• PROJECT 2 – DYNAMICS OF MACHINERY 41514

– Dynamics of Rotor-Bearing System –Lateral Vibrations and Stability Threshold of Rotors Supported On

Hydrodynamic Bearing and Ball Bearing.

Ilmar Ferreira Santos, Prof. Dr.-Ing. dr. techn.Section of Solid Mechanics

Department of Mechanical EngineeringTechnical University of Denmark

2800 Lyngby, Denmarke-mail: ifs@mek.dtu.dkphone: +45 45256269

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1 IntroductionThe occurrence of instability in rotating machines is a very typical problem in high speed machinesand may be caused by fluid-structure interactions due to oil film forces presented in hydrodynamicbearings or aerodynamic forces resulting from seals, among others. A machine designer normallywants to know the range of angular velocity in which the machine will operate stable and withoutvibration problems, i.e. maximum angular velocity and distance from the unstable operational range.

The aim of this second project in the discipline is to make you familiar with the prediction of criticalspeeds in rotating machines, stability threshold and unbalance response and maximum vibrationamplitude while crossing critical speed ranges.

Aiming at dealing with a real rotor-bearing problem, the test rig illustrated in figure 1(a) will beused as example of a rotating machine. It will facilitate the visualization of the problem and aidthe students to link the theoretical results to practical problems and experimental results. The testrig shown in figure 1(a) simulates a large overhung centrifugal compressor. The scheme of a largeoverhung centrifugal compressor can be seen in figure 1(b). The test rig as well as the overhungcentrifugal compressor is composed of one flexible shaft, discs or impellers positioned at the endof the shaft. The shaft is laterally supported by two bearings. In the test rig the bearing closestto the coupling will operate as thrust bearing as well. All technical information needed is given inthis project description. By using all theoretical and experimental tools related to Rotor Dynamicsobtained during the classes, you will win a solid overview and understanding about how real andimaginary parts of eigenvalues – and consequently also eigenvectors – change as a function of theoperational speeds.

The steps of the project are summarized as follows:

• First, you have to obtain mechanical and mathematical models of the flexible shaft element usingFinite Element Method (FEM).

• Second, the disc (impeller) shall be mounted onto the shaft. Mechanical and mathematical modelsshall be expanded to cope with the dynamics of the disc.

• Finally, the flexible shaft connected to the rigid disc is mounted onto a ball bearing and a journalbearing, as it is shown in figure 1(a). The eigenvalues and eigenvectors will be calculated for theglobal rotor-bearing-system. These calculations will serve as the basis for an elaborate analysisinvolving mode shape visualization and stability.

2 Modelling the Lateral Dynamics of Flexible Shaft1. Modelling – In the design phase of a rotating machine, the design engineer will obtain the

drawing of the machine elements in order to simulate the system dynamics. In appendix A,drawings of the flexible shaft are presented.

a) Elaborate a mechanical model with the aim of analysing lateral vibrations of the shaft in afree-free condition (without bearings), stating clearly the assumptions/simplifications used.

b) Elaborate a correspondent mathematical model using the Finite Element Method. Discretizethe flexible shaft using at least 3 different levels of discretization, starting with a very coarsediscrete model.

c) Calculate the natural frequencies and modes shapes of shaft in a free-free condition using theMatLab program presented during the lectures ”flexible-rotor-modal-DTU”.

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(a)

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2 3 4 5 6 7

(b)

Figure 1: (a) Test rig built to simulate the dynamic behavior of an overhung centrifugal com-pressor; 1 : Excitation point; 2 : Disc; 3 : Journal bearing; 4 : Seal (neglect this componentin the project); 5 : Shaft; 6 : Ball bearing (also functioning as thrust bearing); 7 : Flexiblecoupling; (b) Operational scheme of a large overhung centrifugal compressor.

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(a) (b)

Figure 2: (a) Flexible shaft in a free-free condition – (b) Flexible shaft and one disc in a free-freecondition.

d) Elaborate a convergence study (table), illustrating how the first 8 natural frequencies varydepending on the number of shaft elements used. Use at least 3 levels of discretization.

IMPORTANT: In case you face numerical problems while calculating the sys-tem eigenvalues due to zeros associated to rigid body motion, add small valuesof bearing stiffness at the bearing nodes to overcome the numerical problem.Pay also attention to the length of the shaft elements in order to avoid nu-merical ”unbalanced” matrices which makes the calculation of eigenvalues andeigenvectors troublesome.

2. Validation – After simulating the dynamic behaviour of the shaft in a free-free condition, you havethe possibility of checking the model accuracy by comparing the theoretical natural frequencieswith the experimental ones obtained from an experimental modal analysis test and illustratedin figure 3. Present the theoretical and experimental results in a form of table and state clearlythe discrepancies in %.

3. Model Adjustment - Using the results from the experimental modal analysis (natural frequencies)try to adjust the model parameters having as main goal discrepancies among the theoretical andexperimental natural frequencies under 5% in the frequency range of 0 and 2000 Hz. Presentthe theoretical and experimental results in a form of table and state clearly the discrepanciesin %. If you are not able to adjust them under such a precision, try to explain with reasonableargumentation the origin of the discrepancies.

3 Modeling the Lateral Dynamics of Flexible Shaft Coupled tothe Rigid Disc

1. Modeling – In appendix A, the drawings of two rigid discs (impellers) are presented.

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0 0.2 0.4 0.6 0.8 1 1.2 1.4

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

Shaft acceleration at the node 10 − Excitation at node 2

TIME [s]

z 10(

t) [

m/s

2 ]

0 500 1000 1500 2000 2500 30000

0.01

0.02

0.03

0.04

0.05

FREQUENCY [Hz]

fft(z

10)

388 Hz

930 Hz 1689 Hz

0 0.2 0.4 0.6 0.8 1 1.2 1.4

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

Shaft acceleration at the node 10 − Excitation at node 4

TIME [s]

z 10(

t) [

m/s

2 ]

0 500 1000 1500 2000 2500 30000

0.1

0.2

0.3

0.4

0.5

FREQUENCY [Hz]

fft(z

10)

388 Hz

930 Hz

1689 Hz

2624 Hz

Figure 3: Experimental Modal Analysis – flexible shaft in a ”free-free” condition and its naturalfrequencies.

a) Elaborate a mechanical model for the two rigid discs and state clearly the simplificationsmade.

b) Mount the disc 1 (80 mm thickness) onto the flexible shaft as shown in figure 2(b), elaboratea corresponding mathematical model using the Finite Element Method, and calculate the naturalfrequencies and modes shapes of shaft-disc system in a free-free condition, using the MatLabprogram presented during the lectures ”flexible-rotor-modal-DTU”.

c) Mount the disc 2 (100 mm thickness) onto the flexible shaft as shown in figure 2(b) (replace itwith disc 1), elaborate a corresponding mathematical model using the Finite Element Method,and calculate the natural frequencies and modes shapes of shaft-disc system in a free-freecondition, using the MatLab program presented during the lectures ”flexible-rotor-modal-DTU”.

2. Validation – After simulating the dynamic behaviour of the shaft and discs in a free-free con-dition, you have the possibility of checking the model accuracy by comparing the theoreticalnatural frequencies with the experimental ones obtained from an experimental modal analysisin figure 4. Present the theoretical results for both cases, i.e. discs with 80 mm and 100 mmthickness, and the experimental results in a form of table and state clearly the discrepancies in%. Following answer the question: Which disc was used during the experimental test, the onewith 80 mm thickness or the one with 100 mm thickness?

3. Model Adjustment - Using the results from the experimental modal analysis (natural frequencies)try to adjust the model parameters having as main goal discrepancies among the theoretical andexperimental natural frequencies under 5% in the frequency range of 0 and 2000 Hz. Presentthe theoretical and experimental results in a form of table and state clearly the discrepancies in%. If you are not able to adjust them under such a precision, try to explain the origin of thediscrepancies using reasonable argumentation.

4 Modelling the Ball Bearing as Stiffness Element and Couplingto Shaft

Two flexible rotating systems, i.e. system (I) with only the flexible shaft and system (II) withthe flexible shaft and the rigid disc, will now be mounted onto ball bearings located at positions

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0 0.2 0.4 0.6 0.8 1 1.2 1.4

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

Shaft acceleration at the node 10 − Excitation at node 6

TIME [s]

z 10(

t) [

m/s

2 ]

0 500 1000 1500 2000 2500 30000

0.1

0.2

0.3

0.4

0.5

FREQUENCY [Hz]

fft(z

10)

301 Hz

669 Hz

1348 Hz

0 0.2 0.4 0.6 0.8 1 1.2 1.4

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

Shaft acceleration at the node 10 − Excitation at node 8

TIME [s]

z 10(

t) [

m/s

2 ]

0 500 1000 1500 2000 2500 30000

0.05

0.1

0.15

0.2

0.25

FREQUENCY [Hz]

fft(z

10)

301 Hz

669 Hz 1348 Hz

2383 Hz

Figure 4: Experimental Modal Analysis – flexible shaft coupled to one rigid disc in a ”free-free”condition and its natural frequencies.

3 and 6 in figure 1(a). The rotor diameter at these locations are 99.6 mm and 50 mm as seenfrom appendix A. The elasticity of rolling bearings is generally small, even when the loads are high;in most bearing arrangements it is therefore not important. You can assume that its stiffness canbe considered of the order 109 N/m for practical simulation proposes. If you want to calculate thestiffness of the rolling bearing the specification is given hereby: SKF 2210-2RS1. Normally it isassumed the one rolling element is positioned exactly in the direction of the load. The calculationsare therefore concerned only with the effect of the elastic deformations which the loaded rollingelements and raceways undergo at their contact areas. The amount of elasticity changes duringrotation of the rolling element. These changes, however, are so small that they can be neglected inthe calculations. (see reference: Brändlein, J., Eschmann, P., Hasbargen, L. and Weigand, K (1999).”Ball and Roller Bearings - Theory, Design and Application”, John Wiley & Sons Ltd., Third edition,pages 136-142.)

After this short explanation, and considering zero angular velocity, answer the following questions:a) Investigate the behaviour of the first 8 natural frequencies of the rotor-bearing system of systems(I) and (II), varying the two identical bearing stiffness from 101 N/m (almost free-free) to 109 N/m(simply supported by two ball bearings) and plotting in a log-log scale graphically the behaviour ofthe first 8 natural frequencies of the rotor-bearing system (I) and (II) as a function of the bearingstiffness. Neglect the rotor angular velocity, i.e. Ω = 0b) Explain and illustrate with pictures how the mode shapes of the rotor-bearing system (I) and (II)change depending on the bearing stiffness. Neglect the rotor angular velocity, i.e. Ω = 0

4.1 Prediction of Critical Speeds Using Two Ball BearingsPlot the Campbell’s diagram for the rotor-bearing system (I) and (II) and define the first first criticalspeed of the rotating machine for the case in which two ball bearings are used. Use the MatLabprogram presented in the manuscript. Plot the first mode shape of the rotor-bearing systems (I) and(II) at their first critical speeds and explain the differences between them.

4.2 Unbalance Response Using Two Ball BearingsImagine that the rotor-bearing system (II) has to operate between the two first critical speeds wherethe vibration level is normally very small. Nevertheless, to operate between the two critical speeds,

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the rotor-bearing system has to cross the first critical speed. The maximum vibration amplitudeat the disc locations shall not exceed 30 µm while crossing the critical speed. Such a conditionshall not be violated. A violation of such a criteria will cause rubbing problem between rotating andnon-rotating parts of the machine. Assuming that the shaft is perfectly balanced and the unbalanceonly occur in the rigid disc (impeller), what is the maximum unbalance in [g·mm] allowed at the discclosest to the shaft extremity in order to assure the design specification? Is it possible to meet sucha design specification? Explain.

5 Modeling of Hydrodynamic Bearing as Stiffness and DampingElements

In the following, the ball bearing located at position 3 in figure 1 will be replaced with a journalbearing so that the bearing set-up corresponds to the real test rig. As designer, you will haveto choose among several journal bearings and justify your choice based on three dynamic criteria:i) operational range, ii) stability margin and iii) maximum vibration amplitude due to unbalance.Two different journal bearings with different dynamic properties are presented in the course website(lecture 9). Please, download the files

• Journal Bearing Data (project 2)

• Journal Bearing Data (matlab example)

The static equilibrium position, stiffness and damping coefficients of the rotor-bearing system canbe obtained with help of the Sommerfeld number S = η ∗ N ∗ L ∗ D/W ∗ (R/C)2 and tablespresented in the Journal Bearing Data file, if all parameters η, N L D W R and C are known.Recall that D is the bearing inner diameter [m]; L is the bearing width [m]; R = D/2 is therotor radius [m]; C is the bearing clearance [m]; W - external load [N]; η - oil viscosity [N.s/m2];N is rotor angular velocity in [1/s] or [Hz]; ω = 2 ∗ π ∗ N is rotor angular velocity in [rad/s];S = η ∗ N ∗ L ∗ D/W ∗ (R/C)2 is the Sommerfeld number; ϵ = e/C is eccentricity ratio; Φis attitude angle; Kij = (C/W ) ∗ kij (i, j = v, w) is the dimensionless stiffness coefficients;Cij = (C ∗ ω/W ) ∗ dij (i, j = v, w) is the dimensionless damping coefficients.

The bearings will be lubricated using the ISO VG 32 oil. Its viscosity can be described by the expressionη = 0.0277 ∗ e[0.034∗(40−T )], η in [N.s/m2] and T in [C]. In this initial analysis the dependency ofthe oil temperature on the rotor angular velocity will be neglected. Based on experience with othermachines with the same size one can assume that the mean temperature T will be 55 [C]. In otherwords the stability analysis have to be led assuming an isothermal hydrodynamic lubrication. Themaximum angular velocity N has to be defined, and this is the most important part of the modelapplication. The rotor diameter D [m] can be obtained from the technical drawings. The relationshipR = D/2 and the bearing clearance C = 85 [µm] are pre-defined parameters. The relation L/D willbe dependent on the bearing type used and is specified in the journal bearing data tables.

1. Before using/applying any kind of theoretical results, i.e. journal bearing data, a good engineerwill always try to clearly understand the theoretical assumptions/simplifications made in order toobtain the theoretical results, in our case the tables. This engineering practice is very importantand prevents engineers from using wrong and inappropriate theoretical data, in ranges wherethe data should not be used. Violation of the theoretical assumptions leads consequently to awrong analysis. In this framework, answer the following questions: what are the main assump-tions/simplifications made in order to obtain the Reynolds equation, which is the foundation forbuilding the journal bearing data?

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2. Calculate the external loads due to the weight acting on the journal bearing W [N] whenoperating with the system (I) and (II). For this purpose, please use the geometry of the shaftelements and rigid discs elements . The Sommerfeld number will be a function of the angularspeed of the machine N .

3. Using the journal bearing data tables illustrate graphically the behavior of the oil film thicknessin [µ m] as function of the angular velocity N in [Hz] for the two different types of journalbearings. Analyze the results for system (I) and (II).

4. a) Plot the values of stiffness coefficients [N/m] and damping coefficients [N/(m/s)] as a functionof the angular velocity N in [Hz] for the different types of journal bearings. Remember thatthe stiffness coefficients kij given in [N/m] are calculated as a function of kij = (W/C) ∗Kij (i, j = x, y), where Kij are the dimensionless stiffness coefficients obtained from thejournal bearing data tables. The damping coefficients dij given in [N/(m/s)] are calculated asa function of kij = W/(C ∗ ω) ∗ Cij (i, j = x, y), where Cij are the dimensionless dampingcoefficients obtained from the journal bearing data tables.b) Interpolate piecewise functions, for example splines or linear functions, to describe the be-havior of the stiffness and damping coefficients as a function of the angular velocity. Plot thestiffness and damping coefficients and the interpolated functions. Tip: Check out the MatLabfunction interp1.c) Write some conclusions about direct stiffness coefficients, cross coupling stiffness coefficientsand direct damping coefficients.d) Compare the reference frames xyz used for describing the bearing dynamics (Someya)and flexible shaft dynamics (Finite Element Method). They are not same! Find thetransformation matrix between the two reference frames and the relationships betweenKxx, Kxy, Kyx, Kyy and Kvv, Kvw, Kwv, Kww as well as CXxx, Cxy, Cyx, Cyyand Cvv, Cvw, Cwv, Cww.

5.1 Prediction of Critical Speeds1. Define the two first critical speed of the rotor-journal bearing system (I) and (II) based on the

Campbell’s diagram considering the two different hydrodynamic bearing types. Write the twofirst critical speeds for each of the two cases in a table, specifying bearing type, first criticalspeed, and second critical speed.

2. Imagine that the rotor-bearing system has to operate between the two first critical speeds wherethe vibration level is normally very small. What is the bearing type which allows for the largestoperational velocity range? Analyse it for system (I) and (II).

5.2 Prediction of Rotor Bearing Stability Limit1. Based on the stability maps of each of the two rotor-bearing systems, calculate the maximum

angular speed N that system (I) and (II) can operate without instability problems resulting fromthe interaction with the oil film forces? Write in a table, bearing type and maximum angularspeed (instability threshold).

2. Will the stability threshold be the same for system (I) and (II)? What is the best journal bearingfrom the viewpoint of stability threshold for system (I) and (II)?

3. When the rotor-bearing system (I) and (II) become unstable, the system vibration amplitudeincreases. What is the frequency of such an unstable vibration for the two different bearingcases?

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4. What is the relationship between the unstable frequency [Hz] and the rotor angular velocity Ω[Hz] in the stability threshold for system (I) and (II) operating with the two different bearingtypes?

5.3 Unbalance ResponseThe maximum vibration amplitude at the disc locations shall not exceed 29 µm along the wholeoperational speed range. Such a condition shall not be violated. A violation of such a criteria willcause rubbing problem between rotating and non-rotating parts of the machine. Assuming thatthe shaft is perfectly balanced and the unbalance only occur in the rigid disc (impeller), what isthe maximum unbalance in [g·mm] allowed at the disc in order to assure the design specification.Consider only the rotor-bearing system (II) in this case and the two bearing types.

1. Write in a table, bearing type and maximum unbalance in [g·mm] allowed.

2. What is the best journal bearing from the viewpoint of unbalance response and maximumunbalance allowed?

5.4 Engineering Design – Application1

1. Bearing Design – The relationship R = D/2 and the bearing clearance C = 85 [µm] were upto now pre-defined parameters and the relation L/D will be dependent on the bearing typeused and specified in the journal bearing data tables. As design engineer, chose one of thehydrodynamic bearings, and answer the following questions for system (I) and (II):

a) How variations in the bearing clearance C [µm] can influence the critical speeds?

b) How variations in the bearing clearance C [µm] can influence the limit of stability?

c) How variations in the bearing clearance C [µm] can influence the unbalance response?

2. Rotor-Bearing Design – Change now the ball bearing used by a new hydrodynamic bearing withthe same diameter as the ball bearing (see drawing) with a clearance of C = 85 [µm]. What canyou conclude about the stability margin of the new rotor-bearing systems (I) and (II) operatingon two hydrodynamic bearings instead of one ball bearing and one hydrodynamic bearing asoriginally proposed?

1For the students who want to fight for a 12 grade

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6 Geometrical Data6.1 Journal Bearing PropertiesTable 1a : Two-axial-groove bearing, L/D = 0.5

S E Phi Q P T Kxx Kxy Kyx Kyy Bxx Bxy Byx Byy

Table=[5.96 0.0750 82.6 0.0663 1.00 111.5 1.77 13.6 -13.1 2.72 27.2 2.06 2.06 14.94.43 0.100 80.4 0.0880 0.999 83.0 1.75 10.3 -9.66 2.70 20.6 2.08 2.08 11.42.07 0.200 71.6 0.170 0.999 39.5 1.88 5.63 -4.41 2.56 11.2 2.22 2.22 6.431.24 0.300 64.1 0.243 0.997 24.4 2.07 4.27 -2.56 2.34 8.50 2.32 2.32 4.730.798 0.400 57.5 0.309 0.986 16.4 2.39 3.75 -1.57 2.17 7.32 2.24 2.24 3.500.517 0.500 51.1 0.366 0.967 11.3 2.89 3.57 -0.924 2.03 6.81 2.10 2.10 2.600.323 0.600 44.7 0.416 0.939 7.75 3.65 3.62 -0.427 1.92 6.81 2.08 2.08 2.060.187 0.700 38.2 0.459 0.900 5.13 4.92 3.88 0.0235 1.83 7.32 2.16 2.16 1.700.135 0.750 34.9 0.478 0.875 4.07 5.90 4.11 0.258 1.80 7.65 2.10 2.10 1.460.0926 0.800 31.2 0.496 0.846 3.13 7.35 4.46 0.527 1.78 8.17 2.05 2.05 1.240.0582 0.850 27.2 0.510 0.806 2.30 9.56 4.92 0.805 1.73 9.12 2.06 2.06 1.060.0315 0.900 22.7 0.524 0.758 1.56 13.8 5.76 1.24 1.72 10.6 2.03 2.03 0.8460.00499 0.975 12.5 0.543 0.663 0.543 43.3 9.61 4.26 1.99 22.4 2.93 2.93 0.637];

Table 1a : Two-lobe bearing, L/D = 0.5, mp = 1/2

S E Phi Q P T Kxx Kxy Kyx Kyy Bxx Bxy Byx Byy

Table=[4.79 0.0556 87.2 0.292 0.530 104 61.5 33.1 -43.3 2.06 123 -43.3 -42.5 36.63.06 0.0868 87.2 0.293 0.524 65.9 39.4 21.7 -28.0 1.43 80.0 -27.7 -27.7 23.41.93 0.137 86.9 0.196 0.520 41.6 25.1 13.7 -17.7 1.16 50.2 -17.9 -17.5 15.00.993 0.262 85.9 0.312 0.502 21.4 13.5 7.50 -9.20 0.897 26.5 -8.47 -7.34 8.280.496 0.438 84.4 0.343 0.469 10.8 7.88 4.52 -4.60 0.831 14.0 -3.42 -3.36 4.710.306 0.608 80.8 0.370 0.441 6.99 6.06 3.73 -2.82 0.940 10.5 -1.13 -1.11 3.450.209 0.718 77.2 0.402 0.404 4.91 5.66 3.60 -1.67 1.16 8.74 -0.0503 -0.0240 2.720.105 0.888 68.3 0.433 0.364 2.73 6.51 4.19 -0.115 1.57 7.86 1.44 1.45 1.890.0507 0.970 56.8 0.436 0.353 1.61 9.47 5.46 0.945 1.94 9.03 2.29 2.31 1.380.0303 0.995 48.6 0.424 0.359 1.12 13.2 6.45 1.72 2.12 10.4 2.58 2.62 1.160.0207 1.01 42.5 0.407 0.370 0.891 17.2 7.81 2.33 2.24 11.8 2.70 2.73 1.020.0100 1.01 34.1 0.372 0.400 0.590 29.6 9.68 3.28 2.39 15.3 2.88 2.93 0.875];

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A Technical Drawings of Shaft and DiscsOn the following pages you will find technical drawings of:

• Assembly drawing of shaft with two mounted discs.

• Shaft drawing with dimensions.

• Shaft drawing with grinding tolerances.

• 80 mm disc drawing with dimensions.

• 100 mm disc drawing with dimensions.

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