a method for use of cyclic symmetry properties in analysis

8/7/2019 A Method for Use of Cyclic Symmetry Properties in Analysis

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E. P. PetrovCenter for Vibration Engineering,

Mechanical Engineering Department,Imperial College London,

South Kensington Campus,

London SW7 2AZ, UK

A Method for Use of CyclicSymmetry Properties in Analysisof Nonlinear MultiharmonicVibrations of Bladed Disks

An effective method for analysis of periodic forced response of nonlinear cyclically sym-metric structures has been developed. The method allows multiharmonic forced responseto be calculated for a whole bladed disk using a periodic sector model without any loss ofaccuracy in calculations and modeling. A rigorous proof of the validity of the reduction ofthe whole nonlinear structure to a sector is provided. Types of bladed disk forcing forwhich the method may be applied are formulated. A multiharmonic formulation and asolution technique for equations of motion have been derived for two cases of description

for a linear part of the bladed disk model: (i) using sector finite element matrices and (ii)using sector mode shapes and frequencies. Calculations validating the developed methodand a numerical investigation of a realistic high-pressure turbine bladed disk withshrouds have demonstrated the high efficiency of the method. DOI: 10.1115/1.1644558

Introduction

It is well known that in analysis of linear vibration, use of

cyclically symmetric properties of bladed disks allows the analy-

sis of a whole bladed-disk assembly to be reduced to analysis of

its one sector, which usually comprises a blade, a disk sector and,

possibly, parts of shrouds adjoined to the blade. Methods for use

of the cyclic symmetry properties of linear mechanical structures

in analysis of natural frequencies and mode shapes, and in forced

response analysis, have been developed more than 30 years. Many

papers on the subject including papers 1 3, together provide atheoretical basis and methods for analysis of linear vibrations of

cyclically symmetric structures. It has been shown that cyclic

symmetry allows a sector model to be used instead of a whole

bladed disk model, which decreases number of degrees-of-

freedom in the resulting equations by a factor equal to the numberof blades in a bladed disk, without any compromise with model-

ing accuracy. Coupling of vibrations of all sectors can be ac-

counted for by special boundary conditions imposed on the inter-

faces where a sector considered interacts with neighboring ones.

This method of taking into account cyclic symmetry, for linear

systems, has become conventional and numerous applications in

analysis of linear vibrations of bladed disks and other structures

have been developed. An option allowing use of cyclic symmetry

has become a standard in the majority of commercial finite ele-

ment programs.

In contrast to the case of linear systems for nonlinear systems,

up to date, there has been no a rigorously formulated and univer-

sal method available in the literature that would take advantages

of cyclic symmetry for analysis of strongly nonlinear vibrations in

a consistent way.There are very few papers on numerical analysis of strongly

nonlinear vibration of cyclically symmetric structures but calcula-

tion techniques and investigations provided in papers 4,5 for amodels with cubic nonlinear springs should be noticed.

One of first calculations of cyclically symmetric bladed disks

with gaps was proposed in 6, where single degree-of-freedomand beam models of blades are explored. Later papers on analysis

of bladed disks with underplatform dampers by 7,8 andshrouded bladed disks 9, differ from the above-mentioned paperby modeling the bladed disk and friction forces, but use similarapproaches to account for the cyclic symmetry, which are basedon assuming that excitation and vibrations are monoharmonic andleaving aside the question about applicability of the method de-veloped for linear structures with cyclic symmetry to a nonlinearstructure. Multiharmonic vibrations of bladed disks are analyzedin 10,11 although the method for calculation of the cyclicallysymmetric bladed disks is not elaborated in either paper.

In this paper an effective method for analysis of steady-statenonlinear vibrations of cyclically symmetric bladed disks sub-

jected to arbitrary distributed in space and periodic in time loadsis proposed.

At the beginning the possibility for reduction of calculation of

vibrations of a whole bladed disk to analysis of its sector model isrigorously proved and conditions of the method applicability arederived. Types of forcing that can occur in practice of bladed diskanalysis and satisfy to the conditions are discussed. Then the non-linear equations of multiharmonic motion are formulated for asector model for two cases: i using finite element matrices andii using frequency response function matrices. Further decreaseof computational efforts by exclusion of linear degrees-of-freedom is introduced for the latter case. A technique for usingcomplex arithmetic for the calculation of the multiharmonic non-linear equations of motion together with the necessary transforma-tion into domain of real numbers for their solution is developed.Calculations validating the developed method are performed and anumerical investigation of a realistic high-pressure turbine bladeddisk with shrouds is carried out.

Basis of the Method

Bladed Disks of Cyclically Symmetric Design: Sector andOrder of Cyclic Symmetry. Bladed disks are designed in ma-

jority of cases to be cyclically symmetric. Cyclically symmetricbladed disks are such bladed disks for which it is possible to finda rotationally periodic, cyclic part of the structure which canform the whole bladed disk by simple rotations of this part rela-tively to the axis of the symmetry. The rotation angles have to bemultiples of 2/N where integer number N is so-called order ofthe cyclic symmetry, which is equal to the number of cyclic partsin the whole structure. This cyclic part is sometimes called also

Contributed by the International Gas Turbine Institute and presented at the Inter-

national Gas Turbine and Aeroengine Congress and Exhibition, Atlanta, GA, June

1619, 2003. Manuscript received by the IGTI December 2002; final revision March

2003. Paper No. 2003-GT-38480. Review Chair: H. R. Simmons.

Copyright 2004 by ASMEJournal of Turbomachinery JANUARY 2004, Vol. 126 175

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sector reflecting the fact that the cyclic part can be in manycases separated from the whole structure by two planes passingthrough the symmetry axis with angle 2/N between them. Thestructure possesses all cyclic symmetry properties also for a gen-eral case when surfaces separating the cyclic part can have very

complex and curved shapes. Because of that word sector isused in this paper in a general meaning assuming that there is norestrictions on shapes of the sector boundaries.

As evident all co-factors, Nj , of NN1*N2*N3 . . . , if such ex-ist, and all possible combinations of multiplication of these co-factors are also orders of cyclic symmetry. In order to take amaximum of advantages of using cyclic symmetry properties thelargest possible order of cyclic symmetry is used in analysis. Thischoice of the cyclic symmetry order provides the smallest sectorof the cyclically symmetric bladed disk and accordingly the small-est number of degrees-of-freedom in the sector model. Number ofblades that has to be included into one sector depends on a designof a bladed disk. For bladed disks of gas turbine engines a sectorcan usually include only one blade as shown in Fig. 1 and num-ber of sectors, N, is equal to the number of blades, although thereare cases when bladed disks have such design that the sector hasto include two or more blades.

Major Relationships and Applicability Conditions. If abladed disk is cyclically symmetric its equations of motion can bewritten in the following form:

KqjCqjMqjfiqjflqj1 ,qjfrqj ,qj1pj t(1)

where j1,N is a sector number; K, C, and M are conventionallinear stiffness, damping and mass matrices of one sector whichare the same for all sectors of the cyclically symmetric system;qj(t) is a vector of displacements of j-th sector; pj( t) is a vectorof excitation forces; fi(qj) is a vector of nonlinear forces whichare dependent only on displacements of the considered jth sector;

fl(qj1 ,qj) and fr(qj ,qj1) are forces of nonlinear interaction ofjth sector and sectors adjacent to it from the left and from theright, they are dependent on displacements of the considered, jthsector and neighboring (j1)th and (j1) th sectors. Due to thecyclic symmetry the functional dependencies of vectors fi(qj),fl(qj1 ,qj) and fr(qj ,qj1) on the displacements are the same forall sectors, and, moreover, fl(qj ,qj1)fr(qj ,qj1), althoughdifferent displacements can produce different force values.

Equation 1 represents a system of N matrix equations. In thissystem of equations subscript values 0 and N1 appearing inqj1 for j1 and jN are replaced by Nand 1 accordingly sincefirst sector interacts with Nth sector. If arbitrary periodic excita-tion forces, pj( t), applied to each jth sector are similarly distrib-

uted over sector nodes and different only in a fixed phase shift, t,between adjacent sectors, then these excitation forces can be writ-ten in the form

pj tp tj1 t (2)

where tT/N and T is the period of the force variation, ort0 when the phase shift between the forces applied to differentsector is 0.

For this kind of excitation and when linear and nonlinear prop-erties of the bladed disk are cyclically symmetric it is possible towrite explicit relationship between displacements of all sectors.

This relationship has the formqj tq tj1 t (3)

where q is the vector of displacements of first sector, which is alsoperiodic with period T. The relationship given by Eq. 3 is themajor relationship which allows all forces applied to a sector ofthe bladed disk including forces of interaction with adjacent sec-tors to be expressed in terms of its own displacements only. As aresult the dynamic equation for a sector can be written in the formthat describe vibrations of the whole bladed disk. Validity of thisrelationship can be proved by substitution of Eqs. 2 and 3 intoEq. 1. As a result of such substitution one obtains a system of Nequations of identical form:

Kq tjCq tjMq tjfiq tjflq tjt,q tj

frq tj,q tjtp tj (4)where tjt(j1)t and j takes values from 1 to N. One cansee that these equations differ for different sectors by the phase ofthe time variable only and the solution found, q( t), for one sectorfrom Eq. 4 satisfies equations for all sectors.

Moreover, comparing in Eq. 4 fl and fr allows a relationshipfor forces of interaction between the considered sector and adja-cent to it on left and right boundaries to be written in the follow-ing form:

fr tfl tt. (5)

This property together with Eq. 3 provides us a basis forcalculation of forced response of the whole cyclically symmetricbladed disk using its sector model without introduction of anyassumptions and loss of accuracy of the calculations even for the

case of nonlinear vibrations with strong nonlinear forces of anykind.

Types of Forcing That can be Analyzed by the Method.The condition formulated for excitation forces by Eq. 2 togetherwith the cyclic symmetry of the bladed disk are the only condi-tions that have to be satisfied to apply the method proposed. Typesof the excitation forces that occur in analysis of vibration ofbladed disks and satisfy to this condition are briefly discussed inthis section.

Excitation of Traveling Wave Type. This type of excitation isone of the most customary in analysis of vibrations of bladeddisks. It is typical for rotating bladed disks when they are excitedby aerodynamic or other forces that travel relatively to the bladed

disk due to the bladed disk rotation with constant speed whilepreserving their spatial distribution. In cylindrical coordinates, r,z, and , distribution and travelling of these forces can be ex-pressed in the form: pp(r,z,t), where sign corre-sponds to forward travelling wave and sign correspond tobackward travelling wave. All forces of this type satisfy to thecondition given by Eq. 2. In our method no restrictions on dis-tribution of the forces over a bladed disk are imposed and theseforces can be distributed arbitrarily. Being expanded into Fourierseries with respect to bladed disk circumferential coordinate, ,they can comprise many harmonic components. The rotationspeed, , determines the period of the excitation: T2/ andphase shift is determined as tT/N, where sign corre-

Fig. 1 A cyclically symmetric bladed disk; a a whole struc-ture, b a sector

176 Vol. 126, JANUARY 2004 Transactions of the ASME



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spond to the forces travelling forward and sign correspondsto the forces travelling backward with respect to the bladed disk.

The conventional engine-order excitation is a particular kind ofthe travelling wave forcing considered here. For engine-order ex-citation the forces are distributed along the circumferential direc-tion of the bladed disk as a sine or cosine function. A number ofwaves along the circumference is prescribed by the engine order.

Excitation That is Stationary in Space. This type of excitationis characteristic for analysis of bladed disk vibration when thebladed disk is immobile, as in static test rigs with shakers attachedto the bladed disk, although it can also occur in rotating bladed

disks under a special kind of excitation when the excitation forcesare moving together with the rotating bladed disk. The stationaryexcitation forces can be expressed in the form pp s(r,z,)p t(t) where p s(r,z,) is a function of spatial distri-bution of the forces and p t(t) is a periodic function of the forcevariation in time. The conditions of the method applicability canbe satisfied in the following cases.

1. A case when distribution of forces is the same for each of allsectors of the bladed disk, i.e., when the function of the spatialdistribution, p s(r,z,), is periodic with the period equal to a sec-tor angle, T2/N:

psr,z,psr,z,T. (6)

For this case Eq. 2 is satisfied and phase shift between forcesapplied to different sectors is zero, i.e., t0.

2. The method proposed can also be applied for a more com-plex case, when the function of the spatial distribution can differfor several sectors. However, to apply the method for this case aperiod for this spatial distribution has to exist. This period, T ,can stretch over a group that comprises several sectors, i.e., Tns2/N where ns is the number of sectors in the group. In orderto satisfy Eq. 2 the group of ns sectors has to be considered as anew super sector. Number of such super sectors in the whole

system is determined as N*N/ns . This allows this case to bereduced to the case considered in the previous paragraph althoughwith smaller magnitude of the order of the cyclic symmetry.

Multiharmonic Formulation of Equations of Motion Us-

ing a Sector Model

Multiharmonic Expansion of Displacements. In caseswhere the excitation forces are periodic, it is usually desirable tofind steady-state, periodic regimes of response variation. For asearch of the periodic vibration response the variation of alldegrees-of-freedom DOF of the system in time can be repre-sented as a restricted Fourier series, which can contain as manyand such harmonic components as are necessary to approximatethe sought solution, i.e.,

q tQ0j1

n

Qjc cos mjtQj

s sin mjt (7)

where Qj(c) and Qj

(s ) (j1 . . . n) are vectors of harmonic coeffi-

cients for system DOFs upon cosine and sine components marked

by superscripts (c) and (s) accordingly; Q0 is a vector of constantcomponents of the displacements; mj (j1 . . . n) are specificnumbers of harmonics that are kept in the displacement expansionin addition to the constant component. Equation 7 can be rewrit-ten in more concise and convenient for our derivation form:

qHT IQ (8)

where H1,cos m1,sin m1, . . . ,cos mn,sin mnT is a vector of

harmonic functions used in the multiharmonic expansion; Q

Q0 ,Q1(c)

,Q1(s )

, . . . , Qn(c)

,Qn(s )T is a total vector including all

harmonic coefficients for the sector DOFs; t is dimension-less time, I is identity matrix of size equal to the number of DOFs

in the sector; and is a symbol of the Kronecker matrix productoperator, 12. This operator maps here vector H and matrix I intothe following rectangular matrix:

HT

12n1

INqNq

I,cos m1I,sin m1 I, . . . ,Nq2n1 Nq

cos mnI,sin mnI

(9)

where Nq is size of vector q.

Formulation Using a Finite Element Sector Model. In ac-cordance with the multi-harmonic balance method, the expansionfrom Eq. 7 is substituted into the equation of motion 4, afterthis Eq. 4 is sequentially multiplied by (cos mjt) and (sin mjt)for all harmonics from the expansion and integrals over the vibra-tion period, T, are calculated. As a result, the Eq. 4 of motion intime domain is transformed into a frequency domain equation inthe following form:

ZQFQP (10)

where F(Q)F0(Q) ,F1(c)(Q), F1

(s)(Q), . . . ,Fn(s)(Q)T is a vector

of harmonic components of nonlinear forces; P

P0 ,P1(c) ,P1

(s) , . . . , Pn( s)T is a vector of harmonic components

of the excitation forces, and Z() is the dynamic stiffness matrixof the linear part of the system, constructed for all harmonic com-

ponents, i.e.,ZdiagZ0 ,Z1 , . . . , Zn (11)

where

Z0K; and Zj Kmj2M mjCmjC Kmj

2M .

(12)

This equation is nonlinear with respect to the harmonic compo-nents of the displacements, Q.

In order to take into account conditions imposed on the sectorof cyclically symmetric bladed disk by Eqs. 3 and 5 one canpartition the vector of sector displacements into three vectors: i avector of displacements at nodes located at the left boundary ofthe sector, ql ; ii a vector of internal displacements, qi , and iiia vector of displacements at nodes located at the right boundary ofthe sector, qr .

The major relationship, Eq. 3, imposes constraints on dis-placements of the nodes located at left, ql , and right, qr , bound-aries of the sector where this sector interacts with adjacent ones.

qrql (13)

where 2/N and sign or is chosen here accord-ingly to the direction of the rotation of the excitation forces in Eq.2.

These constraints can be formulated with respect to harmoniccoefficients for displacements at the right, Qr , and left, Ql ,boundaries with the help of Eq. 8 in the form

HT IQrHT IQl . (14)

The vectors of the multiharmonic expansion, H, with a constant

phase shift, , are related asHTH (15)

where matrix of the transformation, T, takes the following form:

Tdiagt0 ,t1 , . . . , tn (16)

where

t01 and tjcos mj sin mjsin mj cos mj

for j1 . . . n .From Eq. 14 and Eq. 15 the following relationship betweenharmonic coefficients for the displacements at right and leftboundaries can be derived:

Journal of Turbomachinery JANUARY 2004, Vol. 126 177



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QrT IQl . (17)

Selecting from Eq. 17 expressions for each jth harmonic of thedisplacements one can write a relationship between coefficients ofthe multiharmonic expansion at left and right sector boundaries:

Qrc

Qrs

j

cos mjI sin mjIsin mjI cos mjI

Qlc Ql

sj

(18)

and then harmonic coefficients of all sector displacements can beexpressed through displacements at internal and left boundary

nodes in the form

Ql

c

Qic

Qrc

Qls

Qis

Qrs

j

I 0 0 0

0 I 0 0

cos mjI 0 sin mjI 0

0 0 I 0

0 0 0 I

sin mjI 0 cos mjI 0

Qlc

Qic

Qls

Qis

j

Gj

Qlc

Qic

Qls

Qis

j

. (19)

Relationships for harmonic coefficients of the interaction forcesare derived from Eq. 5 and their form is similar to Eqs. 17 and19 but F should be written instead of Q.

Cyclic symmetry properties are taken into account by substitut-ing Eq. 19 into Eq. 10. This allows equation of motion of thesector to be obtained in terms of displacements of the one sectoronly whereas taking into account forces applied to the sector fromadjacent parts of the bladed disk. Moreover, the number of un-knowns in the equation is reduced by excluding displacements onright boundary of the sector and thus results in the followingequation:

EFEQZQFQP0 (20)

where ZGTZG; F

(Q

)GTF(Q

); PGTP; GdiagG0 ,

Gj , . . . , Gn] and Gj is a matrix used in Eq. 19 for expression ofall sector DOFs through internal and left boundary DOFs for eachharmonic.

Equation 20 represents a nonlinear set of equations with re-

spect to QQl ,QiT which describes vibration of a whole

bladed disk exactly using a finite element model for one sectoronly.

Formulation With Use of Complex Arithmetic. Specialstructure of matrix, Z, allows the computational efforts necessaryfor evaluation of the vector of residuals, EQ to be reduced. Thisis achieved by formulation with a use of complex numbers, whichallows all relationships to be simplified and, moreover, size of thematrices to be reduced by a factor of two. In order to make such

formulation the complex vectors for each jth harmonics of dis-placements, Qjj , nonlinear forces, Fj , and excitation forces, Pjare introduced in the following form:

QjQjc iQj

s ; PjPjc iPj

s ; FjFjc iFj

s

(21)

where i1 and all complex quantities in order to differ themfrom their real counterparts are written here and further in thepaper using a different font. A matrix of complex dynamic stiff-ness for each jth harmonics, Zj , is introduced as

ZjKmj2Mim jC. (22)

Equation 10 can now be rewritten, with taking into account theblock diagonal structure of the matrix Z, separately for each jthharmonics in the following form:

ZjQjFjQPj ; j0,n. (23)

It should be noted that vector of jth harmonics of the nonlinearforces, Fj(Q), is dependent on all multiharmonic components ofdisplacements, Q, because of that all harmonics components ofEq. 23 are interdependent.

The expression given by Eq. 19 for all sector displacementsthrough internal displacements and displacements of nodes lo-

cated at left boundary is transformed into the following form:

QlQiQr

j

I 00 Ie im jI 0

QlQi jGjQlQi j . (24)Matrix of this transformation, Gj , allows Eq. 20 to be rewrittenfor each jth harmonic in the form:

EjFE

QZjQ

jF

jQPj0; j0,n (25)

where ZjGj*ZjGj ; FQG*FQ and PG*P and an asterisk

* denotes Hermitian conjugate. A relationship between vectors

of residuals expressed in real numbers, EjFE

, and in complex

numbers, EjFE

, takes the following form:

EjFE ReEj

FE

ImEjFE . (26)

Mutiharmonic Formulation Using Sector Frequency Re-sponse Function Matrices. For many practical cases a formu-lation based on a use of dynamic compliance or frequency re-sponse function FRF matrices can be significantly more efficientthan the formulation which uses finite element FE matrices di-rectly. These cases are, for example, when number of DOFs wherenonlinear interaction occurs is much smaller that total number ofDOFs in a sector model or when information about dynamic prop-erties of a bladed disk with nonlinear interactions is provided bymode shapes and natural frequencies calculated by conventionalFE software.

The multiharmonic formulation using sector FRF matrices can

be made with a use of complex arithmetic. The dynamic stiffnessmatrices of the sector, Zj(mj), used in Eq. 25 for each of themultiharmonic components describe linear forces. These matricesare independent on vibration amplitudes and Eq. 25 can be mul-

tiplied by AjZ

j1(mj) order to obtain equations in the follow-

ing form:

EjFR F

QQjA

jF

jQPj0; j0,n (27)

where Ajis a sector FRF matrix determined for j-th harmonic ofthe multiharmonic expansion. This is a sector matrix that is cus-tomary used in analysis of linear vibration of bladed disks. It canbe obtained by direction inversion of dynamic stiffness matrix of

the sector, Zj(mj). However for structures with large number ofDOFs such calculation of the FRF matrix require large computa-

tional expenses, because of that special techniques are applied thatare discussed in the following sections.

Exclusion of Linear Degrees-of-Freedom From the NonlinearEquations. Usually in models of practical bladed disks a numberof DOFs where nonlinear interaction forces are applied is muchsmaller than the total number of degrees-of-freedom in a sector. Ause of the FRF matrices allows all DOFs where only linear forcesare applied to be excluded easily and without any loss of accuracyand completeness of the model. As a result the resulting equationis formulated only with respect to DOFs where nonlinear forcesare applied and size of this equation and accordingly computa-tional expense are reduced usually in many times or orders.




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In order to exclude linear DOFs Eq. 27 can be rewritten in the

form where vector Qj is partitioned into a vector of nonlinear

DOFs, Qjnln , and a vector of linear DOFs, Qj

lin :

EjFRFQ Qj

lin

Qjnln 0AjnlnFjQnlnAjPj0 (28)

where Ajnln

is a minor of matrix Aj corresponding to nonlinear

DOFs. Selecting from Eq. 28 equations corresponding to non-linear DOFs we obtain the sought for equation of significantlyreduced size:

EjFRF

QnlnQjnlnAj

nlnFjQ

nlnQP jnln0 (29)

where QnlnQ0nln ,Q1

nln , . . . , Qnnln is a vector comprising all non-

linear harmonic coefficients. QP jnln(AjP

j)

nln is a vector of com-

plex amplitudes determined for nonlinear DOFs of the bladed diskexcited by j-th harmonic of the excitation forces. These ampli-tudes are determined for a completely linear system when thenonlinear forces appearing at contact interfaces of the bladed disk

are not taken into account. The vector of excitation forces, Pj ,can include forces applied to all DOFs of the structure, as linear

and as nonlinear. FRF matrix including all DOFs, Aj , is used to

evaluate vector AjP

j and then components corresponding to non-

linear DOFs are selected from the resulting vector to form QP jnln .

Calculation of Sector Frequency Response Function Matrices.The sector FRF matrix can be efficiently generated from naturalfrequencies, rm

j, and mode shapes, rm

j, calculated for the sec-

tor for a harmonic number needed, mj , in order to avoid verynumerically inefficient operation of the matrix inverse

AjZ

j1mj

r1

Nm rm jrm j*

1irmjrm j2mj

2(30)

where subscript r is a number of the mode shape in a family ofmodes with mj circumferential waves of displacements; rm j

is

damping loss factor determined for rth mode of the family withmj waves, and Nm is the number of modes that are used in themodal expansion. In many cases small numbers of modes, Nm ,

kept in the expansion provides sufficient accuracy in the FRFmatrix calculation analysis of the accuracy obtained for FRF ma-trices of bladed disks models is performed in 13.

It should be noted that Eq. 30 is exact when all sector modesof the family of modes with mj waves are included in the expan-sion and when the damping matrix, C, represents proportionaldamping i.e., it can be expressed as a linear combination of thestiffness matrix and mass matrix.

The following form of presentation for the damping matrix de-scribes as structural, frequency-independent damping and as vis-cous damping:

Cj11

mjK2K

3M

(31)

where first summand describes structural, hysteresis damping and

two other summands correspond to damping of viscous type; herek (k1,2,3) are coefficients that can be determined theoreticallyor experimentally. The modal damping loss factors can be thenexpressed through these coefficients in the form:

rm j1mj23 /rm j

2 . (32)

Assuming here 230 one can describe pure hysteresisdamping, and when 10 the damping is pure viscous. In manycases the modal damping loss factors, rm j

, are determined di-

rectly from experiments for each modes included into Eq. 30.The expansion given by Eq. 30 can be accurate enough in manypractical cases even when damping is not proportional but level

of linear damping described by matrix C and interaction betweendifferent modes caused by this damping are small.

The vector of complex amplitudes, QP jnln

, introduced in Eq. 29can be very efficiently calculated as

QP jnln

r1

Nm rmj

* Pj

1irmjrm j

2mj

2rmj

nln . (33)

Multiplication of large vectors, rmj

* Pj , involving all sector

DOFs provides a single coefficient for each mode shape, which isusually called a modal force. These modal forces take into ac-count arbitrary distribution of the excitation forces over all nodesof the sector model. For a case when distribution of the excitationforces over a sector is preserved for all rotation frequencies addi-tional decrease of computational efforts can be achieved since forthis case calculation of the modal forces involving many degrees-of-freedom is carried out only one time. At the same time for thiscase values of the forces can be varied with variation of the rota-tion frequency when a coefficient of variation is common for

forces applied to all DOFs. As a result calculations of QP jnln for

each current value of the rotation speed, , is performed by sum-

mation of mode shapes, rmjnln , containing only nonlinear DOFs,

which usually represent a very small fraction of all DOFs in a

sector model. In this summation each such mode, rm jnln , is multi-

plied by a simple coefficient which is product of the modal force

and an explicitly given function of the rotation frequency.

Solution of Nonlinear Equations

Nonlinear Equations Formulated in Real Numbers. One ofthe most efficient methods for solution of the nonlinear equationsis the Newton-Raphson method which possesses quadratic conver-gence when an approximation is close enough to the solution. Aniterative solution process is expressed by the following formula:

Qk1 QkEQkQ

1EQk (34)where superscript (k) indicates the number of the current iteration.

Vector of residuals, E(Q(k)) can be taken for this equation as fromFE and as from FRF formulation.

For a case of FE model performing differentiation of Eq. 20with respect to Q, one can obtain the following expression forderivatives of the residuals:

EQk

QZ

FQk

QZKnlnQ (35)

where Knln(Q) represents a so-called tangent stiffness matrix,i.e., a stiffness matrix describing stiffness properties of the non-linear contact interfaces in vicinity of current values of the dis-placements, Q. An efficient method for analytical derivationwhich provides explicit, extremely fast and exact expressions ofthe stiffness matrices and the nonlinear forces at contact interfaceswith friction forces, gaps and interferences can be found in papers,

14,15.The case of a use of FRF sector matrices can be efficiently

analyzed using complex arithmetic and is considered below.

Nonlinear Equations Formulated in Complex NumbersAlthough nonlinear equation of motion can be formulated andconveniently evaluated using complex arithmetic but at the stageof their iterative solution by Newton-Raphson method they haveto be transformed back into domain of real numbers. This is veryimportant and necessary requirement since residuals, Ej , are not

analytic functions of complex vectors, Qj , and as result of this

their derivatives, Ei /Q

j cannot be defined.




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After calculation of complex residuals for a sector, Ej , whichcan be made very efficiently using FRF formulation, vector of realresiduals is calculated from their complex counterparts as

EE0 ,ReE1,ImE1, . . . ,ImEnT. (36)

The matrix of derivatives of the residuals involved into Eq. 34 iscalculated for the case when complex FRF sector matrices areused by the following way:

EFRF

QI

D00 ReD01 ImD02 ImD0,2n

ReD10 ReD11 ReD12 ReD1,2n

ImD10 ImD11 ImD12 ImD1,2n

ImDn0 ImDn1 ImD12 ImDn,2n

(37)

where

D00Ainln

Fi

Q0; Di,2iAi

nlnFi

Qic

; Di ,2i1Ainln

Fi

Qjs

for i0,n. As seen in first row of matrix, EFR F/Q, real and

imaginary parts of D0j are alternating and all the other rows con-tain only real or imaginary parts of Di j (j0,2n) and these rowsare also alternating.

Numerical Results

Comparison of Results Obtained With the ProposedMethod and With a Whole Bladed Disk Model. In order tovalidate the method forced response of cyclically symmetric sys-tems obtained with proposed method was compared with resultsobtained using whole models of structures when cyclic symmetryproperties were disregarded.

As an example of a bladed disk that was calculated using asector model and a whole bladed disk model, is shown in Fig. 2.The whole bladed disk consists of eight sectors and is modeled byhexahedral finite elements with a total number of degrees-of-freedom in the model equal to 5760; the sector model comprises720 DOFs. Excitation by first engine order harmonic is consideredand the damping loss factor was accepted to be 0.003. Nonlinearinterface elements are applied between adjacent blades at a middlenode of blade tip faces and as illustrated in Fig. 2 by boxes witharrows.

The nonlinear interface elements produce forces that are depen-dent on relative displacements of the adjacent blades. Three dif-ferent types of the nonlinear interface elements were examined: ia friction damper with the friction coefficient 0.3; the normal

force 2*105 N and stiffness coefficient 4*10

4 N/mm); ii a

spring with cubic dependence, f(x)cx 3, of the nonlinear inter-

action force, f(x), on relative displacement, x, stiffness coeffi-

cient, c, of the spring is taken to be 103 N/mm); iii a gap ele-ment, which does not produce any force unless the prescribed gapis closed and when the gap is closed then interaction force isproportional to the relative displacement a gap value is taken tobe 0.5 mm and stiffness coefficient when the gap is closed is equal

to 105 N/mm). Explicit expressions for tangent stiffness matrix

and for multiharmonic components of the interaction forces pro-duced by these elements are used in for calculations in the form asthey are analytically derived in papers 14,15. For all these inter-face elements forced response of the bladed disk exhibits stronglynonlinear behavior. First five harmonics from 0 to 4 are kept inmultiharmonic expansion of the forced response and maximumdisplacements for all three coordinates, x, y, and z, of displace-ments at the nodes where the nonlinear interface elements werecalculated. Results obtained by using the sector model and byusing the whole bladed disk are identical. This is demonstrated inFig. 3, where lines plot results obtained with the whole bladeddisk model and results obtained with the proposed method areshown by circles, squares and triangles for x, y, and z coordinatesaccordingly.

Realistic High-Pressure Turbine Bladed Disk. As an ex-

ample of a practical application a turbine high-pressure bladed

disk shown in Fig. 4 is considered. The bladed disk comprises 92

shrouded blades. The damping loss factor is set to 0.003 and ex-citation by 4, 6, 8, and 16 engine orders is considered in theanalysis. Natural frequencies of the high-pressure turbine disk

normalized with respect to the first blade-alone frequency areshown in Fig. 5 for all possible nodal diameter numbers from 0 to46. These frequencies are calculated for the case where there is nocontact between shrouds.

Nonlinear forces can occur under certain circumstances during

vibration as a result of contact interaction between blade shrouds.These nonlinear forces and nonlinear stiffness matrices of the con-tact interface were calculated using friction contact interface ele-ments developed in 14. The contact interface elements allow totake into account unilateral nature of the force normal to the con-tact surface, friction forces with accounting for the normal loadvariation and they also allow consistent determination of all stick-

slip and contact-separation transitions that occur under the vibra-tions.

Eleven of the friction interface elements are distributed over

nodes of left boundary and eleven interface elements on the rightboundary of the sector shroud contacts locations of the elementson the right sector boundary are shown by circles marked by letterA in Fig. 4c.

Full finite element model of the bladed disk comprises about 15

millions DOFs and the finite single-sector model contains 162,708degrees-of-freedom DOFs. Number of DOFs in nonlinear equa-tions is reduced to 66 by applying the developed method for ex-clusion of linear DOFs from resulting equation while preserving

accuracy and all dynamic properties of the initial model sinceduring this reduction there is no any loss of information aboutinfluence of the excluded DOFs. In all calculations performed themaximum displacement is determined as

max0,2

x 2()y 2()z2(). Ratio between amplitudes of all

nodes of the bladed disk is varied with excitation frequency and isdependent on the shroud contact conditions but for the bladed disk

analyzed the displacements have higher levels at the blade tip inthe frequency ranges considered. Because of that the maximumdisplacement is plotted in all figures for a representative nodelocated at the blade tip. The node is shown by a circle marked by

letter B in Fig. 4c.In Fig. 6 the maximum displacement is shown as a function of

excitation frequency for different values of clearances Fig. 6aand interferences Fig. 6b between shrouds of adjacent sectors.Excitation of engine order type by 4EO is applied to the bladeddisk and 4th harmonic component is used in the displacementexpansion. For comparison forced responses of linear systems are

Fig. 2 Models of the bladed disk compared: a a sectormodel, b the whole structure




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plotted also by thin lines: i in Fig. 6a forced response of thebladed disk without shroud contacts is plotted and ii in Fig. 6bforced response of the bladed disk when all nodes are always infull contact without slip is plotted. As seen for the bladed disk

with clearances 102 mm and 103 mm the system starts exhibit-ing strongly nonlinear behavior in vicinities of resonance peakswhere vibration displacement are large enough to close the clear-ance between shrouds. The nonlinear forced response has stiffen-ing characteristics when with increase of amplitude resonance fre-

quency is increasing. Far from resonances the amplitudes are sosmall that shrouds cannot contact during vibrations and the systemhas forced responses identical to the linear system. For clearance

104 mm the shrouds come to contact over the whole excitationfrequency range. This causes significant reduction of the responselevel over the analyzed frequency range and even disappearanceof one of three resonance peaks that the linear system withoutshroud contacts has. Moreover, the forced response characteristicstakes a more complicated shape, as on can see for frequency rangeafter first resonance and in vicinity of second resonance. For thelatter zoomed view is shown to see the curve in more detail. Forthe bladed disk with interferences shown in Fig. 6b one can seethat force response exhibits softening characteristics when in-crease of amplitudes is accompanied by decrease of resonancefrequencies.

This happens because increase of amplitude of displacementscauses more nodes at the shroud contact surfaces to go out of

contact and to slip. For interference values 104, 105, and

106 mm nonlinear effects become evident in the vicinity of reso-

nance and for their values 5*107, 107, and 0 forced response is

different from the linear system compared over the whole fre-quency range. For these small interference values the amplitudesare much higher that those of the linear system in frequenciesranges far from the resonance whereas they are many times lower

Fig. 3 Comparison of results obtained with the proposedmethod and with using the whole bladed disk model: a a caseof friction dampers, b nonlinear spring elements with nonlin-ear cubic dependence of forces on displacements, c gap ele-

ments

Fig. 4 Finite element model: a a sector of the bladed disk, bshroud of the sector model with shrouds adjacent to it, c in-terface nodes where friction and impact forces are considered

Fig. 5 Natural frequencies of a tuned bladed high-pressureturbine disk and the analyzed frequency range




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than those of the linear system at the resonance. This observationis explained by different effects of the shroud nonlinear interac-tion for excitation frequencies far and close to the resonance. Forout-of-resonance regimes the friction damping produced at shroudcontact surfaces does not affect significantly level of amplitudesand major effect on the amplitudes is due to decrease of shroudcontact stiffness when some or all of the contact nodes lose tem-porarily contact. For the resonance regime the friction damping isa major cause of the reduction of the amplitude level while changeof the stiffness is also reflected in the decrease of the resonancefrequency value.

Effects of the clearances and interferences on nonlinear forcedresponse excited by different engine orders is demonstrated in Fig.

7 where cases of the clearance value 10

2

mm Fig. 7a and theinterference value 106 mm Fig. 7b are analyzed. Forced re-sponses of the corresponding linear systems are also plotted forcomparison by thin lines. For the case of the system with theclearances one can see that the higher engine order number, thelower amplitude level when shroud contact occurs.

This is explained by the fact that to close the gap betweenadjacent shrouds their relative displacements have to be largeenough. Amplitudes are the same for adjacent blades of the cycli-cally symmetric bladed disk but a phase shift between the domi-nant harmonics of the displacements is proportional to the excita-tion engine order number. For larger engine order numbers due tolarger phase shift the relative displacements can be larger even

when the absolute blade amplitudes are smaller. For both cases ofthe considered clearances and the interferences the friction damp-ing is significant although for the case with the interferences itseffect much more evident.

Conclusions

An effective method for analysis of periodic forced response ofnonlinear cyclically symmetric structures has been developed.

A rigorous proof of the validity of the reduction of the wholenonlinear structure to a sector is provided for a general case ofnonlinear forces of arbitrary character and nature. Types of bladeddisk forcing permitting application of the method are discussed.

The method allows multiharmonic nonlinear forced responsefor a whole bladed disk to be calculated using a sector modelwithout any loss of accuracy in calculations and modeling. A mul-tiharmonic formulation for equations of motion has been derivedfor two widely used techniques of description of the linear part ofa bladed disk model with the large number of degrees-of-freedom:i using sector finite element matrices; ii using sector naturalmode shapes and frequencies. An approach for a use of complexarithmetic in the multiharmonic formulation and in the solution ofthe nonlinear equations of motion has been examined. A techniquefor reduction of size of the nonlinear multiharmonic equations byexclusion of linear DOFs from the resulting equations has beendeveloped.

Fig. 6 Forced response of the bladed disk for excitation by4EO: a a case of different clearance values, b a case of dif-ferent interference values

Fig. 7 Forced response of the bladed disk for excitation ofdifferent engine orders: a a case with gap between shrouds of

0.01 mm, b a case with interferences of 106 mm




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The numerical investigations show high efficiency of the pro-posed method. The identity of the results obtained with the devel-oped method using a sector model and with a whole bladed diskmodel has been demonstrated for different types of nonlinearforces produced by nonlinear interfaces: i friction dampers; iigaps; iii cubic nonlinear springs. A practical, large finite elementmodel of a shrouded turbine bladed disk has been analyzed withaccounting for friction forces, clearances, and interferences atnodes located at surfaces of shroud contacts.

Acknowledgments

The author is grateful to Rolls-Royce plc. for providing thefinancial support for this project and for giving permission to pub-lish this work.

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a method for use of cyclic symmetry properties in analysis

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