fig. 1 - imperial college · pdf file · 2018-02-28gas-turbine structures, such as...

SPECIAL ISSUE PAPER

State-of-the-art dynamic analysis for non-linear gasturbine structures

E P Petrov* and D J Ewins

Centre of Vibration Engineering, Mechanical Engineering Department, Imperial College of Science, Technology andMedicine, London, UK

Abstract: An accurate and robust method has been developed for the general multiharmonicanalysis of forced periodic vibrations of bladed discs subjected to abrupt changes of elastic anddamping properties at contact nodes located on the interfaces of the components. The approach isbased on an analytical formulation of friction contact interface elements which provides exactexpressions for multiharmonic forces and stiffness matrices of the elements. The analyticalformulation overcomes the numerical difficulties associated with searching for a solution of thenon-linear equations, which have been inherent so far for the considered systems, and provides abreakthrough in the analysis of the non-linear forced response of bladed discs and other structureswith gaps and friction interfaces. An effective reduction method is reported that allows the size of theresolving equations to be decreased to the number of DOFs (degrees of freedom) at the contactsurfaces, while preserving completeness and accuracy of the whole large-scale model.Numerical investigations demonstrate the outstanding properties of the proposed approach with

respect to computational speed, accuracy and stability of calculations for practical gas-turbinestructures.

Keywords: friction contact, non-linear vibration, multiharmonic, damping

NOTATION

aj ¼ z� 1j forced response function matrix for the jth

harmonicfx, fy tangential and normal components of

interaction forces at a friction interfaceFcj ,F

sj vectors of the coefficients for non-linear

interaction forcesiZðoÞ multiharmonic dynamic stiffness matrixkt, kn contact stiffness coefficients due to

roughness of a contact surfaceK, C, M stiffness, viscous damping and mass

matricesPcj ,P

sj vectors of the coefficients for excitation

loadsqðtÞ vector of displacementsQc

j ,Qsj vectors of cosine and sine coefficients for

the jth harmonic of the displacementsqj ,fj,pj vectors of complex harmonic coefficients

R vector of residuals for multiharmonicnon-linear equations of motion

Zj,zzj real and complex dynamic stiffnessmatrices for the jth harmonic

m friction coefficiento principal vibration frequencyog,fg natural frequencies and mode shapes of a

linear structure

1 INTRODUCTION

Gas-turbine structures, such as bladed discs, rotorstructures or casings (Fig. 1) and other machinerystructures usually comprise many components thatinteract through contacting interfaces. In operationconditions these structures are subjected to dynamicforcing, which results in vibratory displacements. Highvibratory displacements and stress levels that can causefailure can occur in such systems at resonance condi-tions when excitation frequencies and natural frequen-cies are close and when the spatial distribution of

The MS was received on 3 July 2003 and was accepted after revision forpublication on 27 April 2004.* Corresponding author: Centre of Vibration Engineering, MechanicalEngineering Department, Imperial College of Science, Technology andMedicine, London, SW7 2BX, UK.

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G02403 # IMechE 2004 Proc. Instn Mech. Engrs Vol. 218 Part G: J. Aerospace Engineering

excitation forces is similar to the deflection mode shapesof vibration of the assembly.

There are many sources of excitation in gas-turbineengines and, particularly for bladed discs where excita-tion forces are caused by highly inhomogeneous gasflow, these are usually a major source of vibration. Suchforces can have a very dense frequency spectrum and,similarly, the spectrum of the natural frequencies ofpractical structures is also customarily very dense. Verydense excitation and natural frequency spectra togethermake the prospect of avoiding all resonance regimesalmost impossible. The problem consists of developingmethods and models that provide accurate, fast androbust predictive tools for the dynamic analysis ofdifferent components of gas-turbine engine structures,and especially of bladed discs, which are one of the mostresponsible and crucial components of the engines.Comprehensive surveys of the dynamic problemsassociated with bladed disc assemblies can be found inreferences [1] to [3].

In many practical cases forces occurring at joints andother contact interfaces of the components of anassembled structure have a significantly non-linearcharacter. Among such primary non-linear forces arethe friction forces that occur as a result of the relativemotion of points of contact belonging to differentcomponents of the assembly at contact interfaces. Theseforces are inevitable and are usually highly beneficialbecause of the damping effects they produce, althoughthe effect on surface fretting has to be taken intoconsideration.

Moreover, special devices, such as so-called ‘frictiondampers’, are used in gas turbines in order to increasedamping and to restrict the level of resonant vibrationamplitudes. The effectiveness of using friction fordamping of vibrations is very much dependent ongeometry and the characteristics of contact surfaces,the parameters of components of the assembly, thechoice of parameters for the friction dampers and onexcitation loads. The problem of modelling frictionforces for the dynamic analysis of different machinerystructures is discussed in the extensive surveys of

references [4] and [5], and the friction models used forthe dynamic analysis of bladed discs are considered inthe review paper of reference [6]. Some more recentinvestigations of dynamics of the bladed discs withfriction elements are described in references [7] to [13]. Avariable normal load at the friction contact interfacechanges not only the friction force levels but also theinstants of slip–stick transitions, significantly affectingthe friction forces, and friction models taking intoaccount the normal load variation have recently beendeveloped and reported in references [14] to [17].

Another major source of non-linear contact forces arethe forces of unilateral interaction at the contactinterfaces. These forces permit the transmission ofcompression stresses but not extension and cannotgenerally be linearized. Furthermore, clearances andinterferences at the contact surfaces are inevitable inpractical assemblies and when these open and closeunder certain vibration conditions, they change theareas of the contact interfaces. The resulting non-linearinteraction significantly affects the resonance frequen-cies, friction damping and forced response levels.

Non-linear forces can also change dynamic propertiesof structures significantly. In many cases, they causequalitatively new phenomena that do not exist in thedynamics of linear systems. Such phenomena includemultivalued forced response levels for a structure withgiven design parameters and excitation, sub- and super-harmonic resonances, dependence of steady state forcedresponses on initial values of displacements andvelocities of the structure, etc. Special methods have tobe developed to solve the resulting non-linear equationsof motion numerically. From the point of view oftemporal behaviour, problems occurring in gas turbinedynamics can be divided into (a) a class of non-periodic,transient dynamics non-linear problems and (b) a classof periodic, steady state vibration. The latter problemusually prevails in practical forced response analysis andamong the most effective methods for calculation of theperiodic solution are the multiharmonic balance meth-ods, which permit effective accounting for periodicity ofthe sought solution and the avoidance of having to

Fig. 1 A gas-turbine engine

E P PETROV AND D J EWINS200

Proc. Instn Mech. Engrs Vol. 218 Part G: J. Aerospace Engineering G02403 # IMechE 2004

undertake time integration of the equations of motion.When non-linear effects are significant, the customaryprinciple of superposition is no longer valid and theharmonic components of vibration displacements can-not be analysed separately. These harmonic componentsof the response displacement interact through non-linearinterface forces and several of them can occur simulta-neously even when a mono-harmonic excitation isapplied. For the common practical case of multi-harmonic excitation, interaction of different harmonicscan affect the response levels significantly. Someformulations and applications of the multiharmonicbalance methods to the non-linear dynamics problemare presented in references [18] to [25].

Notwithstanding the many investigations that therehave been of the dynamic behaviour of structures withfriction contact interfaces, the problem of developingefficient and robust methods for the analysis of multi-harmonic forced response levels of practical structures,using large-scale finite element models, still needs to besolved. In this paper, new reliable and efficientcomputational models and a robust method to providea prediction of response levels of such structuresdeveloped recently by the authors are reported.

Friction contact interface elements are described byallowing for variable normal load and for the existenceof clearances and interferences. The friction interfaceelements used describe unilateral contact along thenormal to a contact surface and the friction forces underthe variable normal load. No restrictions on the level ofthe normal load variation are imposed, which allows themodel to be used for the practically inevitable caseswhen separation of the contact surfaces occurs undervibration conditions.

To guarantee high accuracy and rapid convergence ofthe iterative solution process, the non-linear interfaceelements are derived analytically, including derivation oftheir tangent stiffnessmatrix formultiharmonic vibrationanalysis. The resulting expressions allow the interactionforces and the stiffness matrix to be calculated with theaccuracy of the real number presentation in the computer,while providing extremely fast calculation of the char-acteristics of the contact interaction. The accuracy of theanalytical derivation overcomes common difficultiesencountered in numerical analysis of non-linear struc-tures comprising interfaces with abrupt changes ofcontact conditions (such as contact/absence of contact,slip/stick, etc.). Such difficulties have been inherent so farand in many cases they have even caused a loss ofconvergence and failure to find a solution.

A method for analysing primary non-linear vibrationsof structures using large-scale finite element (FE) modelsis elaborated in this paper. A multiharmonic balanceformulation for the equations of motion is derived and amethod to reduce the size of the resulting non-linearequations is proposed, which preserves the accuracy ofthe initial model.

The outstanding efficiency of the method is demon-strated in a set of representative test cases, including (a)some using large FE models of practical bladed discswith friction dampers and where impacts betweenshrouds are considered and (b) a model of a test piecefor an impact damper test rig.

2 PROBLEM FORMULATION

The equation for the forced response of a non-linearstructure consisting of a linear part, which is indepen-dent of vibration amplitude, and non-linear frictioninterfaces can be written in the following form:

KqðtÞ þ C _qqðtÞ þM€qqðtÞ þ f ðqðtÞ, _qqðtÞÞ � pðtÞ ¼ 0 ð1Þ

where K, C and M are the stiffness, viscous dampingand mass matrices respectively of its linear part, whichcan include terms corresponding to the action ofcentrifugal, Coriolis, thermoelastic and other linearstatic forces; pðtÞ is a vector of external periodicexcitation forces with a known period, T.

In equation (1), qðtÞ is a vector of nodal displacementsfor all the degrees of freedom (DOFs) in the structureconsidered. The vector, qðtÞ, comprises a vector of‘linear DOFs’, qlin, and a vector of ‘non-linear DOFs’(i.e. DOFs at interfaces where interaction forces withnon-linear dependence on displacements can occur atsome instants or over time intervals), qnln. The non-linear DOFs include all those nodes of the FE modelswhere the non-linear forces can act and also the nodeswhere non-linear interactions might occur. Underdynamic loading, clearances in the structure can beclosed and interferences can be opened up and, as aresult, contact areas can vary during vibration (seeFig. 2). The nodes where non-linear forces act in realityare determined as a result of calculation, and the vector,qnln, usually has a larger number of DOFs than thenumber of nodes where non-linear effects occur. In themajority of practical structures, the number of linearDOFs is much larger than the number of non-linearDOFs.

The term f ðqðtÞ, _qqðtÞÞ is a vector of non-linear frictioninterface forces which are dependent on the displace-

Fig. 2 A friction contact interface

ANALYSIS FOR NON-LINEAR GAS TURBINE STRUCTURES 201


ments and velocities of the interacting nodes. Models ofthe non-linear forces occurring at the friction contactinterfaces describe forces of unilateral contact along adirection normal to the contact surface and frictionforces acting in directions that are tangential to thecontact surface. The effects of variations in the normalload on the friction force are modelled, including thecase of partial separation of the contact surfaces. Themodels of the contact forces include provision of initialvalues of clearances and interferences as parameters.

3 METHOD OF CALCULATION

3.1 Transformation of equations of motion to the

frequency domain using the multiharmonic

balance method

Steady state, periodic regimes of vibration response aresought and a vector of forced response, qðtÞ, satisfyingequation (1) is determined. In order to search for aperiodic vibration response, the variation of all DOFs ofthe system in time is represented as a restricted Fourierseries, which can contain as many and such harmoniccomponents as are necessary to approximate the soughtsolution with a desired accuracy, i.e.

qðtÞ ¼ Q0 þXnj¼1

Qcj cosmjotþQs

j sinmjot ð2Þ

where n is a number of harmonics kept in the multi-harmonic solution, Q0,Q

cj and Qs

j ð j ¼ 1, . . . , nÞ arevectors of harmonic coefficients for the system DOFs,mj ð j ¼ 1, . . . , nÞ are specific harmonics that are kept inthe displacement expansion in addition to the constantcomponent ando is the fundamental vibration frequency.

The fundamental frequency is defined by a period ofthe sought vibration response. In an analysis ofvibration gas-turbine structures for most practical cases,major resonance regimes (i.e. when the period of theforced response is equal to the period of excitationforces) are the subject of interest. The fundamentalfrequency is determined for this case to be o ¼ 2p=T .The multiharmonic representation of the forcedresponse allows superharmonic resonances to be deter-mined along with major resonances. For cases whensubharmonic vibrations (i.e. when the period of vibra-tion is a multiple of the period of excitation forces, j1T ,where j1 > 1 is an integer number) or combinationresonance regimes (i.e. when the period of vibration isequal to j1T=j2, where j1 and j2 are integer numbers)need to be determined, the fundamental frequency canrelate to the period of excitation forces in the followingway: o ¼ 2p=ð j1T=j2Þ.

In accordance with the multiharmonic balancemethod, the expansion from equation (2) is substitutedinto equation (1), after which it is sequentially multiplied

by ðcosmjoÞ and ðsinmjoÞ to reveal all the harmonicsfrom the expansion from which the integrals over thevibration period are calculated. As a result, equationsfor determining all the harmonic components areobtained in the form

ZðoÞQþ FðQÞ � P ¼ 0 ð3Þ

where Q ¼ fQ0,Qc1,Q

s1, . . . ,Q

cn,Q

sng

T is a vector ofharmonic coefficients for the system, FðQÞ is a vectorof non-linear forces, P is a vector of harmoniccomponents of exciting forces and ZðoÞ is the dynamicstiffness matrix of the linear part of the system,constructed for all harmonic components, i.e.

Z ¼ diag Z0,Zm1, . . . ,Zmn

½ � ð4Þ

where

Z0 ¼ K and

Zj ¼K� ðmjoÞ2M mjoC

�mjoC K� ðmjoÞ2M

" #ð5Þ

Although matrix Z is block-diagonal, equation (3)represents a set of simultaneous non-linear equationswith respect to Q because the equations for all harmoniccomponents are coupled through the vector of non-linear forces, FðQÞ, which is dependent on all harmoniccomponents of the displacements.

3.2 Condensation and size reduction for the non-linear

equations of motion

Finite element models of practical gas-turbine struc-tures, and even models of the individual components,usually comprise very many degrees of freedom; indeed,hundreds of thousands or millions of DOFs in themodels are customary in such industrial applications.The number of governing equations is equal to thenumber of DOFs multiplied by the number of harmoniccoefficients in the multiharmonic expansion. The largenumber of simultaneous non-linear equations thusobtained cannot be solved directly due to the size ofthe matrices involved and prohibitively high computa-tional expense. In order to make the analysis of a non-linear forced response feasible for such large systems, aspecial effective reduction technique has been developed.This technique reduces the number of DOFs explicitlyretained in the resolving non-linear equations to thenumber of non-linear DOFs, i.e. to those where non-linear forces are applied. All the other, linear, DOFs areexcluded from the reduced equations while the accuracyof the description of the dynamic properties of thestructure is preserved. After determination of the non-linear DOFs at the non-linear contact interfaces, thedisplacements at any node of interest in the whole large-scale FE model can then be easily determined.



In order to perform this reduction efficiently, it isconvenient to rewrite equation (3) using complexarithmetic. Such a formulation allows the relationshipsto be reduced immediately by a factor of two and isuseful for further reduction. In order to make such aformulation, complex vectors for each of the jthharmonic vectors of complex displacements, qj, non-linear forces, fj, and excitation forces, pj, areintroduced in the following form:

qj ¼ Qcj þ iQs

j , pj ¼ Pcj þ iPs

j ,

fj ¼ Fcj þ iFs

j

ð6Þ

where i ¼ffiffiffiffiffiffiffi� 1

pand all the complex quantities are

written here and further in the paper using a differentfont in order to differentiate them from their realcounterparts. A complex dynamic stiffness matrix foreach jth harmonics, zj, is introduced as

zj ¼ K� ðmjoÞ2Mh i

� imjoC ð7Þ

Equation (3) can now be rewritten separately for each ofthe jth harmonics, taking into account the blockdiagonal structure of the dynamic stiffness matrix, Z,in the following form:

zjqj þfjðQÞ �pj ¼ qj þz� 1j ðfjðQÞ �pÞj

¼ qj þajðfjðQÞ �pjÞ¼ 0 ð j ¼ 0, nÞ

ð8Þ

Here aj ¼ z� 1j ðoÞ is an FRF matrix of a linear

structure determined for a case when non-linear inter-action forces are not applied. In order to avoid thenumerically inefficient operation of matrix inversion, theFRF matrix can be efficiently generated from thenatural frequencies, og, and mode shapes, fg, of thislinear structure:

aj ¼ z� 1j ðoÞ^

XNm

r¼1

fgfTg

ð1� iZgÞo2g � o2

ð9Þ

Here the subscript g is the number of the mode shape inthe spectrum, Zg is the damping loss factor determinedfor the gth mode and Nm is the number of modes thatare used in the modal expansion. In many cases,relatively small numbers of modes, Nm, need to bekept in the expansion to provide sufficient accuracy inthe FRF matrix calculation (analysis of the accuracyobtained for FRF matrices of bladed disc models isperformed in reference [26]).

In order to derive equations with excluded linearDOFs, the vector of complex amplitudes for the jthharmonic, qj, is partitioned into a vector of non-linearDOFs, q

nlnj , and a vector of linear DOFs, q

linj ,

i.e. qj ¼ fqlinj ,qnln

j gT, and then equation (8) can be

rewritten in the form

qlinj

qnlnj

( )þ 0

anlnj fjðQnlnÞ

� ��ajpj ¼ 0 ð10Þ

where anlnj is a minor of matrix aj, corresponding to

the non-linear DOFs, and Qnln is a subset of vector Q,which contains all the harmonic coefficients of displace-ments at the non-linear interfaces. Selecting fromequation (10) those equations that correspond to thenon-linear DOFs, the required equation of significantlyreduced size is

rjðQnlnÞ ¼ qnlnj þa

nlnj fjðQnlnÞ � q

nlnPj

¼ 0 ð11Þ

where qnlnj is a vector of complex amplitudes for the jth

harmonic and qnlnPj

¼ ðajpjÞnln is a vector of complexamplitudes determined for the non-linear DOFs whenthe non-linear forces appearing at the contact interfacesof the analysed structure are not taken into account. Thevector of excitation forces, ~ppj, can include forces appliedto all DOFs of the structure, linear as well as non-linear.The FRF matrix including all the DOFs, aj, is used toevaluate vector, ajpj, and then the components thatcorrespond to the non-linear DOFs are selected from theresulting vector to form q

nlnPj

. When the solution ofequation (11) is found, the forced response for all DOFs,qj, is easily determined from Eq. (8).

Equations of form (11) formulated for all theharmonics of the multiharmonic expansion are non-linear. Owing to the dependence of the non-linear contactinterface forces on all these harmonics, they have to besolved simultaneously. One of the most effective methodsfor the solution of non-linear equations is the Newton–Raphson method. The iterative solution process can beexpressed by the following formula:

Qðkþ1Þ ¼ QðkÞ þ qRqQ

� �� 1

RðQðkÞÞ ð12Þ

where

R ¼ fR0, Reðr1Þ, Imðr1Þ, . . . , ReðrmnÞ, Imðrmn

ÞgT

is a real vector of residuals of the non-linear equations(11). The superscript ðkÞ indicates the number ofiterations and the superscript nln is dropped here. Itcan be seen that a non-linear force vector, FðQnlnÞ, and atangent stiffness matrix, KðQnlnÞ ¼ qFðQnlnÞ=qQnln,which is included in qR=qQ, have to be provided toapply this method. It should be noticed that transform-ing the equation in real arithmetic is a necessaryrequirement in order to apply the Newton–Raphsonmethod. This requirement occurs due to the fact thatgenerally rjðqÞ cannot be expressed as an analyticfunction of the complex vector q and hence derivativesqrðkÞ=qq are not determined for complex arithmetic.



3.3 Friction contact interface elements

When the derivatives qFðQÞ=qQ are calculated usingfinite difference formulae, this results in (a) a very largecomputational effort, (b) difficulty with the choice of thestep for the numerical evaluation of the derivatives and,very often, (c) a loss of accuracy and robustness of thesolution. To overcome these problems, analyticalexpressions have been derived for a non-linear forcevector of the friction contact interfaces, FðQÞ, and for itstangent stiffness matrix, qFðQÞ=qQ. The expressionsderived allow exact and extremely fast calculations to bemade for the contact interaction forces and tangentstiffness matrices as functions of displacements of thecontacting components of the structure. This, in turn,results in outstanding robustness and speed of the forcedresponse calculations for the analysed structure.

As an example of the analytical derivation for contactinterface elements, a friction contact element thatmodels friction forces acting along a direction tangentialto the contact surface as well as unilateral forces (i.e.transferring compression but not tension) acting alongthe normal direction have been described. Distributionof such elements over areas where nodes of the finiteelement mesh can come into contact allows the forcedresponse to be calculated for a structure accounting forvariable contact areas, multiple slip–stick transitionsduring periods of multiharmonic vibration and frictionimpacts at contact–separation transitions. A frictionmodel applied in this paper takes into account theeffects of variations in normal load on friction forces.These effects include not only variations of the frictionforce magnitude but also the influence of the normalload variation on time instants when stick–slip transi-tions occur. These transition times can be different fromtimes when the direction of motion is changing.

During relative motion of the contacting surfaces withfriction, several different states are possible. The motionalong the normal direction, yðtÞ, determines whether theinteracting surfaces are in contact or separated. Whenthe surfaces are in contact then two other different statesare possible: slip or stick depending on the tangentialmotion, xðtÞ, normal motion, yðtÞ, and the history ofthe interaction. Expressions for non-linear interactionforces can be defined for all possible states in thefollowing form:

tangential force

fx ¼f 0x þ ktðxðtÞ � x0Þ for stick

xðtÞmfyðtÞ for slip

0 for separation

8><>:normal force

fy ¼N0 þ knyðtÞ for contact

0 for separation

�ð13Þ

where t ¼ ot is non-dimensional time, the contactsurface mechanical properties are described by a frictioncoefficient, m, and the stiffness coefficients for deforma-tion along tangential and normal directions to thecontact surface, kt and kn, and xðtÞ ¼ + 1 is a signfunction of the tangential force at the time instant of slipstate initiation, tslip. For the conventional case of aconstant normal force, the value of the sign isdetermined by that of the tangential velocity:x ¼ sgn½ _xxðtslipÞ�. However, for the case of a variablenormal load, considered here, the sign is dependent onmutual action of the normal load and the tangentialvelocity on the time of slip–stick transitions. It does notnecessarily coincide with the direction of the relativemotion. To guarantee time continuity of the tangentialforce the sign function is determined by that of thetangential force at the end of the preceding stick phase,i.e. x ¼ sgn½ fxðtslipÞ�.

The times of transitions, tj, between all possible statesof the contact interaction are determined from thefollowing conditions:

j fxðtÞj ¼ mfyðtÞ

x _ffxðtÞ ¼ m _ffyðtÞ and x€ffxðtÞ < m€ffyðtÞ

fyðtÞ ¼ 0 and _ffyðtÞ < 0

fyðtÞ ¼ 0 and _ffyðtÞ > 0

for stick--slip transition

for slip--stick transition

for contact--separation

transition

for separation--contact

transition

ð14Þ

where one or two dots above the symbols denote first orsecond time derivatives, and the expression fromequations (13) corresponding to the contact state isused for the normal load, fyðtÞ.

A more detailed description of the friction model isgiven in reference [16] and the effects of the variations ofnormal load and other contact parameters on frictionforces are demonstrated in reference [17].

Multiharmonic motion of each degree of freedom isexpressed in the following form: xðtÞ ¼ HT

�ðtÞX , yðtÞ¼ HT

�ðtÞY , where X and Y are vectors of harmoniccoefficients of the relative motion of the mating contactnodes in the tangential and normal directions respec-tively and H� ¼ f1, cosm1t, sinm1t, . . . , cosmnt,sinmntgT. Vectors of multiharmonic Fourier expansioncoefficients for tangential, Fx, and normal, Fy, forcescan be obtained in the form

Fx

Fy

� �¼ 1

p

Xntj¼1

$tjþ1

tj

HþðtÞfxHþðtÞfy

� �dt

¼Xntj¼1

J ð jÞx

J ð jÞy

( )ð15Þ



where

Hþ ¼ f1=2, cosm1t, sinm1t, . . . , cosmnt, sinmntgT:

Substitution of the expressions for the interaction forcesgiven by equations (13) into equation (15) gives anexpression for the force vector. Introduced here arevectors of integrals J ð jÞ

x and J ð jÞy over each interval of

stick, slip or separation, which are obtained in the form

J ð jÞx ¼

ktWjX þ cjwj stick

xm N0wj þ knWjY� �

slip

0 separation

8<:

J ð jÞy ¼

N0wj þ knWjY contact

0 separation

�ð16Þ

where

cj ¼ f 0x ðtjÞ � kxxðtjÞ ¼ � xmðN0 þ knyðtjÞÞ � ktxðtjÞ

Wjðn6nÞ

¼ 1

p$tjþ1

tjHþðtÞHT

�ðtÞdt

and wjðn61Þ

¼ 1

p$tjþ1

tjHþðtÞdt

ð17Þ

The stiffness matrix of the friction interface element isdetermined by differentiating equation (15) with respectto vectors X and Y. Because of the independence of thenormal force to the tangential displacement, the stiffnessmatrix has the following, inherently unsymmetrical,structure:

Kf ¼qFx

qXqFx

qY

0qFy

qY

" #¼

Xntj¼1

qJ ð jÞx

qXqJ ð jÞ

x

qY

0qJ ð jÞ

y

qY

24

35 ð18Þ

where

qJ ð jÞx

qX¼

knWj þ wjqcjqX

� T

stick

0 slip

0 separation

8>><>>:

qJ ð jÞx

qY¼

wjqcjqY

� T

stick

xmknWj slip

0 separation

8>><>>:

qJ ð jÞy

qY¼

knWj contact

0 separation

�ð19Þ

Since the vector used for the transformation from thetime domain into the frequency domain, Hþ, and thatfor transformation backwards, H�, consists of sine andcosine functions of different orders, then components ofmatrix W and vector w are simple integrals of sine andcosine functions and integrals of products of these

functions. These integrals can be calculated analytically,thus providing an exact and very fast calculation for thevectors of the Fourier expansion coefficients of theinterface forces and the stiffness matrix.

The stiffness coefficients kn and kt characterizestiffness properties for a contact point of the layer ofmicroasperities that are inherent owing to inevitableroughness of the contact surfaces. The multitude offriction contact elements that developed when distrib-uted over an FE mesh of the structure can model contactareas that are varied during vibration, including adescription of closing and opening interferences andclearances. Hence many different realistic frictioninteractions can be modelled with a detailed descriptionof contact conditions and geometry of the contactsurfaces.

4 NUMERICAL RESULTS

The method developed has been realized in a programsuite FORSE (standing for ‘forced response suite’) andis applied to analysis of realistic bladed discs used in thegas-turbine industry, providing good agreement withexperimental measurements.

4.1 A practical high-pressure turbine bladed disc

As an example of application, the friction contactinterface elements reported above have been appliedfor the analysis of the vibration response of a practicalhigh-pressure bladed turbine disc comprising 92 blades.A finite element model of the sector of the bladed disc isshown in Fig. 3a. It contains more than 160 000 DOFsand the whole bladed disc is described by more than 15million DOFs. The damping loss factor of the bladeddisc without friction dampers included is 0.003. Naturalfrequencies of the high-pressure turbine disc are shownin Fig. 3b for all possible nodal diameter numbers. Acase when there are no underplatform damper and bladeshroud friction contacts, i.e. non-linear interactionforces, are not applied and the system is linear withblades interacting through the disc. The naturalfrequencies are normalized with respect to the firstblade-alone frequency. Cyclic symmetry properties ofthe structure considered were introduced into theanalysis of the non-linear vibration behaviour using amethod developed in reference [27].

4.1.1 Tuned bladed disc with underplatformfriction dampers

For an analysis of the influence of friction dampers onthe forced response of the bladed disc, the nodes whereeach of the bladed disc sectors has a friction contact,



marked as Bleft and Bright in Fig. 3a, are located at theblade platform. Another node, A, where displacementsare calculated is selected near the tip of the blade. Thesenodes are marked in Fig. 3a by circles. For each sector,node B interacts through the friction element with thecorresponding node of the adjacent sector of the bladeddisc and its counterpart on the other, left-hand, side ofthe considered sector interacts with the preceding sector.As an example, forcing by 4th, 6th, 8th and 16th engineorder excitations are considered in the frequency rangecorresponding to the family of first predominantly flap-wise blade modes and in the frequency range of the

second family of natural frequencies. In order tocompare the damping effect produced by the frictionelements, the amplitudes of the excitation loads areassumed to be the same for all engine orders studied.

Examples of maximum response levels for displace-ment at a node A are shown in Fig. 4 where, forcomparison, response levels for the bladed disc withoutthe friction damper are also plotted. Damping producedby the friction dampers decreases the response ampli-tudes significantly and, moreover, the friction dampercan be seen to increase the resonance frequencysignificantly. For both the considered families of modes

Fig. 3 Finite element model of a sector of the turbine bladed disc and nodes of friction contact

Fig. 4 Forced response of the high-pressure turbine bladed disc with a friction damper (solid line) andwithout a friction damper (dashed line) for different numbers of engine orders



(first flap-wise modes and first edge-wise modes withdifferent numbers of nodal diameters), greater dampingeffects appear for the higher engine orders when therelative displacements of neighbouring blades arelargest. In all these calculations faultless determinationof the solution with accuracy kRðQÞk < 10� 8 wasobtained, and for each curve calculated for Fig. 4 overa whole range from 800 to 3200Hz, only 60 seconds ofCPU (central processing unit) time was required on apersonal computer with 450MHz CPU frequency.

4.1.2 The bladed disc with contacting shrouds

The forced response of the same bladed disc wasanalysed when contact of integral shrouds was con-sidered to be possible (Fig. 5). Two cases of contactinteraction are explored: (a) the case when each shroudcan contact its neighbours at one node (this node ismarked by the letter A in Fig. 5b at the left-hand-sidecontact interface); (b) the case when many nodesdistributed over the shroud contact faces can be incontact with the adjoining shrouds (marked by the letterB in Fig. 5b). Different values of gap between theshrouds are considered, namely: (a) a case of inter-ference, i.e. negative gap values: g ¼ � 0:001; (b) a caseof zero gaps: g ¼ 0:0; and (c) a case of positive gapvalues: g ¼ 0:001. The forced response was calculated atthe same node as for the case of the bladed disc withfriction dampers (see the node marked by a letter A inFig. 3a) and excitation by the fourth engine order (4EO)is examined.

Results of the calculations for all the consideredcases are shown in Fig. 6, where, for comparison,forced responses of the bladed disc for the case oflinear vibrations are plotted alongside (a) forcedresponse of the bladed disc when shrouds never contactand (b) forced response of the bladed disc whencontact nodes are always in full contact. It can be seenthat for the shrouds with clearances, the forcedresponse in the frequency ranges where the amplitudes

are smaller than these clearances coincides with theforced response of the bladed disc without contact. Forlarger amplitude levels the system exhibits forcedresponse characteristic ‘stiffening’ with an amplitudeincrease. For the shrouds with interferences the forcedresponse in frequency ranges where amplitudes aresmaller than the interferences coincides with theresponse of the bladed disc with full contact, and forlarger amplitude levels the system exhibits a ‘softening’of the forced response curve. For cases with multiple-contact nodes non-linear behaviour starts at a lowerlevel of response and the resonance frequency can varyover much larger ranges than for a case of single-nodecontact. This effect is especially significant for a case ofclearances. For the case of zero gaps, the forcedresponse differs from that of the linear systemcompared over the whole frequency range analysed.For this case, resonance frequencies differ significantly,although neither stiffening nor softening of theresonance curves occurs. To calculate the forcedresponse over the whole excitation frequency range,from 20 to 360 seconds of CPU time was required,depending on the number of contact nodes and thecomplexity of the forced response curve.

The influence of structural damping on the forcedresponse curves was also analysed, and an example ofresults obtained for the case of multiple node contactand for two different values of the structural damping,Z ¼ 0:003 and Z ¼ 0:02, is shown in Fig. 7 for the case ofthe gap equal to 0.001. Here it can be seen that asufficient level of damping for this strongly non-linearsystem has two favourable effects: (a) similar to the caseof linear vibration, it reduces the level of resonanceamplitudes and (b) it also reduces drastically thefrequency range where the resonances (which dependon amplitude for the non-linear system) can be excited:e.g. instead of the frequency range 1.15, . . . , 2.2, wherethe second resonance can occur for a case of Z ¼ 0:003,for larger damping, Z ¼ 0:02, a relatively high level ofamplitudes exhibits only in the small vicinity offrequency 1.15.

Fig. 5 Integral shroud model and nodes of possible contact considered in the calculations



4.2 Impact damper test piece

The capabilities available by using the gap elementsdeveloped above for the analysis of structures withimpact dampers were explored on the example of a test

piece of an impact damper test rig. The test piececonsists of a beam and an impact damper placed in achamber made within the brick (Fig. 8). The beam andthe impact damper are both modelled using solid finiteelements and a finite element model of the test piece

Fig. 6 Forced response of the high-pressure turbine bladed disc: (a) shrouds have a single contact node; (b)shrouds interact through many contact nodes

Fig. 7 Forced response of the turbine bladed disc for different levels of structural damping



consists of more than 96 000 DOFs. The beam isclamped at one end and vibrations are excited by aforce applied at the other end, with proportionalstructural damping assumed to be 0.0016. Frictiondamping and energy dissipation due to inelasticdeformation at impacts are neglected and unilateralinteraction of the damper and the brick during vibrationis modelled by the gap elements only. The possibilities ofreducing the maximum forced response level by varyingthe impact damper parameters (gaps, contact stiffnesscoefficient, mass) were explored. It was found that massvariation was the most effective parameter in thepractically imposed restrictions on the parameter values.

Examples of calculated forced responses for threeimpact damper configurations are shown in Fig. 9.These three different impact dampers (A, B and C) haverelative mass values mA < mB < mC, and the gapsbetween the walls of the internal brick chamber andthe impact damper are 0.2mm in all cases. Forreference, the forced response of the beam without theimpact damper, i.e. when vibrations are linear, is alsoshown. It can be seen that the proper choice ofparameters for the impact damper can reduce resonanceamplitudes significantly, although use of such an impact

damper device can increase the level of amplitudes thatare far from resonances. In contrast to the linear system,the introduction of the impact damper makes the forcedresponse curves multivalued. The choice of whichbranch of the calculated forced response to follow inpractice depends on the direction of the excitationfrequency sweep and on perturbations of displacementsand velocities for the beam and the friction damper. Itwas found that 20 seconds of CPU time were required tocalculate the forced response over the whole frequencyrange considered in Fig. 9 for each case of the damperdesign.

5 CONCLUDING REMARKS

A new method has been developed for the analysis of thesteady state non-linear vibration response of a gas-turbine engine and other machinery structures withjoints and friction contacts. The method developedenables extremely fast and robust calculations to bemade of multiharmonic forced responses for a widerange of problems occurring in the dynamics of complex

Fig. 8 Impact damper test rig: (a) finite element model of the test piece; (b) the impact damper

Fig. 9 Forced responses of the impact damper test rig for impact dampers of different masses



structures using large-scale finite element models. Themethod offers the possibility for a breakthrough in theanalysis of non-linear forced responses of structureswith gaps and friction.

The friction model developed takes into account theeffects of normal load variations of an arbitrary level,including separation of contact surfaces on frictionforces. Analytical formulation of the friction contactelements provides exact expressions for the contactforces and stiffness matrices of the contact interface andallows fast and robust calculations to be made for thenon-linear forced responses.

The high efficiency of the method is demonstratedusing examples of calculations for practical bladed discswith friction and impacts and on specially selected testcases. Finite element models with large numbers ofdegrees of freedom and with multiple friction andimpact contact nodes were analysed.

ACKNOWLEDGEMENT

The authors are grateful to Rolls–Royce plc forproviding the financial support for this project and forgiving permission to publish this work.

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