The Relationship Between Finite ElementAnalysis and Modal AnalysisNeville F. Rieger, Stress Technology Incorporated, Rochester, New York

The properties and techniques of Modal Analysis and of FiniteElement Analysis are identified, together with the present advantagesand shortcomings of both methods. The interrelationship betweenthese techniques is described, and the contributions of modal analysisto efficient finite element analysis are reviewed. It is noted that theterm modal analysis is used to describe, a) a test procedure forobtaining structural data, b) an analytical procedure for efficientsolution of structural dynamics problems, and c) the same solutionprocedure for rotordynamics analysis. Four case histories areincluded which describe the joint use of modal analysis and finiteelement analysis to diagnose and confirm the results obtained inselected practical problems of structural analysis.

During the past ten years, the rapid development of specializedtest equipment and efficient numerical methods for modal calculationof structures has revolutionized vibration analysis. The purpose ofthis article is to discuss the interrelationship between the variousmodal methods and finite element analysis. Certain limitations, whichaffect the results attained by each procedure, are identified, and therestrictions, which these limitations impose on modal analysis and itsapplication are discussed. The extent to which modal analysis andfinite element analysis can be coordinated into an effective diagnosticprocedure for vibration analysis is demonstrated by several casehistories. Some definitions to clarify the terminology of this subjectare discussed in the following section.

Types of Modal AnalysisAs used in the general literature of vibration analysis, modal

analysis may refer to either:a) A formalized test procedure for identifying the dynamical

properties of structures.b) A mathematical procedure for increasing the efficiency of

structural dynamics calculations.c) A technique for rotor balancing.

Modal testing is a formalized method for identification ofnatural frequencies and mode shapes of structures. It utilizesdedicated modal test equipment, and requires a formalized procedurefor disturbing, e.g., rapping, the structure into motion, and thenrecording the distribution of the resulting motions throughout thestructure. The end results of a modal test are the various naturalfrequencies, mode shapes, and impedance data of the structure. Thesedata are identified from the digitized input signals using efficientcurve-fitting routines. The results are subsequently displayed asimpedance plots and mode shapes (possibly animated).

Mathematical modal analysis is an analytical procedure usedto uncouple the structural equations of motion by use of a knowntransformation, as outlined in the following section. Details are givenin standard textbooks such as Hurty and Rubinstein.1 The resultinganalysis is then readily achieved by solution of the uncoupledequations. The modal response of the structure is then found througha reverse transformation, followed by a summing of the respectivemodal responses, in accordance with their degree of participation inthe structural motion.

Modal balancing is a rotor balancing procedure in which therespective modes of a rotor system are first isolated and thencorrected for residual unbalance in sequence. The balance correctionsused for one mode are carefully arranged in accordance with modal

principles so as not to re-introduce the other modes of the rotorsystem.

The above modal procedures have the following items incommon:a) Identification of structural modes and frequencies for further

analysis.b) The theory of each procedure is based on an analytical

technique known as modal analysis, which uncouples theequations of motion to make possible their efficient solution.

c) The orthogonal properties of structural dynamics matrices areutilized either directly in the analysis, or implicitly in (hepractical test procedure.

NotationA1 = First Axial Mode[C] = Damping Matrix

[C∗] = Modal Damping Matrixc = Modal Damping Coefficient

[D] = Dynamical Matrix, [D] = [K] -ω2 [M] + iω [C]E = Young’s Modulus

{F} = Force Vector[I] = Unity Matrix

[K] = Stiffness Matrix[K*] = Modal Stiffness Matrix

k = Modal Stiffness[M] = Mass Matrix

[M*] = Modal Mass Matrixm = Modal MassN = Normal ForceP = Applied Force

R1 = First Torsional Mode{S} = State Vector of DisplacementsT1 = First Tangential ModeY = Displacementb = Distancee = 2.71828…h = Distance

i = 1−

r = Applied Force Location in Modal Tests = Displacement Location in Modal Testt = Timeu = Displacementu& = Velocityu&& = Accelerationa = Real Part of λ (Damping Term)γ = Phase Angleδ = Modal Forceη = Modal Displacementθ = Angular Displacementλ = Complex Eigenvalue (a + i ω)µ = Coefficient of Frictionω = Circular Frequency - Rad/Sec – Imaginary Part of λ

[ϕ] = Modal Displacement Matrix

Practical Modal Analysis ProceduresPractical modal analysis, or modal testing, involves the

following operations:a) The structural response amplitude is acquired in digital format

throughout a prescribed frequency domain, at a givendisplacement point r for excitation applied at a point s.

b) The modal mini-computer automatically develops and storesthis digitized frequency response data in a designated memoryfor subsequent processing.

c) Curve-fit routines are applied to the frequency response data toidentify the natural frequencies within the given frequencyrange. The corresponding mode shapes are extracted from thedigitized amplitude data at the natural frequencies.

d) The mode shapes may be animated in terms of the simplifiedstructural model, corresponding to those locations at which theresponse has been determined.

e) The modal damping is estimated from the magnitude of theresponse at each natural frequency. This is often the mostapproximate structural parameter obtained by modal testing.

f) Modal matrix data are identified for the structure. Output isdeveloped for mass, stiffness, and damping matrices suitablefor further computations, based on the structural modalproperties. These data are printed out for subsequent use.

g) Some software packages permit modifications to be made to thematrix data, to evaluate the influence of possible changes on thenatural frequencies and mode shapes. These packages can be runon certain commercially available modal analyzers.

Finite Element AnalysisFinite element analysis is a computerized procedure for the

analysis of structures and other continua. Rapid engineering analysescan be performed because the structure is represented (modeled)using the known properties of standard geometric shapes, i.e., finiteelements. Efficient, large, general-purpose computer codes now existwith appropriate matrix assembler routines and equation solvers forcalculation of the following structural properties:a) Static displacement and static stress.b) Natural frequencies and mode shapes.c) Forced harmonic response amplitude and dynamic stressd) Transient dynamic response and transient stress.e) Random forced response, random dynamic stress.Finite element analysis used in this manner provides the dynamicproperties of structures, including mode shapes and correspondingnatural frequencies.

General purpose finite element codes such as NASTRAN,ANSYS, SAP, ADINA, etc., are programmed to develop and solvethe matrix equation of motion for the structure, viz:

?)}t(?cos{F{u}[K]}u{[C]}u{[M] +=++ &&& (1)

where the terms used are defined in the Notation section above.The model details are entered by the analyst in a standardized

format. The computer then assembles the matrix equation of thestructure. The first part of the solution to a given problem is to solvethe matrix equation:

0{u}[K]}u{[M] =+&& (2)

for the free vibrations of the structure. The solution to Equation (2)gives the natural frequencies (eigenvalues) and the undamped modeshapes (eigenvectors). These parameters are the basic dynamicalproperties of the structure, and they are needed for use in subsequentanalysis for dynamic displacements and stresses. For harmonicmotions, {ü} = -ω2{u}. Substituting gives the matrix eigenvalueexpression:

{u}[I]{u}[M][K]1

=−

(3)

where [I] is the unity matrix. Standard procedures (Jacobi, QR, Sub-Space iteration) exist for extracting the eigenvalues: see Bathe andWilson.2 The corresponding matrix of eigenvectors, [ϕ] are obtainedby back substitution.

Typically the matrix equations of motion for the structurecontain off-diagonal terms. The matrix equation may be de-coupledby introducing the transformation:

}{?][{u} ϕ= (4)

and writing the following expressions:

[ϕ]T [M] [ϕ] = [M*] (5)

ϕ]T [K] [ϕ] = [K*] (6)

[ϕ]T [C] [ϕ] = [C*] (7)

which contain only diagonal terms.The damping matrix may be uncoupled on the condition that the

damping terms are proportional to either the corresponding stiffnessmatrix terms and/or the corresponding mass matrix terms. Theuncoupled matrix expressions are:

?)}t(?cos{F][}{?][K}?{][C}?{][M T*** +=++ ϕ&&& (8)

Each equation in this expression then has the form:

)?t(?cosd?k?c?m rrrrrrrr +=++ &&& (9)

To which the solution is:

2r

22rr

rrr

? )(c)?m(k

)?tcos(?dn

+−

+= (10)

The dynamic displacements, u at frequency ω may then be obtainedfrom the transformation {u} = [ϕ] {η}.

Modal testing complements finite element analysis, both byobtaining the natural frequencies and mode shapes directly bymeasurement, and by providing the matrix data [M] [C] and [K] forresponse analysis. Modal test results can be used to confirm the modeshapes and natural frequencies predicted by finite element analysisand the test data for the natural modes may be used to obtain themodal mass matrix and the modal stiffness matrix for the structure.

To improve the efficiency of finite element calculations, a so-called ‘eigenvalue economizer’ routine (Guyan reduction) is oftenused. These routines reduce the size of the dynamical matrix:

[C]?i[M]?[K][D] 2 +−= (11)

by ‘condensing’ it around selected ‘master’ nodes having ‘master’degrees of freedom (DOF), for which the modal displacements areneeded. These master DOFs, are critical locations whosedisplacements participate strongly in the modes of interest for a givencase. For example, the tip of a cantilever beam would be a typicalmaster DOF location for most modes of a beam system. Guidance inthe selection of such nodes for analysis can be obtained from theresults of modal testing, where the response of the structure todynamic forcing at such locations has been determined by test.

Advantages of Modal AnalysisThe mode shapes and natural frequencies of a structure are its

basic dynamic properties. Modal testing is used to rapidly identifythese modes and their natural frequencies, and to provide thestructural matrices, which govern the modes and natural frequencies.Thus the basic structural dynamic data, when obtained accuratelyfrom a valid test also provides a true identification of the structuralproperties for the modes of interest. These derived matrices are basedon the measured participation of the mass, stiffness and dampingproperties in the modes of interest, for the actual boundaryconditions, which the structure is experiencing. These data can thenbe used directly in a finite element model for the structure orcomponent, for subsequent problem solving, or re-designing theequipment for more optimum dynamic response.

Modern modal analysis test equipment has been developed toprovide the maximum convenience in testing and data reduction, andto provide the above-mentioned dynamic properties of the structure.All modal analyzers contain dedicated mini-computers for efficient

high-speed data processing, performed in a prescribed manner inaccordance with a specialized test routine. In the hands of anexperienced modal analyst, this leads to economical extraction of thedata mentioned above.

The advantages of modal analysis are, first, that a modal testprovides the most rapid and effective procedure available for theacquisition of data on the dynamic properties of a structure. Suchtesting can often be performed by a skilled technician for laterinterpretation by a dynamics engineer. Second, modal analysis is aneffective analytical procedure for the solution of large sets ofstructural dynamics equations because it reduces coupled matrixequations (which must otherwise be solved by some iterativeprocedure) to a set of independent linear equations, each with thewell-known closed-form solution given above. Modal solutions cantherefore be obtained directly, without further numerical operations.These solutions are then re-combined to form the complete solutionto the structural response problem in question. It should here benoted that solutions to harmonic, transient, and random forcedvibration problems can all be obtained using this modal analyticalprocedure, by means of simple extensions to the theoreticalprocedure outlined above: see Reference [1] for details.

Shortcomings of Modal AnalysisThe output from modal testing consists of natural frequencies,

mode shapes, modal stiffness, modal damping, and modal massmatrices. The main assumption involved in the acquisition of thisinformation is that the structural system is linear, i.e., structuraldisplacements are directly proportional to applied loads. In practicalstructures this condition is not always met. Structural systems may benon-linear to some degree, due to those causes listed below. Non-linearities complicate the extraction of modal data and, where theireffect is strong, they may invalidate the results obtained by linearanalysis. Non-linear effects may be present in a structural system dueto several causes:a) The material properties may be non-linear, e.g., composite

structures, viscoelastic materials, elastic-plastic materials, wheredisplacement is non-linearly related to force.

b) Where large amplitudes are involved, the geometry may result indisplacements, which are non-linearly related to load, e.g., largedeflections of plate and shell-type structures.

c) The structural boundary conditions may introduce non-linearities, e.g., structures where the number of support pointschanges, or where the structure is a rotor mounted in fluid-filmbearings experiencing relatively large whirl amplitudes.Such non-linear effects complicate the analysis and tend to

introduce errors into the data reduction and curve-fitting estimates ofnatural frequencies. Such results cannot always be adequatelyrepresented by a linear analysis, because the properties changeaccording to the magnitude of the applied load. Errors can range fromsmall errors where minor non-linearities are present to large errorswhere the non-linear effects are substantial, such as in multiplesupport structural contact problems (load-dependent indeterminacy).

A further limitation to modal testing is that it does not directlyaddress the forced response problem, nor problems of transient responsenor of random response. For problems in which the response to suchloadings is of interest, modal amplitude data can be obtained bytesting to formulate an efficient structural model for finite elementanalysis. Once the structural model is available in matrix form, theforcing data can be loaded into a finite element program, and theresponse to dynamic loading (harmonic, transient, or random) canthen be obtained by calculation. The accuracy of such analysesdepends of course on the validity of the model, which is generatedfrom the modal test data. It is good practice to make a preliminarynatural frequency/mode shape calculation with such data, to verifythat the test modal data is consistent with the structural modes andfrequencies upon which it is based.

Another limitation of modal testing is that it cannot, by itself,predict threshold conditions for structural stability problems, such asstructural buckling, and rotor whirl stability in fluid-film bearings.

Again, the modal test structural matrix data from such problems canbe developed for subsequent (linear) finite element analysis, such asthe prediction of stability threshold conditions. However, the non-linear limitation again applies to the post- threshold behavior of suchstructures. Following the development of an unstable condition, e.g.,buckling or rotor whirl, the structure characteristically undergoeslarge displacements until a new equilibrium condition is found. Suchbehavior may be highly non-linear, and so beyond the capabilities ofmodal analysis, and of the structural matrices developed by modaltesting.Advantages of Finite Element Analysis

Finite element analysis in conjunction with the high-speeddigital computer permits the efficient solution of large, complexstructural dynamics problems. As the majority of structural dynamicsproblems are linear they can be solved in the frequency domain usinga modal transformation as noted above, subject to certain simplifyingassumptions concerning the nature of damping.*

Many efficient and comprehensive finite element computercodes are now available to perform structural dynamics responsecalculations involving harmonic response, transient response, andrandom response of complex structures. Provision is made in manylarge codes for storing specific solutions on tape and using thesesolutions as input to a second related problem, involving the samestructure. For example dynamics problems where high temperaturescause changes in the elastic properties of the structure may beaddressed by solving for the temperature distribution prior to thenatural frequency calculations. The temperature distribution is firstobtained for known input conditions, and this solution is used tosolve the structural dynamics problem with temperature-dependentelasticity. Similar comments apply to fluid/structural interactions,where the equivalent mass properties of the fluid must beincorporated within the structural mass matrix.

The finite element method therefore offers a very efficientprocedure for the calculation of complex linear structures under avariety of dynamic excitation conditions, and under environmentalconditions, which may include temperature effects and entrainedfluid effects. Where the structure is nonlinear, modal testing may stillbe used (with caution) to estimate initial values for mass, stiffness,and damping parameters, which can then be modified to suit moreadvanced structural models.

Shortcomings of Finite Element AnalysisAlthough most linear structural dynamics problems may now be

solved accurately and economically, it is still costly to solve mostnon-linear problems. For such cases a solution strategy must usuallybe developed on a case-by-case basis. In such instances the structuralgeometry and elasticity may be needed in considerable detail in theinput data, and the formulation time for such cases may be significantunless suitable pre-processors are available within the code.

The finite element analysis of recurrent structures, i.e., where aspecific segment of the structure geometry is repeated a number oftimes, are still costly to solve. No general-purpose codes (or pre-processors) are yet in use, which specifically address this problem.Problems of recurrent geometry are relatively common, e.g., bladedturbomachine structures, axisymmetric structures, building structures,and many types of rotating machinery. The geometry of suchstructures often closes on itself ('ring' structures). The total structuralmatrix is still symmetrical and tri-diagonal, but the dynamical matrixcontains off-diagonal elements, which may substantially increase thelocal matrix bandwidth. This causes a corresponding increase incomputation time. Efficient computation of such recurrentcomponents has been undertaken by special finite differenceprocedures,3 but sub-routines to undertake such computations are notyet in widespread use.

* So-called Rayleigh (or proportional) damping conditions must be used forthe simple modal transformation to apply. Other Techniques are availablewhen the damping matrix is non-symmetrical.

Case History 1 - Vibrations of a Three Blade GroupPurpose. Vibrations of a group of three turbine blades were

studied to determine whether resonance was likely between a naturalfrequency of the group and some per-rev harmonic of running speedin the machine.

Procedure. A modal test was made on the three-blade groupshown in Figure 1. The group was welded to a massive steel blockattached to a concrete floor slab at the level of the first hook contact,to simulate the rim attachment flexibility. The blades were alsowelded together at their cover sections. The cover was attached to theblades by a single large rivet. The structure had no tiewire.

The measured natural frequencies of the group were obtainedwithout centrifugal stiffening, but with some root flexibility. A finiteelement calculation was then made of the blade group naturalfrequencies, without centrifugal stiffening, and then with centrifugaleffect.

The modal test natural frequencies confirmed the finite elementnatural frequencies, as shown in Table 1. This indicated that the finiteelement model was appropriately scaled. Natural frequency valueswere then calculated with stress stiffening added, corresponding to3600 rpm, as shown in Table 1. The influence of the centrifugal riseon the natural frequencies is evident.

Mode Shapes. The first three test mode shapes are shown inFigure 2. These were obtained with Structural Measurement Systems,Inc. (SMS) Modal Analysis System. The first mode is a sideways, ortangential, motion of the group, al T1 = 170.2 Hz. Viewed from thetop (not shown) the motion occurs at about 45° to the tangential. Thesecond mode occurred at AI = 263.5 Hz, mostly in the axial direction,i.e., normal to the tangential direction.

The third mode RI occurred at 369.7 Hz, and is mostly a grouptorsional motion, in which the cover lends to rotate as a rigid bodyaround the tenon of the middle blade.

Figure 1. Three blade group for frequency testing.

Table 1. Natural frequencies of three-blade group.ClosestPer-Rev

Frequency, Hz

Mode ModalFrequency,

Hz

FiniteElement

Frequency(no CF), Hz

FiniteElement

Frequency(with CF), Hz

180 (3x) + 6% 1 170.2 175.4 190.3300 (5x) + 5% 2 263.5 270.7 286.4420 (7x) + 2% 3 369.7 393.1 413.1

Figure 2. First three-mode shapes for three-blade groups.

Comments. The results show that the frequency margins for T1and A1 are 6% and 5% respectively from the closest excitationharmonics. This difference is adequate for effective detuning fromeither resonance condition.

The apparent frequency margin for R1 is only 2%, which inpractice would be less than desirable to ensure safe operation,although the present detuning from resonance is still very significant.Also the attachment boundary conditions at the welded rootattachment and at the welded rivet, as used to calculate this mode arethought to be somewhat stiffer than with actual blades in practice. In

practice this would tend to lower the R1 mode further away form anythird mode resonance.

The correlation between modal results and finite element resultsfor the first two modes is typical for such blade structures withcomplex airfoil shapes and imprecise boundary conditions. Potentialimprovements in correlation through improved knowledge ofboundary conditions are often offset by the statistical variance offrequencies between actual blades, and by the specific boundaryconditions of each attachment under rotating conditions. Havingverified the blade group model, the next step in such studies would beto determine the group response to specified per-rev steam stimulus.

Case History 2 – Vibrations of Sewage Pump and PipingSystem

Problem Description. Excessive rotor vibrations were reportedfor the sewage pump and piping system shown in Figure 3. Thesevibrations were the suspected cause of rapid gland seal wear andfrequent seal replacement, which led to unacceptable restriction ofthe operating speed and pump discharge range. A program ofvibration tests and modal tests, supported by rotor systemcalculations was made to determine the reasons for this problem.4

Figure 3. Sewage pump and piping system.

Pumping and Piping System. The pump and piping systemshown in Figure 3 raises sewage from a wet well sump through avertical riser pipe into a treatment tank. The pump inlet and outletpipes both have a 30 inch I.D., and the vertical pipe has a 42 inch I.D.A check valve is situated 65 ft. above the outlet pipe centerline. Thehorizontal pipe to the treatment tank is 165 ft. above the pumpcenterline. The pump is driven by an 850 Hp motor mounted on a 30ft. vertical pipe column supported from the pump volute casing. Themotor drives the pump through a vertical shaft. The pump rotor alsohas a casing with cantilever bearing supports and a gland packingseal above the rotor, which experienced the wear and sewage leakage.The pump impeller is overhung, with two outlet flow passages (two-vane impeller). The volute casing surrounds the impeller.

Test Program. A series of tests to determine the cause of thepump system vibrations was undertaken as follows:a) Accelerometer tests were made on the motor casing, on the

gland seal casing and pump outlet flange, and at variouslocations on the pipes, including the vertical bend and the checkvalve flange. Data were recorded throughout a matrix of testconditions relating to pump head and operating speeds.

b) Modal tests were made to determine the natural frequencies andmode shapes of the pump and piping system. These tests weremade with and without fluid in the pump, but not with the pumprotating.

c) Displacement sensor tests were made near the gland seal,between the rotor shaft and the pump casing, to determine theconditions under which the shaft vibrations were worst, for thetest matrix.

d) Strain gage telemetry tests were made to determine the locationof the torsional natural frequencies of the motor drive and pumprotor. Strain gages were placed on the pump shaft between thebearings. Induced-power radio telemetry was used to bring outthe shaft torsional signals.

Figure 4. Rotor and support model details.

Figure 5. Amplitude vs. frequency response for drive system.

Calculations. The forced response properties of the pump andcasing structure were investigated using a multi-level rotordynamicscode. The rotor system model details for the pump and casing areshown in Figure 4. An arbitrary unbalance of 1 oz. in. was added tothe impeller to provide unbalance excitation at once per-rev. Themodel also contained a small amount of viscous damping in each ofthe rolling element bearings in the drive system. This calculation wasperformed to determine the location of the natural frequencies of thepump structure, including those of the cantilevered motor, drivecoupling and cantilevered pump rotor (multi-level system). The

frequency response of the system in its first mode is shown in Figure5. The mode shape is shown in Figure 6. These natural frequenciesand mode shapes are compared with the modal test naturalfrequencies and mode shapes in Table 2, and with the systemresonances from he accelerometer tests. Torsional drive systemcalculations were also made to determine the torsional naturalfrequencies of the drive rotating components. Calculated torsionalresults are compared with the telemetry test results in Table 2.

Modal Test Results. The first mode of the pump and pipingsystem is shown in Figure 7. In this mode, the pump structurevibrates against the riser pipe, in the direction of the outlet pipe axis,at 24.5 Hz. The drive rotor system also vibrates against the motorsupport structure. The second mode at 28 Hz is similar. The pumpstructure and riser pipe again vibrate against each other, but this timein a direction transverse to the outlet pipe axis. The rotatingcomponents again vibrate against the motor support structure. Otherpump and piping system modes were found at frequencies given inTable 2.

Figure 6. Mode shape of drive shaft and pump casing at ƒ1 = 35.5 Hz(rotordynamic calculation).

Figure 7. Mode shape of piping and drive at ƒ2 = 24.5 Hz (modal test).

Table 2. Calculated and test natural frequencies.Mode Calculated

Frequency, HzModal

Frequency, HzAccelerometerFrequency, Hz

Reed 1 24 24.5 -Reed 2 28 28.1 28.0Shaft Bending 1 (32)* 32.8 30.5Shaft Bending 2 35.5 35.2 -Torsional 44.5 44.8 46.3Structural 1 - 53.1 54.8Structural 2 - 82.4 81.5Structural 3 - 88.1 86.5Structural 4 - 96.6 98.0

*Depends on assumed casing support stiffness

Table 3. System resonance frequenciesMode Natural

Frequency, HzExcited By Speed, rpm

Drive Shaft(Lateral Mode 1)

32.0 2x3x4x

960640 Op. Range480

Drive Shaft(Lateral Mode 2)

35.5 2x3x4x

1050690 Op. Range510

Reed Mode 2 28.0 2x3x4x

840 Op. Range560420

Figure 8. Typical accelerometer response spectrum (bump test at motor).

Vibration Test Results. Resonant frequencies found duringoperation of the pump are listed in Table 3. These data were obtainedand confirmed at a number of locations. The results indicate thefrequency of the excitation (1×, 2×, etc.) related to pump speed ofrotation. A typical accelerometer response spectrum is shown inFigure 8, which indicates the characteristically strong 2× excitation,which was consistently observed in the accelerometer anddisplacement sensor results. The maximum relative shaft vibratorydisplacements occurred around 700 rpm and especially above 800rpm. Similar results were found for the impeller displacements.

Comments on Test Results. Resonance frequencies aresummarized in Table 3. At 736 rpm the 2× impeller frequencycoincides with the motor support reed frequency of 24.54 Hz, and the3× coincides with the shaft lateral natural frequency at 35 Hz whenthe impeller speed is 700 rpm. The occurrence of these two lightly-damped resonances within this speed range is thought to be the causeof the observed strong vibrations around 700-740 rpm. At 795 rpmthe 4× harmonic coincides with the system structural mode 1 at 53Hz. At 840 rpm the 2× harmonic coincides with the second-motorsupport reed mode at 28 Hz. This response was observed strongly inseveral readings. The shaft torsional mode at 45 Hz is also

approached at 885 rpm. Torsional resonance would occur at 900 rpm,which was beyond the test speed range.

Each of the above resonances appear to have been excited by theimpeller jet transient loadings on the casing and pipe system. Theseresonances appear to have been the cause of the observed structuralvibrations of the pump and piping. It is noted that the system has littleinherent damping, and the resulting transient vibrations will thus besustained by the recurring impeller jet impacts as rotation occurs,especially under resonant conditions.

The conclusions are as follows:• The measured structural, lateral and torsional responses of the

pump and piping system are all caused by forced vibration. Theprimary cause of this forced vibration is the rotating impeller jetdischarge, which impacts on the pump structure at twice perimpeller revolution.

• The time-variation of each jet discharge forcing resembles a sawtooth shape. Two saw tooth loadings correspond to a singlerotation of the impeller.

• This saw tooth forcing is the cause of the observed structuralvibration and the shaft lateral and torsional vibration. The sawtooth Fourier spectrum contains the 1×, 2×, 3×, and 4× andhigher components.

• The strongest source of excitation is the two-per-rev harmonicof pump rotational speed. This excitation causes large vibrationsto develop when it becomes resonant with any of the naturalfrequencies of the pump piping structure.

• The 24.5 Hz reed frequencies can be resonated by the strongtwo-per-rev forcing at 720 rpm.

• The 28 Hz reed frequency can be resonated by the strong two-per-rev forcing at 840 rpm.

• The 32.8 Hz lateral mode was excited but not resonated by the2× pump harmonic. It could be resonated by the 3× at 656 rpm.

• The 45 Hz torsional mode was excited but not resonated by the2× impeller harmonic. It could be resonated by the 4× at 675rpm, which is close to the operating speed range of the pump.

These modal tests and calculations identified and explained the causeof the pump-system vibrations, and allowed replacementmodifications to be specified which are expected to overcome thisproblem, through a generally stiffer system with a five-vane impeller.

Case History 3 – Modal Analysis of a Turbine GeneratorFoundation

Description of Foundation. A 330 MW turbine generator ismounted on the massive reinforced concrete foundation shown inFigure 9. The total weight of the turbine foundation is 2180 tons. Thefoundation is supported on 50 ft. concrete piles which extend down tobedrock. The natural frequencies and mode shapes of the unit wereobtained by modal test, and by finite element analysis to satisfyseismic requirements.5

Figure 9. Turbine generator concrete foundation.

Modal Test. A total of 68 nodal test sites were selected toprovide modal rap test data (selectively) in the x, y, z directionsshown. Analysis of the modal test data led to the identification of 20modes between 3.88 Hz (fundamental, x-longitudinal) and 229 Hz(complex platform bending torsion).

Finite Element Calculation. A finite element calculation wasalso made for the foundation based on the construction drawing, andassuming that the concrete columns were elastic for 3 ft. below theconcrete floor, and were rigidly secured below this depth. Thecalculated natural frequencies are compared with the measuredfrequencies in Table 4.

Table 4. Frequencies of natural modes, modal analysis and finite elementcalculations.

Mode

MeasuredFrequency,

Hz

CalculatedFrequency,

Hz Mode

MeasuredFrequency,

Hz

CalculatedFrequency,

Hz

1 3.88 3.41 7 27.30 29.622 4.12 4.27 8 35.61 40.333 5.05 5.36 9 40.19 45.874 9.52 9.89 10 55.22 56.215 15.86 17.44 11 73.71 79.686 19.11 20.61 12 90.23 99.64

Figure 10. A-Mode 1, 3.88 Hz, x-translational, in-phase, rigid platform.B-Mode 2, 4.12Hz, y-lateral, in-phase, rigid platform. C-Mode 3, 5.05 Hz, y-torsional, platform warping. D-Mode 4, 9.52 Hz, y-lateral, platform bending.

Foundation Modes. The first four calculated modes of thestructure are shown in Figure 10. Mode 1 is a lateral translation of thestructure in the x-direction at 3.88 Hz. Mode 2 is the y-translation ofthe structure at 4.12 Hz. Mode 3 is rigid torsion of the platform on itscolumns at 5.05 Hz, and mode 4 is platform free-free bending at 9.52Hz. Higher modes show further platform warping and column vs.column bending.

Significance of Modes. For seismic considerations the mostsignificant vibration responses are expected to occur in modes 1 and2, because these modes can most readily couple with the strongestearthquake spectral components in the 2 to 5 Hz domain; see thetypical earthquake spectrum in Figure 11. Above this frequency rangethe earthquake excitation strength is less, and the modal forms arealso less likely to couple with the ground motions, e.g., column vs.column modes are unlikely with ground vibration, as are platformwarping modes, and turbine generator vertical modes.

Discussion of Results. The correlation obtained between thefrequencies shown in Table 4 confirms both analyses (modal andF.E.), plus the column fixity assumption. The finite element modelwas formed from solid elements with reinforcement rods situatedalong element interfaces. The turbine generator was modeled withdiscrete masses. The contribution of the turbine generator is expectedto be minor below 30 Hz, at which frequency any excessive rotatingunbalance may couple with mode 7.

The results of this study identify and confirm the location of thenatural frequencies of the foundation structure, and reveal theassociated mode shapes. Further calculations concerning possibleresponse to earthquake excitation can next be made, using the now-confirmed finite element models.

Figure 11. El Centro, CA earthquake, May 18, 1940, NS ground acceleration,velocity, and displacement.5

Case History 4 – Brake Squeal in Subway TrainsProblem Description. An unpleasant squeal problem developed

in the braking system of subway cars in the new Washington, DCtransit system. Sharp squealing occurred as the cars were brought torest. An investigation of the braking system arrangement was made todetermine the cause of this squealing and to seek a remedy.6

Analysis Procedure. Detailed modal tests were made on atypical brake assembly to identify potential natural frequencycomponents in the squeal spectrum. A series of operational tests weremade to determine the brake assembly natural frequencies, and toprovide spectrum data for confirming the squeal model, for which aspecial computer code was developed for parametric studies.

Modal Tests. The brake and wheel arrangement shown inFigure 12 was tested to determine its modal properties. A total of 26modal test locations were used. A typical response spectrum for thebrake pad is shown in Figure 13. The mode at 268 Hz shown in thisspectrum is plotted in Figure 15. These tests identified the principalmodes of the brake structure, and the modal parameters associatedwith them. These parameters are listed in Table 5. They were usedlater in this study for the parametric model studies.

Table 5. Modal parameters identified for disk.

ModeFrequency

Hz Md, KgKd1,

108N/m Id, Kgm2Kdr,

106Nm/rad

1 272 52.78 1.55 0.2457 0.72262 754 11.29 2.55 0.1215 2.74433 1414 5.48 4.35 0.0289 2.29194 1689 7.25 8.22 0.0173 1.95635 2495 12.43 30.70 0.0126 3.10736 2628 5.25 14.40 0.0298 8.17337 3158 733.80 2906.60 0.4929 19.52148 3376 38509.00 174000.00 2.4352 1101.80009 3388 124.76 564.40 0.0101 4.5733

10 4194 104.84 731.30 0.0383 26.6830

Figure 12. Brake and wheel arrangement.

Figure 13. Modal response.

Finite Element Calculations. A finite element model of thebrake disk was made to calculate the disk natural frequencies, and toconfirm the disk modal parameters obtained from the modal tests. A180° segment of this disk model is shown in Figure 14. The 360° diskwas made from 462 plate elements, 126 solid elements and 142 beamelements. The one diameter finite element mode shape for this disk(180° segment) is shown in Figure 15. The correlation obtainedbetween modal frequencies and finite element frequencies is given inTable 6.

Figure 14. Finite element model of disk.

Figure 15. Mode 1, 268 Hz, finite element calculation.

Squeal Model. The modal data in Table 6 as confirmed by thefinite element calculations were used to prepare the squeal modelshown in Figure 16. This model had six degrees of freedom, i.e., alateral and a translatory DOF for the disk, caliper, and wheel.

Writing M, K, and I for mass, stiffness and inertia; Y, θ fordisplacement and rotation; µ, N, h, b for friction coefficient, normalforce, and distance; and l, r for linear and rotational direction leads tothe following equations of motion for the system:

0N?µYK)Y(YKYM dddepdpedd =−+−+&& (12)

Disk

0)Y(YhKµ)?(?K?K?I pdpepdprddrdd =−+−++&& (13)

0P)Y(YKYM dppepp =+−+&& (14)

Pad

0Pa)Y(YdK2µ)?(?K?I pdpedpprpp =+−+−+&& (15)

0PYKYM cclcc =−+&& (16)

Caliper

0bPµPa?K?I bccrcc =+−+&& (17)

This leads to the system of equations {S}[D]}S{ =&& where {S}is astate vector of displacements, and [D] is a matrix of stiffness massand damping coefficients. The solution for {S}is obtained by writing:

?t0}e{S{S}= (18)

which leads to the standard eigenvalue problem }{S?}{S[D] 002= .

When the brake system is stable the real part of the eigenvalues λ isnegative. When the mode in question is unstable the real part ispositive. In general:

?ia?1,2 ±= (19)

where a represents system damping (positive, unstable; negative,stable) and ω is the circular frequency of the mode in question. Acomputer program EIGRDISK was written and used to findinstability threshold conditions for specified conditions of the brakesystem.

Table 6. Correlation between modal and finite element natural frequencies.Mode Frequency Element, Hz Modal Rap Tests, Hz

1 268.45 2252 950.76 7503 1679.78 13624 1984 - 2260 1875 - 19375 2582 - 2599 2462 - 25006 - 3050 -32257 - 37378 - 4400

Figure 16. Squeal model of brake system.

Discussion of Theoretical Results. Young’s modulus of the padfriction material and the coefficient of friction of the pad were variedin the analysis as these values change from time to time depending onbrake pressure, temperature and wear of the pad. The distancebetween pad center and cylinder centerline was also varied. Thisparameter was found to exert a strong effect on the squeal propensity.

Three different groups of squeal frequencies were found in theanalysis: the first around 3000 Hz for large values of Young’smodulus from 0.5 to 3.0 × 108 N/m2, the second was 400 Hz for anarrow range of Young’s modulus around 0.2 × 108 N/m2 and thethird around 4000 Hz for lower values of Young’s modulus around0.1 × 107 N/m2.

3000 Hz Squeal Frequency. This squeal frequency wasobserved with the 6th natural mode of the disk, 2628 Hz. Figure 17shows the squeal propensity and squeal frequency of one brakesystem as a function of pad Young’s modulus, for four values offriction coefficient. The squeal frequency is independent of the

coefficient of friction and increases with Young’s modulus in a linearmanner in the instability zone. For a large value of friction coefficientthe instability regime covers a wide range of Young’s modulus of thepad. This range decreases with reduction in the coefficient of friction.The intensity of squeal, characterized by the squeal propensity, ishigh with large values of coefficient of friction.

Figure 17. Squeal instability regimes for disk brake system.

Figure 18. Schematic of field test instrumentation.

400 Hz Squeal Frequency. Squeal frequency was observedwith the 10th mode of the disk at 4194 Hz, for a pad Young’smodulus of 0.1 × 108 N/m2. An interesting feature observed for thisinstability was that the disk coupled frequencies are also very closetogether which could result in another squeal frequency, 4200 Hz.The natural frequencies obtained for the coupled system with E (pad)= 0.1 × 108 N/m2 were 4204.8, 4114.0, 436.2, 436.2 (squealfrequency) and 442.2 Hz. The squeal propensity for the 436.2 Hzfrequency was only 21.24 as compared to three digit values for the3000 Hz squeal frequency considered before. However, because thisis near to a possible second squeal frequency (4210 Hz), and furtherthat the caliper frequency of 442.2 Hz is also near the 436.2 Hzsqueal frequency, this system instability could clearly be a dangerous

condition, in terms of generating loud unpleasant noise around 400and 4000 Hz. Figure 17 also shows the effect of friction coefficienton the squeal instability.

Figure 19. Disk frequency spectrum during squeal initiation.

Figure 20. Disk frequency spectrum after initiation of squeal.

Field Test Instrumentation. Figure 18 shows theinstrumentation used to collect the field data. PCB Series 303Aquartz accelerometers were used to convert the shock and vibratorymotion into high level, low impedance signals used in these tests.303A02 PCB accelerometers were stud mounted on the disk and onstationary parts. 302A03 PCB accelerometers were also used withadhesive mounts. Coaxial shielded cables were used to eliminatecable noise.

AB&K type 2203 precision sound level meter with one inchcondenser microphone was used to record the noise signature. Thesystem was mounted on a portable floor stand, with the microphonepointing downwards near the front open window of the rail car usedfor testing.

The brake pressure and truck speed was obtained from the carbrake system directly. Two tape recorders, Racal Store 4 and Store 7were used to store the data on magnetic tapes. A special bracket wasused to accommodate slip rings, thermocouple amplifiers,accelerometer amplifiers and connecting leads from the rotatingsection. This bracket was clamped onto the axle using three dustcover bolts. A Rockland Scientific 444A Mini-Ubiquitous FFTcomputing spectrum analyzer was used to analyze the vibration andnoise signatures, along with a Tektronix 4622 interactive digitalplotter to obtain a permanent graphic record.

The two tape recorders and noise measuring equipment werelocated inside the car along with the spectrum analyzer, oscilloscope,and a plotter. The brake pressure and speed signals were taken fromthe circuits in the driving car.

Field Test Results. The disk modal analysis indicated thatsqueal would occur mostly in the range of 2800-3300 Hz.Occasionally squeal was observed around 4250 Hz. In every case, thesqueal frequency was first observed in the vibration signals of disk,pad and caliper limb, and in the noise signal, as the brakes wereapplied and the train began to slow down. After the squeal frequencyappeared in the real time spectra, the amplitudes of vibration around500 Hz grew rapidly as shown in the vibration signals, and in thenoise signals. Two typical sets of the frequency spectra are discussedbelow.

As soon as squeal appeared, the disk vibration was almost aharmonic motion as shown in Figure 19. Immediately following theappearance of the squeal at frequency 2825 Hz, the disk vibrationlevels increased strongly. The corresponding frequency spectrum isshown in Figure 20. The main frequency components in thisspectrum are in the 250-350 Hz band, which is at the fundamentalfrequency of the disk, as determined by calculation and modal test.Higher mode components of the disk vibration are also present.

The pad response during the initiation of squeal is shown inFigure 21 and the response after squeal started in Figure 22 shows thefrequency spectrum of noise when the squeal began to appear with afrequency component of 2950 Hz. The noise spectrum then spreadrapidly to a lower frequency range, which contains lower disk modesand lower system modes. The maximum temperature at the outerradius of the disk front face is 460°F, decreasing to 335°F at the innerradius.

During these tests, it was observed that squeal was present forone direction of train motion but not in the reverse direction. Thiswas also validated by the mathematical model; no instability occurredwhen the kinematic constraint of pad-piston contact point wasreversed.

Another brake system was also calculated and found that nosqueal occurred for a wide range of Young’s modulus values of thepad. Field tests with this system also produced no squeal. This furtherconfirms the analysis and the squeal model.

Figure 21. Pad frequency spectrum during signal initiation.

Comments of the Test Results.• Based on the theoretical results and field data diagnosis, the

mechanism of squeal in disk brakes has been confirmed, anddefined in detail.

• The instability of the brake system was found to initiate in thehigher disk modes (frequencies), typically in the 6th and 10th

modes in the present analysis.

• The squeal instability in the system is dependent on thecoefficient of friction of the pad, Young’s modulus of the pad,and the point of application of brake pressure.

• The coefficient of friction of the pad is a function oftemperature, wear, relative speed, and humidity conditions. TheYoung’s modulus of the pad is a function of brake pressure andtemperature. These conditions vary quite randomly duringoperation and may give rise to different squeal behavior underapparently similar conditions of operation.

• The coefficient of friction exerts a dominant influence on squealand it should be kept as low as possible. The squeal frequency isnot significantly affected by any changes in coefficient offriction.

• For a given set a parameters of the system, squeal instabilityoccurs over certain ranges of Young’s modulus of pad. Thesqueal frequency increases with Young’s modulus of pad in agiven zone.

• For a given system, squeal can be eliminated by reducing thedistance between pad center and cylinder centerline.

• The observations of squeal from the field data are substantiatedby the mathematical model.

The guidance provided by this study established the cause and themechanism of the squeal problem, and provided several alternativepaths for its solution. Washington Metro is now squeal-free.

Figure 22. Pad frequency spectrum after initiation of squeal.

References1. Hurty, W. C., and Rubinstein, M. F., Dynamics of Structures,

Prentice-Hall, Inc., Englewood Cliffs, NJ, 1964.2. Bathe, K. J., Wilson, E. L., Numerical Methods in Finite

Element Analysis, Prentice-Hall, Inc., Englewood Cliffs, NJ,1976.

3. Salama, A. M., Petyt, M., and Mota Soares, C. A., “DynamicAnalysis of Bladed Disks by Wave Propagation and MatrixDifference Techniques,” ASME Monograph, StructuralDynamic Aspects of Bladed Disk Assemblies, presented at theWinter Annual Meeting of the ASME, December 5-10, 1976.

4. Rieger, N. F., and Kimber, A. W., “Dynamic InteractionBetween a Vertical Pump and Its Piping System,” Proceedingsof International Conference on the Hydraulics of PumpingStations, BHRA, The Fluid Engineering Center, Manchester,England, pp. 253-266, September 17-19, 1985.

5. Newmark, N. M., Fundamentals of Earthquake Engineering,Prentice-Hall Inc., Englewood Cliffs, NJ, 1971.

6. Rao, J. S., and Rieger, N. F., “Brake Squeal Problems inUnderground Trains,” Inst. Mech. Engrs. (London) Third Intern.Conf. Vibrations of Rotating Machinery, York, England,September 11-13, 1984.