as the rotor rotation. The second term represents a vector of length br which rotates in the opposite direction, with the same angular speed. Addition of the two circular counter-rotating motions ( ).const=Ω yields an ellipse (Fig. 3.4).

Fig. 3.4

At 0=t , ===+= ayrrr CbfC )0( major semiaxis. At =t π /(2Ω ), ===−= bzrr/r CbfC )2( Ωπ minor semiaxis.

The direction of rotation of vectors Cr (hence of the motion of point C along the ellipse) depends on the relative magnitude of the two vector components.

If ,rr bf > then the point C 'rotates' in the same direction as the disc;

its motion is a forward precession. If ,rr bf < then the point C 'rotates' in the

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3. SIMPLE ROTORS IN FLEXIBLE BEARINGS 107

opposite direction, the motion is a backward precession. If bf rr = , the ellipse

degenerates to a line and point C has a rectilinear harmonic motion.

Based on Fig. 3.3, where ,yz ωω < and on equations (3.12), the following can be said. For speeds zωΩ < and ,yωΩ > the precession is forward when

bf rr > . For speeds ,yz ωΩω << the precession is backward when

bf rr < . At zωΩ = the major semiaxis becomes (theoretically) infinite and as

the running speed traverses the rotor first critical speed, the motion changes from forward to backward precession. Analogously, at yωΩ = , when the running speed traverses the second critical speed, the backward precession changes into a forward precession.

The motion along elliptical orbits produces variable stresses in the shaft even at constant running speed. During the synchronous forward precession, the part of the cross-section in tension remains in tension, and the part in compression remains in compression, but the bending stresses vary cyclically due to the variation of the orbit radius. During the backward precession, the bending stresses vary in an alternating non-symmetric cycle, having two reversals per rotation (Fig. 3.3).

The motion of journal centres is defined by the variation in time of the coordinates of point B (Fig. 3.1). Equations (3.2) yield

.zkzkk,ykykk CBCB =+=+ )2( )2( 21 (3.13)

Based on equations (3.9), the steady-speed solution is

.tekk

ktz

,tekk

kty

zB

yB

sin2

)(

cos2

)(

22

2

2

22

2

1

ΩΩω

Ω

ΩΩω

Ω

−+=

−+=

(3.14)

Point B has an elliptic orbit, whose semiaxes are smaller than those of the point C. Points B and C have a synchronous motion, the largest amplitudes occurring at yω and zω , the rotor critical speeds.

Equations (3.3), (3.9) and (3.14) show that the points O, B, C and G are collinear. This is due to the neglecting of damping. As will be shown in the following, in damped rotors the lines BO , CB and GC are not collinear.

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DYNAMICS OF MACHINERY 108

When the shaft is much stiffer than the bearings, it can be considered that k→∞ and 21 2 2 kk,kk zy == . Equations (3.13) give CBCB zz,yy == . The disc centre precession orbit is identical to the precession orbit of the journal centres.

3.1.1.3 Natural modes of precession

Dropping the index C, the equations of the free precession, obtained for 0=e in (3.5), are

.zkzm

,ykym

z

y

0

0

=+

=+

&&

&& (3.15)

Using the complex representation

zyr i+= , zyr i−= , (3.16)

equations (3.15) become

,rkrkrm 0 =++ Δ&& (3.17)

where

2

zy kkk

+= , 0

2>

−= zy kk

kΔ . (3.18)

The precession behaviour can be analyzed in terms of the forward and backward componets of the motion. Substituting

tb

tf rrr i i ee ωω −+= , t

ft

b rrr i i ee ωω −+= (3.19)

into (3.17), we obtain the homogeneous set of equations

( )

( ) .rmkrk

,rkrmk

bf

bf

0

02

2

=−+

=+−

ωΔ

Δω (3.20)

The characteristic equation is

( ) ( ) 0222 =−− kmk Δω , (3.21)

m

kk Δω m=2 . (3.21, a)

The natural frequencies are

2221 z

z, m

km

kk ωΔω ==−

= , 2243 y

y, m

km

kk ωΔω ==+

= . (3.22)

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3. SIMPLE ROTORS IN FLEXIBLE BEARINGS 109

From the first equation (3.20) we obtain the amplitude ratio

k

mkrr

f

bΔω2−

−= . (3.23)

For z, ωω ±=21 , omitting indices, we obtain

fb rr −= , 0=+= bf rry , brrz bf =−= . (3.24, a)

The motion is a (horizontal) vibration along the z-axis, with amplitude b.

For y, ωω ±=43 , omitting indices, we obtain

fb rr += , arry bf =+= , 0=−= bf rrz . (3.24, b)

The motion is a (vertical) vibration along the y-axis, with amplitude a.

The modal orbits of the four natural modes of precession degenerate into straight lines. They may be thought of as being made up of two circular orbits of equal radii, where one has forward motion and the other one has backward motion.

3.1.2 Effect of external damping

In this section, the effect of external damping on the response of the flexibly supported rotor is considered. In a first approximation, it is assumed that the external damping is isotropic and viscous, giving rise to forces which are proportional to the disc absolute velocity. The main effects are the finite amplitude steady state response to unbalance and the inclination of precession elliptical orbits.

3.1.2.1 Unbalance response

For the calculation of the rotor damped precession, new terms proportional to the disc centre velocity CC zc,yc && are added in equations (3.5).

The following equations of motion are obtained:

.temzkzczm

,temykycym

CzCC

CyCC

sin

cos 2

2

ΩΩ

ΩΩ

=++

=++

&&&

&&& (3.25)

In steady motion, the solutions of equations (3.25) are

,tzz

,tyy

zCC

yCC

) (sin

) (cos

θΩ

θΩ

+=

+= (3.26)

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DYNAMICS OF MACHINERY 110

where

22

2

2

2

2

22

2

2

2

2

21

21

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛−

=

⎟⎟⎠

⎞⎜⎜⎝

⎛+

⎟⎟

⎜⎜

⎛−

=

zz

z

zC

yy

y

yC

ez,

e

y

ωΩζ

ωΩ

ωΩ

ωΩζ

ωΩ

ωΩ

(3.27)

2

2

2

21

2 tan

1

2tan

z

zz

z

y

yy

y ,

ωΩωΩζ

θ

ωΩ

ωΩζ

θ−

−=

= (3.28)

In equations (3.27) and (3.28), the notations (3.8) and

zz

zyy

y mc

mkc,

mc

mkc

ωζ

ωζ

22

22==== (3.29)

have been used.

In the following, for the simplicity of presentation, the bearing vertical stiffness is taken four times larger than the horizontal stiffness [2]:

kk,kk41 21 == .

This gives

kk,kk zy 31

32

== .

The system vertical total stiffness is two times the horizontal total stiffness.

The external viscous damping coefficient is taken [2]

.mk.m.c n 210210 ⋅=⋅= ω

The rotor undamped natural frequencies are

.577031 8160

32

nzny .mk,.

mk ωωωω ====

The damping ratios are

...

.mc,.

..

mc

zz

yy 170

577010

2 120

816010

2======

ωζ

ωζ

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3. SIMPLE ROTORS IN FLEXIBLE BEARINGS 111

Using the dimensionless frequency

nωΩη = , (3.30)

equations (3.27) and (3.28) become

2

22

2

22

2

2

)20(31

) 20(32 ηη

η

ηη

η

.e

z,

.e

y CC

+⎟⎠⎞

⎜⎝⎛ −

=

+⎟⎠⎞

⎜⎝⎛ −

= (3.31)

..,.zy

22

31

20 tan

32

20tanη

ηθη

ηθ−

−=

−= (3.32)

Figure 3.5, a illustrates the speed-dependence of the disc unbalance response components, based on equations (3.31). Unlike the curves from Fig. 3.3, plotted for the undamped rotor, finite amplitudes result at the critical speeds:

..e

z,.e

y

.

C

.

C 8852 08457708160

=⎟⎠⎞

⎜⎝⎛=⎟

⎠⎞

⎜⎝⎛

== ηη

a b

Fig. 3.5

The speed-dependence of the phase angles yθ and zθ is illustrated

in Fig. 3.5, b. The speeds where the phase difference is 90o are denoted ∗1Ω and

∗2Ω . As shown in the following, the phase difference yz θθθΔ −= yields

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DYNAMICS OF MACHINERY 112

inclined elliptical orbits for the precession of damped rotors, in contrast with the undamped rotors whose elliptical orbits have vertical and horizontal semiaxes.

3.1.2.2 Disc precession orbit

Substitution of Cy and Cz by y and z in equations (3.26) yields

.tztztztzz

,tytytytyy

sczCzC

scyCyC

sin cos cossin sincos

sin cos sinsin coscos

ΩΩΩθΩθ

ΩΩΩθΩθ

+=+=

+=−= (3.33)

Equations (3.33) define an ellipse. Elimination of time gives the orbit equation

.zyzyzyyzyzyzyyzz sccsscssccsc2222222 )()()(2)( −=+++−+ (3.34)

Equation (3.34) is more often expressed in terms of the major and minor semiaxes, a and b, and the inclination angle α .

In a principal coordinate frame 11Ozy , taking the 1Oy and 1Oz axes along the ellipse axes (Fig. 3.6), the motion is described by

,tbz,tay

) (sin ) ( cos

1

1

αγΩαγΩ

−+=−+=

(3.35)

where γ is the phase angle at 0=t .

Fig. 3.6

The ellipse equation in principal coordinates is

.bz

ay 1

21

21 =⎟

⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛ (3.36)

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3. SIMPLE ROTORS IN FLEXIBLE BEARINGS 113

The coordinate transformation

,zyz,zyy

αααα

cossinsincos

11

11

+=−=

(3.37)

leads to parametric equations of the form (3.33).

Combining equations (3.33), (3.35) and (3.37) it is possible to obtain a, b, and α in terms of .z,z,y,y scsc The result is

222222

22222

)()(41+

)(21

csscscsc

scsc

zyzyzzyy

zzyya

−−+++

++++=, (3.38)

,zyzya

b cssc )( 1−= (3.39)

.zzyy

zyzy

scsc

sscc

)()(22tan 2222 +−+

+=α (3.40)

Using notations (3.33), the equations of motion (3.25) become

⎪⎪⎭

⎪⎪⎬

⎪⎪⎩

⎪⎪⎨

=

⎪⎪⎭

⎪⎪⎬

⎪⎪⎩

⎪⎪⎨

⎥⎥⎥⎥⎥

⎢⎢⎢⎢⎢

−−

−−−

1001

0000

0000

2

2

2

2

2

Ωem

zyzy

mΩkcΩmΩkcΩ

cΩmΩkcΩmΩk

s

s

c

c

z

y

z

y

. (3.41)

They are two-by-two decoupled.

In the considered particular case, equations (3.41) become

⎪⎪⎭

⎪⎪⎬

⎪⎪⎩

⎪⎪⎨

=

⎪⎪⎭

⎪⎪⎬

⎪⎪⎩

⎪⎪⎨

⎥⎥⎥⎥⎥

⎢⎢⎢⎢⎢

−−−−

−−

1001

)31(02000)32(020200)31(00200)32(

2

2

2

2

2

e

zyzy

ηη.ηη.

η.ηη.η

s

s

c

c

η .

The four ellipse parameters are

1

22

32

Δ

ηη ⎟⎠⎞

⎜⎝⎛ −

=eyc ,

1

220Δ

ηη ⋅=

.eys ,

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DYNAMICS OF MACHINERY 114

2

220Δ

ηη ⋅−=

.ezc ,

2

22

31

Δ

ηη ⎟⎠⎞

⎜⎝⎛ −

=ezs ,

where

)20()31( )20()32( 2222

2221 ηηΔηηΔ .,. +−=+−= .

The inclination of the major axis is given by

..221

402tan ηηα

−=

The minor semiaxis is zero, ,b 0= when

.yy

zz

c

s

c

s = (3.42)

Equation (3.33) gives

,yy,

zz

yc

sz

s

c θθ tan tan −==

and condition (3.42) becomes

yz θ

θtan

1tan −= (3.43)

or

.yz 2πθθ =−

When the phase difference yz θθθΔ −= between the projections of

the precession motion on the axes Oy and Oz is 090 , the elliptic orbit degenerates into a straight line. In fact the two motions are in phase and the 090 angle shows the spatial lag between the two directions. Condition (3.43) defines the limits between forward and backward precession. On inserting (3.28) into (3.43) we obtain the threshold angular speeds ∗

1Ω and ∗2Ω .

Figure 3.5, b shows that there are two speeds at which condition (3.43) holds and these are different from the peak response critical speeds. For undamped rotors (Fig. 3.3), the change from forward to backward precession and vice versa takes place at the system undamped natural frequencies, hence at the undamped critical speeds. For damped rotors, the precession reversal, possible only when the orbit degenerates into a straight line, occurs at speeds which are different from the

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3. SIMPLE ROTORS IN FLEXIBLE BEARINGS 115

peak response critical speeds, where the motion components have maximum amplitude.

Fig. 3.7 (from [2])

In the considered particular case, substitution of (3.32) into (3.43) yields

092960 24 =+− ηη , ,

with solutions

.,, 7550 , 6240 21 == ∗∗ ηη

Figure 3.7 depicts the orbits at several rotor speeds.

Figure 3.8 shows the rotor unbalance response presented as diagrams of the ellipse semiaxes as a function of running speed.

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DYNAMICS OF MACHINERY 116

Peak response critical speeds are located at the peaks in the major semiaxis curve e/a . The minor semiaxis curve, e/b , crosses the speed axis at the threshold speeds between forward and backward precession, ∗

1Ω and ∗2Ω .

Fig. 3.8

3.1.2.3 Decomposition into two circular motions

If the motion along the ellipse is represented as the sum of two counter-rotating circular motions, as in equation (3.11, a), then

2

2bar,bar bf

−=

+= (3.44)

and, unlike the Fig. 3.4, vectors fr and br have non-zero phase angles at .t 0=

Figure 3.9 shows the diagrams of radii fr and br as a function of

speed for the analyzed system. Because for bf rr > the precession is forward,

and for bf rr < the precession is backward, the intersections of the two curves

locate (for bf rr = ) the threshold (dimensionless) speeds ∗1η and ∗

2η .

Resuming, the unbalance response can be illustrated by three kinds of frequency response diagrams: a) diagrams of the motion projections y and z onto the coordinate axes (Fig. 3.5, a); b) diagrams of the semiaxes a and b of the elliptic

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3. SIMPLE ROTORS IN FLEXIBLE BEARINGS 117

orbit (Fig. 3.8); and c) diagrams of the radii of circular motions which generate the ellipse (Fig. 3.9).

Fig. 3.9

Because the maximum relative displacement between rotor and stator is given by the major semiaxis a, the diagram from Fig. 3.8 is the most useful in practice. It is used together with Fig. 3.7 which represents the evolution of the precession orbit with the change of speed.

3.1.3 Effect of external and internal damping

Considering both external and internal damping, the equations of motion of the disc centre with respect to the stationary coordinate system become

.tΩΩemzkyΩczcczm

,tΩΩemykzΩcyccym

CzCiCieC

CyCiCieC

)(sin)(

)(cos)(

02

02

θ

θ

+=+−++

+=++++

&&&

&&& (3.45)

Denoting

2zy kk

k+

= , 2

zy kkk

−=Δ ,

kkq Δ

= , mk

n =2ω ,

n

ii m

ζ2

= , n

ee m

ζ2

= , ie ζζζ += , (3.46)

equations (3.45) can be written in matrix form as

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DYNAMICS OF MACHINERY 118

( )( ) ⎭

⎬⎫

⎩⎨⎧

++

=⎭⎬⎫

⎩⎨⎧

⎥⎥⎦

⎢⎢⎣

−−+

+⎭⎬⎫

⎩⎨⎧

+⎭⎬⎫

⎩⎨⎧

) (sin ) ( cos

1 2 212

0

022

2

θΩθΩ

ΩωΩωζ

Ωωζωωζtt

ezy

qq

zy

zy

C

C

nni

nin

C

Cn

C

C

&

&

&&

&&.

(3.47)

Denoting

nωΛΛ = ,

nωΩη = , (3.48)

the study of the motion of the perfectly balanced rotor ( )0=e , leads to the characteristic equation

( ) ( ) 0 4141224 2222234 =−++++++ qi ηζΛζΛζΛζΛ . (3.49)

A comparison of equation (3.49) with (2.66) shows a difference only in the last term, due to the stiffness asymmetry coefficient q. Application of the Routh-Hourwitz criterion [1], yields the stability condition

0 44 2222 ≥+− qi ηζζ . (3.50)

The onset speed of instability is (Smith, 1933)

.q

ii

ens

22

21 ⎟⎟

⎞⎜⎜⎝

⎛+⎟⎟

⎞⎜⎜⎝

⎛+=

ζζζωΩ (3.51)

A comparison with equation (2.68) shows that the bearing support stiffness orthotropy can be used to increase the rotor onset speed of instability. For rotors supported in rolling bearings this is achieved with unequal support stiffnesses in two directions, while for hydrodynamic bearings - by increasing the eccentricity ratio.

The physical explanation of the effect of bearing stiffness orthotropy in restraining instability due to rotating damping is that “since the natural frequencies of the rotor system are different in the two principal transverse directions, there is no tendency to set up a whirl of the type which can be dragged forward by rotating damping until the rotating damping forces have been so far increased by rising speed that they are commensurate with the difference between elastic restoring forces in the two principal directions”.

Analysis of the unbalance response reveals that, with asymmetrical bearing stiffness, the amplitude of steady motion due to unbalance is restricted by both internal and external damping, but internal damping has smaller influence in this respect, especially if there is only slight dissymmetry of bearing stiffness.

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3. SIMPLE ROTORS IN FLEXIBLE BEARINGS 119

3.1.4 Effect of bearing damping

In order to reveal the effect of bearing damping on the dynamics of rotors, the rotor from Fig. 3.1 is supported in damped isotropic bearings as in Fig. 3.10. The bearings are assumed to have the same stiffness constant 1k in all radial directions. The bearing damping forces are assumed to be proportional to the journal absolute velocity. The viscous damping coefficients c are the same in all radial directions [3].

Fig. 3.10

At constant running speed Ω =const., the equations of motion are written as for undamped bearings (see § 3.1) but adding the damping forces. For the shaft:

)zz(kzkzc),yy(kykyc

BCBB

BCBB

−=+−=+

1

1

2 22 2

&

& (3.52)

and, for the disc

,zzkzm,yykym

BCG

BCG

0 )( 0 )(

=−+=−+

&&

&& (3.53)

where

.tΩezz,tΩeyy

CG

CG

sin cos

+=+=

(3.54)

Using complex notation

,zyr,zyr,zyr GGGCCCBBB i i i +=+=+= (3.55)

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DYNAMICS OF MACHINERY 120

equations (3.52)-(3.54) produce the equations of motion of the disc centre and journal centre

.Ωemrrkrm

,rrkrkrctΩ

BCC

CBBB i2

1

e )(

0 )(22

=−+

=−++

&&

& (3.56)

The natural frequency nω of the rotor in rigid bearings and the ratio N between the shaft stiffness and the support (bearings in parallel) stiffness are

12

kkN,

mk

n ==ω . (3.57)

The damping ratio

mkc

mc

n 22

22

==ω

ζ (3.58)

is defined with respect to the critical damping of the rigidly supported rotor.

The resulting equations of motion are

.Ωerrr

,rrrN

r

tΩBCnC

CBnBnBn

i22

22

e )(

0 )( 1 2

=−+

=−++

ω

ωωωζ

&&

& (3.59)

3.1.4.1 Damped natural frequency

For zero right-hand side in (3.59) and substituting solutions of the form

,Rr,Rr tCC

tBB

e e λλ == (3.60)

we obtain the homogeneous algebraic set of equations

( ) .RR

,RRN

CnBn

CnBnnn

0

012

222

222

=++−

=−⎟⎠⎞

⎜⎝⎛ ++

ωλω

ωωωλζω (3.61)

The requirement for non-trivial solutions is

011 2

222

22=

+−

−⎟⎠⎞

⎜⎝⎛ ++

nn

nnn Nωλω

ωωλζω.

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3. SIMPLE ROTORS IN FLEXIBLE BEARINGS 121

This yields the characteristic equation

.NN

N

nnn0

21

21

23

=++⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎞⎜⎜⎝

⎛ζω

λωλ

ζωλ (3.62)

If 0=ζ , then 1

1i+

=Nnω

λ . The critical speed of the rotor supported in

undamped flexible bearings is

.N

nel 1+=

ωω (3.63)

If ,0≠ζ then equation (3.62) has positive coefficients and can be written

.CBBAnnn

0 2 222

2=⎟

⎟⎠

⎞⎜⎜⎝

⎛+++⎟⎟

⎞⎜⎜⎝

⎛+

ωλ

ωλ

ωλ (3.64)

There is a negative real root A)( n −=1ωλ and two complex conjugate roots with negative real part CB)( ,n i32 ±−=ωλ , so that the system motion is always stable.

The free damped motion of point C is described by a solution of the form

.RRRtr tCtBC

tCtBC

tACC

nnnnn ωωωωω ii eeeee)(321

−−−− ++= (3.65)

The frequency of the damped free precession (of the perfectly balanced rotor) is

nd Cωω = (3.66)

where C is the imaginary part of the complex roots of the characteristic equation (3.62).

3.1.4.2 Unbalance response

For the steady-state motion due to mass unbalance, the solutions are of the form

,r~tr,r~tr tCC

tBB

ΩΩ ii e )( e )( == (3.67)

where Cr~ and Br~ are complex amplitudes.

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DYNAMICS OF MACHINERY 122

Using the dimensionless frequency (3.30) and substituting (3.67) into (3.59) we obtain

( ) .er~r~

,r~r~N

CB

CB

22 1

0 2 i 11

ηη

ζη

=−+−

=−⎥⎦

⎤⎢⎣

⎡+⎟

⎠⎞

⎜⎝⎛ +

(3.68)

The solutions are

( )

,

NN

er~B22

2

12 i 111 ηζηη

η

−+⎟⎠⎞

⎜⎝⎛ +−

= (3.69)

( )

.

NN

Ner~

n

C22

32

12 i 111

2 i 11

ηωΩζη

ζηη

−+⎟⎠⎞

⎜⎝⎛ +−

+⎟⎠⎞

⎜⎝⎛ +

= (3.70)

The motion of the journal centre B in the complex plane (Fig. 3.11) is represented by the vector BO .

Fig. 3.11

Its magnitude is

( )2222

2

2

1 411 ηηζη

η

−+⎟⎠⎞

⎜⎝⎛ +

=

NN

N

er~B (3.71)

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3. SIMPLE ROTORS IN FLEXIBLE BEARINGS 123

and the phase shift with respect to the unbalance vector GC is

( ).N

NN

B2

21

111 2tan

η

ηηζθ+

−−= − (3.72)

The motion of the disc centre C is represented by the vector CO , of magnitude

( )22222

2

6242

2

1 4 11

4)1(

ηηζη

ηζη

−+⎟⎠⎞

⎜⎝⎛ +

++

=

NN

N

NNe

r~C (3.73)

and phase angle

( )

.

NNζ

NN

N

C2222

2

1

1 1

4111

11 2tan

ηηη

ηηθ

−+

+⎟⎠⎞

⎜⎝⎛ +

⎟⎠⎞

⎜⎝⎛

+−−

= − (3.74)

The points B and C have circular orbits around the point O, but the points O, B, C and G are not collinear. The vector CO has a phase lag Cθ with respect to the excitation vector GC and the vector BO has a phase lag Bθ with respect to

GC (Fig. 3.11).

If the radii of precession orbits (3.71) and (3.73) are plotted against the dimensionless speed nωΩ , for given values of N and ζ , the peak values of the displacements of points B and C occur at speeds ,BΩ and CΩ respectively, different from elω and dω (Fig. 3.12).

Differentiating with respect to 2η the expressions of these displacements, yields two different equations.

The condition of maximum journal displacement

0)(d

d2 =Br~η

gives the equation

,NNN

01 11122 22262 =−⎥

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ +−− ηζηζ (3.75)

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DYNAMICS OF MACHINERY 124

Fig. 3.12 (from [3])

and the condition of maximum disc centre displacement

0)(d

d2 =Cr~η

gives the equation

.NN

NN

ζNN

NN

ζζζ

01111112 421

1124416

2

22

32

2

42264

=⎟⎠⎞

⎜⎝⎛ ++

⎥⎥⎦

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ +−⎟

⎠⎞

⎜⎝⎛ +++

+⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ +−+−

η

ηη (3.76)

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3. SIMPLE ROTORS IN FLEXIBLE BEARINGS 125

The physically acceptable solution of equation (3.75) is denoted )( nB ωΩ and that of equation (3.76) is denoted )( nC ωΩ .

Generally, the different critical speeds are in the following order

.nCBael ωΩΩωω <<<<

If the damping in the shaft is taken into account, it is possible that .nC ωΩ >

The angular speeds CΩ and ,BΩ at which the radial displacements of the damped rotor have maximum values, are referred to as peak response critical speeds. Sometimes, they considerably differ from the critical speed elω of the undamped system, being much larger. Therefore, a computation neglecting the bearing damping can result in erroneous values of the critical speeds.

Example 3.1

Consider the rotor from Fig. 3.10, with the following characteristics: disc mass m = 500 kg, shaft stiffness constant 102 5⋅=k N/mm, bearing stiffness k1

510= N/mm, bearing viscous damping coefficient c = 316.225 Ns/mm [3].

The computations yield:

1=N , 1=ζ ,

== mknω 632.45 rad/sec,

ω ωel n= =/ 2 447.2 rad/sec.

The critical speed of the rigidly supported rotor is

=nn rpm6040 .

The undamped critical speed of the rotor supported in flexible bearings is

=eln rpm4270 .

Equation (3.62) is written as

050)()()( 23 =+++ .nnn ωλωλωλ

and the roots are

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DYNAMICS OF MACHINERY 126

=nωλ1 64780.− , ...n, 86070 i1761032 ±−=ωλ

From the imaginary part we obtain

3554486070 .. nd == ωω rad/sec.

The rotor damped critical speed is

rpm5198=dn .

Equation (3.75) is written

01)(2 6 =−nωΩ ,

hence

4.5638909.0 == nB ωΩ rad/sec.

The peak response critical speed computed from the journal unbalance response is

rpm5380=Bn = 1.26 eln .

Equation (3.76) is written

02)(3)(8 26 =++− nn ωΩωΩ ,

hence

5749076.0 == nC ωΩ rad/sec.

The peak response critical speed computed from the disc centre unbalance response is

rpm5481=Cn = 1.28 eln .

3.1.4.3 Equivalent model

The dependence of critical speeds on the bearing damping can be simply explained noticing that the rotor-bearing system can be represented by the simplified model from Fig. 3.13.

The mass m is supported by a spring of stiffness constant k (representing the shaft), connected in series to an element consisting of the dashpot of constant 2c and the spring 12 k connected in parallel (representing the bearings). The mass

is acted upon by a force .Ωem)t(F tΩi2e=

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3. SIMPLE ROTORS IN FLEXIBLE BEARINGS 127

Fig. 3.13

Two limit cases are first considered. If the damping coefficient is infinite, ∞=c , the lower spring is blocked and the journal displacement is zero, .rB 0=

The frequency response curve of the disc centre (Fig. 3.14) has a(n infinite) peak at the natural frequency of the system consisting of the mass and the upper spring,

.mkn =ω The case corresponds to the rigidly supported rotor.

Fig. 3.14

If the damping coefficient is zero, 0=c , the springs of stiffness constants k and N/kk =12 are connected in series, and the equivalent spring rate is

.)N(k 1+

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DYNAMICS OF MACHINERY 128

The frequency response curve has a(n infinite) peak at the undamped critical speed .Nnel 1+=ωω The case corresponds to the rotor supported in undamped flexible bearings.

For intermediary values of the bearing damping coefficient, denoted c', and c", respectively, the frequency response curves have peaks at the peak response critical speeds crΩ′ , and crΩ ′′ , respectively, within the range ] [ nel ,ωω . They correspond to the rotor supported in damped flexible bearings.

Fig. 3.15

There is an optimum value optc of the bearing damping coefficient, for which the maximum precession amplitude has the lower value, equal to the ordinate of the crossing point of all frequency response curves, plotted for different values of the bearing damping coefficient.

The model from Fig. 3.13 can be replaced by an equivalent model, having a single spring in parallel with a dashpot (Fig. 3.15).

The equivalent stiffness constant echk and the equivalent viscous damping coefficient echc can be expressed in terms of the parameters k, c and N of the initial system as follows:

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3. SIMPLE ROTORS IN FLEXIBLE BEARINGS 129

,cΩ

NNk

cΩNNk

kkech22

2

22

222

2

4)1(

4)1(

++

++= (3.77)

.cΩ

NNk

kccech22

2

22

2

4)1(2

++

= (3.78)

Figure 3.15 shows the variation of these quantities as a function of the damping coefficient c.

The stiffness constant echk increases with c, hence the natural frequency mkech also increases with c, fact that explains the increase of critical speeds

with the bearing damping. The stiffness increase is higher when the natural frequencies elω and nω are relatively more distanced, hence when the ratio N (of the shaft stiffness to the bearing stiffness) is larger.

The equivalent viscous damping coefficient echc has a maximum value for the optimal c, fact that explains the lowest value of the maximum amplitude in this case.

Example 3.2

Consider a rigid rotor )( ∞→k supported by identical orthotropic bearings with the following characteristics:

rpm,60030==

mk

n yy π

rpm,50030==

mkn z

z π

161

2=

mk

c

y

y , 201

2=

mkc

z

z .

Plot the unbalance response diagrams and several precession orbits for an eccentricity μm10=e [4].

Figure 3.16, a shows the plot of major and minor semiaxes a and b, and forward and backward circle radii fr and br as a function of speed. Figure 3.16,

b shows the plot of the y and z displacement components and the minor semiaxis b versus speed, as well as the precession orbit at eight different speeds.

The range with backward precession is marked by the threshold speeds ∗1n and ∗

2n , where the orbit degenerates into straight lines.

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DYNAMICS OF MACHINERY 130

a

b

Fig. 3.16 (from [4])

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3. SIMPLE ROTORS IN FLEXIBLE BEARINGS 131

3.1.5 Combined effect of bearing damping and shaft mass

In the following, the effect of bearing damping on the dynamics of rotors is analyzed taking into account the shaft distributed mass. The analysis is simplified lumping the shaft mass at the ends of each half. This results in a three mass model, with a quarter of shaft mass at each bearing and half of the shaft mass at the disc location (Fig. 3.17).

At constant running speed Ω =const., the equations of motion are written as for undamped bearings (see § 3.1) but adding the damping forces. For the shaft:

)zz(kzmzkzc),yy(kymykyc

BCBBB

BCBBB

−=++−=++

&&&

&&&

11

11

22 222 2

(3.79)

and, for the disc

,zzkzmzm,yykymym

BCGC

BCGC

0 )(2 0 )(2

1

1

=−++=−++

&&&&

&&&& (3.80)

where

.tΩezz,tΩeyy

CG

CG

sin cos

+=+=

(3.81)

Fig. 3.17

Using the complex notation (3.55), equations (3.79)-(3.81) yield

( ) .Ωemrrkrmm

,rrkrkrcrmtΩ

BCC

CBBBB

i21

11

e )(2

0 )(222

=−++

=−+++

&&

&&& (3.82)

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DYNAMICS OF MACHINERY 132

Denoting the ratio of half the shaft mass to the disc mass

mm

mms 122

==μ , (3.83)

and using (3.57) and (3.58), we obtain the equations of motion

( ) .Ωerrr

,rrrN

rr

tΩBCnC

CBnBnBnB

i22

22

e )(1

0 )( 1 2

=−++

=−+++

ωμ

ωωωζμ

&&

&&& (3.84)

3.1.5.1 Damped natural frequency

For zero right-hand side in (3.84) and substituting solutions of the form (3.60), we obtain the homogeneous algebraic set of equations

( )[ ] .RR

,RRN

CnBn

CnBnnn

01

012

222

2222

=+++−

=−⎟⎠⎞

⎜⎝⎛ +++

λμωω

ωωωλζωλμ (3.85)

The requirement for non-trivial solutions yields the characteristic equation

( ) ( ) ( ) .NN

N

nnnn012 11121

234

=++⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡++

++⎟⎟

⎞⎜⎜⎝

⎛++⎟⎟

⎞⎜⎜⎝

⎛+

ωλζ

ωλμμ

ωλμζ

ωλμμ

(3.86)

We denote

dωαλ i+= , (3.87)

where α is a negative attenuation factor and dω is the damped natural frequency.

Two particular cases are considered: 1=N , 1=μ , and 52.N = , 1=μ , respectively, both corresponding to a relatively heavy shaft. Root locus diagrams are presented in Fig. 3.18, using dimensionless coordinates nωα and nd ωω .

For the rotor with 1=N (Fig. 3.18, a), the first damped natural frequency 1dω increases from nel . ωω 468201 = (for 0=ζ ) to nrig . ωω 7070= (for ∞=ζ ). The second natural frequency 2dω decreases from nel . ωω 510212 = (for 0=ζ ) to zero (for 3251.≅ζ ) when the second mode becomes overdamped.

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3. SIMPLE ROTORS IN FLEXIBLE BEARINGS 133

a b

Fig. 3.18 (from [5])

Fig. 3.19 (from [5])

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DYNAMICS OF MACHINERY 134

The rotor with 52.N = (Fig. 3.18, b) has a different behaviour. The second damped natural frequency, 2dω , decreases first from nel . ωω 337212 = (for

0=ζ ) to n. ω 6760 (for 1=ζ ) then increases to nrig . ωω 7070= (for ∞=ζ ). The first damped natural frequency 1dω diminishes from nel . ωω 334401 = (for 0=ζ ) to zero (for 980.≅ζ ) when the first precession mode becomes overdamped.

Figure 3.19 shows (for 1=μ ) the variation of the dimensionless damped natural frequencies nd ωω 1 (solid lines) and nd ωω 2 (broken lines) as a function of the bearing stiffness, for different values of bearing damping. Generally, the increase of ζ and N1 makes the first mode overdamped for values 1<ζ , and the second mode overdamped for 1>ζ . For 2=N and 1=ζ the system has equal eigenvalues.

3.1.5.2 Unbalance response

For the steady-state motion due to mass unbalance, substituting the solutions (3.67) into (3.84), we obtain the algebraic set of equations

( )[ ] .~1~

,0~~12i

2222

2222

ΩΩμωω

ωωωΩωζΩμ

err

rrN

CnBn

CnBnnn

=+−+−

=−⎟⎠⎞

⎜⎝⎛ +++−

(3.88)

The solutions to (3.88) are

( ),

N

er~

nnn

nB

111 2 i112

2

2

2

2

2

−⎥⎥⎦

⎢⎢⎣

⎡+−⎟

⎟⎠

⎞⎜⎜⎝

⎛+−+

=

ωΩμ

ωΩζ

ωΩμ

ωΩ

(3.89)

( ).

N

Ne

r~

nnn

nnnC

111 2 i11

2 i11

2

2

2

2

2

2

2

2

−⎥⎥⎦

⎢⎢⎣

⎡+−⎟

⎟⎠

⎞⎜⎜⎝

⎛+−+

⎟⎟⎠

⎞⎜⎜⎝

⎛+−+

=

ωΩμ

ωΩζ

ωΩμ

ωΩζ

ωΩμ

ωΩ

(3.90)

Figures 3.20 illustrate the variation of er~C and er~B as a function of

nωΩ , for 1=μ and 52.N = . With increasing ζ , the first response peak is shifted to higher speeds, while the second response peak is shifted to lower speeds.

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3. SIMPLE ROTORS IN FLEXIBLE BEARINGS 135

The peaks of the unbalance response diagrams occur at the angular speeds BΩ and CΩ . The peak response speeds BΩ are obtained from condition

( ) 0dd 2 =nBr~ ωΩ . The peak response speeds CΩ are obtained from condition

( ) 0dd 2 =nCr~ ωΩ .

For 1=μ , 1=N and 40.=ζ , we obtain nB . ωΩ 485101 = ,

nB . ωΩ 374512 = , nC . ωΩ 483701 = , nC . ωΩ 558912 = .

For these values of system parameters, the different critical angular speeds can be ordered as follows

22221111 CeldBrigBCdel ΩωωΩωΩΩωω <<<<<<<< .

For other values of N, μ and ζ , the order can be different. For 31.=ζ ,

12 dd ωω < .

a b Fig. 3.20 (from [5])

For relatively high damping, the angular speeds 1dω and 1BΩ tend to

rigω . This explains why the measured critical speeds are nearer those calculated for the rigidly supported rotor than those determined for the rotor on undamped flexible bearings.

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3.2 Symmetric rotors in fluid film bearings

As mentioned earlier, the linear theory of hydrodynamic bearings allows expressing the force exerted by the lubricant film on the rotor journal, resolved into two components

zy BB f,f along the coordinate axes, under the form [6]

⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡+

⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡=

⎭⎬⎫

⎩⎨⎧

B

B

zzyz

zyyy

B

B

zzyz

zyyy

B

B

zy

cccc

zy

kkkk

ff

z

y

&

&, (3.91)

where BB z,y are the projections along the axes of the fixed coordinate frame of the journal centre displacement, and BB z,y && are the corresponding velocities.

For many types of radial bearings, the stiffness matrix is non-symmetric, zyyz kk ≠ . It is not possible to determine stiffness principal directions, with respect

to which the off-diagonal elements of the stiffness matrix vanish. The bearings are anisotropic, zzyy kk ≠ , and the stiffness matrix non-symmetry produces unstable precession motions. In the following, only Laval-Jeffcott rotors are considered, neglecting the disc rotary inertia.

The bearing damping matrix is generally symmetric, zyyz cc = .

3.2.1 Unbalance response

Consider a Laval-Jeffcott rotor as in Fig. 3.10, but supported in hydrodynamic bearings, characterized by the eight dynamic coefficients defined by equation (3.91). The steady-state motion produced by the disc unbalance is examined.

The equilibrium equations for the shaft are

,)(2

,)(2

BCB

BCB

zzkf

yykf

z

y

−=

−= (3.92)

and the motion equations for the disc are

,zzkzm,yykym

BCG

BCG

0)(0)(

=−+=−+

&&

&& (3.93)

where

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3. SIMPLE ROTORS IN FLEXIBLE BEARINGS 137

t.Ωezzt,Ωeyy

CG

CG

sin cos

+=+=

(3.94)

Substituting (3.94) into (3.93) and (3.91) into (3.92) we obtain

,tΩΩemzzkzm

t,ΩΩemyykym

BCC

BCC

sin)(

cos)( 2

2

=−+

=−+

&&

&& (3.95)

and

.zcyczkykzzk

,zcyczkykyyk

BzzByzBzzByzBC

BzyByyBzyByyBC

&&

&&

+++=−

+++=−

)( 2

)( 2 (3.96)

The steady-state solutions have the form

t,ΩBtΩAtyB sin cos)( += (3.97)

,tFtEtzB sin cos )( ΩΩ += (3.98)

,tDtCtyC sin cos )( ΩΩ += (3.99)

.tHtGtzC sin cos )( ΩΩ += (3.100)

On inserting expressions (3.97) and (3.99) into the first equation (3.95) and identifying the coefficients of the terms in t cosΩ and t sinΩ , we obtain a non-homogeneous algebraic set of equations, in which C and D are expressed in terms of A and B. Substitution into (3.99) yields

,tBteAyn

n

n

nC sin cos 22

2

22

22Ω

ΩωωΩ

ΩωΩω

−+

−+

= (3.101)

where

mkn =ω (3.102)

is the critical speed of the rigidly supported rotor.

Analogously, inserting (3.98) and (3.100) into the second equation (3.95), identifying the coefficients and solving the algebraic equations, G and H are expressed in terms of E and F, yielding

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DYNAMICS OF MACHINERY 138

,teFtEzn

n

n

nC sin cos 22

22

22

ΩωΩωΩ

Ωωω

−+

+−

= (3.103)

The solutions (3.97), (3.98), (3.101) and (3.103) are then substituted into (3.96). Identifying the coefficients of the terms in t cosΩ and t sinΩ , we obtain the algebraic set of equations

,eFkEcBkAc

,FcEkBcAk

,FkEcBkAc

,eFcEkBcAk

zzzzyzyz

zzzzyzyz

zyzyyyyy

zyzyyyyy

)(

0 )(

0 )(

)(

χχΩΩ

ΩχΩ

ΩχΩ

χΩΩχ

=−+−+−

=+−++

=+−−+−

=+++−

(3.104)

where

.k

n22

2

2 ΩωΩχ−

= (3.105)

In the following, in order to simplify the solution, the eight dynamic bearing coefficients are reduced to four coefficients zyzy c,c,k,k , defined by the equations:

⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡+

⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡=

⎭⎬⎫

⎩⎨⎧

B

B

z

y

B

B

z

y

B

B

zy

cc

zy

kk

ff

z

y

&

&

00

00

(3.106)

The four equations (3.104) become

.eFkEc,FcEk

,BkAc

,eBcAk

zz

zz

yy

yy

)( 0 )(

0)(

)(

χχΩΩχ

χΩ

χΩχ

=−+−=+−

=−+−

=+−

(3.107)

It is necessary now to establish the relationships between the four equivalent bearing coefficients and the eight dynamic coefficients defined by (3.91).

In (3.104) the second equation is added to the third and the fourth equation is subtracted from the first equation. This yields

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3. SIMPLE ROTORS IN FLEXIBLE BEARINGS 139

.FckEck

BckAck

,FckEck

BckAck

zyzzzzzy

yyyzyzyy

zzzyzyzz

yzyyyyyz

) () (

) () (

) () (

) () (

ΩχΩ

ΩΩχ

ΩΩχ

ΩχΩ

−−++−=

=−−+−

+−−−−=

=+−+−

(3.108)

Solving equations (3.108) in terms of A and B we obtain

yy

BAF,BAEχμν

χνμ −

=+

−= (3.109)

where

.ckck

,ckck

ckck

,ckck

ckck

zzzyzyzzy

zzzyyyyz

zyzzyzyy

zzzyyzyy

zyzzyyyz

22 ) () (

) () (

) () (

) () (

) () (

ΩΩχχ

ΩΩ

ΩχΩχν

ΩΩχ

ΩχΩμ

++−−=

+−−

−−−+−=

++−+

+−−−= (3.110)

Putting (3.109) into the first (second) equation (3.104), by identification to the first (second) equation (3.110), we find

.ckcc

,ckkk

yzyzy

yyy

yzyzy

yyy

) (1

) (1

Ωμνχ

ΩΩ

Ωνμχ

+−=

−−=

(3.111)

Solving equations (3.108) in terms of E and F yields

,EFB,EFAzz χνμ

χμν +

−=−

= (3.112)

where

.ckck yyzyzyyyy

z22

22) () ( ΩΩχ

χνμχ ++−−=

+= (3.113)

Substituting (3.112) into the third (or the fourth) equation (3.104), by identification to the third (or the fourth) equation (3.107), we obtain

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DYNAMICS OF MACHINERY 140

.ckcc

,ckkk

yzyzz

zzz

yzyzz

zzz

) (1

) (1

Ωμνχ

ΩΩ

Ωνμχ

−+=

+−= (3.114)

Equations (3.111) and (3.114), together with (3.110) and (3.113), allow the reduction of the eight bearing coefficients to only four, equations (3.107) being coupled two by two.

Solving equations (3.107), the coefficients A, B, E, F are determined as:

) ()(

) ()(

)(2222 ,

ck

ceB,

ck

keA

yy

y

yy

y

Ωχ

Ωχ

Ωχ

χχ

+−=

+−

−= (3.115)

.ck

keF,ck

ceEzz

z

zz

z2222 ) ()(

)( ) ()(

Ωχχχ

ΩχΩχ

+−−

=+−

−= (3.116)

The solutions (3.97) and (3.98) define the journal centre motion and can be written as

,tΩztz

,tΩyty

B

B

zBB

yBB

) (sin )(

) ( cos )(

θ

θ

+=

+= (3.117)

where

,k

cΩAB

,cΩk

eBAy

y

yy

yyB

B χθ

χ

χ

−−=−=

+−=+=

tan

) ()( 2222

(3.118, a)

and

.

kcΩ

FE

,cΩk

eFEz

z

zz

zzB

B χθ

χ

χ

−−==

+−=+=

tan

) ()( 2222

(3.118, b)

The parametric equations (3.117) define an ellipse. Eliminating the time, the orbit equation is obtained as (3.34)

.EBAFzBAzyBFAEyFE BBBB 0)()()(2)( 2222222 =−−+++−+

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3. SIMPLE ROTORS IN FLEXIBLE BEARINGS 141

The distance from the origin of the stationary coordinate frame O to the journal centre B is represented in the complex plane by the vector BO = Br , which can be written as the sum of two counter-rotating vectors

.rreBEFAeBEFA

tΩFBtΩEA zyr

tΩb

tΩf

tΩtΩ

BBB

iiii ee 2

i2

2

i2

sin )i( cos )i(i

−− +=⎟⎠⎞

⎜⎝⎛ +

+−

+⎟⎠⎞

⎜⎝⎛ −

++

=

=+++=+=

(3.119)

The magnitudes of the two components are

,EBAFEFBABEFAr f 2221)()(

21 222222 −++++=−++= (3.120)

respectively

.EBAFEFBABEFArb 2221)()(

21 222222 +−+++=++−= (3.121)

They rotate in opposite directions with the same speed Ω .

As shown in sections 3.1.1.2 and 3.1.2.2, the end of vector Br moves along an elliptic orbit, of major semiaxis

bf rra += (3.122)

and minor semiaxis

bf rrb −= (3.123)

where a and b are functions of the running speed Ω .

The inclination of the major axis on the Oy axis is defined by (3.40)

.FEBA

FBEA)()(

)(22tan 2222 +−++

=α (3.124)

If 0>b , the journal centre has a forward precession, and if 0<b , it has a backward precession.

Similar conclusions are obtained from the analysis of the motion of point C, the disc geometric centre.

The forces acting on the bearing supports have the following components

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DYNAMICS OF MACHINERY 142

( )( ) ( ) ( ),t

ck

cke

ycykf

yyByy

yy

ByByBy

φθΩΩχ

Ωχ ++

+−

+=

=+=

cos 22

22

&

(3.125, a)

( )( ) ( )

( ),tck

cke

zczkf

zzBzz

zz

BzBzBz

φθΩΩχ

Ωχ +++−

+=

=+=

sin 22

22

&

(3.125, b)

where

y

yy k

cΩφ =tan ,

z

zz k

cΩφ =tan . (3.126)

3.2.2 Stability of precession motion

Experience has shown that the rotor synchronous precession in hydrodynamic bearings becomes unstable at a given value sΩ of the running speed, when the orbit radius has a sudden increase.

Analytically, this is studied using the equations of motion (3.95) and (3.96) for the perfectly balanced rotor ( )0=e .

For 0=e , hence for ,yy CG = equations (3.95) become

.zkzkzm,ykykym

BCC

BCC

=+=+

&&

&& (3.127)

The solutions have the form

,Zz,Yy

,Zz,Yyt

BBt

BB

tCC

tCC

nn

nn

ωνων

ωνων

e e

e e

==

== (3.128)

where

mk

n =ω (3.129)

is the critical angular speed of the rigidly supported rotor (3.102).

Substituting solutions (3.128) into equations (3.127) yields

BCBC ZZ,YY 22 11

11

νν +=

+= . (3.130)

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3. SIMPLE ROTORS IN FLEXIBLE BEARINGS 143

Inserting (3.128) and (3.130) into (3.96) we obtain the homogeneous algebraic equations

0) ( ) (

0) ( ) (

,ZckXYck

,ZckYckX

BnzzzzBnyzyz

BnzyzyBnyyyy

=++++

=++++

ωνων

ωνων (3.131)

where

.kX 2

2

12 νν+

= (3.132)

At the stability threshold, ν is pure imaginary. Substituting

Λi=ν (3.133)

into equations (3.131), the requirement for non-trivial solutions produces

0ii

ii=

++++++

nzzzznyzyz

nzyzynyyyy

cΛkXcΛkcΛkcΛkX

ωωωω

. (3.134)

Canceling the real and the imaginary parts of the determinant (3.134) gives the equations

,kckckckc

ccXΛ

,ccccΛ

kkkkXkkX

yyzzzzyyyzzyzyyz

zzyyn

yzzyzzyyn

zyyzzzyyzzyy

0])(

)([

0)(

)()(22

2

=−−+−

−+

=−−

−−+++

ω

ω (3.135)

which can also be written as

,cc

kckckckcX

,cccc

kkkkXkkX

zzyy

yyzzzzyyyzzyzyyz

yzzyzzyy

zyyzzzyyzzyyn

+

+−+=

−+++=

)()(

)()(222 ωΛ

(3.136)

where

.kX12 2

2

−=

ΛΛ (3.137)

It is useful to use the dimensionless coefficients

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DYNAMICS OF MACHINERY 144

,mg

RSocC,mg

RSokK jijijiji ΩΔΔ 2 2== (3.138)

where So is the inverse of the usual Sommerfeld number S [6], ΔR is the bearing clearance (difference between the bearing radius and the journal radius) and g is the acceleration of gravity.

Equations (3.136) become

,CCCC

KKKKXKKX

yzzyzzyy

zyyzzzyyzzyy

ns

22

222

)()( Ω

ωΛΩ

−+++=

== (3.139)

.CC

KCKCKCKCRSo

gmXzzyy

yyzzzzyyyzzyzyyz

+

−−+=

)()(2 Δ

(3.140)

The onset speed of instability sΩ can be computed using an iterative approach. A value Ω is first selected. The corresponding Sommerfeld number So and the eight bearing coefficients are then computed. They are given in tabular or graphic form, as functions of So for given values of the bearing clearance and length-to-diameter ratio (see Chapter 6).

Fig. 3.21

Equation (3.140) delivers X, which is inserted into (3.139), wherefrom sΩ is obtained. If ,sΩΩ < a new value Ω is considered and the computations are repeated until .sΩΩ = The results are plotted as in Fig. 3.21 [7].

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3. SIMPLE ROTORS IN FLEXIBLE BEARINGS 145

A more detailed analysis of the stability of precession for rotors supported in hydrodynamic bearings is presented in Chapter 7.

3.3 Asymmetric rotors in flexible bearings

The previous sections 3.1 and 3.2 examined single-disc Laval-Jeffcott rotors, considering only the disc motion in the rotor plane of symmetry, hence neglecting the effect of disc rotary inertia. In the following, the two single-disc rotor models from Table 3.1 are analyzed. The bearings are orthotropic and dissimilar.

3.3.1 Equations of motion

Consider the asymmetric rotor from Fig. 3.22, supported in orthotropic flexible bearings.

Fig. 3.22

In this case, the flexibility influence coefficients ijδ in equations (2.78) are different from those used in (2.79).

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DYNAMICS OF MACHINERY 146

The force-deflection equations can be written:

C4441

3332

2322

1411

C

000000

00

⎪⎪⎭

⎪⎪⎬

⎪⎪⎩

⎪⎪⎨

⎥⎥⎥⎥

⎢⎢⎢⎢

=

⎪⎪⎭

⎪⎪⎬

⎪⎪⎩

⎪⎪⎨

z

y

z

y

MMFF

zy

δδδδδδ

δδ

ψϕ

(3.141)

Table 3.1

Model I Model II

1

2

1

222

3

11 3 BA kkEIl αββαδ ++=

1

2

1

22

3

1111

3 BA k)(

k)(

EIl γγγγδ +

+++=

lklkEIl

BA 11

2

14 )(3

αβαβαβδ −+−−= lklkEI

l

BA 11

2

141)32(

6γγγγδ +

−−+−=

21

21

3344

11)(3 lklkEI

l

BA+++= βαδ 2

12

144

11)31(lklkEI

l

BA+++= γδ

2

2

2

222

3

22 3 BA kkEIl αββαδ ++=

2

2

2

22

3

22)1()1(

3 BA kkEIl γγγγδ +

+++=

lklkEIl

BA 22

2

23 )(3

αβαβαβδ +−−= lklkEI

l

BA 22

2

231)32(

6γγγγδ +

+++=

22

22

3333

11)(3 lklkEI

l

BA+++= βαδ

22

22

3311)31(

lklkEIl

BA+++= γδ

lb,

la

== βα lc

The flexibility influence coefficients jiij δδ = of the two rotor models are listed in Table 3.1 [7].

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3. SIMPLE ROTORS IN FLEXIBLE BEARINGS 147

The inverse of the flexibility matrix is the stiffness matrix given by

.δδδ, Δ

Δδ, k

Δδ, k

Δδk

,δδδ, ΔΔδ, k

Δδ, k

Δδk

22333222

2

2233

2

2323

2

3322

21444111

1

1144

1

1414

1

4411

−==−==

−==−==

(3.142)

The equations of motion (2.80) are written in the form

),t(Fkykym CCG 11411 =++ ψ&&

),t(Fkzkzm CCG 22322 =++ ϕ&&

),t(MkykJJ CCGPGT 14441 =++− ψϕΩψ &&&

).t(MkzkJJ CCGPGT 23332 =+++ ϕψΩϕ &&&

(3.143)

Using expressions (3.54) and (2.123) to eliminate the coordinates of the mass centre G, equations (3.143) are written in matrix form as

⎪⎪⎭

⎪⎪⎬

⎪⎪⎩

⎪⎪⎨

=

⎪⎪⎭

⎪⎪⎬

⎪⎪⎩

⎪⎪⎨

−⎥⎥⎥⎥

⎢⎢⎢⎢

−−

+

+

⎪⎪⎭

⎪⎪⎬

⎪⎪⎩

⎪⎪⎨

−⎥⎥⎥⎥

⎢⎢⎢⎢

+

⎪⎪⎭

⎪⎪⎬

⎪⎪⎩

⎪⎪⎨

−⎥⎥⎥⎥

⎢⎢⎢⎢

2

2

1

1

3332

2322

4441

1411

000

000

00

00

MFMF

zψy

kkkk

kkkk

zψy

J

zψy

Jm

Jm

C

C

C

C

C

C

C

C

P

P

C

C

C

C

T

T

ϕ

ϕϕ &

&

&

&

&&

&&

&&

&&

(3.144)

or

[ ][ ]

[ ][ ]

[ ][ ]

00

00

00

⎭⎬⎫

⎩⎨⎧

⎭⎬⎫

⎩⎨⎧

⎥⎥⎦

⎢⎢⎣

⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡−⎭

⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

z

y

z

yff

= zy

k

k + zy

gg

+zy

m

m&

&

&&

&&

(3.145)

which in shorthand has the form

.fxKxGxM ][][][ =++ &&& (3.146)

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DYNAMICS OF MACHINERY 148

The vector in the right-hand side has the form

,

MFMF

JJemJJem

ΩtΩ

JJemJJem

= Ω

FtΩFtΩFf

cPT

c

sPT

s

sPT

s

cPT

c

sc

⎪⎪⎭

⎪⎪⎬

⎪⎪⎩

⎪⎪⎨

+

⎪⎪⎭

⎪⎪⎬

⎪⎪⎩

⎪⎪⎨

−−−

+

⎪⎪⎭

⎪⎪⎬

⎪⎪⎩

⎪⎪⎨

=++=

2

2

1

1

22 sin

)(

)( cos

)(

)(

sin cos

α

α

α

α (3.147)

where sc e,e and sc ,αα are the projections of e and α along the coordinate axes.

3.3.2 Natural frequencies of precession

The displacements y and z can be written as in (3.33)

( )( ),tztztzz

,tytytyy

zsc

ysc

θωωω

θωωω

+=+=

+=+=

cossincos

cossincos

( ) ( ) .ezzeeezez

,eyyeeeyeyt

sct

tsc

t

z

y

ωωθ

ωωθ

iii

iii

i

i

−ℜ=ℜ=

−ℜ=ℜ=

For 0 =f , equations (3.146) describe the rotor free precession. The solutions of these equations can be expressed in terms of complex phasors as [8]

zzyy

zy

x ωtt

sc

sc i i e e ii

= =⎭⎬⎫

⎩⎨⎧

−−

⎭⎬⎫

⎩⎨⎧

= ω (3.148)

Substituting (3.148) into (3.145) with zero right-hand side, leads to the eigenvalue problem

[ ] [ ] [ ]

[ ] [ ] [ ]

⎬⎫

⎩⎨⎧

⎭⎬⎫

⎩⎨⎧

−−

⎥⎥⎦

⎢⎢⎣

−−

−00

ii

i i

2

2 =

zzyy

mωkgΩ

gΩmωk

sc

sc

z

y

ωω

(3.149)

which delivers the following four equations coupled two by two

[ ] [ ]( ) [ ] ,zgΩymk scy 0 2 =+− ωω (3.150)

[ ] [ ]( ) [ ] ,zgΩymk csy 0 2 =−− ωω (3.151)

[ ] [ ]( ) [ ] ,ygΩzmk scz 0 2 =−− ωω (3.152)

[ ] [ ]( ) [ ] 0 2 =+− csz ygΩzmk ωω . (3.153)

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3. SIMPLE ROTORS IN FLEXIBLE BEARINGS 149

Equations (3.150) and (3.151) yield

[ ] [ ]( ) [ ] ,zgΩmky syc 12 −

−−= ωω (3.154)

[ ] [ ]( ) [ ] cys zgΩmky 12 −

−−= ωω . (3.155)

Inserting (3.154) and (3.155) into (3.152) and (3.153) we obtain

[ ] [ ] [ ] [ ] [ ]( ) [ ] ,zgΩmkgΩmk cyz 0 1222 =⎟

⎠⎞⎜

⎝⎛ −−−

−ωωω (3.156)

[ ] [ ] [ ] [ ] [ ]( ) [ ] .zgΩmkgΩmk syz 0 1222 =⎟

⎠⎞⎜

⎝⎛ −−−

−ωωω (3.157)

Comparing equations (3.156) and (3.157) it can be noticed that the two solutions are proportional to one another

,zz sc β= (3.158) where β is a real constant.

Substituting equation (3.158) into equation (3.154) and comparing the result with equation (3.155) yields

.yy cs β−= (3.159)

Inserting (3.158) and (3.159) into (3.148) we obtain

⎬⎫

⎩⎨⎧−

=⎭⎬⎫

⎩⎨⎧−

+⎭⎬⎫

⎩⎨⎧

−−

=z

y

s

c

sc

sc

aa

zy

zzyy

i e

i )i1( =

ii

iγβΦ (3.160)

where ya and za are real vectors.

Equation (3.160) shows that, by proper normalization, the elements of vectors Φ become real in the xOy plane and pure imaginary in the xOz plane, hence the precession modes are planar.

From equations (3.148) and (3.149) we obtain

s

c

c

szz

yy

=−=β (3.161)

.zyzy sscc 0=+

According to expression (3.40), the inclination angle 0=α and the ellipse axes are collinear with the coordinate axes.

For undamped rotors, the eigenvectors are complex quantities due to the spatial character of the precession and the gyroscopic coupling, but the natural modes of precession are planar.

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DYNAMICS OF MACHINERY 150

Using the transformation to real vectors

⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡−⎭

⎬⎫

⎩⎨⎧− z

y

z

y

aa

II

aa

][i]0[

]0[][ =

i, (3.162)

where [ ]I is an identity matrix, and pre-multiplying by ⎥⎦

⎤⎢⎣

⎡][i]0[]0[][

II

, equation (3.149)

becomes

⎭⎬⎫

⎩⎨⎧

⎭⎬⎫

⎩⎨⎧

⎥⎥⎦

⎢⎢⎣

−00

=

][][][

][ ][][2

2

z

y

z

yaa

mkgΩgΩmk

ωωωω . (3.163)

The requirement for non-trivial solutions is

0 ][][][

][ ][][ 2

2=

mkgΩgΩmk

z

y

ωωωω

(3.164)

or

.

JkkJΩkmk

JΩJkkkmk

TP

PT 0

000

000

23332

232

22

24441

142

11

=

−−−−

−−

ωωω

ωωω

The frequency equation has the form

.AωΩBAωΩBAωΩBAω 0)()()( 022

2242

4462

668 =++−+++− (3.165)

Figure 3.23 presents a plot of the natural frequencies of precession ω as a function of the running speed Ω , for TP JJ > . The symmetry with respect to the ω and Ω axes is due to the odd powers in equation (3.165). The horizontal asymptotes correspond to zero angular precession of the disc for ∞→Ω .

Inserting Ωω = into (3.163), the synchronous precession critical speeds are obtained from

⎭⎬⎫

⎩⎨⎧

⎭⎬⎫

⎩⎨⎧

⎥⎥⎦

⎢⎢⎣

−00

=

][][][

][][][22

22

z

y

z

yaa

mkggmk

ΩΩΩΩ (3.166)

which can be written as a generalized eigenvalue problem

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3. SIMPLE ROTORS IN FLEXIBLE BEARINGS 151

),..,rmggm

kk

rrrz

y 41( . ][][][][

][]0[]0[][ 2 =⎥

⎤⎢⎣

⎡−

−=⎥

⎤⎢⎣

⎡ΨΩΨ (3.167)

The eigenvalues rΩ give the synchronous critical speeds. The eigenvectors rΨ define the semiaxes of the disc precession orbit at the related critical speed and the directivity of precession (forward or backward).

Fig. 3.23

Substituting Ωω −= into (3.163), the off-diagonal elements of the matrix from equation (3.166) become negative, but the same critical speeds are obtained. Unlike the rotor supported in isotropic bearings, the unbalance will also excite the backward critical speeds.

The Campbell diagram is shown in Fig. 3.24, b (first quadrant of Fig. 3.23). For comparison, Fig. 3.24, a depicts the Campbell diagram for the same rotor supported in rigid bearings. The motions with angular speeds 1ω and 3ω are backward precessions, while the motions with angular speeds 2ω and 4ω are forward precessions. The points where the line Ωω = intersects the precession natural frequency lines define the critical speeds.

The shape of the curves in the Campbell diagram from Fig. 3.24, b is typical for lightly damped rotors. As will be shown in section 3.3.4, the consideration of damping can substantially modify the shape of the diagram.

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DYNAMICS OF MACHINERY 152

a b

Fig. 3.24

Figure 3.25 presents separately the effects of bearing flexibility, disc diametral mass moment of inertia and gyroscopic coupling on the Campbell diagram of single-disc asymmetric rotors, supported in isotropic bearings (Fig. 3.25, a) and in orthotropic bearings (Fig. 3.25, b) [2]. The rigid disc has TP JJ > . The horizontal asymptotes correspond to natural frequencies of pure translatory precession.

3.3.3 Unbalance response

Considering equations (3.146), for a synchronous excitation

tFtFf sc sin cos ΩΩ += , (3.168)

the steady-state response has the form

.tXtXx sc sin cos ΩΩ += (3.169)

Substituting (3.168) and (3.169) into (3.146) we obtain the algebraic set of equations

( )( ) .][][][

][][][ 2

2

ssc

csc

FXMKXG

,FXGXMK

=−+−

=+−

ΩΩ

ΩΩ (3.170)

The two components of the disc translational displacements are given by equations of the form (3.33). They are utilized for the calculation of the elliptic orbit parameters of the unbalance response, using equations (3.38) to (3.40) and (3.44).

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3. SIMPLE ROTORS IN FLEXIBLE BEARINGS 153

Fig. 3.25

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DYNAMICS OF MACHINERY 154

In order to calculate the finite amplitudes of motion at the peak response critical speeds, some form of damping has to be taken into account. In the left-hand side of equation (3.146), a diagonal damping matrix

] [ diag][ 33224411 ccccD =

is added to the gyroscopic matrix. Its elements are calculated assuming given values of the damping ratios

.Jk

cJk

c,mk

c,mk

c

TT 44

444

33

333

22

222

11

111 2

,2

2

2

==== ζζζζ (3.171)

Usually, it is considered that .ζζζζζ ==== 4321

Example 3.2a

Consider a rotor with an overhung disc (Model II) with the following parameters: kg 8000=m , m kg 8520 2 ,JP = ,JT

2m kg 4260= m, 4 =l m, 80.c = ,E GPa 210 = m, 30 .d = 020 .=ζ , mN/ 333 1 μ=Ak ,

m,N/ 6672 μ=Ak m,N/ 383 1 μ.kB = mN/ 167 2 μ=Bk .

Fig. 3.26

Figure 3.26 presents the Campbell diagram with the running speed on the horizontal axis. The intersections with the synchronous line determine the damped critical speeds: rpm, 4371 =n rpm, 7612 =n rpm 12823 .n = The second critical speed is in forward (synchronous) precession. The others are in backward (asynchronous) precession.

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3. SIMPLE ROTORS IN FLEXIBLE BEARINGS 155

Figure 3.27 shows the unbalance response diagrams. The peaks in the major semiaxis diagram (Fig. 3.27, a) locate the peak response critical speeds. Although the first and the third critical speed correspond to backward precession modes, they are excited by the rotating unbalance. The maximum value of the major semiaxis in the operating speed range is usually compared to admissible limits.

a b

c d

Fig. 3.27

In figure 3.27, b the minor semiaxis diagram is added. The two ranges with negative values define the operation speeds with backward precession produced by the unbalance. The crossing points with the horizontal axis locate the threshold speeds where the precession orbit degenerates to a straight line. At these speeds the orbit changes from forward to backward precession and vice versa. Note that the threshold speeds are different from the critical speeds.

Figure 3.27, c shows the diagrams of the y and z components of the disc centre displacement. Figure 3.27, d presents the diagrams of the radii fr and br

of the two circular counter-rotating motions that generate the ellipse. The ranges where the radius br of the circle with backward motion is larger than the radius

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DYNAMICS OF MACHINERY 156

fr of the circle with forward precession define the speed ranges with backward

precession. It is easy to see that they correspond to ranges with negative minor semiaxis in Fig. 3.27, b.

3.3.4 Effect of bearing damping

Including the effect of bearing damping, equations (3.146) become

.fxKxCxM ][][][ =++ &&& (3.172)

where

⎥⎦

⎤⎢⎣

⎡−

+⎥⎦

⎤⎢⎣

⎡=

g g

Ω cc cc

Czzzy

yzyy

]0[][][]0[

][][][][

][ (3.173)

is the sum of the damping and gyroscopic matrices.

For 0 =f , trying solutions of the form

te λux = , (3.174)

the following quadratic eigenvalue problem is obtained

)41( 0)][][][( 2 ,..,ruKCM rrr ==++ λλ . (3.175)

The eigenvalues rλ are real numbers for overdamped modes and complex numbers for underdamped modes. The complex eigenvalues have the form

i i rrrrrr , ωαλωαλ −=+= (3.176)

and are functions of the running speed Ω .

The imaginary part rω is the damped natural frequency (of precession) and the real part rα is an attenuation ( or growing) constant.

Usually, the damping is expressed in terms of the modal damping ratio

.r

r

rr

rr ω

α

ωα

αζ −≅+

−=22

(3.177)

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3. SIMPLE ROTORS IN FLEXIBLE BEARINGS 157

The Campbell diagram is a plot of the dependence .rr )(Ωωω = Sometimes it is presented together with the stability diagram )(Ωαα rr = or the damping ratio diagram )( Ωζζ rr = .

For underdamped systems, the complex eigenvectors have the form

i i .bau,bau rrrrrr −=+= (3.178)

The solution for the free precession can be written as

( )tbtatx rrrrt

rr ωωα sincose2)( −= , (3.179)

and describes spiralling orbits. However it is agreed to represent the orbits as incomplete (open) ellipses, considering 0=rα and approximating the expression (3.179) by

tututx rsrcr ωω sincos)( += (3.180) where

.umbu,ueau rrsrrc −ℑ=−=ℜ==

Fig. 3.28

Example 3.3

Consider the cantilevered rotor from Fig. 3.28, with m, 40.l = m, 020.d = GPa 210=E . The end carrying a thin rigid disc with kg, 516.m =

,.JT2mkg 0940 ⋅= 2mkg 1860 ⋅= .J P , is flexibly supported. The support has

principal stiffnesses N/m105 51 ⋅=k and N/m,102 5

2 ⋅=k and damping coefficients proportional to the related stiffness, 11 kc β= and .kc 22 β=

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DYNAMICS OF MACHINERY 158

Figures 3.29 depict the Campbell diagrams and the diagrams of the modal damping ratio for three values of the coefficient β .

a b c

Fig. 3.29

It can be seen that the curves corresponding to the first mode of precession change significantly with the bearing damping. For large damping levels (Fig. 3.29, c), the first precession mode becomes overdamped within a certain speed range. This mode is not 'seen' in the unbalance response diagrams, where the corresponding peak is missing.

3.3.5 Mixed modes of precession

The rotor precession is usually described by modal characteristics associated with forward and backward modes. However, the directivity of precession is a local, not a global property. A rotor can have mixed modes, with both forward and backward precession coexistent at different stations, at a given speed.

Hopefully, the precession of rotors in almost isotropic bearing systems can be classified as pure forward or pure backward, the motion at all stations of interest having the same direction. This enables a logical mode labeling. In most cases, mixed modes are predominantly forward or backward, with limited zones of reverse precession, so that if the number of stations in the model is small, then the mixed character of the precession is lost.

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3. SIMPLE ROTORS IN FLEXIBLE BEARINGS 159

In anisotropic rotor-bearing systems, bearing orthotropy yields different rotor deflected shapes in two orthogonal directions, at the same rotational speed. The coupling of two (even slightly) different rotor orthogonal eigenforms yields mixed precession modes.

This can be easier explained for conservative rotor-bearing systems that have planar precession modes. In this case, the deflected shapes in two orthogonal planes correspond to mode shapes plotted at two instants with a quarter of period time difference. At the station where only one modal form crosses the rotor longitudinal axis, the precession orbit degenerates into a straight line, separating portions of forward and backward motion along the rotor [9].

In a rotor with oil-lubricated bearings, the slightest asymmetry yields small differences in load and oil temperature between the two bearings. Even with physically identical bearings, the stiffness and damping coefficients are different at the both ends. For reasonable amounts of dissymmetry and coupling effects, the damped natural frequency curves in the Campbell diagram do not cross, giving rise to curve veering, denoting modal coupling and compound modes. The abrupt continuous change of mode shapes within the speed interval of natural frequency curve veering yields mixed modes. Along a natural frequency curve, a mode can be forward over a given speed interval, then mixed in the region of curve veering, changing to backward away of that region. Simultaneous plotting of the speed dependence of modal damping ratios helps understanding the nature of mixed modes.

For a large class of actual rotor systems, precession modes occur in pairs, the mode at lower frequency has backward precession and the mode at higher frequency has forward precession. Inclusion of bearing damping can change the sequence. Overdamped modes can transform into underdamped modes and appear in the Campbell diagram only in limited speed regions.

The bearing cross-coupling stiffnesses increase the gap between the natural frequencies of a backward-forward pair. This way, a forward mode from a lower pair approaches a backward mode from a higher pair, yielding either a crossing or a curve veering in the Campbell diagram. The compounding of two different modes gives rise to mixed precession.

In some academic examples, the mixed nature of some precession modes is lost if the motion is analyzed at a relatively reduced number of stations along the rotor.

As a first example, a simple rotor system is taken, consisting of a rigid disc attached to a massless rigid shaft supported by two identical bearings at the ends. Three cases are considered: a) a symmetric rotor with isotropic bearings; b) a symmetric rotor with orthotropic bearings; and c) an asymmetric rotor with orthotropic bearings. The massless rigid shaft was modeled with values of

Pa 102 15⋅=E and 3mkg1=ρ in the computer simulation [9].

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DYNAMICS OF MACHINERY 160

Example 3.4 a

A rigid disc is mounted at the centre ( )m35021 .== ll of a massless rigid shaft and the shaft is supported by identical isotropic bearings at both ends. The disc mass and mass moments of inertia are kg, 30=m ,.JT

2mkg 21= 2mkg 81.JP = . The bearing stiffness and damping coefficients are

N/m107 6⋅== zzyy kk and Ns/m200== zzyy cc .

The Campbell diagram is shown in Fig. 3.30. Forward modes are labeled ‘F’ while backward modes are labeled ‘B’. The two ‘cylindrical’ modes at

Hz48103. have natural frequencies independent of the rotational speed, hence overlaid straight lines. The disc has a translational motion not influenced by gyroscopic effects and decoupled from the angular motion.

The third and fourth ‘conical’ modes, labeled 2B and 2F, are decoupled from the cylindrical modes. As the rotor speed increases, the natural frequency of the backward mode decreases and crosses the line of the cylindrical modes, due to gyroscopic effects. The natural frequency of the forward mode increases with rotor speed. Due to bearing isotropy the two curves start from the same point at zero rotational speed.

Fig. 3.30

The synchronous excitation line is plotted with dotted line. The critical speeds are determined as the abscissae of the crossing points with the natural frequency lines, at 6209 rpm and 6876 rpm. In the case of unbalance excitation, the only one critical speed is located at the intersection with the line of mode 1F. For rotor systems with isotropic bearings, backward modes cannot be excited by synchronous excitation.

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3. SIMPLE ROTORS IN FLEXIBLE BEARINGS 161

a b

c d

Fig. 3.31

The precession mode shapes at 10000 rpm are shown in Fig. 3.31. Due to bearing isotropy, the orbits at any station are circles. They are plotted as incomplete (“open”) orbits to help recognizing the motion directivity. The mode shape at 0=t is plotted with solid line and the mode shape at Ωπ 2=t is drawn with broken line, so that the motion along the orbit takes place from the point lying on the solid line, at 0=t , to the point lying on the broken line, a quarter of a period later.

Fig. 3.32

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DYNAMICS OF MACHINERY 162

The radius of the unbalance response orbit at the disc location is plotted in Fig. 3.32 as a function of the rotational speed, for a mmg30 unbalance of the disc. As expected, only one peak occurs in the diagram, at the natural frequency of mode 1F.

Example 3.4 b

Consider the symmetric rotor of Example 3.4 a, but with orthotropic identical bearings . The bearing vertical stiffness coefficients are N/m105 6⋅=yyk ,

the horizontal stiffness coefficients are N/m107 6⋅=zzk , and the damping

coefficients are Ns/m102 2⋅== zzyy cc [10].

Fig. 3.33

The Campbell diagram is shown in Fig. 3.33. The two ‘cylindrical’ modes 1B and 1F have different natural frequencies at Hz6388. and Hz48103. due to the bearing anisotropy. They are independent of the rotational speed due to system symmetry.

The third and fourth ‘conical’ modes, are decoupled from the cylindrical modes. As the rotor speed increases, the natural frequency of the mode 2B decreases and crosses the lines of the cylindrical modes, due to gyroscopic effects. The natural frequency of the mode 2F increases with rotor speed. Due to bearing anisotropy the two curves in a pair start from different points at zero rotational speed.

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3. SIMPLE ROTORS IN FLEXIBLE BEARINGS 163

The synchronous excitation line intersects the natural frequency lines at the points whose abscissae determine the damped critical speeds at rpm5318 , 6209 rpm and 6341 rpm.

Fig. 3.34

The damping ratio diagram is shown in Fig. 3.34. Backward modes are more damped than the forward modes of the same pair. The curves for the conical modes cross those of cylindrical modes, denoting no coupling effects.

a b

c d

Fig. 3.35

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DYNAMICS OF MACHINERY 164

The precession mode shapes at 10000 rpm are shown in Fig. 3.35. As before, the mode shape at 0=t is plotted with solid line and the mode shape at

Ωπ 2=t is drawn with broken line. The motion along the orbit takes place from the point lying on the solid line, to the point lying on the broken line.

The orbits of the two ‘cylindrical’ modes 1B and 1F at Hz6388. and Hz48103. are almost straight lines due to the strong bearing anisotropy and

decoupling of the two motions. The orbits of modes 2B and 2F are elliptical.

a b

Fig. 3.36

Fig. 3.37

The unbalance response curves calculated at the disc station are shown in Fig. 3.36, for a mmg30 unbalance of the disc. In Fig. 3.36, a , curve a is for the major semiaxis and curve b is for the minor semiaxis. In Fig. 3.36, b , curve fr is

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3. SIMPLE ROTORS IN FLEXIBLE BEARINGS 165

for the forward circle radius and br is for the backward circle radius. The two peaks indicate that only two of the three possible damped critical speeds become peak response critical speeds due to the high damping of mode 2B. Between the two peaks there is a speed range with backward precession indicated by negative values of the orbit minor semiaxis, or by fb rr > .

Useful information is given by the root locus diagram (Fig. 3.37). This is a plot of the damped natural frequency versus negative damping ratio, for each mode of precession. When the curves are distant of each other, as in Fig. 3.37, there is no coupling between modes and no compound or mixed modes of precession can occur.

Example 3.4 c

Consider the rotor of Example 3.4 b, but with the rigid disc mounted off the shaft centre ( )m40 m,30 21 .. == ll . The shaft is rigid and massless. The

bearing stiffness coefficients are N/m105 6⋅=yyk , N/m107 6⋅=zzk , and the

damping coefficients are Ns/m102 2⋅== zzyy cc [10].

Fig. 3.38

The Campbell diagram is shown in Fig. 3.38. Curve 2B no more crosses the lines 1B and 1F and, near the rotational speed of rpm8000 , veers away from the line 1F. The rotor translational and angular motions are coupled. With increasing rotational speed, mode 2B becomes a mixed mode and tends to change into the first forward mode, while mode 1F becomes a mixed mode and tends to change into the first backward mode. The synchronous excitation line intersects the

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DYNAMICS OF MACHINERY 166

natural frequency lines at the points whose abscissae determine the damped critical speeds rpm5236 , 6051 rpm and 6532 rpm.

The damping ratio diagram is shown in Fig. 3.39. With increasing rotational speed, curve 2B transforms into the former 1F, while 1F transforms into the former 1B and curve 1B follows the former line 2B. These transformations take place in the speed range with curve veering in the Campbell diagram.

Fig. 3.39

The root locus diagram is presented in Fig. 3.40. Modes are labeled as before, according to their shapes at low rotational speeds. When the root loci are close to each other, two modes with nearly the same natural frequency and different mode shapes can combine to yield a compound mode which has mixed backward and forward precession due to the coupling between modes.

Fig. 3.40

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3. SIMPLE ROTORS IN FLEXIBLE BEARINGS 167

a b

c d

Fig. 3.41

The precession mode shapes at 10000 rpm are shown in Fig. 3.41. For mixed modes, the precession along the ellipse is marked by B (backward) or F (forward) and takes place from the point lying on the solid line, at 0=t , to the point lying on the broken line, a quarter of a period later.

a b

Fig. 3.42

Along the rotor, the portions of backward and forward motion are separated by a location where the precession orbit degenerates into a straight line. Such lines do not appear in Fig. 3.41 due to the small number of stations where orbits have been drawn.

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DYNAMICS OF MACHINERY 168

The unbalance response curves calculated at the disc station are shown in Fig. 3.42, for a mmg30 unbalance of the disc. The abscissae of the three peaks indicate the peak response critical speeds. Again, the speed range with backward precession is indicated by negative values of the orbit minor semiaxis or values of

br larger than fr . When 0=b and fb rr = the orbit degenerates into a straight

line.

3.4 Simulation examples

Example 3.5 a

A rigid disc is mounted at the middle of a uniform shaft (Fig. 3.43) of length m440. , diameter mm90 , Young’s modulus 211 mN102 ⋅ and mass

density 3mkg7800 .

Fig. 3.43

The mass of the disc is 560 kg, while the diametral and polar mass moments of inertia are 2kgm18 and 2kgm32 , respectively. The shaft is supported at the ends by identical bearings with the following constant coefficients:

mN1022 8⋅=′′=′ .kk yyyy , mN1011 8⋅=′′=′ .kk zzzz , msN1022 4⋅=′′=′ .cc yyyy , and

msN10111 4⋅=′′=′ .cc zzzz [9].

The Campbell diagram is shown in Fig. 3.44. Due to the system symmetry, the disc translational and angular motions are decoupled. Modes 1B and 1F have natural frequencies independent of rotational speed. As the rotor speed increases, the natural frequency of the mode 2B decreases and crosses the lines of the cylindrical modes, due to gyroscopic effects. Due to bearing orthotropy, the two curves in a pair start from different points at zero rotational speed.

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3. SIMPLE ROTORS IN FLEXIBLE BEARINGS 169

Fig. 3.44

The damping ratio diagram is shown in Fig. 3.45. As in Example 3.4 a, backward modes are more damped than the forward modes of the same pair. The curves for the conical modes do not cross those of cylindrical modes.

Fig. 3.45

Example 3.5 b

Consider the rotor of Example 3.5 a supported by bearings with slightly different stiffness and damping coefficients (Fig. 3.43): mN10152 8⋅=′ .kyy ,

mN10151 8⋅=′ .kzz , msN10152 4⋅=′ .cyy , msN10151 4⋅=′ .czz , mN10252 8⋅=′′ .k yy ,

mN10051 8⋅=′′ .kzz , msN10252 4⋅=′′ .cyy and msN10051 4⋅=′′ .czz [9].

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The Campbell diagram is shown in Fig. 3.46 for the first four modes of precession. The curves in the diagram are labeled in the usual way, 1B, 1F, 2B and 2F, as for Example 3.5 a, though there are speed intervals with mixed modes. Curve 2B crosses the line 1F at 600 rpm and veers away from line 1B at 2065 rpm.

Fig. 3.46

In Fig. 3.47 the damping ratio curves of modes 2B and 1F have a trough, respectively a peak, at 600 rpm, not crossing each other, while curves 2B and 1B do cross each other at about 2065 rpm.

Fig. 3.47

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3. SIMPLE ROTORS IN FLEXIBLE BEARINGS 171

With increasing rotational speed, mode 2B becomes a mixed mode and changes into the first backward mode, while mode 1B changes into 2B.

The root locus diagram (Fig. 3.48) indicates the possible coupling of modes 1B and 2B, whose loci are close to each other. Also modes 2B and 1F have a range of equal natural frequencies and this can also produce compound modes with mixed precession.

Fig. 3.48

A closer look at the shape of precession modes is useful, especially at their evolution within the speed intervals with modal interaction.

a b c

Fig. 3.49

Figure 3.49 shows the evolution of mode 1M between 1700 and 3000 rpm. Mode 1M results from the coupling of a vertical conical mode 1B with a horizontal cylindrical mode 1F. With increasing rotational speed, the latter becomes a conical horizontal mode.

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DYNAMICS OF MACHINERY 172

a b c

Fig. 3.50

Figure 3.50 shows the evolution of mode 2M between 200 and 1000 rpm. Despite the crossing of natural frequency curves (Fig. 3.46) the mode is mixed. It is the result of the compounding of a cylindrical vertical mode and a conical horizontal mode. Mixed modes exist even when there is no curve veering in the Campbell diagram.

a b c

Fig. 3.51

Figure 3.51 presents the evolution of mode 3M between 250 and 1000 rpm. It is basically the second backward mode 2B, but at low speeds, the vertical and horizontal conical components cross the rotor longitudinal axis at different locations. At these points the precession orbit degenerates into straight lines that mark the change from backward to forward or vice versa.

a b c

Fig. 3.52

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3. SIMPLE ROTORS IN FLEXIBLE BEARINGS 173

Figure 3.52 shows the evolution of mode 3M between 1700 and 2500 rpm. At 1700 rpm the mode is apparently still backward 2B. Its mixed nature is overlooked due to the small number of stations at which the orbit is drawn.

A closer look at Fig. 3.52, a shows that the vertical and horizontal conical components cross the rotor longitudinal axis at different locations so that there is a portion with forward precession not revealed with only five stations. At 2100 rpm the horizontal mode becomes cylindrical. Because the vertical component remains conical, the precession mode is mixed.

A similar rotor system with slightly different parameters is presented in the following, to illustrate the above statements. Horizontal stiffnesses are larger in this case than the vertical stiffnesses.

Example 3.6

A uniform shaft of length m4370. , diameter mm91 , Young’s modulus 211 mN102 ⋅ and mass density 3mkg7750 carries at the middle a rigid disc of

mass kg566 , diametral and polar mass moments of inertia 2mkg118. and 2mkg236. , respectively.

The shaft is supported at the ends by orthotropic bearings with the following constant stiffness and damping coefficients: (Fig. 3.43): mN10141 8⋅=′ .kyy ,

mN10142 8⋅=′ .kzz , mN10041 8⋅=′′ .kyy , mN10242 8⋅=′′ .kzz , msN10141 4⋅=′ .cyy ,

msN10142 4⋅=′ .czz , msN10041 4⋅=′′ .cyy and msN10242 4⋅=′′ .czz [11].

Fig. 3.53

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DYNAMICS OF MACHINERY 174

The Campbell diagram is shown in Fig. 3.53 for the first four modes of precession.

In Fig. 3.54 the damping ratio curves of modes 2B and 1F have a trough, respectively a peak, at 400 rpm, not crossing each other, while curves 2B and 1B do cross each other at about 1800 rpm.

Fig. 3.54

With increasing rotational speed, mode 2B becomes a mixed mode and changes into the first backward mode 1B, while mode 1B changes into 2B.

a b c

d e f

Fig. 3.55

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3. SIMPLE ROTORS IN FLEXIBLE BEARINGS 175

Figure 3.55 shows the evolution of mode 3M between 1000 and 2000 rpm. Mode 3M is obtained from the coupling of a horizontal conical mode 2B with a vertical cylindrical mode 1B. With increasing rotational speed, the latter becomes a vertical cylindrical mode 1F.

a b c

Fig. 3.56

Figure 3.56 shows the evolution of mode 1M between 1900 and 2400 rpm.

Fig. 3.57

Example 3.7

Consider a rotor with two bearings and a single disc overhung at one end (Fig. 3.57). The rigid disc, with mass kg8000 , polar mass moment of inertia

2kgm8520 and diametral mass moment of inertia 2kgm4260 , is located at station

7, at the right end. The shaft with Young’s modulus 211 mN1012 ⋅. and mass

density 3mkg7800 has four different sections with the following lengths and diameters: m701 .=l , m101 .d = , m922 .=l , m302 .d = , m403 .=l ,

m3203 .d = , m804 .=l , m3404 .d = , and is modeled by 6 beam elements. The bearings are located at stations 1 and 6 having the following constant stiffness and damping coefficients: at station 1, ( ) mN1061 9⋅=′yyk , ( ) mN10121 9⋅=′zzk ,

msN105=′=′ zzyy cc ; at station 6, ( ) mN1032 9⋅=′′yyk , ( ) mN1031 9⋅=′′zzk ,

msN105=′′=′′ zzyy cc [7].

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DYNAMICS OF MACHINERY 176

The Campbell diagram is presented in Fig. 3.58 for the first six natural modes. Modes are numbered in ascending order and labeled with their index without mentioning the directivity.

Fig. 3.58

The damping ratio diagram is shown in Fig. 3.59 for the same six modes.

Fig. 3.59

The shape of the first six modes of precession at 2400 rpm is shown in Fig. 3.60. The system has 4 mixed modes, although there is neither curve veering nor curve crossing at 2400 rpm in figure 3.58. The natural frequencies of modes 2

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3. SIMPLE ROTORS IN FLEXIBLE BEARINGS 177

and 3, as well as those of modes 4 and 5, belonging to different pairs, are approaching each other. Mode 2 is predominantly forward (2F) and its mixed character is the result of the different crossing points of the vertical and horizontal component modes with the rotor axis.

a b c

d e f

Fig. 3.60

The unbalance response curves calculated at the bearing stations are shown in Fig. 3.61, for a mmg80 unbalance of the disc. The abscissae of the five peaks indicate the peak response critical speeds. The peak due to the first mode is barely noticeable at 413 rpm.

a b

Fig. 3.61

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Example 3.8

Consider a rotor with two bearings and an overhung disc (Fig. 3.62). The rigid disc, with mass kg57. , polar mass moment of inertia 2kgm040. and

diametral mass moment of inertia 2kgm020. , is located at station 5, at the right

end. The shaft with Young’s modulus 211 mN102 ⋅ and mass density 3mkg8000 has diameter mm50=d and total length m1=l , and is modeled by

4 equal length beam elements. The identical bearings are located at stations 1 and 3 having the following constant stiffness and damping coefficients

mN1052 7⋅= .kyy , mN104 7⋅=zzk , and msN105 3⋅== zzyy cc [12].

Fig. 3.62

Fig. 3.63

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3. SIMPLE ROTORS IN FLEXIBLE BEARINGS 179

The Campbell diagram for the first six modes is presented in Fig. 3.63. Mode 4 is mixed, due to the interaction of modes 2F and 3B. There is a curve veering in Fig. 3.63 and a curve crossing in Fig. 3.64, around 13000 rpm.

The damping ratio diagram is shown in Fig. 3.64 for eight modes.

Fig. 3.64

The root locus diagram is shown in Fig. 3.65 for the first six modes and speeds up to 30000 rpm.

Fig. 3.65

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DYNAMICS OF MACHINERY 180

The evolution of the mixed mode with the rotor speed is shown in Fig. 3.66.

a b c

d e f

Fig. 3.66

The first six mode shapes at 10000 rpm are presented in Fig. 3.67.

a b c

d e f

Fig. 3.67

The unbalance response curves at the bearing stations 1 and 3 are shown in Fig. 3.68 for a disc unbalance of mmg15 . Peaks occur at the eigenfrequencies of forward modes, because backward modes are relatively highly damped.

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3. SIMPLE ROTORS IN FLEXIBLE BEARINGS 181

a b

Fig. 3.68

Figure 3.69 shows the variation of the undamped natural frequencies of modes 3 to 6, as a function of the bearing stiffness, at 12000 rpm. The vertical lines indicate the bearing vertical and horizontal stiffness coefficients

mN1052 7⋅= .kyy and mN104 7⋅=zzk . Modes 4 (2F) and 5 (3B) have different shapes but almost equal natural frequencies. They interact, giving rise to a compounded mixed mode.

Fig. 3.69

In the following examples, the rotors are carried in oil-film journal bearings. The Campbell diagrams of these systems have specific features. The first two ‘rigid body’ backward modes are overdamped and do not appear in the diagram. The curves of the first two forward modes follow closely the half-frequency excitation line. The two-node flexural forward mode interacts with the

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DYNAMICS OF MACHINERY 182

cylindrical forward rigid body mode giving rise to compounded modes, sometimes referred to as ‘convex cylindrical’ and ‘concave cylindrical’. In some cases, even the forward modes become overdamped and disappear from the diagram.

The stability diagrams are useful to locate the onset speed of instability. The damping ratio diagrams help locating this threshold speed and show also when some modes are overdamped. The root locus diagrams give an overview of the eigenvalue variation with the rotor speed and can be used to explain the occurrence of mixed modes of precession.

Generally, the mode labeling for these systems is more difficult than for rotors carried by supports with constant coefficients, and the pattern of mode pairs with backward and forward precession is either changed or difficult to recognize.

Example 3.9 a

Consider the rotor from Fig. 3.70 supported in two identical journal bearings. The rigid disc has the mass kg20 , the polar mass moment of inertia

2mkg1 and the diametral mass moment of inertia 2mkg70. . The massless flexible

shaft of diameter mm425. and Young’s modulus 211 mN1012 ⋅. has lengths mm8512 =l and mm25523 =l [13] and is modeled with only two elements.

Fig. 3.70

The bearings have diameter mm 425.D = , length mm 16=L , radial clearance μm 235.C = , and oil dynamic viscosity 2mNs 020.=μ . The static loads on bearings are N41421 .W = and N8532 .W = . The speed dependence of the stiffness and damping coefficients, calculated based on Ocvirk’s short bearing assumptions [14], with a fully cavitated film, i.e. with the oil film extending only

0180 , is shown in Fig. 3.71.

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3. SIMPLE ROTORS IN FLEXIBLE BEARINGS 183

a b

Fig. 3.71

The Campbell diagram is presented in Fig. 3.72 for the first six natural modes of precession. At the crossing points with the synchronous excitation line, the damped critical speeds are determined as 2186, 6047 and 9442 rpm.

Fig. 3.72

Modes 1B and 2B are overdamped and do not appear in the diagram. Modes 1F and 2F are ‘rigid body’ modes controlled by the hydrodynamic bearings and follow closely the half-frequency excitation line 2Ωω = . If one sliding bearing is replaced by a rigid bearing, one of these lines disappears. If both sliding bearings are replaced by rigid bearings then both lines disappear. The curves of modes 3F and 4B cross each other at about 12600 rpm but the two modes do not interact.

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DYNAMICS OF MACHINERY 184

Fig. 3.73

The damping ratio diagram is shown in Fig. 3.73 for the same 6 modes.

Fig. 3.74

The stability diagram is plotted in Fig. 3.74 for only four modes. Mode 1F becomes unstable at 10331 rpm. Looking at the associated point in the Campbell diagram, it can be seen that the whirling takes place at a frequency of about half the spin speed, describing the bearing instability known as the ‘oil whirl’.

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3. SIMPLE ROTORS IN FLEXIBLE BEARINGS 185

a b c

d e f

Fig. 3.75

The shape of the first six eigenmodes at 15000 rpm is shown in Fig. 3.75. The forward modes, with larger relative displacements in bearings, have higher damping ratio values.

Fig. 3.76

The root locus diagram for the first six modes and for speeds up to 15000 rpm is shown in Fig. 3.76. The curve of mode 1F crosses the vertical at zero damping ratio, indicating the loss of stability.

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DYNAMICS OF MACHINERY 186

Example 3.9 b

Solve the problem of Example 3.9 a using the Moes’ impedance model [15] for plain cylindrical bearings.

The speed dependence of the bearing stiffness and damping coefficients is shown in Fig. 3.77.

a b

Fig. 3.77

The Campbell diagram is presented in Fig. 3.78 and the damping ratio diagram in Fig. 3.79. The damped critical speeds are 2179, 6047 and 9322 rpm.

Fig. 3.78

Modes 1B and 2B are overdamped and do not show up in the Campbell diagram. Modes 1F and 2F are ‘rigid body’ modes and their curves follow closely

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3. SIMPLE ROTORS IN FLEXIBLE BEARINGS 187

the half-frequency excitation line 2Ωω = . The curves of modes 3F and 4B cross each other twice, but the modes do not interact.

Fig. 3.79

The stability diagram for only four modes is given in Fig. 3.80. Mode 1F becomes unstable at 10016 rpm, which is lower than the onset speed of instability calculated for Ocvirk bearings. Thus, use of the short bearing approximation is not recommended in stability analyses.

Fig. 3.80

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DYNAMICS OF MACHINERY 188

The shapes of the first six eigenmodes at a rotor speed of 15000 rpm are shown in Fig. 3.81. With only three nodes in the model, their shape is approximate.

a b c

d e f

Fig. 3.81

The root locus diagram in presented in Fig. 3.82 for the first six modes and speeds up to 15000 rpm.

Fig. 3.82

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3. SIMPLE ROTORS IN FLEXIBLE BEARINGS 189

Example 3.9 c

Consider the rotor of Example 3.9 a supported now by two-lobe bearings with mm 712.L = .

The speed dependence of the bearing stiffness and damping coefficients, calculated based on data from Someya’s book [16] for 50.DL = and a preload factor 43=pm , is shown in Fig. 3.83.

a b

Fig. 3.83

The Campbell diagram is presented in Fig. 3.84. The damped critical speeds are 2182, 7638 and 9619 rpm. Modes 1F and 2F do not follow the half-frequency line. The curves of modes 3F and 4B do not cross each other.

Fig. 3.84

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DYNAMICS OF MACHINERY 190

The damping ratio diagram in presented in Fig. 3.85 for only two modes. Mode 3F becomes unstable at 13854 rpm, which is much higher than the onset speed of instability for cylindrical bearings.

Fig. 3.85

The same information is given by the stability diagram from Fig. 3.86.

Fig. 3.86

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The root locus diagram in presented in Fig. 3.87 for the first six modes and speeds up to 16000 rpm. Modes 1F and 2F are highly damped. The curve of mode 3F intersects the zero damping line at the point marking the damped natural frequency at the instability threshold.

Fig. 3.87

The unbalance response curves calculated at the left bearing and disc locations are presented in Fig. 3.88 for an unbalance of mmg20 on the disc.

a b

Fig. 3.88

Example 3.10 a

Consider the rotor of Fig. 3.43 supported in two identical plain cylindrical bearings. The rigid disc has the mass kg079. , the polar mass moment of inertia

2mkg04680. and the diametral mass moment of inertia 2mkg03050. . The

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DYNAMICS OF MACHINERY 192

massless flexible shaft of diameter mm22 and Young’s modulus 211 mN101452 ⋅. has the total length m5080. and is divided into four equal

elements [17].

The bearings have diameter mm 425. , length mm 425. , clearance μm 2203. , and oil viscosity 2mNs 02410. . The static loads on bearings are

N494421 .WW == . The speed dependence of the stiffness and damping coefficients, calculated based on Moes’ impedance model, is shown in Fig. 3.89.

Fig. 3.89

The Campbell diagram for the first four modes is presented in Fig. 3.90. The damped critical speeds are 384, 1141, 1739 and 2790 rpm.

Fig. 3.90

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3. SIMPLE ROTORS IN FLEXIBLE BEARINGS 193

The damping ratio diagram in presented in Fig. 3.91. Mode ∗F3 becomes unstable at 4061 rpm.

Fig. 3.91

The same information is given by the stability diagram from Fig. 3.92.

Fig. 3.92

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DYNAMICS OF MACHINERY 194

The root locus diagram in presented in Fig. 3.93 for speeds up to 6000 rpm. The curve of mode ∗F3 intersects the zero damping line at the point marking the damped natural frequency at the instability threshold.

Fig. 3.93

The first six mode shapes at 3000 rpm are presented in Fig. 3.94.

a b c

d e f

Fig. 3.94

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3. SIMPLE ROTORS IN FLEXIBLE BEARINGS 195

Mode 3 ( ∗F3 ) is a convex-cylindrical mode, while mode 2 (2F) is a concave-cylindrical mode.

a b

Fig. 3.95

a b

Fig. 3.96

a b

Fig. 3.97

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DYNAMICS OF MACHINERY 196

The unbalance response curves are calculated at locations 1 and 3 for an eccentricity of m100841 4−⋅. of the disc mass. Figures 3.95 show the diagrams of the ellipse semiaxes, Fig. 3.96 presents the diagrams of the vertical and horizontal components, while Fig. 3.97 gives the diagrams of the radii of the forward and backward generating circles. Around 3000 rpm, the orbits in bearings are circular, while the disc orbit is elliptical.

Example 3.10 b

Consider the rotor of Example 3.10 a, with small modifications. The massless flexible shaft has the diameter mm222. and Young’s modulus

211 mN100382 ⋅. . The plain cylindrical bearings have diameter mm 425.D = ,

length mm 425.L = , radial clearance m 1087961 4−⋅= .C , and oil dynamic viscosity 25 mNs 10960 −⋅=μ , as in [18].

The speed dependence of the stiffness and damping coefficients, calculated based on Moes’ impedance model, is shown in Fig. 3.98.

Fig. 3.98

The Campbell diagram for the first four modes of precession is presented in Fig. 3.99. The damped critical speeds are 805, 876, 1778 and 2817 rpm. Mode 1F becomes overdamped beyond 1000 rpm.

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3. SIMPLE ROTORS IN FLEXIBLE BEARINGS 197

Fig. 3.99

The damping ratio diagram in presented in Fig. 3.100 for only three modes. Mode ∗F3 becomes unstable at 5180 rpm.

Fig. 3.100

The same information is given by the stability diagram from Fig. 3.101.

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Fig. 3.101

The root locus diagram in presented in Fig. 3.102 for speeds up to 6000 rpm. The curve of mode ∗F3 intersects the zero damping line at the point marking the damped natural frequency at the instability threshold.

Fig. 3.102

The first three mode shapes at 2500 rpm are presented in Fig. 3.103.

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3. SIMPLE ROTORS IN FLEXIBLE BEARINGS 199

a b c

Fig. 3.103

The unbalance response curves are calculated at locations 1 and 3, for an eccentricity of m100841 4−⋅. of the disc mass. Figures 3.104 show the diagrams of the ellipse semiaxes, Figs. 3.105 present the diagrams of the vertical and horizontal components, while Figs. 3.106 give the diagrams of the radii of the forward and backward generating circles.

a b

Fig. 3.104

a b

Fig. 3.105

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DYNAMICS OF MACHINERY 200

Around 3000 rpm, the orbits in bearings are circular, while the disc orbit is elliptical.

a b

Fig. 3.106

Between about 2200 and 2800 rpm, the steady state response due to unbalance is a mixed mode (Fig. 3.107, a), with backward precession at the disc and forward precession at bearings. At 5200 rpm the steady state precession is forward (Fig. 3.107, b).

a b

Fig. 3.107

Example 3.11

The rotor rig of Fig. 3.108 is carried by an Oilite (oil impregnated, sintered bronze) bush supported on a rubber O-ring at the left inboard end, and by an oil lubricated Lucite plain cylindrical journal bearing at the outboard end. The rigid disc has the mass kg810. , the polar mass moment of inertia

24 mkg1078355 −⋅. , the diametral mass moment of inertia 24 mkg1035723 −⋅. and is located at mm422.=l from the right end. The flexible shaft of diameter

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3. SIMPLE ROTORS IN FLEXIBLE BEARINGS 201

mm5259. , density 3mkg7860 and Young’s modulus 211 mN10062 ⋅. has total length m590. and is divided into six elements [19].

Fig. 3.108 (from [19])

The parameters of the left bearing are mN10764 5⋅= .k yy ,

mN10544 5⋅= .kzz , msN8726.cyy = and msN123.czz = . The journal bearing

parameters are mm 9124.D = , mm 13=L , μm 120=C , 2mNs 027840.=μ . The static loads on bearings are N3821 .W = and N882 .W = .

a b

Fig. 3.109

The speed dependence of the stiffness and damping coefficients, calculated based on Moes’ impedance model, is shown in Fig. 3.109, b.

The Campbell diagram for the first three modes of precession is presented in Fig. 3.110. The damped critical speeds are 2648, and 2867 rpm.

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DYNAMICS OF MACHINERY 202

Fig. 3.110

The damping ratio diagram in presented in Fig. 3.111. Mode 3F becomes unstable at 5070 rpm.

Fig. 3.111

The same information is given by the stability diagram from Fig. 3.112.

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3. SIMPLE ROTORS IN FLEXIBLE BEARINGS 203

Fig. 3.112

The root locus diagram in presented in Fig. 3.113 for speeds up to 8000 rpm. The curve of mode 3F intersects the zero damping line at the point marking the damped natural frequency at the instability threshold.

Fig. 3.113

The first four mode shapes at 4000 rpm are shown in Fig. 3.114.

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DYNAMICS OF MACHINERY 204

a b

c d

Fig. 3.114

Mode 3 is a mixed mode, predominantly backward.

a b Fig. 3.115

The unbalance response curves are calculated at the disc location 5 (Fig. 3.115, a) and at the oil-film bearing 7 (Fig. 3.115, b) for an eccentricity of

m03050. of the disc mass.

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References

1. Gasch, R. and Pfützner, H., Rotordynamik, Springer, Berlin, 1975.

2. Wölfel, H. P., Maschinendynamik, Umdruck zur Vorlesung, T. H. Darmstadt, 1989/90.

3. Radeş, M., On the effect of bearing damping on the critical speeds of flexible rotors, Buletinul Inst. Politehnic Bucureşti, Vol.42, No.3, pp.101-112, 1980.

4. Kellenberger, W., Elastisches Wuchten, Springer, Berlin, 1987, p.59.

5. Radeş, M., Influenţa amortizării lagărelor asupra turaţiilor critice ale rotorilor elastici, St. Cerc. Mec. Apl., Vol.30, No.6, pp.903-911, 1980.

6. Constantinescu, V. N., Nica, Al., Pascovici, M. D., Ceptureanu, Gh., and Nedelcu, Şt., Lagăre cu alunecare, Editura tehnică, Bucureşti, 1980.

7. Krämer, E., Maschinendynamik, Springer, Berlin, 1984.

8. Wang, W., and Kirckhope, J., New eigensolutions and modal analysis for gyroscopic/rotor systems. Part 1: Undamped systems, J. Sound Vib., Vol.175, No.2, pp 159-170, 1994.

9. Radeş, M., Mixed precession modes of rotor-bearing systems, Schwingungen in rotierenden Maschinen III, (Irretier, H., Nordmann, R. and Springer, H., eds.), Vieweg, Braunschweig, pp. 153-164, 1995.

10. Jei, Y.-G. and Kim, Y.-J., Modal testing theory of rotor-bearing systems, ASME J. of Vibration and Acoustics, Vol.115, pp.165-176, April 1993.

11. Radeş, M., Dynamics of Machinery, Vol.2, Univ. Politehnica Bucureşti, 1995.

12. Lee, C.-W., Vibration Analysis of Rotors, Kluwer Academic Publ., Dordrecht, 1993.

13. Genta, G. and Vatta, F., A lubricated bearing element for FEM rotor dynamics, Proc. Int. Modal Analysis Conf., pp 969-975, 1991.

14. Ocvirk, F., Short bearing approximation for full journal bearings, NACA TN 20808, 1952.

15. Childs, D., Moes, H., and van Leeuwen, H., Journal bearing impedance descriptions for rotordynamic applications, J of Lubrication Technology, pp.198-219, 1977.

16. Someya, T., (ed.), Journal-Bearing Databook, Springer, Berlin, 1988.

17. Bhat, R. B, Subbiah, R., and Sankar, T. S., Dynamic behavior of a simple rotor with dissimilar hydrodynamic bearings by modal analysis, ASME Journal of Vibration, Acoustics, Stress, and Reliability in Design, Vol.107, pp.267-269, April 1985.

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DYNAMICS OF MACHINERY 206

18. Subbiah, R., Bhat, R. B., Sankar, T. S., and Rao, J. S., Backward whirl in a simple rotor supported on hydrodynamic bearings, NASA CP 2409, Instability in Rotating Machinery, 1985.

19. Van de Vorst, E. L. B., Fey, R. H. B., De Kraker, A., and Van Campen, D. H., Steady-state behaviour of flexible rotor dynamic systems with oil journal bearings, Proc. WAM of ASME, Symposium on Nonlinear and Stochastic Dynamics, (A.K.Bajaj, N.S. Namachchivaya, R.A.Ibrahim, eds.), AMD-Vol.192, DE-Vol.78, New York, pp.107-114, 1994.

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4. ROTOR DYNAMIC ANALYSIS

In order to understand the dynamic response of a rotating machine it is necessary to have information on the following aspects of its behavior: 1) the lateral critical speeds of the rotor-bearing-pedestal-foundation system; 2) the precession orbits as a response to different unbalance distributions, over the whole operating range of the machine; 3) the rotor onset speed of instability, i.e. the threshold speed for stable whirling due to the rotor/bearing and/or working fluid interaction; and 4) the response to transient excitation such as blade loss.

In the prediction (design) phase of rotor dynamic analysis, the main concern is the placement of critical speeds with respect to the machine operating speed. In order to ensure smooth and safe operation, most standards require at least

%15 separation margin between the operating speed and the critical speeds.

When the critical speeds are within the undesirable range, they can be shifted outside this range by modifying bearing and support stiffnesses, the bearing span, the disc mass properties or the shaft geometry.

4.1 Undamped critical speeds

For lightly damped rotor systems, the associated undamped isotropic system is used for critical speed computation. When the bearings have different stiffnesses, a single mean value is considered.

4.1.1 Effect of support flexibility

The simplest rotor with distributed mass consists of a shaft with uniform cross-section, simply supported at the ends on flexible bearings of equal stiffness

2Bk . Figure 4.1 shows the influence of bearing flexibility on the first four lateral critical speeds of the undamped rotor system.

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In Fig. 4.1, the ratio of the rotor critical speed on flexible bearings to the first critical speed on rigid bearings 1ω ( )∞=Bk is plotted against the

dimensionless flexibility parameter ( )[ ] 2121ωδ gB , where Bδ is the static

deflection of bearings.

The first and second critical speed ratios show a continuing decrease with increasing bearing flexibility. At the same time, the associated mode shapes gradually transform, from those corresponding to rigid bearings to the ‘cylindrical’ and ‘conical’ rigid-body mode shapes.

Fig. 4.1

The third and fourth critical speed ratios first exhibit a rapid decrease with increasing bearing flexibility and then level out and asymptotically approach values of 2.27 and 6.25. These two modes are referred to as the first and second flexible “free-free” modes of the system.

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4. ROTORDYNAMIC ANALYSIS 209

In Fig. 4.2, the first four critical speed ratios are plotted against the

dimensionless stiffness parameter ( )[ ] 2121

−ωδ gB .

Fig. 4.2

4.1.2 Critical speed map

A convenient means for analyzing the influence of rotor support dynamic properties on the dynamic performance of a rotor-bearing system is by a critical speed map, as shown in Fig. 4.3.

In such a chart, the horizontal scale represents support stiffness, mN , and the vertical scale is rotor speed, rpm. The curves are drawn calculating the first few lateral critical speeds for different values of an average constant stiffness of the bearings. For relatively highly damped rotors, an equivalent dynamic stiffness is used, calculated as

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( )22 ckkd Ω+= , (4.1)

where k and c are the average stiffness and damping coefficients, respectively, and Ω is the rotor angular speed.

But few actual bearings retain constant stiffness with speed change. To determine the actual critical speeds, it is necessary to plot the bearing stiffness versus speed characteristic over the rotor critical speed lines. The intersections between the rotor curves and the bearing characteristic are the critical speeds for the actual rotor in its bearings.

Figure 4.3 shows a rotor having a “soft” support in the horizontal direction, zzk , and a “harder” stiffness in the vertical direction, yyk . The lower stiffness

mode shape in the horizontal direction at the first critical speed is almost cylindrical. In the vertical direction, higher support stiffness causes this rotor to be almost simply supported.

Fig. 4.3 (from [1])

In order to minimize the rotor precession radius, hence the vibration amplitude, one should not design the machine to operate at the critical speed. There are two ways that this can be avoided: 1) change bearing stiffness; 2) change rotor geometry.

The first method works when the support stiffness curves cross the critical speed curves in the sloped region (Fig. 4.4, a). In this position, an undesirable resonance can be shifted upward and downward in frequency by changing the support stiffness. Such a shift can be accomplished by a small geometric change in

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4. ROTORDYNAMIC ANALYSIS 211

the bearings such as decreasing the clearance or increasing the preload to raise the critical speed (10 to 20%).

a b

Fig. 4.4 (from [1])

High bearing stiffness (Fig. 4.4, b) is undesirable. The support stiffness curves are in the flat part of the critical speed curves, where simple geometric bearing changes have minor effects on resonance. The shaft is much more flexible than the bearing supports.

Fig. 4.5 (from [1])

Rotors with stiff bearings operating near criticals change the engineer options to the rotor shaft. He can modify the shaft diameter (Fig. 4.5) or change the

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DYNAMICS OF MACHINERY 212

bearing span. An increase in resonance in the asymptotic region of stiffness-speed curves is dependent on rotor construction. Attempts to alter resonance conditions in this region by bearing stiffness changes are often doomed to failure.

In some applications, the insensitivity of the third critical speed to support stiffness permits a range of operating speeds that does not traverse any of the critical speeds. Many machines are designed to work between the second and the third critical speed where, in the region of soft bearings, there is the largest speed range between criticals.

The critical speed map provides a useful guide to the dynamic performance of rotor-bearing systems, especially for lightly damped machines such as those in rolling element bearings, and for externally pressurized gas bearings. Apart from showing just undamped critical speeds, it shows only possible critical speeds.

For other bearings having high damping, e.g. oil lubricated hydrodynamic journal bearings, bearing stiffness alone does not determine rotor behaviour. Damping plays an equally important part, especially at high speeds. The first two ‘rigid-body’ precession modes have relatively large orbit radii in soft bearings so that the motion at the critical speeds can be completely damped and the rotor passes through the critical without noticeable vibrations.

The critical speed map is particularly useful for rotors in tilting pad bearings. Bearings with vertical preloading on the pad are ‘more anisotropic’ than the bearings with loading between pads, as shown in Fig. 4.6.

Fig. 4.6 (from [1])

Critical speed maps are also constructed for machines on more than two bearings. Figure 4.7 shows such a diagram for a 15 MW synchronous electrical

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4. ROTORDYNAMIC ANALYSIS 213

motor without slip rings. In the expected range of bearing flexibility MNmm21−=α , the first critical speed is 24002100− rpm and the other critical

speeds are above the maximum operating speed of 5000 rpm [2].

Fig. 4.7 (from [2])

Another critical speed map for a generator rotor on three bearings (used for balancing) is presented in Fig. 4.8, where the mode shapes are also shown for various bearing flexibilities. The hatched area marks the speed range below one-third the first critical speed for rigid supports ( 0=α ).

Figure 4.9 shows the differences in the mode shape forms of a turbo-generator, coupled and uncoupled. The first five mode shape forms and the corresponding critical speeds of the shaft line of the generator G coupled with the turbine T are shown in Fig. 4.9, a. The first two mode shapes of the generator in three bearings (as it is balanced) are shown in Fig. 4.9, b. The first and second mode shapes of the generator alone are recognized in the first and fifth mode shapes of the shaft line, but occur at different speeds. The rigid coupling places a firm restraint on the shaft end, changing the critical speed, but the difference between the corresponding mode shapes is small.

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Fig. 4.8 (from [3])

Fig. 4.9 (from [3])

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4. ROTORDYNAMIC ANALYSIS 215

Figure 4.10 shows the critical speed map for an industrial turbine rated 50 MW. The mode shapes of the shaft line show the predominance of each component – generator G, turbine T and exciter E - at the respective critical speed. Comparing the operating speed of 3600 rpm with the critical speed lines it is concluded that the practical region of bearing flexibility is between 5 and 10 tfμm .

Fig. 4.10 (from [4])

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DYNAMICS OF MACHINERY 216

In practice, the following important conclusions are useful.

Undamped critical speeds calculated taking into account the bearing support stiffness are lower than those predicted by rigid support analysis. The less stiff the bearings are relative to the shaft, the more likely the mode shape nodes will be displaced from the bearings. With the nodes displaced from the bearings, relative motion between the bearing and shaft will occur at the critical speeds. This relative motion produces a velocity-dependent force. The larger the displacement, the larger the velocity and, thus, the larger the damping force.

Bearing damping (inherent in hydrodynamic bearings) has the effect of raising the undamped critical speeds. Damped critical speeds will be determined using the Campbell diagrams. The amount of increment depends on the degree to which the undamped critical speeds are depressed from the rigid support criticals and on the location of the nodal points with respect to the bearing centre lines. Bearing damping does not raise the undamped critical speeds if these are close to rigid support critical speeds.

It is of paramount importance to distinguish among undamped critical speeds, damped critical speeds and peak response speeds. Undamped and damped critical speeds are different from the speeds of peak steady-state unbalance response. This requires unbalance response analyses. In general, the percentage deviation of the first critical lies within a range of %6± . For higher critical speeds, the variations may be significantly greater than 6 percent.

The plotting of undamped critical speeds versus static stiffness in order to predict peak response speeds can be very misleading and should be avoided. This is especially true when trying to specify the location of second and third modes for the purpose of meeting specification on their closeness to maximum continuous operating speeds, as required by, for example, API Standards.

According to the API Standard 617 [5], to avoid excessive vibration, the first lateral critical speed of rigid-shaft compressors must be at least 20 per cent higher than the maximum continuous speed. Flexible-shaft compressors must operate with the first critical speed at least 15 per cent below any operating speed. The second lateral critical speed must be 20 per cent above the maximum continuous speed. The same conditions are imposed by API Standard 613 [6] for gear units.

For special-purpose steam turbines, the API Standard 612 [7] requires a 10 per cent margin for rigid-shaft rotors while the first critical speed of a flexible-shaft rotor should not exceed 60 per cent of the maximum continuous speed, nor should it be within 10 per cent of any operating speed.

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4. ROTORDYNAMIC ANALYSIS 217

4.1.3 Influence of stator inertia

An interesting application of a critical speed map is the selection of bearings for a balancing facility [3]. Figure 4.11 shows a resonance curve measured on a bearing pedestal using a vibration exciter. The driving point displacement is plotted against the excitation frequency for constant amplitude of the sinusoidal excitation force. The narrow bandwidth and the sharp phase change at resonance indicate practically negligible damping, the quality factor being 14=Q .

Fig. 4.11 (from [3])

Neglecting the damping, the bearing dynamic flexibility (ratio of deflection to force amplitude) can be plotted versus speed, instead of frequency, as in Fig. 4.12, a. Plotting the speed versus flexibility in semi-logarithmic coordinates (Fig. 4.12, b), the bearing dynamic characteristic encompassing both stiffness and mass effects is obtained. It has the same format as the critical speed map of a rotor mounted in idealized elastic bearings (springs), so that they can be overlaid to obtain the critical speeds of the combined system.

Figure 4.13, a shows the critical speed map for the rotor of a turbo generator rated ( )Hz60MVA130 . The speed-flexibility curve of the bearing from Fig. 4.11, transformed as in Fig. 4.12, is overlaid.

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a b

Fig. 4.12

The cross-over points of the two curves give the critical speeds for the combined rotor-bearing system. In the range from zero to slightly above overspeed (4320 rpm) there are four critical speeds, at 1000, 2900, 4100 and 4400 rpm. Unfortunately, two critical speeds lie very close to the overspeed, which is undesirable, making the balancing difficult.

a b

Fig. 4.13 (from [3])

The bearings are too ‘soft’. If the bearing pedestal is the softest element in the chain, then it should be stiffened.

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4. ROTORDYNAMIC ANALYSIS 219

Figure 4.13, b shows the critical speed map of the same rotor in which the speed-flexibility characteristic of another stiffer bearing is overlaid, having a higher natural frequency of 128 Hz (7690 rpm). In this case, the bearing dynamic flexibility is almost constant ftμm5 . This leaves only two critical speeds, at 1000 and 2900 rpm, well below the operating speed and the overspeed.

Operating a rotor at or near a forward critical speed causes large and potentially damaging deflections and should be avoided.

4.2 Damped critical speeds

Bearing damping shifts the critical speeds to larger values. When the damping is significant, as for most hydrodynamic bearings, the main concern is the computation of damped critical speeds and their associated damping ratios. They are determined solving the eigenvalue problem of the linearized damped rotor system.

4.2.1 Linear bearing models

The nonlinear characteristics of the sliding bearings can be linearized at the static equilibrium position, as shown in Chapter 6. The dynamic characteristics of a bearing can be represented by four stiffness and four damping coefficients. The amplitude of the forces acting on the shaft journal can be expressed as in (1.1)

⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡+

⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡=

⎭⎬⎫

⎩⎨⎧

zy

cccc

zy

kkkk

ff

zzzy

yzyy

zzzy

yzyy

z

y&

&, (4.2)

where zy f,f are the components acting in the z,y directions, respectively, z,y and z,y && are the journal displacements and velocities, in the same directions.

For journal bearings, the eight dynamic coefficients zzyyzzyy c,..,c,k,...,k are functions of the Sommerfeld number (see Chapter 6). For rolling element bearings, the bearing coefficients can be assumed to be constant and with the cross coupling coefficients equal to zero.

In the following applications, three kinds of data will be used for hydrodynamic bearings: a) the Moes’ impedance model [8] and b) the Ocvirk short bearing model [9] for plain circular bearings, and c) tabular values of the eight dynamic dimensionless coefficients as a function of either speed or the Sommerfeld number, as given in Someya’s book [10]. Spline interpolation is used in the last case for calculations at speeds not included in the initial data.

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4.2.2 Equations of damped motion

Rotor-bearing systems are modelled as an assemblage of rigid discs, distributed mass and stiffness shaft elements, and discrete bearings and seals. Once the element mass, stiffness and damping matrices are established (see Chapter 5), they can be assembled to obtain the global system matrices.

The damped critical speeds and system stability limits are determined from the homogeneous form of the equations of motion

0][][][ =++ xKxCxM &&& , (4.3)

where x is the global vector of nodal coordinates and [ ] [ ] [ ]GCC D Ω+= , where [ ]G is the skew-symmetric gyroscopic matrix, Ω is the rotational speed, and [ ]DC is the damping matrix.

In (4.3), [ ]M is the symmetric mass matrix, the stiffness matrix [ ]K is usually unsymmetrical due to the cross coupling terms of hydrodynamic bearings and the clearance excitation factors. The damping matrix [ ]DC is usually symmetric, but it can be unsymmetrical when the internal damping or structural damping due to shrink fits is taken into account.

4.2.3 Eigenvalue problem of damped rotor systems

It is convenient to write the system equation (4.3) in the state space form

0][][ =+ qBqA & , (4.4)

where the matrices [ ]A , [ ]B and the vector q are defined as

[ ] ⎥⎦

⎤⎢⎣

⎡=

IM0

0A , [ ] ⎥

⎤⎢⎣

⎡−

=0IKC

B , ⎭⎬⎫

⎩⎨⎧

=xx&

q . (4.5)

On trying a solution to equation (4.4) of the form

te λyq = (4.6)

we obtain the linear eigenvalue problem

( ) 0][][ =+ yBAλ . (4.7)

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Since matrix [ ]A is a positive definite real symmetric matrix and [ ]B is an arbitrary real matrix, the generalized eigenvalue problem (4.7) can be reduced to the standard form

[ ] [ ]( ) yyBA λ=− −1 , (4.8)

where

[ ] [ ]( ) [ ] [ ] [ ] [ ][ ] [ ] ⎥

⎥⎦

⎢⎢⎣

⎡ −−=−−−

0

111

IKMCMBA (4.9)

is an unsymmetrical real matrix.

Due to the generally unsymmetrical matrices [ ]C and [ ]K , the eigenvalues rλ of equation (4.8) are real numbers for overdamped modes and complex numbers for underdamped modes. Because the matrix has real elements, the complex eigenvalues must occur in complex conjugate pairs and have the form

i i rrrrrr , ωαλωαλ −=+= ( ),....,,r 321= (4.10)

and are functions of the rotational speed Ω .

The imaginary part rω is the damped natural frequency (of precession) and the real part rα is an attenuation (or growth) constant.

Usually, the damping is expressed in terms of the modal damping ratio

.r

r

rr

rr ω

α

ωα

αζ −≅+

−=22

(4.11)

For underdamped systems, the complex eigenvectors have the form

⎬⎫

⎩⎨⎧

=r

rrr u

uy

λ,

⎭⎬⎫

⎩⎨⎧

=

r

rrr u

uy

λ, (4.12)

where

i i rrrrrr bau,bau −=+= (4.13)

so that only the lower half is considered.

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4.2.4 Campbell diagrams

Plots of the damped natural frequencies rω as a function of the rotor speed Ω are called precession speed maps. When these plots contain the excitation lines overlaid they are referred to as interference diagrams or Campbell diagrams.

It is common practice to plot also the damping ratios versus the rotor speed. These plots will be referred to as the damping ratio diagrams.

Figure 4.14 shows a typical Campbell diagram for a multi-mass flexible rotor.

As shown in Section 3.3.5, the rotor precession is usually described by forward (F) and backward (B) modes. The stator anisotropy gives rise to pairs of backward and forward modes. The gyroscopic effect splits a B and F mode pair, softening the B mode (lowering its natural frequency) and stiffening the F mode (increasing its natural frequency).

For actual rotors, the normal sequence of B and F modes can change. The speed-dependent bearing coefficients, the high damping levels and the formation of compounded modes, with mixed B and F precession, require the calculation and use of precession mode shape forms for the proper labeling of modes in the Campbell diagram.

Fig. 4.14

In some cases, it is better to use a mode numbering based on the index of the eigenfrequencies (sorted in ascending order) and not based on the mode directivity B or F. Mixed modes (M) are difficult to label. One can either mention

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4. ROTORDYNAMIC ANALYSIS 223

the percent of F and B motion along the rotor stations, or indicate the two basic components of a compounded mode which influence each other to give rise to the M mode.

A critical speed of order κ is defined as the rotor speed for which a multiple of that speed coincides with one of the system natural frequencies of precession.

An excitation frequency line has an equation Ωκω = . It is a line of slope κ passing through the origin of the Campbell diagram. The intersection of this line with the damped natural frequency curve rω defines the damped critical speed rΩ . When Ω equals rΩ , the excitation frequency rΩκ creates a resonance (critical) condition.

One approach for determining critical speeds is to use the diagram of damped natural frequencies versus speed and overlap all excitation frequency lines of interest, marking the intersection points of the two families of curves. Their abscissae determine the damped critical speeds.

For 1=κ , Ωω = is the synchronous excitation line, usually due to mass unbalance. For 2=κ , Ωω 2= is the misalignment excitation line. For 21≅κ ,

2Ωω = is the half-frequency subharmonic excitation line due to oil whirl in plain bearings.

4.2.5 Orbits and precession mode shapes

A damped mode shape can be solved from equation (4.4) with a particular root substituted

)sincos(e2)( tbtatx rrrrt

rr ωωα −= , (4.14)

which describes spiralling orbits. However it is agreed to represent the orbits as incomplete (open) ellipses, considering 0=rα and approximating the expression (4.14) by

tumtuetx rrrrr ωω sincos)( ℑ−ℜ= . (4.15)

It is common practice to plot spatial precession mode shapes by first drawing an ellipse at each station, then connecting the points on the ellipses at all stations along a rotor (at a given time t). Figure 4.15 shows a typical precession mode shape.

In the following examples, the points corresponding to 0=t are connected by a solid line, while the points at a quarter of a period later are connected by a broken line. This way, the directivity of the motion along each orbit is from the solid line to the broken line.

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Fig. 4.15

There are ellipses with forward precession (denoted F), ellipses with backward precession (denoted B), and ellipses degenerated into straight lines. Mixed modes are described in Section 3.3.5.

4.3 Peak response critical speeds

Undamped and damped critical speeds have been defined based on the coincidence of an excitation frequency with a rotor natural frequency. Both are ‘possible’ critical speeds, as far as nothing is said about the level of damping.

Of practical interest are the rotational speeds at which the rotor response has the largest value. The abscissae of peaks in the plots of the unbalance response at a rotor station as a function of the rotational speed determine the so-called peak response critical speeds. As shown in Example 3.1, these criticals are slightly different from the damped critical speeds and depend on the location along the rotor where they are calculated.

In machines with journal bearings, the relative motion between journal and bearing is measured with proximity transducers. The largest orbit radius (major semiaxis for ellipses) is an indication of the severity of rotor precession. The peaks in the diagrams of the steady state synchronous response at a rotor section, for a given unbalance magnitude and location, indicate the peak response critical speeds.

Figure 4.16 shows an example of unbalance response numerical simulation for an industrial turbine rotor.

First, the total unbalance is estimated based on existing standards on permissible residual unbalance values (ISO 1940) [11]. For a turbine shaft, a quality grade G2.5 is usually selected. [12].

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The value =G 2.5 mm/s corresponds to a centre of gravity offset

[ ]μm 0002488523

30 π100052

NNN n,

n,

n.Ge ≈=⋅

==ω

(4.16)

where Nn is the operating speed , rpm.

The total unbalance is

[ ]gmm 00024⋅

⋅==

Nnm,emU , (4.17)

where m is the mass of the rotor section between two bearings, kg.

For a turbine operating at 3000 rpm, the permissible residual unbalance is 8mU = [mm g].

Using a finite element model of the rotor, the first modes of lateral vibration are calculated, usually all modes below the trip speed and the mode just above the trip speed. Then, the worst unbalance distribution for each mode is considered, as in Figs. 4.16, da − , subdividing the total unbalance into suitable individual unbalance components. Apart from the first three modes of vibration, the vibration produced by the overhang half couplings is also considered.

Fig. 4.16 (from [12])

The result of the unbalance response calculation at the left bearing location is shown in Fig. 4.16, e. Each unbalance distribution results in a different amplitude versus speed curve. The amplitude A is calculated as the major semiaxis of the precession elliptical orbit.

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The positions of the amplitude peaks along the horizontal axis indicate the peak response critical speeds. In figure 4.16, e, 11,n and 21,n are the critical speeds for the first unbalance distribution, 12,n and 22,n are the critical speeds for the second unbalance distribution. The amplitudes of the unbalance response have to be compared with limit values given by guidelines and standards for the operating speed Nn .

The guideline ISO 7919-2 [13] indicates, as a limit of 'good' vibration performance, a maximum value of the journal orbit radius

[ ] μm 2400

Nmax n

sA= . (4.18)

Taking a safety factor of 1.7, the limit value is established at

[ ]μm 140071 N

maxlim n.

sA A == . (4.19)

For a turbine operating at 3000 rpm, μm26=limA . The condition

limAA ≤ must be satisfied at all speeds up to Nn . In Fig. 4.16, e, the peak at 22,n higher than limA is beyond the operating speed range.

Fig. 4.17 (from [12])

The effectiveness of the unbalance response calculation is illustrated in the following by means of an example concerning a turbine rotor used for mechanical drive with variable operating speeds [12].

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The critical speed map for the first two modes of vibration is shown in Fig. 4.17, a. The average value of oil film flexibilities of both bearings for the first mode is vα =1.3 mm/MN in the vertical direction and hα =3.7 mm/MN in the horizontal direction. According to Fig. 4.17, a, there are two undamped critical speeds within the speed range from 2600 to 5800 rpm, the first at 3250 rpm with a predominant horizontal response, and the second at 5000 rpm with a predominant vertical response. The turbine must therefore be operated at both critical speeds which is inadmissible according to the conventional critical speed design considerations.

The commissioning report, however, stated: "Turbine operating behavior is very good irrespective of load and speed. Turbine shaft amplitudes are μm 108− , bearing housing vibrations are μm 31− , in the horizontal and vertical directions. Critical speeds could not be determined" [12].

The unbalance response calculation confirmed the observed operating behavior. Shaft vibration amplitudes of both bearings show absolutely no resonance peaks near the critical speeds (solid lines in Fig. 4.17, b). Despite this, there are two critical speeds in the operating speed range which can be determined if the unbalance response is calculated with the oil film damping reduced to 10% (dotted lines). As shown in Fig. 4.17, b, there are two unbalance response peaks, the first at 3250 rpm and the second at approximately 5000 rpm.

The operating behavior of the shaft was smooth, since in this case it was short and quite rigid and was also provided with a soft oil film. The softer the oil film is in relation to the shaft rigidity, the more the shaft moves in its bearing so that oil film damping becomes fully effective.

4.4 Stability analysis

The stability of a linear rotor-bearing system depends on the damping exponent rα (4.10). A positive damping exponent indicates instability. The solution (4.14) shows that 0>rα is a growth factor, the whirling motion is along a spiral with growing radius, while 0<rα is a decaying factor, the whirl being along a spiral with decreasing radius.

When the real part of one of the roots changes from negative to positive, 0=rα , the rotor reaches the instability threshold. The corresponding imaginary

part defines the precession frequency of the incipient rotor instability. The rotational speed at the threshold of instability, sn , is called the onset speed of instability.

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In most applications, when the system becomes unstable, the rotor whirls in its first forward precession mode. The motion associated with an instability becomes unbounded in time. Below sn the rotor motion is stable and synchronous. Above this speed, there is a subsynchronous component to the rotor motion whose amplitude diverges exponentially with time and is a forward motion. The onset speed of instability always exceeds the rotor first critical speed.

Most of the destabilizing forces in rotor systems are “cross-coupled” in two directions. A radial deflection of the shaft, away from its equilibrium position, gives rise to a tangential force which, if it is larger than the opposite damping force, drives the rotor in an orbital motion.

The destabilizing force is proportional to the shaft deflection and grows larger as the radius of the whirl grows. The self-excited motion is along a spiral with increasing radius until is limited by nonlinear effects. The results of the linear theory predict the onset speed of instability but do not indicate the degree of instability, i.e. neither the violence of the motion at onset nor the growth of the unstable motion with increasing speed.

A common type of instability for rotors with oil-lubricated bearings is the “oil whirl”. The oil film wedge drives the journal within the bearing at slightly less than half the running speed, hence the name of “half-frequency whirl”.

A sliding bearing can be made stable by increasing its natural frequency or by increasing the bearing eccentricity ratio Ce=ε , where e is the journal eccentricity and C is the radial clearance.

Changing the natural frequency of the bearing is considerably more difficult than adjusting the ε ratio. The latter can be increased by decreasing the Sommerfeld number (see Chapter 6) or by decreasing the bearing length/diameter ratio.

The Sommerfeld number is defined as

2

⎟⎠⎞

⎜⎝⎛=

CR

WDLNS μ , (4.20)

where 2DR = is the bearing radius, L is the bearing length, C is the bearing radial clearance, μ is the oil dynamic viscosity and πΩ 2=N is the journal rotational frequency.

The following changes have a tendency to decrease the Sommerfeld number:

1) increasing the bearing average pressure DLWp = by: a) grooving the bearing circumferentially to reduce the surface loading area, using pressure dams and pockets; b) misaligning the bearing purposely to achieve greater loading; and

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4. ROTORDYNAMIC ANALYSIS 229

c) changing the valve-opening sequence to increase the loading due to partial admission bearing reaction.

2) increasing the bearing clearance;

3) increasing the bearing temperature which reduces the oil viscosity.

For plain cylindrical bearings, the difference of cross-stiffness coefficients zyyz kk − decreases with decreasing Sommerfeld number, which improves the

stability (see Chapter 6). At high Sommerfeld numbers, the coefficient zyk has negative values, increasing the difference. Bearings operating at speeds with positive zyk values are recommended.

The eccentricity ratio can be increased by preloading. This observation led to the construction of multilobe bearings, where the circular arcs are displaced towards the bearing centre to obtain the preloading. For each circular sector an oil wedge is formed producing a radial pressure distribution giving rise to forces that centre the rotor, stabilizing it. Plain cylindrical bearings have been replaced by two-, three- or four-lobe bearings with improved stability.

Changing from sleeve type journal bearings to tilting-pad bearings also eliminates oil whirl. This type of bearing has zero cross-coupling stiffness coefficients so that it is theoretically completely stable.

With increasing speed, the oil whirl frequency increases until it reaches the natural frequency of the rotor-bearing system when it dwells on this frequency, becoming an oil whip, also called resonant whirl. The identifying frequency of this condition is either slightly less than or slightly greater than half the running-speed frequency.

Fig. 4.18

The increase of the onset speed of instability by changing the bearing type was studied [14] for the rotor from Fig. 4.18.

The uniform shaft of diameter mm80 has lengths m3031 .== ll ,

m2042 .== ll , Young’s modulus 211 mN102 ⋅ and mass density 3mkg8000 .

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The three identical discs have the following mass and mass moments of inertia: kg, 15=m ,.JT

2mkg 050= 2mkg 10.JP = .

The bearings have the length/diameter ratio 0.5, radial clearance μm300=pC and oil dynamic viscosity 23 msN105 −⋅ . When applicable, the

preload factor is 431 =−= pbp CCm , where bC is the assembled clearance and

pC is the machined clearance. The speed dependence of the bearing stiffness and damping coefficients was taken from [10].

Fig. 4.19 (from [14])

The stability diagrams for six different bearings are overlaid in Fig. 4.19 for the given rotor configuration. The onset speeds of instability are 9,155 rpm for cylindrical bearings without axial grooves, 9,220 rpm for cylindrical bearings with two axial grooves, 11,760 rpm for 2-lobe bearings, 12,244 rpm for 4-lobe bearings, 12,389 rpm for 3-lobe bearings and 14,332 rpm for cylindrical bearings with pressure dam.

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4.5 Simulation examples

In the following, examples of rotor dynamic analysis are given for selected rotor models taken as benchmark examples. The numerical results are dependent on the spatial discretization error (number of finite elements in the model) and the speed resolution (number of speeds in a given interval).

Example 4.1

A simply supported rotor with three discs (Fig. 4.20) is considered in this example [15]. The shaft of length m31. and diameter m10. has the Young’s

modulus 211 mN102 ⋅ and the mass density 3mkg7800 . It was divided into 13 beam finite elements of length m10. each.

Fig. 4.20

The three rigid discs, located at stations 3, 6 and 11, have the following masses and mass moments of inertia: kg, 58141 .m = ,.JT

2mkg 064601=

2mkg 12301

.J P = , kg, 94452 .m = ,.JT2mkg 4980

2= 2mkg 9760

2.J P = ,

kg, 13553 .m = ,.JT2mkg 6020

3= 2mkg 1711

3.J P = .

The two identical orthotropic bearings are located at stations 1 and 14, and have the following stiffness and damping coefficients: mN107 7⋅=yyk ,

mN105 7⋅=zzk , 0== zyyz kk , msN107 2⋅=yyc , msN105 2⋅=zzc , 0== zyyz cc .

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Fig. 4.21

The Campbell diagram for the first 10 natural modes is shown in Fig. 4.21. The damping ratio diagram is shown in Fig. 4.22.

Fig. 4.22

The precession modes occur in pairs, the lower mode in a pair with backward precession, and the upper with forward precession. With increasing mode index, the gyroscopic effect makes the two lines in a pair to become more

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4. ROTORDYNAMIC ANALYSIS 233

divergent. The line 5B crosses the line 4F so that beyond 20000 rpm the mode ranking is changed. Modes are labeled according to their order at low rotational speeds. The synchronous excitation line (dotted line) intersects the natural frequency lines at the points whose abscissae determine the damped critical speeds 3620 , 3798 , 10017, 11278, 16769, 24397, 26604 rpm, etc.

a b c

d e f

g h i

j k l

Fig. 4.23

Twelve precession mode shapes at 25000 rpm are shown in Fig. 4.23. The orbits at any station are ellipses due to bearing anisotropy. The mode shape at

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0=t is plotted with solid line and the mode shape at Ωπ 2=t is drawn with broken line, so that the motion along the orbit takes place from the point lying on the solid line, at 0=t , to the point lying on the broken line, a quarter of a period later. Modes 1B and 1F are almost ‘cylindrical’. Modes 2B and 2F are almost ‘conical’. Modes 3B and 3F are ‘two-node’ flexural, modes 4B and 4F are ‘three-node’ flexural, etc.

Fig. 4.24

The unbalance response curves calculated at station 6 are shown in Fig. 4.24 for the orbit major semiaxis (solid line) and minor semiaxis (broken line). A mass unbalance of mmg200 on the disc at station 6 was considered.

Example 4.2

Consider the rotor of Example 4.1, but with the following bearing stiffness and damping coefficients [16]:

mN107 7⋅=′yyk , mN105 7⋅=′zzk , mN104 7⋅−=′=′ zyyz kk ,

msN107 3⋅=′yyc , msN104 3⋅=′zzc , 0=′=′ zyyz cc ,

mN106 7⋅=′′yyk , mN104 7⋅=′′zzk , mN1054 7⋅−=′′=′′ .kk zyyz ,

msN106 3⋅=′′yyc , msN105 3⋅=′′zzc , 0=′′=′′ zyyz cc .

The Campbell diagram for the first eight modes is shown in Fig. 4.25. The conservative cross-stiffness increases the interval between the eigenvalues in a pair corresponding to the same modal index. When a lower index forward mode approaches a higher index backward mode, as for modes 4 and 5, the result is a

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4. ROTORDYNAMIC ANALYSIS 235

curve veering in the Campbell diagram (at about 28000 rpm) with a corresponding crossing of damping ratio curves. The relative departure of the two curves in a pair gives rise to mixed modes. Because all modes are mixed, they are labeled in ascending order, with their eigenvalue index at very low rotational speeds.

Fig. 4.25

Fig. 4.26

The damping ratio diagram is shown in Fig. 4.26. Predominantly backward modes (like 1 and 3) are more damped that their (predominantly) forward pair (2 and 4).

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a b c

d e f

g h i

j k l

Fig. 4.27

Twelve precession mode shapes at 25000 rpm are shown in Fig. 4.27. For mixed modes, the precession along the ellipse takes place from the point lying on the solid line, at 0=t , to the point lying on the broken line, a quarter of a period later. The bearings have the principal axes of stiffness oriented at 045+ and 045− , respectively, relative to the vertical axis, so that the elliptical precession orbits have inclined axes.

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The root locus diagram is presented in Fig. 4.28. When the root loci are close to each other, two modes with nearly the same natural frequency (4 and 5) and different deflected shapes can combine to yield a compounded mode which has mixed backward and forward precession due to the coupling between modes.

Fig. 4.28

The unbalance response curves calculated at station 6 are shown in Fig. 4.29 for the orbit major semiaxis (solid line) and minor semiaxis (broken line). A mass unbalance of mmg200 on the disc at station 6 was considered. There is no peak corresponding to the 5th mode, due to the relatively high damping. The ordinate in Fig. 4.29, a is logarithmic and in Fig. 4.29, b is linear. The latter shows better the regions with backward precession, where the minor semiaxis has negative values.

a b

Fig. 4.29

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DYNAMICS OF MACHINERY 238

Example 4.3

A multi-stepped rotor (Fig. 4.30) has Young’s modulus 211 mN100782 ⋅. and mass density 3mkg7806 . At station 5 it carries a rigid

disc of mass kg4011. , diametral and polar mass moments of inertia 2kgm001360.

and 2kgm002030. , respectively. The shaft is supported at stations 11 and 15 by isotropic bearings with the following constant stiffness and damping coefficients:

mN103784 7⋅== .kk zzyy , and msN107521 3⋅== .cc zzyy . The geometric data are given in Table 4.1 [17].

Fig. 4.30

Table 4.1

Element no.

Length, mm

Outer radius,

mm

Inner radius,

mm

Element no.

Length, mm

Outer radius,

mm

Inner radius,

mm 1 12.7 5.1 0 10 30.5 12.7 0 2 38.1 10.2 0 11 25.4 12.7 0 3 25.4 7.6 0 12 38.1 15.2 0 4 12.7 20.3 0 13 38.1 15.2 0 5 12.7 20.3 0 14 20.3 12.7 0 6 5.1 30.3 0 15 17.8 12.7 0 7 7.6 30.3 15.2 16 10.2 38.1 0 8 12.7 25.4 17.8 17 30.4 20.3 0 9 7.6 25.4 0 18 12.7 20.3 15.2

The Campbell diagram is shown in Fig. 4.31 for the first six modes of precession. The damped critical speeds, determined at the intersections with the synchronous excitation line, are also shown in the diagram.

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4. ROTORDYNAMIC ANALYSIS 239

Fig. 4.31

The damping ratio diagram is presented in Fig. 4.32.

Fig. 4.32

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DYNAMICS OF MACHINERY 240

The first six precession mode shapes at 50000 rpm are shown in Fig. 4.33.

a b c

d e f

Fig. 4.33

Figure 4.34 shows the unbalance response orbit radius versus speed at station 15 (right bearing) for an unbalance of mmg200 on the disc at station 5. In the considered speed range, there are only two peak response critical speeds at the frequencies of modes 1F and 2F.

Fig. 4.34

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4. ROTORDYNAMIC ANALYSIS 241

Example 4.4

A rotor is modeled as a 13 station (12 elements) assembly with stations as indicated in Fig. 4.35 [18]. It is supported by bearings at stations 3, 6 and 13. The four rigid discs which represent the fan, the low and high pressure compressors and the turbine are located at the stations 1, 4, 5, and 12. Details of the rotor configuration are listed in Table 4.2.

Fig. 4.35

Table 4.2

Element no.

Length,

mm

Outer diameter,

mm

Inner diameter,

mm

Element

no.

Length,

mm

Outer diameter,

mm

Inner diameter,

mm 1 42.9 59 28.4 7 152.4 59 53.8 2 46.0 59 28.4 8 152.4 59 53.8 3 16.0 59 28.4 9 152.4 59 53.8 4 96.8 59 28.4 10 152.4 59 45.2 5 75.2 59 39.2 11 149.8 59 28.4 6 165.1 59 53.8 12 78.0 59 46.2

The shaft has Young’s modulus 211 mN100692 ⋅. and mass density 3mkg8193 . The disc data are given in Table 4.3.

Table 4.3

Disc no.

Station

Mass,

kg

Polar mass moment of inertia,

22 mkg10 ⋅

Diametral mass moment of inertia,

22 mkg10 ⋅ 1 1 11.38 19.53 9.82 2 4 7.88 16.70 8.35 3 5 7.7 17.61 8.80 4 12 21.7 44.48 22.44

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DYNAMICS OF MACHINERY 242

The isotropic bearing data are given in Table 4.4.

Table 4.4

Bearing no.

Station

Stiffness coefficients,

mN10 6 ⋅−

Damping coefficients,

msN10 3 ⋅−

1 3 1.751 1 2 6 96.95 1 3 13 13.368 1

The Campbell diagram is presented in Fig. 4.36 for the first 8 natural modes. At the crossing points with the synchronous excitation line, the damped critical speeds are determined as 2807, 3670, 10631, 10841 and 17278 rpm. The line 3F crosses the lines 4B and 4F.

Fig. 4.36

The damping ratio diagram is shown in Fig. 4.37 for the same 8 modes. The curve 3F has a peak and the curve 4F has a trough at the speed where the corresponding lines cross each other in the Campbell diagram. The two modes do not interact.

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4. ROTORDYNAMIC ANALYSIS 243

Fig. 4.37

The shape of the first six modes of precession at 25000 rpm is shown in Fig. 4.38.

a b c

d e f

Fig. 4.38

The unbalance response curves calculated at the three bearing stations 3, 6 and 13 are shown in Fig. 4.39, for a mmg200 unbalance on disc 1.

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DYNAMICS OF MACHINERY 244

a b c

Fig. 4.39

Example 4.5

A uniform shaft (Fig. 4.40) is supported by plain cylindrical bearings at the ends, at the stations 1 and 9 [19].

Fig. 4.40

Fig. 4.41

The shaft has a material with Young’s modulus 211 mN100682 ⋅. and

mass density 3mkg67833. . The plain cylindrical bearings have length mm425. ,

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4. ROTORDYNAMIC ANALYSIS 245

diameter mm6101. , radial clearance μm51 and oil dynamic viscosity 23 msN10946 −⋅. . The static loads on bearings are N6395. . The speed

dependence of the bearing stiffness and damping coefficients is shown in Fig. 4.41.

Fig. 4.42

Fig. 4.43

The Campbell diagram is presented in Fig. 4.42 for the first 4 natural modes. Modes 1B and 2B are overdamped and do not show up. Modes 1F and 2F

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DYNAMICS OF MACHINERY 246

follow closely the half-frequency excitation line (lower dotted line). The damping ratio diagram is shown in Fig. 4.43 for the same 4 modes. For mode 1F it becomes negative at 9060 rpm, the onset speed of instability.

The shape of the first four modes of precession at 6000 rpm is shown in Fig. 4.44.

a b

c d

e f

Fig. 4.44

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4. ROTORDYNAMIC ANALYSIS 247

Fig. 4.45

The stability diagram is shown in Fig. 4.45. At 9060 rpm, mode 1F becomes unstable. The precession frequency at the onset of instability is very close to one-half of the rotor speed (lower dotted line in Fig. 4.42), which is characteristic for the type of instability called ‘oil whirl’ or ‘half-frequency whirl’.

Fig. 4.46

The root locus diagram is presented in Fig. 4.46.

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DYNAMICS OF MACHINERY 248

The unbalance response curves calculated at the stations 1 and 5 are shown in Fig. 4.47, for an unbalance of mmg512 at station 2. The major semiaxis has maximum values at different speeds in the two bearings. The peak response critical speeds cannot be predicted from the Campbell diagram.

a b

Fig. 4.47

Example 4.6

A solid rotor is mounted in plain cylindrical bearings. The shaft consists of 12 elements (Fig. 4.48) and has Young’s modulus 211 mN10062 ⋅. and mass

density 3mkg7850 [20].

Fig. 4.48

The shaft dimensions are given in Table 4.5

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4. ROTORDYNAMIC ANALYSIS 249

Table 4.5

Element Diameter, mm

Length, mm Element Diameter,

mm Length,

mm 1 and 12 660 260 4 and 9 657 340 2 and 11 590 570 5 and 8 970 1100 3 and 10 550 280 6 and 7 1100 990

The bearings are located at nodes 3 and 11. The eight bearing dynamic coefficients are given in Table 4.6 [21] and Fig. 4.49. Note that zyyz cc ≠ !

Table 4.6

n yyk yzk zyk zzk yyc yzc zyc zzc

rpm mN10 9− msN10-6

800 5.17 2.67 0.712 1.4 38.7 7.15 13.7 11.0 1000 4.72 2.55 0.57 1.36 29.7 6.12 11.4 9.44 1300 4.28 2.44 0.422 1.34 22.5 4.97 8.98 7.84 1500 3.90 2.38 0.3 1.33 19.3 4.5 7.75 7.0 1700 3.7 2.32 0.232 1.325 16.76 4.02 6.82 6.52 2100 3.17 2.16 0.0416 1.322 13.25 3.49 5.5 5.86 2600 2.7 2.0 -0.175 1.32 10.34 3.02 4.58 5.49 3000 2.5 1.9 -0.3 1.315 9.0 2.7 4.0 5.25 3400 2.3 1.81 -0.437 1.31 8.08 2.49 3.51 5.08 3500 2.28 1.75 -0.45 1.306 7.8 2.4 3.4 5.0 3600 2.25 1.74 -0.5 1.303 7.65 2.35 3.3 4.9 4000 2.14 1.71 -0.56 1.3 7.0 2.21 2.98 4.81 4200 2.05 1.68 -0.6 1.295 6.7 2.15 2.85 4.65 4500 2.0 1.65 -0.65 1.29 6.4 2.05 2.7 4.5 5000 1.91 1.66 -0.75 1.295 5.9 1.91 2.38 4.37 5500 1.8 1.67 -0.8 1.3 5.45 1.8 2.2 4.2 6000 1.75 1.685 -0.9 1.32 5.1 1.7 2.05 4.0 6600 1.7 1.7 -0.95 1.35 4.63 1.59 1.955 3.84

The rotor is symmetrical, so that the physically identical bearings have also identical dynamic properties. They have different cross stiffnesses, zyyz kk ≠ , which produce unstable whirling.

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DYNAMICS OF MACHINERY 250

Fig. 4.49

For the first six modes of precession, the Campbell diagram is shown in Fig. 4.50. Modes 1B and 2B are overdamped at low rotational speeds.

Fig. 4.50

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4. ROTORDYNAMIC ANALYSIS 251

The damping ratio diagram is given in Fig. 4.51. Mode 1F becomes unstable at 4911 rpm.

Fig. 4.51

The root locus diagram is shown in Fig. 4.52. Curves are labeled with both the eigenvalue index and the mode index showing the directivity.

Fig. 4.52

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DYNAMICS OF MACHINERY 252

Despite the absence of discs on the shaft (no gyroscopic effects), the damped natural frequencies have a strong variation with the rotor speed, due to the speed variation of bearing coefficients.

a b

c d

e f

Fig. 4.53

In Fig. 4.50, the synchronous excitation line, Ωω = , is drawn with dotted line. The abscissae of its crossing points with the curves in the Campbell diagram give the damped critical speeds at 1865, 2823, 3324 and 4171 rpm. Mode 1B is highly damped so that it will not produce a peak in the unbalance response.

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4. ROTORDYNAMIC ANALYSIS 253

The shape of the first six precession modes at 5576 rpm is shown in Fig. 4.53. Modes 1B and 1F are ‘cylindrical’, while modes 2B and 2F are ‘conical’. They correspond to the rotor so-called rigid body precession in flexible bearings. Modes 3B and 3F are ‘two-node flexural’.

Fig. 4.54

Mode 1F becomes unstable at 4911 rpm, where the damping ratio becomes negative. This can also be seen in the stability diagram from Fig. 4.54, where the real part of four relevant eigenvalues is plotted versus speed. Curve 1F crosses the zero ordinate line at 4911 rpm – the onset speed of instability.

In the Campbell diagram, the point on 1F at 4911 rpm (81.8 Hz) has a damped natural frequency of about 39 Hz, which is a little less than half the driving frequency 40.9 Hz. This is known as the ‘half-frequency’ or ‘oil-whirl’ type of instability. It transforms into ‘oil-whip’ at the natural frequency of mode 1F and remains almost constant with increasing rotational speed.

Note the atypical behaviour of this rotor, for which the lines 1F and 2F follow closely the synchronous excitation line and not the half-frequency excitation line. This is probably due to the values of the bearing coefficients.

Example 4.7 a

A multi-stepped rotor used in laboratory experiments (Fig. 4.55) is supported by hydrodynamic bearings at stations 6 and 23. Two balancing discs are located at stations 12 and 17 [22].

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DYNAMICS OF MACHINERY 254

Fig. 4.55

The shaft has Young’s modulus 211 mN102 ⋅ and mass density 3mkg7850 . The plain cylindrical bearings have length mm30 , diameter

mm100 , radial clearance μm125 and oil dynamic viscosity 23 msN109 −⋅ . The static loads on bearings are N45272. and N77251. , respectively.

Table 4.7

Element no.

Length, mm

Outer diameter,

mm

Element

no.

Length,

mm

Outer diameter,

mm 1 6.35 38.1 13 76.2 38.1 2 25.4 77.6 14 76.2 109.7 3 25.4 38.1 15 76.2 38.1 4 25.4 38.1 16 25.4 102.9 5 101.6 100 17 25.4 102.9 6 101.6 100 18 44.45 38.1 7 44.45 38.1 19 44.45 38.1 8 44.45 38.1 20 44.45 38.1 9 44.45 38.1 21 44.45 38.1

10 44.45 38.1 22 101.6 100 11 25.4 116.8 23 101.6 100 12 25.4 116.8

Details of the rotor configuration are listed in Table 4.7.

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4. ROTORDYNAMIC ANALYSIS 255

The disc data are given in Table 4.8. Table 4.8

Disc no.

Station

Mass,

kg

Polar mass moment of inertia,

22 mkg10 ⋅

Diametral mass moment of inertia,

22 mkg10 ⋅

1 12 4.33 2.97 1.51

2 17 4.808 3.118 1.585

The speed dependence of the stiffness and damping coefficients for bearing 1 is shown in Fig. 4.56.

Fig. 4.56

The Campbell diagram is presented in Fig. 4.57 for the first 6 natural modes. Modes 1B and 2B show up only above 5700 rpm. At the crossing points with the synchronous excitation line, the damped critical speeds are determined as 549, 2400, 2949, and 4446 rpm.

The first forward mode is denoted ∗F1 because, with increasing rotational speed, it changes from cylindrical to a two-node flexural mode. The third forward mode is denoted ∗F3 because it changes from a two-node flexural to an almost cylindrical mode.

Note again the lines ∗F1 and 2F following closely the synchronous excitation line.

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DYNAMICS OF MACHINERY 256

Fig. 4.57

The damping ratio diagram is shown in Fig. 4.58 for the same 6 modes.

Fig. 4.58

The shape of the first six modes of precession at 7000 rpm is shown in Fig. 4.59. Modes ∗F1 and ∗F3 , whose lines intersect in the Campbell diagram, are in fact almost interchanged.

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4. ROTORDYNAMIC ANALYSIS 257

a b

c d

e f

Fig. 4.59

Fig. 4.60

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DYNAMICS OF MACHINERY 258

Mode ∗F3 becomes unstable at 4876 rpm, where the damping ratio becomes negative. This can also be seen in the stability diagram from Fig. 4.60, where curve ∗F3 crosses the zero ordinate line at 4876 rpm.

a b

c d

Fig. 4.61

Fig. 4.62

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4. ROTORDYNAMIC ANALYSIS 259

The unbalance response curves calculated at the bearing stations 6 and 23 are presented in Fig. 4.61, for a mmg433 unbalance on disc 1. Figures 4.61, a and b show the speed variation of the major and minor ellipse semiaxes. The peak value of the major semiaxis defines the peak response critical speed. It is compared with admissible limits given in standards and recommendations. Figures 4.61, c and d show the speed-variation of the radii of forward and backward circles which generate the elliptical precession.

The root locus diagram is presented in Fig. 4.62 for speeds up to 7000 rpm. Curve ∗F3 crosses the zero damping vertical, marking the threshold of instability.

Example 4.7 b

An alternate set of simulation results has been obtained for the rotor of Example 4.7, a using a ten times smaller oil viscosity 24 msN109 −⋅ .

The speed dependence of the bearing stiffness and damping coefficients is shown in Fig. 4.63. The stiffness coefficient yzk has only positive values. This means that the selected speed range corresponds to relatively low values of the Sommerfeld number. The difference ( )zyyz kk − being smaller than in the previous example, the rotor is more stable with these bearings.

a b

Fig. 4.63

The Campbell diagram for the rotor with modified oil viscosity is presented in Fig. 4.64. The damped critical speeds are located at 2241, 2961, 3551, 4733, 4947 and 5143 rpm. Modes 1B and 2B are overdamped at relatively low running speeds. The curves of modes 1F and 2F start with slopes higher than the synchronous excitation line.

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DYNAMICS OF MACHINERY 260

Fig. 4.64

The damping ratio diagram is shown in Fig. 4.65 for the same 6 modes. There is no negative value in the considered speed range.

Fig. 4.65

The unbalance response curves calculated at the bearing stations 6 and 23 are shown in Fig. 4.66, for a mmg433 unbalance on disc 1.

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4. ROTORDYNAMIC ANALYSIS 261

a b

c d Fig. 4.66

The three peaks occur near the natural frequencies of modes 1F, 3B and 3F. The second peak is narrower due to the low damping of mode 3B.

Example 4.8

A rotor test rig, designed for rotor dynamic experiments, is presented in Fig. 4.67 [23]. The finite element model of the rotor-bearing system is shown in Fig. 4.68.

The shaft is modeled with 19 axisymmetric beam elements (4 DOFs/node) with consistent mass and gyroscopic matrices. The shaft has a material with Young’s modulus 211 mN102 ⋅ and mass density 3mkg7850 .

The two discs, located at nodes 6 and 19, and the coupling located at node 1 have masses kg1935. , kg6561. , kg082. , respectively, polar mass

moments of inertia 2mkg64220. , 2mkg93891. , 2mkg002080. , and diametral

mass moments of inertia 2mkg32580. , 2mkg97890. , 2mkg001920. .

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DYNAMICS OF MACHINERY 262

Fig. 4.67

Fig. 4.68

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4. ROTORDYNAMIC ANALYSIS 263

The two journal bearings, located at nodes 2 and 17, are plain cylindrical, with length mm30 , diameter mm40 , radial clearance μm517. , and oil dynamic viscosity smkg003450. . The static loads on bearings are N37179. and

N44925. , respectively.

The speed dependence of the bearing stiffness and damping coefficients, calculated using Moes’ [5] impedance model, is given in Fig. 4.69.

a b

Fig. 4.69

The Campbell diagram is presented in Fig. 4.70 for the first 6 natural modes. Modes 1F and 2F are controlled by bearings. The associated backward modes are overdamped. At the crossing points with the synchronous excitation line, the damped critical speeds are determined at 847, 1304, 2210, and 2687 rpm.

Fig. 4.70

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DYNAMICS OF MACHINERY 264

The damping ratio diagram is shown in Fig. 4.71 for only four modes.

Fig. 4.71

Mode F3 becomes unstable at 3146 rpm, where the damping ratio becomes negative. This can also be seen in the stability diagram from Fig. 4.72, where curve F3 crosses the zero ordinate line at 3146 rpm.

Fig. 4.72

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4. ROTORDYNAMIC ANALYSIS 265

The shapes of the first six modes of precession at 10000 rpm are shown in Fig. 4.73.

a b

c d

e f

Fig. 4.73

Unbalance response diagrams at bearing 2 are shown in Fig. 4.74 for eccentricities of μm35 at 090 on the disc at location 6, and μm44 at 090− on the disc at location 19.

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DYNAMICS OF MACHINERY 266

a b

Fig. 4.74

The root locus diagram for the first six modes is presented in Fig. 4.75, a. Modes 1F and 2F are highly damped. The root locus diagram for selected four modes is presented in Fig. 4.75, b. Mode 3F becomes unstable at the point marked by a circle.

a b

Fig. 4.75

Example 4.9 a

Consider the rotor with three discs from Fig. 4.76 carried by journal bearings at the ends. The uniform shaft of diameter mm80 has lengths

m3031 .== ll , m2042 .== ll , Young’s modulus 211 mN102 ⋅ and mass

density 3mkg8000 . The three identical discs have the following mass and mass

moments of inertia: kg, 15=m ,.JT2mkg 050= 2mkg 10.JP = .

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4. ROTORDYNAMIC ANALYSIS 267

The bearings have the length/diameter ratio 0.25, radial clearance μm50=C and oil dynamic viscosity 23 msN100387 −⋅. [24]. The static loads on

bearings are N25403. and N68432. , respectively.

Fig. 4.76

The speed dependence of the bearing stiffness and damping coefficients, calculated using Ocvirk’s short bearing model [6], is given in Fig. 4.77.

a b

Fig. 4.77

The Campbell diagram for the first four natural modes is shown in Fig. 4.78. Curves 1F and 2F start along the synchronous excitation line. Curves 1F and 3F merge at a speed about 2700 rpm, where there is an apparent switching. This is due to the short bearing approximation and will be explained in Example 4.9 b.

The damping ratio diagram is presented in Fig. 4.79. The curves 1F and 3F cross each other at the speed where their corresponding pairs merge in the Campbell diagram. The damping ratio of mode 1F becomes negative at 7964 rpm,

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DYNAMICS OF MACHINERY 268

the threshold of instability. The corresponding point in the Campbell diagram is very near the 66.6 Hz point on the half-frequency excitation line.

Fig. 4.78

Fig. 4.79

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4. ROTORDYNAMIC ANALYSIS 269

Fig. 4.80

The stability diagram is shown in Fig. 4.80. The onset speed of instability is marked at 7964 rpm, where the damping constant becomes positive.

Fig. 4.81

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DYNAMICS OF MACHINERY 270

The root locus diagram is shown in Fig. 4.81 for speeds up to 10000 rpm. Curve 1F crosses the zero damping vertical at the onset speed of instability.

a b c

d e f

Fig. 4.82

Six precession mode shapes at 3000 rpm are shown in Fig. 4.82. Modes 1F and 3F are compounded cylindrical and two-node forward. Mode 2F is ‘conical’. Mode 3B is ‘two-node’ backward. Modes 4B and 4F are three-node flexural.

a b

Fig. 4.83

The unbalance response curves calculated at stations 1 and 3 are shown in Fig. 4.83 for a mass unbalance of mmg150 on the disc at station 3.

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4. ROTORDYNAMIC ANALYSIS 271

Example 4.9 b

Consider the rotor of Example 4.9 a and use Moes’ impedance model [8] for the calculation of bearing stiffness and damping coefficients.

Fig. 4.84

The Campbell diagram for the first four natural modes is shown in Fig. 4.84. Note that the curves 1F and ∗F3 are clearly separated now.

Fig. 4.85

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DYNAMICS OF MACHINERY 272

The damping ratio diagram is shown in Fig. 4.85. Curves 1F and ∗F3 are crossing each other. Curve 1F crosses the zero damping line at 7826 rpm.

Fig. 4.86

The stability diagram is shown in Fig. 4.86. The onset speed of instability is marked at 7826 rpm, where the damping exponent of mode 1F becomes positive.

Fig. 4.87

The root locus diagram is shown in Fig. 4.87 for speeds up to 10000 rpm.

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4. ROTORDYNAMIC ANALYSIS 273

4.6 Planar modes of precession

Undamped rotors in isotropic bearings exhibit planar modes of precession with circular orbits. A planar unbalance distribution yields planar deflected shapes. Undamped rotors in orthotropic bearings have spatial precession modes with elliptical orbits. With suitable scaling, the precession modes are planar in the principal planes of orthotropy, and so is the deflected shape due to planar unbalance.

Damped gyroscopic systems have complex modal vectors describing spatial deflected shapes with “damped” elliptical orbits, rotated with respect to each other at different rotor stations.

It was shown [25] that a certain class of damped gyroscopic systems has planar modal vectors describing elliptical orbits with coinciding principal directions and the same phase angle of the motions at different rotor stations. The characteristic phase angles and the mode shapes are speed-dependent.

By a proper transformation, the complex monophase modal vectors can be transformed into equivalent real vectors, solutions of a real eigenvalue problem in the configuration space. There is no need to expand the problem and to solve it in the state space. The planar modes of precession form an orthogonal basis and can be used to decouple the equations of motion. This approach gives an alternative modal analysis solution for the steady-state response of the considered damped gyroscopic systems, otherwise treated by a perturbation technique.

4.6.1 Response of undamped gyroscopic systems

For rotor-bearing systems having: a) axi-symmetric rotor; b) conservative cross-coupling forces; and c) orthotropic bearings with coincident principal directions of stiffness and damping, the equations of the free precession can be written in the form (3.145). The solutions have the form (3.148)

ωtΦx ie =

where (3.160)

⎬⎫

⎩⎨⎧−

=⎭⎬⎫

⎩⎨⎧−

+⎭⎬⎫

⎩⎨⎧

−−

=z

y

s

c

sc

sc

aa

zy

zzyy

i e

i )i1( =

ii

iγβΦ .

The above equation shows that, with appropriate scaling, the elements of vectors Φ become real in the xOy plane and pure imaginary in the xOz plane, hence the precession modes are planar.

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DYNAMICS OF MACHINERY 274

The rotor deflected line in the xOy plane has a 090+ or 090− phase shift with respect to the deflected line in the xOz plane, which corresponds to a quarter of rotation of the rotor. This implies that, as a phasor, zai− has a 090 phase lag with respect to ya . The orbits of the rotor stations are ellipses with axes

coincident with the y-z axes. The inclination angle is either 00 or 090 . By proper scaling of eigenvectors, the ellipse points at the reference time 0=t are on the y-axis, where the phase angle 0=γ .

4.6.2 Response of damped gyroscopic systems

The equation of motion of a damped gyroscopic system can be written as

⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡+

⎭⎬⎫

⎩⎨⎧⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡−

+⎥⎦

⎤⎢⎣

⎡+

⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

z

y

z

y

z

y

ff

zy

kk

zy

gg

cc

zy

mm

][]0[]0[][

]0[][][]0[

][]0[]0[][

][]0[]0[][

&

&

&&

&&Ω

(4.21)

where yf and zf are the forcing vectors in the xOy and xOz planes, respectively, ][ yc and ][ zc are positive definite damping matrices.

Using phasor notation, consider the unbalance excitation

,

ii2 t

z

y eU

Uff ΩΩ

⎭⎬⎫

⎩⎨⎧−

=⎭⎬⎫

⎩⎨⎧

(4.22)

where U is the unbalance complex subvector.

The synchronous unbalance response is

tt eX~e

z~y~

zy ΩΩ ii =

⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧

(4.23)

where X~ is a complex vector.

It is of interest to find an excitation of the form (4.22), with U a real vector, able to produce a synchronous planar response. This is a particular kind of precession, expressed by equation (3.160), in which all displacements have the same phase angle γ with respect to the unbalance plane:

.e

aa

zy

Xz

y γi

i ⎭⎬⎫

⎩⎨⎧−

=⎭⎬⎫

⎩⎨⎧

= (4.24)

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4. ROTORDYNAMIC ANALYSIS 275

The response vector (4.24) is a planar mode of precession, defined by elliptical orbits whose axes coincide with the y-z coordinate axes, and whose generating radius at 0=t has a phase angle γ with respect to the plane of unbalance.

Substituting equations (4.22)-(4.24), and using the transformation (3.162), equation (4.21) becomes

[ ] [ ][ ] [ ]

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

=⎭⎬⎫

⎩⎨⎧⎟⎟

⎜⎜

⎛⎥⎦

⎤⎢⎣

⎡+

⎥⎥⎦

⎢⎢⎣

UUe

aa

cc

kggk y

z

y

z

y2

2i

z22

22

][]0[]0[][

im ][

m ][ΩΩ

ΩΩ

ΩΩΩΩ γ

(4.25)

or

( )[ ] ( )[ ]( ) feqBB IR =+ γΩΩ ii , (4.26)

where q and f are real vectors.

Separation of real and imaginary parts in equation (4.26) yields

[ ] [ ]( ) ,fqBB IR =− γγ sincos (4.27, a)

[ ] [ ]( ) .qBB RI 0sincos =+ γγ (4.27, b)

If 0cos ≠γ , denoting

,γλ 1tan−= (4.28)

the homogeneous equation (4.27, b) can be written as a generalized eigenvalue problem

[ ] [ ] rIrrR BB ΦλΦ −= . (4.29)

Both the eigenvalues rλ and the modal vectors rΦ are real and speed dependent. Vectors rΦ , referred to as planar response modal vectors, represent a specific type of precession, in which all stations execute synchronous motions along elliptical orbits, having the same phase shift rγ with respect to a reference unbalance plane. Their spatial shape varies with the speed. They are produced only by the external forcing defined by the planar excitation modal vectors rF derived from equation (4.27, a):

[ ] .BF rIrr Φλ21+= (4.30)

The planar response vectors satisfy the bi-orthogonality conditions

[ ] ,B rIT

s 0=ΦΦ ( )sr ≠ (4.31)

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DYNAMICS OF MACHINERY 276

[ ] ,B rRT

s 0=ΦΦ 0=rT

s FΦ .

They can be conveniently normalized so that

[ ] ,rrIT

r B γΦΦ sin−= (4.32)

[ ] ,B rrRT

r γΦΦ cos= .FrT

r 1=Φ

The coordinate transformation

∑=

=n

rrrq

4

1νΦ (4.33)

simultaneously diagonalizes the matrices [ ]RB and [ ]IB . This way, a spectral decomposition of the system response is obtained in terms of the planar response vectors

.feqn

r

Trr

r∑=

=4

1

i ΦΦγ (4.34)

The transformation (3.162) is then used to obtain X from q .

If cos γ = 0, then °−= 90rγ and .r 0=λ Equations (4.29) and (4.30) become

[ ] ,B rR 0=Φ (4.35, a)

[ ] .FB rrI =Φ (4.35, b)

Equation (4.35, a) coincides with equation (3.166) so that the undamped normal mode rΨ is the r-th planar response vector rΦ calculated at rΩΩ = , which corresponds to 0=rλ in equation (4.29).

The eigenvalues rλ of the generalized problem (4.29) vary with the rotor angular speed. Each eigenvalue cancels at, and only at, the corresponding undamped critical speed, i.e. ( ) 0=rr Ωλ . Plotting rλ against speed, each curve crosses only once the speed axis, so undamped critical speeds can be easily located. This diagram can be used as a Real Mode Indicator Function (RMIF).

4.6.3 Planar precession modes

A planar precession mode is defined by three speed-dependent elements: a) a planar response vector, whose elements are the semiaxes of the elliptical orbits, and the corresponding slopes at the rotor stations; b) a planar forcing vector, whose elements give the planar unbalance distribution which produces the planar

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4. ROTORDYNAMIC ANALYSIS 277

response; and c) a characteristic phase angle, the same at all stations, between the response plane and the unbalance plane.

A planar response mode is shown in Fig. 4.88. The orbit axes are along the coordinate axes. The phase angle γ is measured in the positive direction of rotational speed Ω , between the point in the plane of unbalance and the point at

0=t on the major generating circle. The construction presented in the next section helps understanding the physical meaning of the characteristic phase angle.

Fig. 4.88

The rotor finite element model has 4n degrees of freedom and is excited by planar unbalance forces applied at the n rotor stations. At any rotational speed Ω , there exist 2n independent sets of unbalance distributions, each of which excites the corresponding planar precession response, in which all points have the same phase lag with respect to the unbalance plane. The phase angle between the unbalance and the generating radius is different for each mode. As the rotor speed changes, so do the characteristic phase angles, planar response vectors and planar forcing vectors.

At an undamped critical speed, one of the characteristic phase angles becomes 090− and the corresponding planar response mode coincides with the undamped normal mode of precession.

4.6.4 Ellipse from two concentric circles

The method of two concentric circles for constructing an ellipse is illustrated in Fig. 4.89.

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DYNAMICS OF MACHINERY 278

Consider an ellipse with semiaxes a and b, and the angle α between the major semiaxis and the y-axis. The 11Ozy coordinate system has the axes along the ellipse axes.

Fig. 4.89

First, two circles are drawn, with the centre at the origin of the coordinate system 11Ozy and radii equal to the ellipse semiaxes. Second, the circles are intersected with a line MO passing through the origin and which rotates anticlockwise with an angular speed Ω , equal to the rotor speed.

From the crossing point with the small circle, 1P or 2P , a line is drawn

perpendicular to the 1Oz axis. From the crossing point with the large circle, M, a line is drawn perpendicular to the 1Oy axis. These two orthogonal lines cross each other at a point on the ellipse, 1C or 2C .

When the crossing points M and 1P are on the same side of the origin, point 1C moves along the ellipse in a forward precession. When the crossing points M and 2P are on both sides of the origin, point 2C has a retrograde motion called backward precession.

In principal coordinates 11Ozy , the ellipse is defined by the parametric equations (3.35). If 1DO is the ellipse vector radius at the reference time 0=t ,

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4. ROTORDYNAMIC ANALYSIS 279

then the phase angle 1γ defines the position of the generating line 1NO , at 0=t , for forward precession. The phase angle 2γ defines the position of the generating line 2NO , at 0=t , for backward precession.

Note that the angular frequency of the precession motion is equal to the angular speed of the generating points M and 1P or 2P on the two circles, and not to the angular speed of the ellipse vector radius, which is not constant.

Example 4.10

Consider the three-disc rotor of Example 4.1 with mNs500== zzyy cc .

Fig. 4.90 (from [25])

The Campbell diagram of the associated undamped system is presented in Fig. 4.90. The undamped critical speeds are determined at the crossing points with the synchronous excitation line.

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DYNAMICS OF MACHINERY 280

For comparison, the RMIF diagram (eigenvalues rλ versus speed) is shown in Fig. 4.91. Undamped critical speeds are located at the intercepts with the horizontal axis.

Fig. 4.91 (from [25])

Fig. 4.92 (from [25])

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4. ROTORDYNAMIC ANALYSIS 281

Precession mode shapes at the first six undamped critical speeds are illustrated in Fig. 4.92. They have been determined from the monophase modal vectors calculated at the corresponding natural frequency.

References

1. Jackson, Ch., and Leader, M. E., Turbomachines: How to avoid operating problems, Hydrocarbon Processing, Nov. 1979.

2. Meyer, A., Schweickardt, H., and Strozzi, P., The converter-fed synchronous motor as a variable-speed drive system, Brown Boveri Rev., Vol.69, No.415, pp.151-156, 1982.

3. Kellenberger, W., Weber, H., and Meyer, H., Overspeed testing and balancing of large rotors, Brown Boveri Rev., Vol.63, No.6, pp.399-411, 1976.

4. Hohn, A., The mechanical design of steam turbosets, Brown Boveri Rev., Vol.63, No.6, pp.379-391, 1976.

5. API Standard 617, Centrifugal Compressors for Petroleum, Chemical and Gas Industry Services, 1995.

6. API Standard 613, Special-Purpose Gear Units for Refinery Services, 1979.

7. API Standard 612, Special Purpose Steam Turbines for Petroleum, Chemical and Gas Industry Services, 1995.

8. Childs, D., Moes, H., and van Leeuwen, H., Journal bearing impedance descriptions for rotordynamic applications, J of Lubrication Technology, pp.198-219, 1977.

9. Ocvirk, F., Short bearing approximation for full journal bearings, NACA TN 20808, 1952.

10. Someya, T., (ed.), Journal-Bearing Databook, Springer, Berlin, 1988.

11. ISO 1940, Balance Quality of Rotating Rigid Bodies, 1973.

12. Busse, L., and Heiberger, D., Aspects of shaft dynamics for industrial turbines, Brown Boveri Rev., Vol.67, No.5, pp 292-299, 1980.

13. ISO 7919-2, Mechanical Vibration of Non-Reciprocating Machines - Measurements on Rotating Shafts and Evaluation Criteria - Part 2: Large Land-Based Steam Generator Sets, 1996.

14. Scarlat, G., Predicţia stabilităţii rotorilor în diferite tipuri de lagăre hidrodinamice cu ajutorul analizei modale, Buletinul Conferinţei Naţionale de Dinamica Maşinilor CDM97, Braşov, 29-31 mai 1997.

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DYNAMICS OF MACHINERY 282

15. Lalanne, M., and Ferraris, G., Rotordynamics Prediction in Engineering, 2nd ed, Wiley, Chichester, 1998, p.125.

16. Radeş, M., Mixed precession modes of rotor-bearing systems, Schwingungen in rotierenden Maschinen III, (Irretier, H., Nordmann, R. and Springer, H., eds.), Vieweg, Braunschweig, pp. 153-164, 1995.

17. Nelson, H. D., and Meacham, W. L., Transient analysis of rotor-bearing systems using component mode synthesis, ASME Paper No.81-GT-110, 1981.

18. Chen, W. J., Rajan, M., Rajan, S. D., and Nelson, H. D., The optimal design of squeeze film dampers for flexible rotor systems, ASME J. of Mechanism, Transmission and Automation in Design, Vol.110, pp.166-174, 1988.

19. Lund, J. W., Stability and damped critical speeds of a flexible rotor in fluid-film bearings, ASME J. of Engineering for Industry, Series B, Vol.96, No.2, pp.509-517, 1974.

20. Radeş, M., Analiza modală a rotorilor elastici în lagăre cu alunecare, Bul. Conf. Naţ. Dinamica Maşinilor CDM94, Braşov, 24-25 Nov 1994, pp 17-24.

21. Bigret, R., Vibrations des machines tournantes et des structures, tome 2, ch.10, Technique et Documentation, Paris, p.40, 1980.

22. Friswell, M. I., Garwey, S. D., Penny, J. E. T., and Smart, M. G., Computing critical speeds for rotating machines with speed dependent bearing properties, J. Sound Vib., Vol.213, No.1, pp.139-158, 1998.

23. Kreuzinger-Janik, T., and Irretier, H., On modal testing of flexible rotors for unbalance identification, Proc. 16th Int. Modal Analysis Conf., Santa Barbara, California, pp.1533-1539, 1998.

24. Lee, C.-W., Vibration Analysis of Rotors, Kluwer Academic Publ., Dordrecht, 1993.

25. Radeş, M., Use of monophase modal vectors in rotordynamics, Schwingungen in Rotierenden Maschinen IV, (Irretier, H., Nordmann, R. and Springer, H., eds.), Vieweg, Braunschweig, pp.105-112, 1997.

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Index Angular momentum 70

− precession 68 − speed 42

Axial compressor 9

Backward precession 77, 105, 116 Bearing 26

− damping 119, 131, 156 − flexibility 102

Blowers 14

Campbell diagram 87, 152, 222 Centrifugal compressors 10

− pumps 15 Complex eigenvalues 221 Coordinate system 154

− rotating 56 − stationary 56, 58

Critical speed 67 − backward 84 − damped 219 − forward 82 − map 209 − peak response 82, 114, 224 − undamped 45, 207

Damped rotors 158 − asymmetric 154 − symmetric 46

Damping 62 − coefficient 128 − external 47, 62, 109, 117 − internal 54, 62, 117 − hysteretic 50 − optimum 127 − ratio 47, 60, 110, 154 − rotating 54 − stationary 67 − viscous 47

Decay factor 48

Eigenvalue problem 220 Eight bearing coefficients 138

Elliptical orbit 105, 112 Ellipse 104 Equations of motion 41, 72, 102, 147 Equivalent stiffness 103 External damping 47, 62, 109, 117

Fans 14 Forward precession 77, 105, 116 Free precession 47

−− damped 47, 121

Gas turbines 6 Gravity loading 65 Gyroscopic torques 71, 97

Harmonic force 84 Hydraulic turbines 18 Hysteretic damping 62

Inertia torques 69 Influence coefficients 146 Internal damping force 54 − − ratio 60

Major semiaxis 86, 106 Mass unbalance 43 Minor semiaxis 86, 106 Mixed modes of precession 158 Matrix, damping 59, 154 − flexibility 75 − stiffness 59, 75 Mode of precession 75 − shapes 90, 92 Model 29 − Laval-Jeffcott 39 − Stodola-Green 39

Natural frequency 67, 111, 148 − damped 48, 120, 132, 221 − undamped 47 Nyquist plot 51

Onset speed of instability 64, 118, 144 Orbits 25, 111, 223 Overdamped modes 183, 186

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MECHANICAL VIBRATIONS 284

Phase angle 50, 123 Planar modes 273 Polar diadram 52 Precession 23, 67

− backward 78, 105 − forward 77, 105 − free damped 47 − radius 50 − synchronous 44, 52

Reference frames 69 Rotor 6

− asymmetric 145 − rigid 26 − symmetric 101

Rotor bearing dynamics 22 Routh-Hourwitz criterion 63, 118 Rigid body modes 208 Rotating damping 54

Shaft 26 − bow 66 − mass 131

Sommerfeld number 228 Spiral 48 Spot, heavy 53

− high 53 Stability 59, 142, 227 Stator inertia 217 Steady state precession 49

− response 43 Stiffness coefficients 40

− matrix 59, 75 Symmetric rotors 101

− damped 46 − in flexible bearings 101 − in fluid film bearings 136 − in rigid bearings 40 − undamped 40

Synchronous excitation 88 − precession 44, 52

Threshold speed 113 Turbo-generators 18

Unbalance response 104, 121, 134, 152 − diagrams 50, 225

Undamped critical speed 45, 207 − natural frequency 47

Undamped rotors 40

− asymmetric 68 − symmetric 40 Underdamped modes 221

Viscous damping 47, 62 − coefficient 47

Whirling 61

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