3.1_General Principles of Dynamic Design

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KBR Technical Portfolio Infrastructure Course 3.1

3.1 General principles of dynamic design

and specific applications

Contents

Introduction 1Units 2Sources of vibration 3Vibrational response of structures 4Design considerations 7Guides to specifications 11Design of pad foundations for machinery 12Spring mounted equipment 17Systems with two degrees of freedom 18Beats with multiple equipment 19Appendix A Use of general-purpose analysis software 20Appendix B Manual checking of dynamic response 21Appendix C Dynamic analysis and design for mining and other plant 25Appendix D Dynamic design issues for footbridges 28

Appendix E Wind induced dynamics 29Appendix F Dynamic loads imposed by pneumatically-tyred vehicles 31Document revision history 36

Introduction

This document was originally published as Kinhills Technical Bulletin No 7/81,with some additional information separately published as addenda later. It has now

been further supplemented with new material, and assembled as a complete document

in a new form for use by KBR staff.

The purpose of this document, Infrastructure Course 3.1, is to acquaint structural

engineers with those aspects of vibration theory which are necessary for the design of

structures subject to sources of excitation. Particular applications are addressed in

Appendix Cfor mining and other machinery; Appendix Dfor footbridges or

walkways; Appendix Efor wind sensitive structures; and Appendix Ffor the effect

of pneumatic tyres.

General principles of dynamic design, Rev. E 1

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Only the essence of the theory is presented, and the document explains the

fundamental issues most relevant to day-to-day problems. It begins with an

explanation of the sources of vibration and explains the difference between forced

vibrations and resonance. The response of structures to vibration is then discussed,

leading to a formulation of the principles of anti-vibration design and practical

solutions. The acceptability of different levels of vibration is considered.

The particular approaches to design of pad foundations for vibrating machinery are

presented. This is a situation that is frequently encountered but poorly understood,

and the preferred method of solution has been prepared as a spreadsheet.

Practical advice is offered in terms of what should be included in specifications when

issuing documents for purchase of machinery, with respect to information to be

supplied and the boundary of responsibility for design of supporting structures. The

question of what geotechnical information is required is also discussed.

Appendix A offers some general guidance in the application of modern structural

analysis software to the modelling of dynamics problems. Appendix B gives guidanceon manual checking of dynamic response, useful for a first assessment or for

checking the sense of a computer analysis. Appendix F discusses the forces likely to

be imposed on supporting structures by vehicles with pneumatic tyres.

It should be noted that wherever the term amplitude is used in this document it refers

to the mean-to-extreme excursion.

Units

When evaluating formulae in structural dynamics, it is essential that a consistent set of

units is used throughout. Particular attention must be paid to the distinction between

force and mass, with forces always expressed in absolute units rather than

gravitational units.

Table 1 below presents some alternative sets of consistent units. Only the

fundamental units and some key derived units are given: any other derived units

can be expressed in terms of the fundamental units.

Table 1 Possible sets of consistent units

Quantity Set 1 Set 2 Set 3

Fundamental units:

Length metre metre millimetre

Mass kilogram tonne tonne

Time second second second

Rotational displacement radian radian radian

Key derived units:

Force newton kilonewton newton

Stress / pressure / elastic modulus pascal kilopascal megapascal


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Sources of vibration

Any elastic system will vibrate if displaced from its rest position by a force and then

released. This is called natural vibration, and the frequency of the vibration is called

the natural frequency. Easily understood examples from the physics of our schooldaysare the tuning fork, or a weight on a spring, or a pendulum. Because some damping

(i.e. absorption of energy) is always present, the amplitude of the vibration

progressively dies down from its maximum value (the initial displacement) towards

zero.

In order to maintain an ongoing state of vibration, an input of energy is required to

replace the energy dissipated by damping. Lightweight structures such as walkways

can be excited merely by pedestrian traffic or the wind. However, the most common

source of vibration with which engineers in industry are concerned is a pulsating force

of regular frequency, usually caused by reciprocating machinery, out-of-balance parts

of rotating machinery, or a combination of both. The types of machinery causing suchpulsating forces include:

Motors and generators

Turbines

Compressors

Vibrating screens, apron feeders or grizzlys

Crushers (jaw and gyratory)

Hammer mills

Motor drives for drum screens, conveyors, etc.

Most rotating machinery is intended to be dynamically balanced. However, with large

machines it will be necessary to allow for some eccentricity of the centre of gravity of

the rotating parts due to manufacturing tolerances or wear.

At the other extreme, vibrating screens have a mass which oscillates, and this motion

is generated by a deliberately out-of-balance mass on a motor-driven flywheel. Jaw

crushers and gyratory crushers also have deliberate out-of-balance forces.

In rotating machinery, the centripetal force set up is2

meF=

where is the out-of-balance massmis its eccentricitye

is the rotational velocity (in radians/sec)The frequency in cycles per second (cps) is

2=f

It is normal, for structural design, to examine the structure or foundation for the effects

of horizontal and vertical loads. Assuming the rotation is about a horizontal axis, the

horizontal and vertical components of the dynamic forces vary with time as follows:

( )tmeFx cos2=

( )tmeFy sin2=


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Since the peak force is the same in each case, and equal to the centripetal force, the

above equations are of interest mainly in further theoretical analysis. They do,

however, show that the two forces follow a simple harmonic pattern, rising to a peak,

dropping to zero, then reversing in sign.

The above are the forces acting at the shaft of the motor or flywheel, and with a rigidlymounted machine these are transmitted to the supporting structure. If the system is

further suspended on a spring frame, as is the case for a vibrating screen, or on rubber

mounting pads, then the force transmitted to the supporting structure will be modified.

Vibrational response of structures

FORCED VIBRATIONS

A structure subjected to any fluctuating force will deflect in response to changes in

that force, at the same frequency as the changes in the applied force. However, the

amplitude of oscillation will differ from the static deflection which would have

occurred under a steady application of the force. This condition is called forced

vibration, and the designers primary concern is to ensure that the amplitude of the

resulting vibrations will be at an acceptable level.

HARMONIC VIBRATIONS

Harmonic vibrations are a special case of forced vibrations. If all components of thefluctuating force are varying sinusoidally and at the same frequency, then the structure

will settle into a steady state response in which every component of its displacement is

varying sinusoidally. The frequency of these sinusoidally-varying displacements will

be the same as that of the fluctuating force. This situation is known as harmonic

vibration. It may take some time to establish itself, because any transient vibrations

associated with start-up need to die away first.

NATURAL VIBRATIONS

As explained above, all elastic systems have one or more natural frequencies of

vibration. For a concentrated mass attached to a mass-less elastic support without

damping (a system with a single degree of freedom):

M

kfnn == 2

where n is the natural frequency in radian/sec,is the natural frequency in cycles/sec,nf

is the spring stiffness of the elastic support (the static force required to

produce a unit deflection),

k

is the concentrated mass.


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With damping, the natural frequency is modified slightly so that

21 Dff nnd =

where is the damping ratio as defined below.D

RESONANCE

As stated previously, the amplitude of vibration is not the same as the deflection that

would occur if the applied load was static. As the frequency of the applied force

approaches the natural frequency of the structural system the amplitude is greatly

magnified, and the mathematical expressions show that without damping the

amplitude would become infinite.

There is always some energy loss or damping in the system, which prevents this

situation from arising. However, for systems with low damping the amplitude can be

magnified many-fold. This phenomenon is called resonance, and a primary aim of all

dynamic design is to avoid it altogether by ensuring that the natural frequency of the

system is well removed from the frequency of the oscillating applied forces.

DAMPING

Damping is dissipation of energy, from hysteresis within the material of the

supporting structure itself, from frictional movement in bolted joints, slip between

reinforcing rods and concrete, friction between soil particles, etc. An additional form

of damping is applicable to pad foundations, where the vibrational energy is carried

away to infinity by compression and shear waves in the soil: this is known as

geometric or radiation damping.

Critical damping is defined as the smallest amount of damping for which no

oscillation occurs in the free response. The actual damping present in any system is

then expressed as the ratio of the actual damping to the critical damping. This

damping ratio ( ) is useful in assessment of likely oscillation amplitudes, and if it is

not known safe approximations can usually be made.

D

Typical published values of damping ratios are given in Table 2 below, where only a

broad range can be given for each type of structure since the damping is influenced by

many factors and design details. It is useful to distinguish between damping in the

bare structure and damping in non-structural elements (such as flooring, partition

walls, furniture, mechanical services, equipment, etc.). This distinction underlies the

Table's differences between general building structures (which usually have a variety

of non structural elements), and pedestrian bridges (which usually have almost no

non-structural elements).


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Table 2 Typical published values of damping ratio

Structure type Damping ratio

General building structures:

Continuous steel structure 0.020.04

Bolted steel structure 0.040.07

Prestressed concrete structure 0.020.05

Reinforced concrete structure 0.040.07

Lightweight pedestrian bridges:

Steel 0.0030.005

Composite 0.0040.008

Reinforced concrete 0.0080.020

Prestressed concrete 0.0050.017

Miscellaneous:

Small diameter piping system 0.010.02

Large diameter pipes 0.020.03

MAGNIFICATION OF AMPLITUDE

As a starting point, it is necessary to calculate the deflection that would be caused by

the out-of-balance force ( ) if it were statically applied. The dynamic displacement

amplitude varies from this static deflection as follows:

F

staticdstatic MDrr

a =+

=222 )2()1(

1

where ris the ratio of the operating frequency to the undamped natural frequency,

D is the damping ratio as already defined,

Md is called the dynamic magnification factor.

At low operating frequency, the amplitude is the same as the static deflection, with the

structural response being in phase with the applied force. As the frequency of the

applied force increases towards the natural frequency of the system, the amplitude

becomes greatly magnified. Because of the presence of damping, the peak amplitude

occurs at a slightly lesser frequency than the natural frequency. As the frequency of

the applied force increases beyond the resonant frequency, the magnification

decreases. By about 1.4 times resonance the magnification has dropped below 1. The

physical explanation of this is that the oscillating force cannot accelerate the mass

quickly enough to achieve the static deflection before the force starts to act in the

other direction.

As a rough rule, undesirable amplification occurs if the frequency ratio rlies between

0.5 and about 1.3. The upper range requires more careful interpretation. We are not

usually looking at the case of a machine whose operating speed is increasing. Usually

we are concerned with designing for a fixed, or nominal, operating speed, and then we

vary the design of the structure or foundation to change its natural frequency, aiming

to move it further away from the operating frequency.

If the frequency ratio is low, i.e. below 1, the natural frequency is higher than the

applied frequency, so design modifications usually aim to make the structure stiffer.

This decreases both the static deflection and the dynamic magnification that applies to


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that static deflection. In addition, the machine will not pass through resonance during

start-up and close-down.

On the other hand, if the frequency ratio is high it means that the natural frequency is

low relative to the operating frequency, probably because the stiffness of the structure

is also low. Any further reduction in structural stiffness will increase the staticdeflection, and this increase in static deflection will usually more than offset any

benefit obtained from a reduction in dynamic magnification. (It is generally difficult

to change the mass significantly, except for foundations as discussed later in this

document.) An added problem in the case of a high frequency ratio is that resonance

can occur during start-up, or if the machinery is operated below its nominal speed:

this factor is sometimes overlooked by a designer given a simple set of criteria to work

to.

Thus we can conclude that the best means of guarding against excessive amplitude of

vibration is to design the supporting structure to have a natural frequency more than

double the applied frequency (i.e. r< 0.5). If this cannot be achieved, we should aimfor a frequency ratio of 1.5 to 2.0 or higher (1.3 is too close to resonance in view of

possible errors, or changes in parameters, given the steepness of the magnification

curve). If this latter approach is achieved by reducing stiffness, then the amplitude

must be carefully checked: in this context, note that to halve the natural frequency

means having to quarter the stiffness, which will lead to four times the static

deflection.

Although the amount of damping has a significant effect on the amplitude near

resonance, it does not have as much effect for frequency ratios in the desirable ranges

described above. Hence, whilst it was necessary to discuss its effect, damping can

usually be ignored for the purposes of designing the supporting structure.

Refer also to Section 8 for comments on how the magnitude of force transmitted to a

supporting structure can be reduced by spring mountings and to Section 7 for

transmissibility from pad footings to the supporting soil.

Design considerations

STRUCTURAL FORM AND STIFFNESS

As stated in the previous section, the approach to controlling vibrations should always

involve first the elimination of resonance and then a check of the predicted amplitude.

Since n = (k/M) it can be seen that large changes in stiffness and/or mass arerequired to shift the natural frequency by significant amounts. It is usually impossible

to increase the natural frequency to any extent by reducing the mass, and the primary

target is the stiffness. To double the natural frequency requires quadrupling the

stiffness. Thus if it is found that resonance occurs with a 460UB67 supporting beam,

then a 760UB173 will be required to give satisfactory performance (r< 0.5).


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On the other hand, it should be noted that for a central point load =Pl3/(48EI) and

so can be halved by dividing the span by3 2 = 1.26. Thus if the initial span was

4 m and resonance occurred, satisfactory performance would be achieved by reducing

the span to 3.17 m with the same size of beam. Because of the layout of feeders,

conveyors, etc, a pair of beams supporting a screen or small crusher are often

supported in turn on two other beams running to the columns. This increases the

overall deflection considerably, and requires all four to be stiffened. It is apparent

from this that the most advantage can be gained at the layout stage, rather than giving

the structural designer a committed layout to size up.

If the span cannot be shortened and increased beam depth cannot be accommodated,

another approach theoretically available is to drop the frequency by reducing stiffness.

If this is done simply by reducing the inertia of the beam it is unlikely that it will have

enough strength to carry the load. However, increasing the span will increase the

deflection at a much faster rate than it increases the stress. This approach can be

adopted if the span cannot be shortened, but is not the preferred approach for reasons

outlined previously.

The last approach available is to move the frequency as far as possible from

resonance, and then to introduce sufficient damping in the mounting of the machine.

This is a last resort, and requires expert advice. It is not covered in this document.

SECONDARY ELEMENTS AND EFFECTS

So far the discussion has been aimed primarily at the main structural elements

supporting the vibrating machine. In mining applications, a vibrating machine is often

contained within a larger complex, such as a screen building having several levels, and

containing walkways and stairs. It is advisable to keep the structure that supportsmain items of equipment as separate as possible from other elements. However,

sometimes other beams will be connected to the same columns. The forced oscillation

of the main supporting beams can be transmitted into the columns as bending

moments, particularly through end plate connections, and then back into other beams.

In such a case, it is important to check these beams and avoid a resonant natural

frequency. The further one goes from the source, the less energy from the vibration

will be available to excite the various elements. However, a high energy source can

affect quite remote areas, and resonant vibrations of even small amplitude can be very

annoying in, for example, a control room where instruments have to be read.

Despite attempts at isolation, it is likely that there will be directly connected bracingmembers which could be affected. Although these have low bending stiffness they

also have low mass. This combination means that they can have a high natural

frequency, and also vibrate at large amplitude in response to a low energy input. They

should therefore be checked and designed to avoid resonance. Again, changes in

configuration should be considered, for example -bracing instead of -bracing.

If -bracing is added after the dynamic assessment has been performed, the apex

should not be at a beam subjected to forced vibration unless the dynamic assessment is

re-done. This is because the presence of the apex at the midspan of the beam will

change its dynamic characteristics.


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ACCEPTABILITY OF VIBRATIONS

The method of calculating vibration amplitudes has been discussed previously.

Having arrived at a figure, it is necessary to decide whether it is acceptable or not.

For some applications it will be necessary to establish the acceptability level as

briefing data or from an equipment supplier. For example, very strict criteria apply to

printing presses for colour printing where registration of three superimposed primary

colours is important.

Guidance on acceptable levels for machines is provided in Figure 1. Figure 2 gives

acceptability levels for comfort of persons. However, more comprehensive

information is contained in ISO 2631-1. Figure 1 is believed to be based on data to

indicate vibrations caused by faults within the machine rather than the effect of

external vibrations transmitted to the machine.

Figure 1

Acceptable amplitudes of vibration for rotating machines


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Figure 2

General limits for vibrations

In all cases the acceptability of an amplitude is dependent upon frequency, since what

we are really concerned about is acceleration and the inertia forces resulting from it.The figures show this relationship. Often criteria are expressed as a limit on velocity.

This is an artificial device. One can travel at 1000 km an hour without discomfort.

The velocity device is an attempt to express the criterion as a combination of

acceleration and amplitude, having the units but not the true meaning of velocity, i.e.

one can tolerate a higher acceleration at very low amplitude, or higher amplitude at

low acceleration.

For most machinery, it is the bearings that determine the level of vibration that can be

tolerated.

The sensitivity of persons to vibration is somewhat subjective, and depends not only

on body-shape, but also on the level of expectation. Relatively large deflections will

be accepted on a conveyor gallery without concern, and the acceptance level of

vibration on the operating platforms of a crusher station will differ from that tolerated

in a control room or sampling laboratory. Figure 2 sets out some guidelines, but

judgement must be applied.

For the types of structure we are concerned with (bare steel or reinforced concrete),

the purely structural consideration is that of fatigue. Since the dynamic magnification

factor will have been calculated to determine the vibration amplitude, it is an easy task

to apply this magnification factor to the nominal static stresses to arrive at the peak

stresses and stress ranges. The allowable stress range is determined by the fabrication

details (refer Section 11 of the Steel Structures code AS41001998). Note that a


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stress range allowable may be as low as 45 MPa. Foundations are a special case

discussed later.

VIBRATION OF LIGHTWEIGHT STRUCTURES

Another source of annoying vibrations in addition to vibrations caused by machineryis the low frequency but high amplitude natural vibration of lightweight structures

such as conveyor galleries, walkways, and handrails.

People walking slowly make about 1.5 footfalls per second, a figure that rises to about

3 footfalls per second for moderate running. Therefore structures having a natural

frequency in the range 1.53 cps can be excited accidentally simply by having people

crossing them or deliberately shaking them. This can cause great concern.

Furthermore, if a group of malfeasers is so inclined, deliberate excitations of up to

about 4 cps can be achieved. The structure's natural frequency should therefore be

checked, and the range 1.54 cps should be avoided by changing the span or otherwise

stiffening (it being usually impractical to change the mass).Whilst the main dynamic effect of walking is a vertical oscillating force at a frequency

of 1.53 cps, there is also a smaller lateral oscillating force at half that frequency.

Lightweight pedestrian walkways should be checked for this as well. It has been

observed that once a perceptible lateral oscillation has been established in a walkway

people feel compelled firstly to fall into step with it and secondly to widen their gait,

both of which will exacerbate the oscillation.

Handrails sometimes give trouble due to inadequate fixity of the stanchions at their

bases. Stanchions should be fixed to cross members and not to the longitudinal

stringers. The latter are usually very flexible in torsion and will allow large

deflections of the handrails to occur.

Guides to specifications

MACHINERY

If the design of the structure is undertaken as a separate assignment from the

machinery supply, then it is important to obtain the correct information for thedesigners. The actual information required will depend upon the nature of the

machinery and its mountings, and the enquiry documents must ask for it to be clearly

spelled out. The information required by the designer includes:

The geometric layout and particular positioning of mountings.

The mass of the machine and all attachments (mounting frame, etc.). Also, the

maximum and minimum additional mass that can be present, such as ore on a

screen.

The possible range of operating speeds. Peak speed is not sufficient.


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The out-of-balance dynamic forces transmitted to the structure (i.e. below the

mountings), and their centroid of action. These forces are required in the vertical

and both horizontal directions if applicable, and any rocking effects should be

stated (such as force alternating between front and rear mountings).

If the machinery supplier cannot provide this force information, then more basicinformation is required to enable it to be calculated, viz:

The out-of-balance mass, the position of its axis of rotation, and the worst possible

eccentricity of the centroid of the out-of-balance mass from that axis.

The nature of the mountings, and if flexible then their spring constants both

vertically and horizontally (which may be quite different).

If the spring constants of the mountings are known, and also the maximum

amplitude of the oscillation that occurs, then this can serve as an alternative to the

information on masses, eccentricities and axes (if that information is not available).

Finally, the specification should state that none of the above parameters may be variedwithout informing the purchaser of the changes and their implications to the dynamic

design.

STRUCTURES

If the design of the structure is made the responsibility of the machinery supplier, then

the need for and submission of calculations for dynamic design should be specified.

The acceptable limits on natural frequency should be specified, i.e. preferably more

than twice the highest operating speed or failing that less than half of the lowest speed,

also the limits on amplitude selected to suit the operating frequency. The extent to

which suppliers are expected to check dynamic response should also be specified toset some limits on their responsibility, e.g. main supporting beams, columns,

secondary beams attached thereto, bracing, foundations.

The brief for the soil testing program should include reference to the information

required for the dynamic design, if foundations for major rotating or vibrating

machinery are required. This information is:

Classification of the soil type.

The allowable static bearing pressure.

The dynamic shear modulus G (i.e. that which is applicable to short term

deformations).

Poissons ratio .

Design of pad foundations for machinery

The fundamental concepts of dynamic behaviour of foundations are the same as those

already discussed for structures generally. The machine and foundation can beregarded as a mass supported on a series of coil springs, and the system has a natural


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frequency and hence suffers resonance or response to forced vibrations. However,

there are obvious differences between a mass supported on a large volume of soil and

the case of a point load supported on a steel beam.

In this section the discussion on foundations is limited to a stiff pad or block

foundation, resting on or partially embedded in the ground. Friction piles and raftfoundations are not discussed, and if these are involved specialist advice should be

sought.

Attempts to assess the dynamic performance of foundations by use of a Finite Element

(FE) model are not recommended. This is because of the effects of boundary

conditions and the inability of an FE model to include radiation damping which is of

great significance. Even a near resonant situation can often be tolerated because the

high level of damping available can limit amplitudes to acceptable levels.

A spreadsheet has been developed to implement the recommended analysis method.

To obtain access to this spreadsheet, see the details at the end of this section.

GENERAL APPROACH TO DYNAMICS OF PAD FOUNDATIONS

Whilst a pad foundation can be thought of as a mass supported on a series of coil

springs, and the general expression ( )= kn still applies, there are severaldifferences in interpretation between this simplified model and the actual situation.A number of investigators have modelled this system mathematically as a body

supported on an elastic half-space. As a result of their investigations, workable

solutions have been found in terms of the previously explained parameters (equivalent

mass, equivalent spring constant and damping factor), thereby enabling natural

frequency, resonance and amplitude magnification to be determined.

A foundation supporting an unbalanced machine is potentially subjected to fluctuating

loads in the vertical, horizontal and rocking modes, as the horizontal forces will act in

a plane well above the foundation base. There could be a rocking (pitching) motion as

well, from loads on mountings fluctuating to and fro. These rocking modes can be

more significant than the translational modes, and must be handled properly. The text

book solutions also deal with a yawing motion (oscillations about a vertical axis), but

this is not considered necessary in the scope of the present discussion.

A geotechnical engineers advice on soil parameters is needed for all but approximate

assessments. These assessments are valuable to gain an understanding of the factors

involved, and should be made at the layout stage whilst there is time to influence the

general arrangements before they are frozen.

EQUIVALENT MASS

Although a substantial mass of soil would appear to be involved, D.D. Barkan, in his

fundamental reference work Dynamics of Bases and Foundations (McGraw-

Hill, 1962), has shown that in practical footings its equivalent effect never

exceeds 23% of the mass of the footing, because the amplitude of its vibrations is

rapidly dying down with increasing distance from the footing. Thus even to ignore it

altogether will only cause an error never exceeding 11% in the calculation of the

natural frequency.


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SPRING CONSTANT

Methods for establishing equivalent spring constants have been developed, with the

results differing for each of the modes of vibration. They depend upon the dimensions

of the foundation, the elastic properties of the soil, and the length of effective

embedment of the foundation in the soil. The procedure is spelled out in full in thebook Design of Structures and Foundations for Vibrating Machines, by Suresh Arya,

Michael ONeill and George Pincus (Gulf Publishing Company, Houston, 1979). It is

quite a complicated procedure to carry out, involving jumps between graphs, tables

and formulae. The spreadsheet developed for foundation design (mentioned above)

automates it.

SOIL SHEAR MODULUS

The elastic shear modulus of the soil, G , is the value applicable to dynamic

vibrations, so it is a short term figure applying to low values of shear strain, rather

than the secant modulus that would be derived from large deformation tests.It can be measured from field tests (either a surface oscillator test or a cross-hole test),

by laboratory tests (the resonant column test), or deduced from published equations

for various types of soil. The method adopted will depend on circumstances and the

relative importance of the application. This could be decided after undertaking a trial

analysis using published generalised data for various soil types, such as that in Table 3

below. Such data must be regarded as only approximate, but could indicate whether

dynamics are a factor.

Table 3 Typical soil properties for dynamic calculations

Soil typeShear modulus

(MPa) Poissons ratio

Soft clay 20 to 35

Stiff clay 70 to 140 0.35 to 0.45

Very stiff clay >150

Medium dense sand* 35 to 100

Dense sand* 70 to 140 0.4 to 0.5

Medium dense gravel* 100 to 170 0.3 to 0.4

Dense gravel* 140 to 270 0.4 to 0.5

*at shallow depths

For pads founded on sound rock, there is no theoretical dynamic issue. However, pad

size must still be adequate to ensure that no tension will develop due to loads which

could cause rocking motions.

SOIL POISSONS RATIO

If the Youngs modulus and shear modulus have been determined, Poissons ratio can

be calculated from the standard elastic relationship

12

=G

Typical values are given in Table 3 above.


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EMBEDMENT

Embedment has a significant effect on the damping by the soil, but care must be taken

before relying upon its benefits. The footing should be poured directly against the

sides of the excavation, which should be in good condition. Alternatively, backfill

should be well compacted and not of a type that will shrink away from the footing.

It is suggested that even when these conditions are met, the upper 300 mm of

embedment should be ignored.

DAMPING

Elastic analysis shows that even if the soil is purely elastic and has no inbuilt material

damping of any kind, considerable damping occurs. This is because the supporting

soil is of infinite extent, and so vibrational energy is constantly dissipated by being

carried away in elastic waves. This form of damping is referred to as geometric or

radiation damping. A method for deriving it is described in the book by Arya et. al.

(op.cit.), and is automated in the spreadsheet.

Of course the soil also has inbuilt material damping. This is expressed as the ratio of

the area of the hysteresis loop (the energy lost) to the energy input. For sands and

clays, an average value of 0.03 can usually be used for the material damping ratio.

The total damping ratio is the arithmetic sum of geometric damping ratio and the

material damping ratio. For the translational vibration modes the geometric damping

is the more important and the material damping can be disregarded. However, for the

rotational modes the material damping is more significant.

Note that if a hard stratum of soil or rock exists at a shallow depth below the footing,

the value of geometric damping should be reduced: the spreadsheet does not do thisautomatically, and at the present stage the effect will have to be fiddled by using an

appropriate negative value for the material damping.

RESONANCE

Total damping is much more significant for foundations than for steel structures, and

so its effect must be considered. Firstly, it lowers the resonant frequency below the

undamped natural frequency according to

221 Dff nr =

However, it also lowers the amplitude magnification factor to such an extent that,whilst the design should seek to avoid resonance, it is permissible to design to be

much closer to it. The general rule is to check that the magnification factor is less than

1.5, and that the operating frequency is outside the range 0.81.2 times the resonant

frequency (not the undamped natural frequencysee above).

There will, however, be times when it is impossible to avoid even this narrowed range.

Given the degree of abstraction in the idealised model adopted, and the inherent

inaccuracies in the parameter values used, it is considered prudent that if the design

lies within this range, the design should be performed as if full resonance had been

predicted.


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TRANSMISSIBILITY

Magnification of displacement amplitude has already been discussed. In a similar

way, the dynamic force transmitted by the footing to the soil differs from the applied

out-of-balance force.

2)2(1 DrMFF dt +=

where the term2)2(1 DrMd + is called the transmissibility factor.

Here is the magnification factor which has been previously defined byd

222 )2()1(

1

DrrMd

+= .

The soil must be checked for the increase in pressure due to this effect. Texts suggest

that the total pressure from static loads plus the transmitted dynamic loads (from all

the simultaneous modes of vibration) should not exceed 0.75 times the allowable

static bearing pressure given by the soil consultant. The writer believes this can be amisleading concept however, as the allowable pressures are often quoted for small

footings. For large footings, whilst the factor of safety against failure rises rapidly, it

is the total load rather than pressure that determines settlements, and the allowable

pressure recommendations should be made in this light.

RULES OF THUMB FOR TRIAL SIZES

Dynamic analysis can be performed only after a foundation size has been selected. It

is appropriate, therefore, to use the following rules to make a trial selection:

Keep in mind these general considerations:

A broad flat footing is usually cheaper and more effective than a heavy

deep one.

It is a mistake to believe that a heavier footing will automatically reduce the

displacement amplitude. Additional mass will lower the natural frequency,

probably moving it closer to the operating frequency. This increases the

magnification factor, and hence the amplitude (which is a function of soil

stiffness and footing area, not footing mass). Thus, whilst there is some

advantage in setting a footing deeper into the ground, it is the depth of

embedment which helps, not the increased mass.

Calculate footing area so that the bearing pressure under the static loads is about0.5 times the allowable bearing pressure.

Select width to be at least 1 to 1.5 times the height of any laterally applied forces,

measured from the footing base.

Arrange dimensions so that the centroid of the static loads and the centroid of the

dynamic vertical loads are both within 5% of the footing centroid (to avoid tilting

due to settlement).

A recognised rule of thumb is to calculate footing mass on basis of 23 times mass

of machine for centrifugal machines, and 35 times mass of machine for

reciprocating machines. However, this is frequently misused. Performance isimproved by increasing the area of the footing, which rapidly increases its


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rotational moment of inertia. Increasing the mass by increasing thickness can

move the footing closer to resonance. A good design does not have to be so heavy.

Select footing thickness to achieve required mass, but keep it at least one-fifth of

least horizontal dimension or one-tenth long horizontal dimension.

PERFORMING THE DYNAMIC ANALYSIS

The calculations described in this section can be performed most easily by using the

specially prepared Excel spreadsheet, which can be found on the

Infrastructure/APAC section of the KBRconnect intranet. It is spreadsheet SSN003,

in the structural engineering pages. The spreadsheet contains its own specific

instructions.

It is frequently found that the rocking modes due to fluctuating horizontal forces

having a considerable lever arm above the base of the footing are the most critical,

rather than vertical modes,

Spring mounted equipment

The principles and formulae set out in this document can be applied to the situation of

equipment mounted on springs or special pads. The natural frequency of the upper

system can be calculated from the mass of the vibrating machine and the spring

constant. (This is the sum of the value of each spring for the vertical mode, and can

easily be derived for a rocking mode in terms of restoring moment per radian ofrotation.) The natural frequency is deliberately selected to be very low so that the

frequency ratio r is high. For springs the damping is close to zero, and so the formula

for the transmissibility factor as given above simplifies to

21

1

r.

Thus the force transmitted can be reduced to 2% by selecting r to be 7. This reduced

force will still be transmitted to the supporting beam with a pulsation equal to the

operating speed, and it is still advisable to avoid resonance in those beams since

damping in steel beams is low.

The system is no longer a one degree of freedom system: it has two, or perhaps even

more, independent degrees of freedom, and in general it should be analysed

accordingly (see below). However, provided the natural frequency of the supporting

beam is well removed from the natural frequency of the spring system, the beam can

be considered separately. For this calculation the mass applicable is that directly

fastened to the beam, and the spring-supported mass is not included.

In summary, the natural frequency of the spring system is selected to be a small

fraction of the operating frequency. The spring constant is then calculated from

M

k

fn 2

1=


1 September 2008


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and the springs are chosen to give this spring constant. Then select the supporting

structure to have a natural frequency of (say) double the operating frequency. Finally,

check the amplitude of vibration of the supporting beam, i.e. the deflection caused by

the out-of-balance force times211 r .

Another reason why the natural frequency of the spring system should be low is thatduring start-up the system will pass through resonance. The out-of-balance force is

proportional to the square of the rotational velocity. Therefore if r is (say) 7, the out-

of-balance force at the time resonance occurs will be only 2% of its operating value.

It is of course necessary to check the deflection of the springs caused by the out-of-

balance force at the operating speed, and ensure that this is acceptable in terms of

clearances, connection of electrical cables, air lines, etc.

Vibrating screens are usually excited by a pair of coupled but counter-revolving

flywheels with out-of-balance weights that are opposite each other in the horizontal

position. Thus they cancel in the horizontal direction while adding in the vertical.

Nevertheless some horizontal and longitudinal bracing should be provided, and naturalfrequencies of members should be kept away from the operating frequency.

The above approach reduces the force transmitted, but it may be necessary to check

that the vibration amplitudes of the machine itself will not be detrimental to it. For

example, fuel lines from a separate supply point may need to be flexible.

Systems with two degrees of freedom

The behaviour of systems with two degrees of freedom is not able to be described

through explicit algebraic formulae. Such systems need to be modelled numerically,

with numeric results being extracted from the model.

To aid in this process, an Excel spreadsheet has been developed. This models a

generalised two degree of freedom dynamic system under loadings that can include

both applied forces and foundation motion, and can be used for:

natural frequencies and mode shapes;

harmonic response to harmonic loadings;

time-history response to more arbitrary loadings.

The spreadsheet is available on the Infrastructure/APAC section of the KBRconnect

intranet. It is spreadsheet SSN002, in the structural engineering pages. The

spreadsheet contains its own specific instructions.

To get an approximate estimate of the lowest natural frequency for a 2-dof system,

you can use Dunkerley's formula (also sometimes known as the Southwell-Dunkerley

formula, or the "inverted squares" formula). This formula is

22

2

2

1

2

1111

nffff

+++= K


1 September 2008


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where is defined as the frequency of a modified form of the structure in which all

the springs remain present, but all masses except M

if

i are set to zero

Points to note with this formula:

It applies only to lumped mass systems. Thus the structure consists of a heap of

lumped masses, interconnected by a tangle of linear springs.

It always gives an underestimate of the true structural frequency. (This is not to

say it is necessarily inaccurate, just to alert you to the direction any error will take.)

You need to be careful in how you calculate your "component frequencies", and

adhere to the above definition.

Beats with multiple equipment

It should be noted that if a foundation or structure supports two pieces of equipment

operating at different speeds, then beating will occur. The frequency of the beats is

equal to the difference between the operating frequencies of the two sources, and the

peak out-of-balance force is the sum of the two effects (but these will probably be

separated by some substantial distance).

This is an added requirement to be checked if the natural frequency of the supporting

structure or foundation has been made lower than the higher of the two operating

frequencies.

This situation can arise with a large motor driving another piece of equipment through

a gearbox, or if two or more items of variable speed equipment are used.


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Appendix A Use of general-purpose analysis software

Many of the modern structural analysis programs include at least a limited ability to

perform dynamic analysis. This, of course, in no way obviates the need for the

designer to understand the general principles described in this document. Suchunderstanding will help the designer to:

create better structural models more quickly;

check the sensibility of any results produced from a model;

move from the analysis process to the design process.

LIMITATIONS OF SUCH PROGRAMS

These programs, as opposed to highly specialised dynamics programs such as Strand7,

do not usually offer a full repertoire of options.

Types of analysis: nearly all the programs offer natural frequency and mode shape

extraction. However, not all will follow that up with either time-domain analysis

(time history) and/or frequency-domain analysis (spectral analysis). Furthermore,

most will not be able to include any non-linear effects, material or geometric, in

their dynamic analysis.

Harmonic analysis: only some programs offer harmonic analysis, where the

program directly determines the harmonic response to some form of harmonic

stimulus. If you require this but it is not available, then you will have to run a

time-history analysis, for a simulated time that is long enough for the transients to

have died away.

Damping: most programs allow only simple viscous damping, with the damping

for the structure as a whole described by a single damping ratio. For different

damping types, and for the ability to concentrate the damping in discrete locations,

specialised programs (and probably specialised advice) are required.

CREATING AND ANALYSING YOUR DYNAMIC MODEL

Advice on the creation, running and verification of models for the dynamic behaviour

of structures is provided in a companion document in the KBR Technical Portfolio.

See "Infrastructure course 3.4: Advanced finite element analysis with the Strand7

software".

This companion document, despite its title, has a section that contains a lot of advice

that is applicable regardless of the analysis software being used.


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Appendix B Manual checking of dynamic response

NATURAL FREQUENCY

As stated in Section 4, for a single degree of freedom system, which can berepresented as a concentrated mass attached to a mass-less elastic support,

M

kfnn == 2 or

M

kfn

2

1=

It is surprisingly easy to represent a number of real situations by an equivalent simple

model (a weight on a spring) with sufficient accuracy. Some examples:

A machine mounted on springs allowing motion in one direction. The above

equation applies directly, where is the mass of the machine and its platform

(spring mass is negligible), and the total stiffness of the springs.k

A machine mounted centrally on a pair of parallel beams. As the beams deflect

elastically in response to a vertical force, they can be treated as a spring where is

the reciprocal of the deflection caused (to the beam pair) by a unit load. Thus

k

3

48

lk

=

The mass of the beams is distributed, but it can be shown (see below) that it can be

adequately treated as a concentrated mass, having half the value of the distributed

mass, and added to the machine mass.

A machine supported on a beam grillage. In a similar manner to above, one can

calculate the deflection caused by a unit force at the machine position (not

necessarily central) and apply an equivalent mass.

Tower like structures subject to wind loading. These can still be addressed by

calculating the horizontal deflection caused by a unit force and considering

equivalent mass. An approximate calculation can show whether more precise

analysis is warranted or not.

The methodology for calculating equivalent mass is explained in the following

section.

EQUIVALENT SYSTEMS

To consider the response of structures to vibration, it is first necessary to convert them

to a mathematical model which will have similar characteristics but will be amenable

to analysis. A simple example is illustrated below in Figure B1.


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Figure B1

Example of an equivalent mathematical model

The system to be examined is a simply supported beam of uniform mass per unit

length, carrying at its centre a massive machine subjected to out-of-balance forces.

The equivalent model suitable for analysis is a concentrated mass hanging on a

massless spring.

In this system, the out-of-balance force acts at the concentrated mass, and the response

is also required at this point. It is first necessary to determine the beams equivalent

mass, modelled as all concentrated at one point. Since the central part of the beam

will oscillate with the same amplitude as the machine whilst the other parts of thebeam will oscillate with a lesser amplitude, it is obvious that the beams equivalent

mass will be less than its total mass.

The technique for quantifying this equivalent mass is to equate the kinetic energies of

the real and equivalent systems. In the real system the concentrated mass of the

machine oscillates with amplitude equal to the maximum deflection, but each element

of the distributed load oscillates with an amplitude proportional to the deflected shape.

It is sufficiently accurate, but not strictly correct, to assume that this shape is the same

as that caused by a concentrated static force of magnitude in the middle of thebeam. (It would also be valid to assume that the deflected shape was the same as that

caused by a uniform distributed load: the results are surprisingly insensitive to thechoice of deflected shape provided the shape is reasonable.) Under this assumption

the deflected shape is defined by the following equation (with the origin located at the

centre of the beam).

)46(48

332 lxlxy

=

=

48

3

max

ly

The peak velocity at any pointx along the beam will be CyVx = where the constant

Cis defined by .maxmax CyV =


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By substitution we get3

332

max

46

l

lxlxVVx

=

Kinetic energy of the beam = dxVm

l

xu

2/

0

2

22

With substitution and integration this becomes2

max280

68lVmu

and by definition this must equal2

max2

1Vequiv

so lmuequiv35

17=

The total concentrated effective mass e is then the mass of the machine ( ) plus

0.486 times the total uniformly distributed mass (m

ul).

This type of calculation can be performed for different beam support conditions. It

can be shown that for a fully fixed beam the equivalent mass is 0.371 times the

uniformly distributed mass. For a partially fixed beam a value between these two

could be selected, say 0.45 for a nominally simply supported beam and 0.40 for a

nominally fixed beam such as an encastr concrete beam. The equivalent kinetic

energy approach can also be used for masses concentrated at points other than the

centre. Table B1 lists the results of some such calculations. It also gives the factor by

which the total dynamic loads should be multiplied to make them equivalent to a

single centrally applied force.

As described above, for a point load on a simply supported beam (or pair of parallel

beams) the equivalent spring constant is equal to the force required to produce unit

deflection. Therefore, since

=

48

3

max

ly we get

3

48

lyke

=

=

This calculation can also be performed for other cases, or extracted in numerical terms

from the deflection calculations for the supporting structure. For example, in the case

of a beam supported on other beams, the relevant deflection is the sum of the

deflection within the inner beam and the deflection of the supporting beam that

occurs when a load is applied to the point under consideration. The spring constant is

then the load that is required to cause a total deflection of one unit.


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Table B1 Equivalent mass and force for various configurations

Equivalent concentrated mass

Case

Equivalent

concentrated

dynamic force

Lumped mass

at load points

Uniformly

distributed massSpring

constant

Simply supported beam,uniformly distributed massmu per unit length, dynamicforce F applied at midpoint,lumped mass M at midpoint.

1.00 F 1.00 M 0.49 mul 48EI/l

3

Simply supported beam,uniformly distributed mass

mu per unit length, dynamicforce F/2 applied at third-points, lumped mass M/2 at

third-quarter points.

0.87 F 0.76 M 0.52 mul 49.1EI/l3

Simply supported beam,

uniformly distributed mass

mu per unit length, uniformlydistributed dynamic force fuper unit length.

0.64 ful 0.50 mul 49.2EI/l3

Fixed ended beam,uniformly distributed mass

mu per unit length, dynamicforce F applied at midpoint,

lumped mass M at midpoint.

1.00 F 1.00 M 0.37 mul 192EI/l3

Fixed ended beam,

uniformly distributed massmu per unit length, uniformly

distributed dynamic force fuper unit length.

0.53 ful 0.41 mul 203.5EI/l3


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Appendix C

Dynamic analysis and design for mining and other plant

DISTINGUISHING CHARACTERISTICS

Whilst the approach to dynamic design of structures (including foundations)

supporting machinery in various types of plant is common for each of a few generic

machine types, there is a big difference in the quantum of the problem when

addressing some types of plant in the mining industry.

In most types of plant and in bridges or tall towers the dynamic loads are an unwanted

or parasitic side effect and efforts are made to eliminate or reduce the loads. However,

crushers and screens in mining plant require dynamic loads for their fundamental

operationthe larger the dynamic load, the more effective the plant. Handling these

dynamic loads becomes the principal structural consideration, not a mere side effect.

The main categories of equipment, classed in accordance with the nature of the

dynamic loads are:

rotating machinery (rotary crushers, motors, turbines, pumps, rotary engines)

reciprocating machinery (diesel engines)

combined action equipment (screens).

BRIEFING CRITERIA

LoadingFor most types of rotating machines manufacturers strive to produce machinery as

balanced as possible, i.e. so that the axis of rotationthe centre of the bearingsand

the axis of rotational inertia, or centre of mass, coincide. The required briefing data

is the mass of the rotating part and its possible eccentricity. This is the initial

tolerance plus an allowance for later wear of the bearings or shafts. This information

should be sought from the vendor. Some guidelines are provided in Figure 1.

Most rotary machines have a horizontal axis, so the dynamic force produces a vertical

component pulsating sinusoidally, and a horizontal component (at right angles to the

axis of rotation). As the axis is above the level of the supports (or base of foundation)

this horizontal component creates a rocking motion. This can be the critical case forfoundation design.

The situation with reciprocating machinery is more complex, and detailed vendor

supplied data must be sought listing out of balance forces and their directions. At the

very least, the designer must be aware that the motion of the reciprocator is not

harmonic, and so there will be dynamic force applied at frequencies higher than the

fundamental operating frequency. These forces must be considered.

Screens in mining plant are vibrated as the result of an eccentric flywheel. They are

spring mounted, so the forces transmitted to the supporting structure, both vertical and

longitudinal, are not the same as the primary effect of the flywheel eccentricity

which is acting on the equipment above the springs. These forces could be calculated


1 September 2008


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using the principles set out in the main document, but are usually supplied by the

manufacturer.

It is necessary to establish the normal or maximum operating speed and also the

possible range of speeds, particularly if an under-tuned solution is adopted.

The static mass of the machine as a whole is also needed for the natural frequency

assessment.

ACCEPTABILITY CRITERIA

As stated in the reference document, the acceptability criteria are a composite of

amplitude and acceleration, but sometimes given as a velocity. The acceptable level is

dependant on where, in the plant as a whole, the vibrations are being felt.

An international standard, ISO 2631-1, addresses the effect on persons as related to

the time of exposure. A much lower level is acceptable at a continuously staffed

operator position than the level acceptable for a point which may be visited only forshort periods.

The acceptability level for an item of equipment can be obtained from the

manufacturers but this criterion is sometimes misunderstood or wrongly quoted. For

example, if a motor exhibited vibration amplitudes of 0.1 mm, due to its own

operation, this would be seen as a sign of deteriorating bearings or worn shaft and

deemed unacceptable. But a motor mounted on a platform which is vibrating in

response to a crusher or screen could be subject to this vibration amplitude without

any damage being caused. To demonstrate the point; if amplitude is 0.1 mm at a

frequency of 900 RPM the peak acceleration is , i.e. 0.9 m/s2

a 2, i.e. the load felt by

the bearing would be 10% higher than the static weight of the rotating parts onlyaninsignificant increase. Refer to the explanation in the main reference document

Section 5.3.

MITIGATION

The methods of analysis of response to vibration sources are addressed in the main

document, as are suggestions for mitigation.

Mining plant

In mining industry plant the main issues are:

Vertical vibration of beams directly supporting machines; these vibrations can also

be transmitted through connections to columns, other beams, platforms, etc. The

objective is to not only ensure that resonance does not occur, but also to limit the

dynamic magnification.

The available methods are to increase the support stiffness to about double that

corresponding to resonance. Stiffness is most effectively increased by reducing the

span of beams and/or re-configuring supports to avoid a grillage situation. To

reduce the beam stiffness and hence magnification is not usually practical, but for

some types of machines the transmissibility of force can be reduced by the use of

resilient mounts. This can be done for large screens as used in coal plants by

double spring mounting as follows. A screen and its exciting eccentric flywheel


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and motor are always spring mounted. It is possible to spring mount the screen on

a frame and then provide further spring mounts between that frame and the main

structure. See Section 8 of the main reference document.

Vibration of secondary elements: platforms and handrails often vibrate in

resonance (sometimes in higher modes); this is often due to flexible fixing detailsso ensure stanchions are as rigidly fixed as possible.

Horizontal vibration of multilevel frames (particularly in screen buildings).

Usually caused by incomplete bracing systems. However, do not ignore the

possibility of providing an undertuned solution if additional stiffness is

impractical.

Vibration of foundations for crushers: the main issue will be the rocking motions

due to horizontal loads at a significant height above the underside of the

foundation pad (which can be aggravated if the pulsating vertical dynamic load is

off centre). These are best mitigated by increasing the moment of area of the

footing/soil interface. Increasing the length by 26% (in the relevant direction) willdouble the stiffness in this mode.

TURBINE FOUNDATIONS

Gas turbines are very high speed machines. They are in consequence manufactured to

high standards and well balanced, but with increasing wear significant forces can

develop due to the squared effect of speed. It is usually impractical to provide a

foundation with a natural frequency higher than the turbine operating speed, (except

perhaps if founding on strong rock) so an undertuned solution is used which can

reduce the amplitude below the static value. This is done by having a deep foundation

block, i.e. large mass.

PRINTING PRESSES

Colour printing presses are very sensitive. Full colour is achieved by superimposing

three separate layers of the primary colours. Vibrations cause lack of registration

with poor definition and edges showing the primary colours. The out of balance

forces are caused by large, rapidly rotating reels of paper which develop eccentricity

relative to the shafts on which they relate (they oval), and the machines are relatively

high so rocking motions can occur. It may be necessary to add piles under the

foundation or bored piers if rock is at reasonable depth.

THE MYTH OF ISOLATION OF FOUNDATIONS

Engineers sometimes seek to isolate the foundation of a machine from an adjoining

foundationeither another machine or an adjacent structure. This is based on a

misconception. If a machines foundation pad is vibrating significantly the movement

is not within the pad itself but in the ground below it. The zone of influence is

significant and a purely nominal separation (e.g. a joint) will not prevent significant

movements being caused in the adjacent foundation. However, if the pads were

monolithic, it is likely that the overall stiffness would be greatly increased and soil

movement reduced. This option should therefore be considered.


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Appendix D Dynamic design issues for footbridges

The design of footbridges is the subject of its own document in the KBR Technical

Portfolio. Material that was previously in this Appendix has been merged into that

document.

See "Infrastructure course 2.3: Elemental Reference Design for footbridges".


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Appendix E Wind induced dynamics

DISTINGUISHING CHARACTERISTICS

The distinguishing issue in the analysis of structures for possible dynamic sensitivityto wind is the complexity of the loading and the range of wind speeds at which

exciting forces can occur.

For tall towerlike structures there are two basic, separately caused forms of

excitationalong wind and across wind.

The former is caused simply by the fluctuation of gust velocities. A simplified

explanation is as follows. Suppose the wind was blowing at a fixed speed, causing

your structure to deflect; then it stops blowing or blows at a slower speed; the

structure will spring back, wholly or part-way, and oscillate at its natural frequency;

while still doing so, a wind gust of increased speed hits it and the process goes on.

Cross wind excitation is quite different. All structures create downstream eddies. In

the case of some structures, particularly of circular shape in plan such as chimneys,

eddies break off the structure alternating from side to side, setting up lateral exciting

forces. If these coincide with, or are close to, the lateral natural frequency a resonant

response occurs. The lateral oscillations of the structure reinforce the eddy shedding

pattern. However, in practice both effects can combine, resulting in elliptical (and

even figure-eight) motions.

Free edged roofs of buildings like sports stadia and similar structures can be subject to

flutter due to eddy effects. Expert advice and/or wind tunnel testing of aerodynamic

aeroelastic models is needed.

BRIEFING AND CRITERIA

There are no issues to be provided by the client, other than possibly the amplitude of

vibration that lights or delicate equipment (such as communications antennae or

dishes) can tolerate. Usually the client cannot provide such criteria as part of the brief

and the designer has to research it from equipment manufacturers and interpret it.

For tower-like structures (including multistorey buildings) the structural issue is that

along-wind oscillations magnify the base moments and shear forces. The normal

structural design rules then apply to the magnified moments. However, this effect is

not very great for the reason explained in Section E3, Analysis.

Cross-wind excitation is often more of a nuisance than a cause of structural distress.

ANALYSIS

The principles of dynamic analysis are as set out in the main document, and some of

its practicalities are discussed in the companion document "Infrastructure course 3.4:

Advanced finite element analysis with the Strand7 software".

AS 1170, Part 2 Section 6 deals with the dynamic loading effects in wind. The

methods set out include equations whose derivation is far from obvious. It is essential


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to read section C6 of the Commentary to AS 1170 Part 2 in parallel with the code

methodology.

The code states, in effect, that the overall load and bending moments will not be

increased above the value obtained by the routine quasi static methods of Sections 1

to 5 of the code if the first mode natural frequency is greater then 1 Hz. It is not clearwhether this statement applies to cross wind effects as well as along wind effects.

Whilst the load and stress effects may not be significant, the statement should not be

taken to mean that dynamic deflections should not be considered if a tower like

structure is supporting sensitive equipment.

If in doubt, seek advice from the Monash University Mechanical Engineering

Department, which specialises in wind dynamics and will carry out model testing if

required.

MITIGATION

For tower-like structures, if the assessments indicate unsatisfactory performance, theobvious solution is to increase the stiffness if practicable. However, this is not always

possible and another option is to increase damping. This is effective for wind induced

vibrations and quite modest increases in damping often suffice. Very tall buildings,

observation towers and the like have sometimes been fitted with large mass dampers.

Cross wind excitation of steel chimneys is frequently dealt with by the use of spiral

strakes around the chimney which prevent the eddy pattern from forming.

Cross wind excitation of tubular diagonal braces on a container crane were damped

out by placing a substantial chain inside the tubethe friction between links and

between the chain and the tube provided the necessary damping.


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Appendix F

Dynamic loads imposed by pneumatically-tyred vehicles

INTRODUCTION

This section is intended to provide guidance to engineers in assessing the likely

dynamic loads imposed on supporting structures by the passage of vehicles with

pneumatic tyres. Simplified formulae are presented for two situations:

a single discontinuity in the road surface;

continuous sinusoidal corrugations in the road surface.

Whilst it is not directly applicable to the subject of short-term dynamic forces, it is

worth pointing out that significant vehicle loadings occur when the vehicle brakes.

Obviously, horizontal forces are generated, via friction between the tyres and the road

surface. Less obviously, the front wheels impose additional vertical forces since the

weight of the vehicle tends to pitch forward. Less obviously again, the effective

stiffness of the front suspension might be increased by this increased force, since

many suspension systems exhibit deliberate non-linear behaviour.

SINGLE DISCONTINUITY

Basic model

The road surface is assumed to consist of two semi-infinite level portions whose

vertical separation is H. The vehicle is travelling along the road at a steady speed

such that a wheel will traverse the discontinuity in time T.

The vehicle suspension system consists of an axle of total mass supported on a

number of tyres whose total stiffness is . The axle supports a portion

m

tk of the

total vehicle mass and load, via a number of springs of total stiffness . In some

instances there is also a hydraulic arrangement between the axle and the body whose

effect is to impose a constant force : the purpose of this is to provide a smoother

ride by allowing softer springs to be used. Damping ratios of and are

applicable to the springs and tyres respectively.

sk

sP

sD tD


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Diagrammatically we have

and

Analysis

The above system has two degrees of freedom. To reduce it to a single degree offreedom we assume that over the period of interest the vehicle body does not move

vertically. This assumption is usually acceptable, because generally is

considerably greater than , and is considerably greater than , so the axle

responds to a disturbance much more rapidly than does the vehicle body.

m tk sk

The damped natural frequency (in radian/sec) of the axle under these assumptions is

[ ] 22 )(1)(1 tststsd DDDDm

kk+=+

+=

where is the undamped natural frequency.


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If damping is ignored, it can be shown that the effect of the disturbance is to

superimpose on the static force between the tyres and the road, a dynamic force that

oscillates between the two values

( )tsts

t kk

kk

Hk

+

where is a function ofT, and is given by( )

T

T

d

d

2sin2= for


34/36


discontinuity whose effects will be additive to those of the first (having due regard

for the sign ofH in the formulae).

The axle traverses the discontinuity in time T . The estimate for the value ofT is

crucial to the entire exercise, and it will be under-estimated if obtained merely by

dividing the horizontal extent of the discontinuity by the horizontal speed of thevehicle. The vehicle has pneumatic tyres, which will tend to flow over the

discontinuity (the caterpillar effect). Approximate allowance for this can be

made by adding half the length of the tyre footprint to the horizontal extent of the

discontinuity before dividing by the speed.

Damping is ignored except insofar as it affects the value of d. This has the

theoretical effect that the independent vibrations set up at the start and the end of

the discontinuity, do not die away, but continue forever. In practice they will die

away within a few cycles. The greater the value of Td the more the above

simplified approach will tend to over-estimate the dynamic forces. For small

values of Td the error is insignificant, and this is the case that is usually of

interest because it gives rise to the larger dynamic forces. (As an aside, for very

large values of Td the simplified approach gives answers that are too large by a

factor of two. The physical reason for this is that the vibrations set up at the start

of the discontinuity will actually have died away completely when the end of the

discontinuity is reached.)

CONTINUOUS CORRUGATIONS

Basic model

The vehicle, its axle and its tyres are idealised in the same manner as before. The road

surface elevation is assumed to be sinusoidal, with amplitude H and a wavelength

such that the angular frequency of the disturbance imposed on the vehicle is

(radian/sec).Analysis

As in the case of the single discontinuity, the problem is reduced to a single degree of

freedom by assuming that the vehicle body does not move vertically. Under this

assumption, and with much algebraic manipulation, the following formulae can be

derived.

Letm

kk ts += (undamped natural frequency of axle)

=r (frequency ratio)

(total damping ratio)ts DDD +=


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Then:

The amplitude of vibration of the axle is given by

( )

( ) ( )222

2

2

21

2

Drr

rD

kk

k

H

t

ts

t

+

+

+

The dynamic force applied to the road surface has an amplitude given by

( )

( ) ( )222

2

2

2

2

21

2121

Drr

k

kkrDrD

kk

kr

Hkt

tsts

ts

t

t

+

++

+

+

which, if the spring stiffness is very much less than the tyre stiffness, simplifies to

( )

( ) ( )222

2

2

2

21

212

1

Drr

rDr

D

rHk

ts

t

+

+

+

The dynamic force applied by the suspension to the vehicle body has an amplitude

given by

( )

( ) ( )222

2

2

2

21

212

Drr

k

kkrDrD

kk

k

Hks

tsst

ts

t

s

+

++

+

+

Discussion

Most of the assumptions made for the analysis of a single discontinuity apply in the

analysis of continuous corrugations. However, damping has been included.

A feature of corrugation-induced vibrations is that they are usually imposed at a

frequency close to the natural frequency of the suspension system. This is because the

corrugations are themselves caused by resonance in the suspensions of the larger

vehicles which travel the road.

The corrugation profile will probably not be sinusoidal. The actual profile could be

measured, then analysed into its Fourier components. This would almost certainly

show that the fundamental component was the predominant one, a predominance that

is enhanced by the fact that this fundamental component will probably be exciting the

system at close to its natural frequency.

The caterpillar effect of the tyres footprint will serve to lessen the effective value of

the corrugation amplitude. The likely magnitude of this can be estimated by drawing

a full corrugation to scale, then superimposing a type footprint on that drawing.

(When doing this, remember that the actual profile of the corrugations will not besinusoidal.)


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TYRE STIFFNESS

In cases where the tyre stiffness is not directly available, an estimate can be made

from knowledge of the tyres radius and operating pressure. Consider a single tyre of

radius R and internal air pressure . The tyre has been flattened by an amountp ,

acquiring in the process a footprint length L . Assume, in the absence of any betterinformation, that the footprint width is 2L .

The footprint area is , and so the load being supported is .25.0 L

25.0 pL

The values of L and are related through the geometry of a circle by

( ) 42 2LR = , which for small deflections simplifies to ( )RL 82= .Hence, the load being supported is equal to pR4 , and so the stiffness of the single

tyre is equal to .pR4

Document revision history

Version Description Release

date

Released by

Initial paper-based document produced. 1981 Tony Dawson

Various bits and pieces added in

loose-leaf form over the years.

Converted to electronic format. ~1998 Rob Niall

More bits & pieces added over the years.

D Becomes part of KBR Technical

Portfolio.

2006 Tony Dawson

Rob Niall

E Cosmetic improvements. Dunkerley's

method described.

Sep 2008 Rob Niall

3.1_General Principles of Dynamic Design

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