Nonlinear bearing stiffness parameter estimation in ﬂexible ...home.iitk.ac.in/~vyas/publication/7-Probabilistic...Nonlinear bearing stiffness parameter estimation in ﬂexible rotor–bearing

Nonlinear bearing stiffness parameter estimation in ¯exible rotor±bearingsystems using Volterra and Wiener approach

A.A. Khan*, N.S. Vyas

Department of Mechanical Engineering, Indian Institute of Technology, Kanpur, India

Accepted 14 July 2000

Abstract

Higher order frequency response functions based on Volterra and Wiener series are explored for the inverse problem of stiffness estimation

of a ¯exible rotor supported in nonlinear bearings. The Volterra series has been employed by researchers earlier for the identi®cation of

higher order kernels of nonlinear systems through a non-parametric approach. The present study investigates the possibility of employing

these kernels for parameter estimation of the system. Numerical simulation has been carried out for a system with three degrees of freedom

and with cubic nonlinearity in stiffness. A frequency domain has been adopted for the identi®cation of higher order kernels. The procedure

involves extraction of Wiener kernels from the response of the system to a Gaussian white noise excitation. Volterra kernels are in turn

synthesised from the Wiener kernels. In addition to direct kernels, the system under consideration, requires de®nitions of cross-kernels and their

estimation. Expressions for the cross and direct kernels are constructed in the frequency domain. A set of third-order kernel factors are

algebraically and graphically synthesised from the measured ®rst-order kernels. These third-order kernel factors are then processed with the

measured third-order kernels for nonlinear parameter estimation. Damping is taken to be linear in the analysis. The procedure is illustrated

through numerical simulation. The assumptions involved and the approximations are discussed. q 2001 Elsevier Science Ltd. All rights reserved.

Keywords: Rotor; Bearing; Nonlinear stiffness estimation; Volterra and Wiener series

1. Introduction

The characterisation of nonlinear dynamic systems, from

input±output data, is broadly categorised into parametric

and non-parametric identi®cations. Parametric methods

require that the mathematical structure and order of the

system are known. The identi®cation procedure, in such

cases reduces to estimation of system parameters through

a search in parameter space. Mechanical and structural

system nonlinearities, occurring in stiffness or in damping

forces are modelled generally in polynomial form. Identi®-

cation of non-polynomial forms, such as drag force type

quadratic damping, hysteretic damping, coulomb damping,

bilinear stiffness have been discussed by Nayfeh [1], Choi

et al. [2], Bendat and Piersol [3]. Bendat et al. [4] developed

a general identi®cation technique from measured input±

output stochastic data for a wide range of nonlinearities

including Duf®ng oscillator, Van-der Pol oscillator, dead

band and clearance nonlinearity. Linear system identi®ca-

tion procedure using spectral density functions were

extended by Bendat and Peirsol [5,6] for nonlinear systems,

through de®nitions of higher order spectral density

functions. Rice and Fritzpatrick [7,8] have also presented

a similar procedure. Mohammad et al. [9] presented a para-

metric method, which works directly with the differential

equation, through measurement of forcing function and

response (acceleration) at each time-step.

Non-parametric identi®cation concerns modelling in a

function space by input±output mapping, for systems

where suf®cient information on the mathematical structure

or class is not available. The input±output mapping, under

non-parametric methods, is done through a series of func-

tionals or a series of orthogonal functions. In orthogonal

series representation, such as in Restoring Force Mapping

techniques [10±14], the response variables are represented

in terms of some orthogonal series and identi®cation

attempts to determine the coef®cients of the terms in the

series. These series are generally in®nite in nature and are

to be truncated to ®nite number of terms for the purpose of

identi®cation. The restoring force mapping techniques

provide general identi®cation methods applicable for a

wide class of nonlinear systems, but the methods have the

limitation that acceleration data is integrated to give velo-

city and displacement records, which may introduce mean

bias and drift error. In the functional series representation,

Probabilistic Engineering Mechanics 16 (2001) 137±157

0266-8920/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved.

PII: S0266-8920(00)00016-3

www.elsevier.com/locate/probengmech

* Corresponding author.

such as in Volterra or Wiener series, the identi®cation

consists of determining the kernels of the functionals either

in the time- or frequency-domain. The Volterra series

[15,16] represents the response of a system in a functional

series form. The ®rst term in the series is the well-known

convolution integral used for input±output mapping of

linear systems. Subsequent series terms consist of higher

order convolution integral operating on the input function

through higher order impulse response function known as

Volterra kernels. Two basic dif®culties associated with

practical application of Volterra series are the convergence

of the series and the measurement of individual Volterra

kernels of the given system. These problems are circum-

vented by construction of Wiener G-functionals [17] with

Gaussian white noise excitation. Wiener kernels can be

related to Volterra kernels, using their orthogonality prop-

erty [18]. Barret [19,20] and Flake [21] have presented

various details on the use of Volterra series in nonlinear

system analysis. Schetzen [22] presented a method for

measuring the Volterra kernels as multi-dimensional

impulse responses of a ®nite order nonlinear systems. Sand-

berg and Stark [23] measured the kernels of the Pupillary

system up to second-order using Schetzen's cross-correla-

tion technique. Bedrosian and Rice [24] illustrated

frequency domain methods for response characterisation

of nonlinear systems under harmonic excitation as well as

random excitation along with effect of noise on the

measured response. French and Butz [25,26] developed a

general frequency domain method for calculation of higher

order Wiener kernels using exponential functions as a set of

orthogonal functions for expanding the kernels. Gifford and

Tomlinson [27] illustrated the technique of calculating

higher order frequency response functions (FRFs) for

nonlinear structural systems. Haber [28] has discussed the

kernel estimation of the discrete Volterra series for quad-

ratic block oriented models. Nam et al. [29] presented a

frequency domain approach for the identi®cation of a

discrete third-order Volterra system with random input.

Koukoulas and Kalouptsidis [30] proposed an approach

for identi®cation of Volterra kernels based on cross-cumu-

lants and their spectra, rather than cross-correlation.

Tomlinson et al. [31] studied the convergence of ®rst-

order FRF of a Duf®ng oscillator under harmonic excitation

and presented a simple formula for determining the upper

limit of excitation level.

In a recent study, Khan and Vyas [32] attempted to extend

the non-parametric Volterra kernel identi®cation procedure,

for further estimation of the system parameters of a single-

degree-of-freedom system. The present study discusses the

case of a more general multi-degrees-of-freedom model.

With polynomial form nonlinearity in stiffness or damping

forces, analytical expressions for higher order kernels can

be built, in terms of the ®rst-order kernel and coef®cients of

the polynomial representing the stiffness or damping forces.

These expressions are processed along with the measured

kernels for parameter estimation. The problem of conver-

gence of the Volterra series is sought to be circumvented

through estimation of Wiener kernels, under white noise

excitation. The Volterra kernels are developed, in turn,

from the Wiener kernels. The problem has been discussed

for a multi-degrees-of-freedom system, through an idealised

¯exible rotor±bearing model. Direct and cross-kernels have

been de®ned for the system and the procedure is illustrated

through numerical simulation.

2. Governing equations and Volterra series responserepresentation

The equations of motion for a balanced rotor with a

centrally located disc on a massless ¯exible shaft supported

in bearings (shown in Fig. 1) are written as

�M�{ �x} 1 �C�{ _x} 1 �K�{x} 1 �Kn�{x3} � {f �t�}

with

�M� �m1 0 0

0 m2 0

0 0 m3

26643775 �C� �

c11 c12 c13

c21 c22 c23

c31 c32 c33

26643775

�K� �k11 1 kL

b1k12 k13

k21 k22 1 kLb2

k23

k31 k32 k33

2666437775

A.A. Khan, N.S. Vyas / Probabilistic Engineering Mechanics 16 (2001) 137±157138

Fig. 1. Flexible rotor in bearings.

�Kn� �kN

b10 0

0 kNb2

0

0 0 0

2666437775 {x} �

x1

x2

x3

8>><>>:9>>=>>;

{f �t�} �f1�t�f2�t�

0

8>><>>:9>>=>>; (1)

m1, m2 and m3 are the system masses at stations 1, 2 and 3,

respectively. External white noise excitation forces,

provided at the bearing stations are f1�t�; and f2�t�: kLb1

and

kLb2

are the unknown linear stiffness terms of the two bear-

ings and kNb1

and kNb2

are their unknown nonlinear stiffness

terms. The shaft stiffness kij is the resistive force at the ith

station corresponding to a unit de¯ection at the jth station,

with all other de¯ections held to zero. kij values can be

obtained from the strength of materials formulae. c11, c12,

c13, etc. are linear damping terms.

De®ning

t � pt p � ��k33=m3

pxih�t� � xi=

iXst

iXst � Fmax=mip2 �f i�t� � fi�t�=Fmax j ij � cij=2mip

lij � kij=mip2 i; j � 1; 2; 3 lL

ij � kLbi=mip

2

lNbi� kN

biF2

max=m3i p6 for i � 1; 2 (2)

and considering only direct damping terms, to keep the

algebra simple, Eq. (1) becomes

{h 00} 1 2�j�{h 0} 1 �l�{h} 1 �lN�{h3} � { �f �t�}

�j� �j11 0 0

0 j22 0

0 0 j33

26643775

�l� �l11 1 lL

b1l12 l13

l21 l22 1 lLb2

l23

l31 l32 l33

2666437775

�lN� �lN

b10 0

0 lNb2

0

0 0 0

2666437775 {h 00} �

x1h 00

x2h 00

x3h 00

8>><>>:9>>=>>;

{ �f �t�} ��f 1�t��f 2�t�

0

8>><>>:9>>=>>;

�3�

The solution of the Eq. (3) is represented in terms of

Volterra operators as

kh�t� �X1n�1

kHn� �f 1�t�; �f 2�t�� 4�

with k denoting x1, x2 or x3. The individual operators kHn are

given in kernel form by

kH�i�1 � �f i�t�� Z1

2 1kh�i�1 �t1��f i�t 2 t1� dt1 for i � 1; 2

kH�i;j�2 ��f i�t�; �f j�t��

�Z1

2 1

Z1

2 1kh�i;j�2 �t1; t2� �f i�t 2 t1��f j�t 2 t2� dt1 dt2

for i � 1; 2; j � 1; 2

kH�i;j;k�3 � �f i�t�; �f j�t�; �f k�t��

�Z1

2 1

Z1

2 1

Z1

2 1kh�i;j;k�3 �t1; t2; t3�

� �f i�t 2 t1� �f j�t 2 t2� �f k�t 2 t3� dt1 dt2 dt3

for i � 1; 2; j � 1; 2; k � 1; 2 (5)

In the above kh�i�1 �t�; kh�i;j�2 �t1; t2� and kh

�i;j;k�3 �t1; t2; t3� are

the ®rst-, second- and third-order Volterra kernels, respec-

tively. For convenience, writing the Volterra operators as

kh�i�1 � kH�i�1 � �f i�t�� kh�i;j�2 � kH�i;j�2 � �f i�t�; �f j�t��

kh�i;j;k�3 � kH�i;j;k�3 ��f i�t�; �f j�t�; �f k�t��

�6�

the response of Eq. (3) can be written as

kh�t� �X

i�1;2

kh�i�1 1X

i�1;2; j�1;2

kh�i;j�2

1X

i�1;2; j�1;2; k�1;2

kh�i;j;k�3 1 ¼ �7�

3. Synthesis of higher order Volterra kernel factors

The Volterra operators are now determined as follows.

The excitation forces �f 1�t� and �f 2�t� are replaced by c �f 1�t�and c �f 2�t�; respectively, c being a constant. Noting Eq. (5),

the resulting response of the system becomes

kh�t� �X

i�1;2

ckh�i�1 1X

i�1;2; j�1;2

c2kh�i;j�2

1X

i�1;2; j�1;2; k�1;2

c3kh�i;j;k�3 1 ¼ �8�

Eq. (8) and the derivatives of its terms are substituted in

A.A. Khan, N.S. Vyas / Probabilistic Engineering Mechanics 16 (2001) 137±157 139

Eq. (3). Summing up the responses of equal order as

kh1 �X

i�1;2

h�i�1 ; kh2 �X

i�1;2; j�1;2

kh�i;j�2 ;

kh3 �X

i�1;2; j�1;2; k�1;2

kh�i;j;k�3 �k being x1; x2; x3��9�

and noting the following symmetry of kernels [18]

kh�i;j�2 � kh�j;i�2 ; kh�i;j;j�3 � kh�j;i;j�3 � kh�j;j;i�3 etc: �10�the governing equations can be written as

�cx1h 001 1 c2x1h 002 1 c3x1h 003��cx2h 001 1 c2x2h 002 1 c3x2h 003��cx3h 001 1 c2x3h 002 1 c3x3h 003�

8>><>>:9>>=>>; 1 2

j11 0 0

0 j22 0

0 0 j33

26643775

��cx1h 01 1 c2x1h 02 1 c3x1h 03��cx2h 01 1 c2x2h 02 1 c3x2h 03��cx3h 01 1 c2x3h 02 1 c3x3h 03�

8>><>>:9>>=>>;

1

l11 1 lLb1

l12 l13

l21 l22 1 lLb2

l23

l31 l32 l33

2666437775

��cx1h1 1 c2x1h2 1 c3x1h3��cx2h1 1 c2x2h2 1 c3x2h3��cx3h1 1 c2x3h2 1 c3x3h3�

8>><>>:9>>=>>; 1

lNb1

0 0

0 lNb2

0

0 0 0

2666437775

��cx1h1 1 c2x1h2 1 c3x1h3�3

�cx2h1 1 c2x2h2 1 c3x2h3�3

�cx3h1 1 c2x3h2 1 c3x3h3�3

8>><>>:9>>=>>;

� c

�f 1�t��f 2�t�

0

8>><>>:9>>=>>; �11�

Eq. (11) is a power series in c with coef®cients of cn beingx1hn;

x2hn or x3hn �n � 1; 2; 3�: The responses, x1hn;x2hn and

x3hn have been worked out in Appendix A, by equating the

like powers of c. The Laplace transform of the ®rst-order

response becomes

kh1�s� �X3

1

kH�i�1 �s�Fi�s� �12�

where kH�i�1 �s� is the Laplace transforms of the ®rst-order

kernel (Appendix A). The second-order response is found to

be identically equal to zero for the system under considera-

tion, i.e. kh2�s1; s2� � 0; which gives the second-order

kernel transforms as

kH�i�2 �s1; s2� � 0 �13�

The third-order response kh3; resulting due to the operation

of the third-order kernel on the excitation forces �f 1�t� and�f 2�t� can be shown (Appendix A) to be equal to a operation

of ®rst-order kernels on q1�t� and q2�t� as

kh3 � kH3� �f 1�t�; �f 2�t�� kH1�q1; q2� �14�where

q1�t� � 2lN11

x1h31; q2�t� � 2lN

22x2h3

1 �15�Since the ®rst-order kernel operator, kH1; is linear Eq. (14)

can be written as

kH3� �f 1�t�; �f 2�t�� 2lN11

kH�1�1 �kh31�2 lN

22kH�2�1 �kh3

1� �16�Noting that

kh1 � kH�1�1 � �f 1�t��1 kH�2�1 ��f 2�t�� 17�the Laplace transform of Eq. (16) provides an expression for

the third-order kernel transforms kH3�s1; s2; s3�; for k denot-

ing x1, x2 or x3 as

x1 H3�s1; s2; s3� � lN11�x1x1C 1±1;1;1

3 1 3x1x1C 1±1;1;23

1 3x1x1C 1±1;2;23 1 x1x1C 1±2;2;2

3 �

1 lN22�x1x2C 2±1;1;1

3 1 3x1x2C 2±1;1;23

1 3x1x2C 2±1;2;23 1 x1x2C 2±2;2;2

3 �x2 H3�s1; s2; s3� � lN

11�x2x1C 1±1;1;13 1 3x2x1C 1±1;1;2

3

1 3x2x1C 1±1;2;23 1 x2x1C 1±2;2;2

3 �

1 lN22�x2x2C 2±1;1;1

3 1 3x2x2C 2±1;1;23

1 3x2x2C 2±1;2;23 1 x2x2C 2±2;2;2

3 �x3 H3�s1; s2; s3� � lN

11�x3x1C 1±1;1;13 1 3x3x1C 1±1;1;2

3

1 3x3x1C 1±1;2;23 1 x3x1C 1±2;2;2

3 �

1 lN22�x3x2C 2±1;1;1

3 1 3x3x2C 2±1;1;23

1 3x3x2C 2±1;2;23 1 x3x2C 2±2;2;2

3 �

�18�

where x1x1C �i±j;k;l�3 �s1; s2; s3� etc. have been termed as `third-

order kernel factors' and are given in Appendix B. The

third-order kernel factors given above can be readily

constructed using equations from the ®rst-order kernelsx1 H�1�1 �s�; x1 H�2�1 �s�; x2 H�1�1 �s�; x2 H�2�1 �s�; x3 H�1�1 �s� andx3 H�2�1 �s�:

4. Measurement of Wiener kernels

Two major problems with the practical application of

Volterra Series analysis, namely, measurement of individual


Volterra kernels and convergence of Volterra series are

circumvented through usage of Wiener functionals. Wiener

kernels can be extracted from the measured response for

a Gaussian white noise excitation. These Wiener kernels

are then employed to generate the Volterra kernels. The

higher order `measured' Volterra kernels thus obtained

are equated to those obtained from the synthesis pro-

cedure described previously for nonlinear parameter

estimation.

In the present case, the Wiener kernels of the nonlinear

system are extracted by the application of white Gaussian

forces �f 1�t� and �f 2�t�; one at a time, i.e. ®rst a white Gaus-

sian force �f 1�t� with variance A1 is applied at bearing 1,

while keeping the force at bearing 2, �f 2�t� � 0: The result-

ing responses x1h; x2h and x3h are employed to extract the

direct, x1±x1 (at bearing 1), Wiener kernels and the cross

(x1±x2 and x1±x3) kernels. In the next instance, a white

Gaussian force �f 2�t� with variance A2 is applied at bearing

2 while keeping the force �f 1�t� � 0: The system responses,

in this instance, are employed to extract the direct x2-coor-

dinate Wiener kernels and the cross (x2±x1, x2±x3) kernels.

The system response in terms of Wiener operators can be

expressed in the two individual cases (with k denoting x1, x2

or x3) as

kh�t� � kW� �f 1�t�; 0� � kW��f 1�t��

when �f 1�t� is Gaussian white;

with variance A1 and �f 2�t� � 0

�19�

kh�t� � kW�0; �f 2�t�� kW��f 2�t��

when �f 2�t� is Gaussian white;

with variance A2 and �f 1�t� � 0

�20�

In either of the two cases, (Eqs. (19) and (20)), the response

gets represented as

kh�t� � kW0 1Z1

2 1kW �i�

1 �v1� �Fi�v1�ejv1t dv1

1

"Z1 1

2 1

Z1 1

2 1kW �i�

2 �v1;v2� �Fi�v1� �Fi�v2� ej�v11v2�t dv1 dv2

2AZ1 1

2 1kW �i�

2 �v1;2v2� dv2

1Z1 1

2 1

Z1 1

2 1

Z1 1

2 1kW �i�

3 �v1;v2;v3�

£ �Fi�v1� �Fi�v2� �Fi�v3� ej�v11v21v3�t dv1 dv2 dv3

2 3AZ1 1

2 1

Z1 1

2 1kW �i�

3 �v1;v2;v2� dv1 dv2

#1 ¼

k � x1; x2; x3; i � 1; 2 (21)

where kW0;kW �i�

1 ;kW �i�

2 �v1;v2� and kW �i�3 �v1;v2;v3� are

the frequency-domain Wiener kernels of the zeroth-, ®rst-,

second- and third-order, respectively. A scheme involving a

complex exponential ®lter suggested by French and Butz

[25] as shown in Fig. 2 is employed for the extraction of

Wiener kernels from the measured response. The ®lter

output is

z�i��t� �Z1

2 1ejvt1 �f i�t 2 t1� dt1 � �Fp

i �v� e2jvt1 �22�

and the ensemble averages of the outputs of the circuit, for k


Fig. 2. Scheme for evaluation of the ®rst-order direct and cross Wiener kernel transforms, xW �i�1 :

equal to x1, x2 and x3 are

kkh�t�z�i��t�l � kW0k �Fpi �v�l e2jvt

1Z1

2 1kW �i�

1 �v1�k �Fi�v1� �Fpi �v�l e2jt�v12v� dv1

1

"Z1

2 1

Z1

2 1kW �i�

2 �v1;v2�

£ k �Fi�v1� �Fi�v2� �Fpi �v�l e2jt�v11v22v� dv1 dv2

2AikFpi �v�l e2jv1t

Z1

2 1kW �i�

2 �v2;2v2� dv2

#

1

"Z1

2 1

Z1

2 1

Z1

2 1kW �i�

3 �v1;v2;v3�

£ k �Fi�v1� �Fi�v2� �Fi�v3� �Fpi �v�l e2jt�v11v21v32v� dv1 dv2 dv3

23Ai

Z1

2 1

Z1

2 1kW �i�

3 �v1;v2;2v2�

£ k �Fpi �v1� �Fp

i �v�l e2jt�v12v� dv1 dv2

#1 ¼ (23)

Employing the properties [33] of the ensemble averages of

products of the stationary Gaussian white noise process�Fi�v�; Eq. (23) gets reduced, after some algebra, to

kkh�t�z�i��t�l � AikW �i�

1 �v� �24�However, due to the equivalence of time and ensemble

averages, the ensemble average kkh�t�z�i��t�l can also be

written as

kkh�t�z�i��t�l � limT!1

1

T

ZT=2

2 T =2

kh�t�z�i��t� dt � �Fpi �v�kh�v�

�25�Eqs. (24) and (25) give

AikW �i�

1 �v� � �Fpi �v�kh�v� �26�

from which the expression for the ®rst-order Wiener kernel

transform is obtained as

kW �i�1 �v� � �Fi�v�kh�v�=Ai �27�

Since k takes values x1, x2, x3 and i takes values 1 and 2

the direct kernels x1 W �1�1 �v�; x2 W �2�

1 �v� and the cross kernelsx1 W �2�

1 �v�; x2 W �1�1 �v�; x3 W �1�

1 �v� and x3 W �2�1 �v� can be

extracted from the measured responses x1h�v�; x2h�v�;x3h�v� and the applied force � �F1�v� and its variance A1 or�F2�v� and its variance A2), through Eq. (27).

For measurement of the third-order kernel, a circuit invol-

ving three exponential delay ®lters as shown in Fig. 3 is

considered. The output, z�i��t�; of the exponential ®lters is

z�i��t� �Z1

2 1e2jv1t1 �f i�t 2 t1� dt1

�Z1

2 1e2jv2t2 �f i�t 2 t2� dt2

�Z1

2 1e2jv3t3 �f i�t 2 t3� dt3

� �Fi�2v1� �Fi�2v2� �Fi�2v3� e2j�v11v21v3�t (29)


Fig. 3. Scheme for evaluation of the third-order direct and cross Wiener kernel transforms, xW �i�3 �v;v;v�:

and the ensemble averages of the outputs of the circuit are

kkh�t�z�i��t�l � �A2i �kW �i�

1 �v1�d�2v2 2 v3� ej�2v22v3�t

1 kW �i�1 �v2�d�2v1 2 v3� ej�2v12v3�t

1 kW �i�1 �v3�d�2v1 2 v2� ej�2v12v2�t�

1 �6A3ikW �i�

3 �v1;v2;v3�� 29�

However, the equivalence of time and ensemble averages

gives

kkh�t�z�i��t�l � limT!1

1

T

ZT=2

2 T=2

kh�t�z�i��t� dti

� �Fi�2v1� �Fi�2v2� �Fi�2v3�kh�v1 1 v2 1 v3� �30�

Eqs. (29) and (30) give the expression for the measurement

of the third-order Wiener kernel transform as

kW �i�3 �v1;v2;v3�

� 1

6A3i

� �Fpi �v1� �Fp

i �v2� �Fpi �v3�x1h�v1 1 v2 1 v3��

21

6Ai

�kW �i�1 �v1�d�v2 1 v3�1 kW �i�

1 �v2�d�v1 1 v3�

1 kW �i�1 �v3�d�v1 1 v2�� (31)

The third-order kernels, x1 W �i�3 �v1;v2;v3�; x2 W �i�

3 �v1;v2;v3�and x3 W �i�

3 �v1;v2;v3� form multi-dimensional surfaces on

the �v1;v2;v3� axes. Measurements are made for special

tri-spectral kernels with v1 � v2 � v3 � v: These kernels

being functions of only one variable v , are easier to

compute and interpret. For such tri-spectral kernels the

expression (31) reduces to

kW �i�3 �v;v;v� �

1

6A3i

�{ �Fpi �v�}3kh�3v��

21

2Ai

�kW �i�1 �v��d�v� �32�

5. Parameter estimation

The ®rst-order Wiener kernels kW �i�1 �v� and the third-

order Special Tri-spectral Wiener kernels kW �i�3 �v;v;v�

are estimated using Eqs. (27) and (32), respectively, from

the measurements of spectral component F�v� of the excita-

tion force, its variance Ai and the spectral componentskh�v�; kh�3v� of the corresponding response. Subsequently,

for a third-order representation of the system response,

noting the orhogonality property [18] between Volterra

and Wiener kernels, the ®rst and Special Tri-spectral

Volterra kernel transforms can be computed as

kH�i�3 �v;v;v� � kW �i�3 �v;v;v�

kH�i�1 �v� � kW �i�1 �v�1 kW �i�

1�3��v��33�

where

kW �i�1�3��v� �

Z1

2 1kW �i�

3 �v;v2;2v2� dv2

The linear parameters can be obtained from the above esti-

mates of the ®rst-order Volterra kernels kH�i�1 �v�: Noting

the algebraic expressions of these kernels from Appendix

A, the linear stiffness parameters of the bearings lb1and lb2

are estimated through a complex curve ®tting routine [35].

In addition, the shaft stiffnesses and damping ratios lij; j ij

i � 1; 2; 3; j � 1; 2; 3 are also obtained from the curve

®t routine.

The nonlinear parameters are computed from the

estimates of the Special Tri-spectral Volterra kernelsx1 H�1�3 �v;v;v�; x1 H�2�3 �v;v;v�; x2 H�1�3 �v;v;v�; x2 H�2�3

�v;v;v�; x3 H�1�3 �v;v;v� and x3 H�2�3 �v;v;v�: Since these

kernels are estimated by application of a single white Gaus-

sian force at a time, the synthesised expressions (18), for the

third-order Volterra kernels also get reduced in the two

individual cases to�f 1�t� Gaussian white, with variance A1 and �f 2�t� � 0 :

x1 H3�v;v;v� � lN11�x1x1C 1±1;1;1

3 �1 lN22�x1x2C 2±1;1;1

3 �x2 H3�v;v;v� � lN

11�x2x1C 1±1;1;13 �1 lN

22�x2x2C 2±1;1;13 �

x3 H3�v;v;v� � lN11�x3x1C 1±1;1;1

3 �1 lN22�x3x2C 2±1;1;1

3 �

�34�

�f 2�t� Gaussian white, with variance A2 and �f 1�t� � 0 :

x1 H3�v;v;v� � lN11�x1x1C 1±2;2;2

3 �1 lN22�x1x2C 2±2;2;2

3 �x2 H3�v;v;v� � lN

11�x2x1C 1±2;2;23 �1 lN

22�x2x2C 2±2;2;23 �

x3 H3�v;v;v� � lN11�x3x1C 1±2;2;2

3 �1 lN22�x3x2C 2±2;2;2

3 �

�35�

While application of two forces (individually) is required

for estimation of the linear cross coupling terms, only two

equations out of the set of six in expressions (34) and (35)

are suf®cient for the estimation of the two nonlinear

unknowns l 11N and l 22

N (cross-coupled nonlinear parameters

l 12N , l 13

N and l 23N being taken zero). Estimation of l 11

N and l 22N

has been carried out here, using the ®rst two equations from

the set (34). For these two equations, we have

1

6A31

�{ �Fp1�v�}3x1h�3v��2

1

2A1

�x1 W �1�1 �v�d�v��

� lN11�x1x1C 1±1;1;1

3 �1 lN22�x1x2C 2±1;1;1

3 �1

6A31

�{ �Fp1�v�}3x2h�3v��2

1

2A1

�x2 W �1�1 �v�d�v��

� lN11�x2x1C 1±1;1;1

3 �1 lN22�x2x2C 2±1;1;1

3 �

�36�


The above are solved simultaneously for l 11N and l 22

N (the

third-order kernel factors, C , are known from the estimates

of the ®rst-order kernels and relationships given in

Appendix B).

6. Computer simulation

The procedure is illustrated through numerical simu-

lation of the response of the non-dimensional Eq. (3).

The shaft stiffness matrix for the simply supported system,

carrying a centrally located disc as (refer Fig. 1 for

station numbering) is computed using strength of materials

formulae [34]

k11 k12 k13

k21 k22 k23

k31 k32 k33

26643775 � 12EI

l3

1 0 21

0 1 21

21 21 2

26643775 �37�

which gives the non-dimensionalised matrix

l11 1 lb1l12 l13

l21 l22 1 lb2l23

l31 l32 l33

26643775

��0:5 1 lb1

�=m1 0 2�0:5=m1�0 �0:5 1 lb2

�=m2 2�0:5=m2�20:5 20:5 1

26643775 �38�

where m 1 and m 2 are equal to the mass ratios m1=m3 and

m2=m3; respectively. In the rotor con®guration shown in Fig.

1, m1 and m2 are the masses effectively seen by the sensors at

the bearing ends and will be small in magnitude in compar-

ison to m3. However, for simplicity in numerical simulation

the mass ratios m1=m3 and m2=m3 are chosen to be each equal

to 1.0. The values of the individual elements of the above


Fig. 4. (a) Typical sample of input force. (b) Power-spectrum of the input force (averaged over 2000 samples).

lambda matrix therefore become

l11 � l22 � 0:5 l33 � 1:0 l12 � l21 � 0

l13 � l23 � 20:5 l31 � l32 � 20:5

In order to illustrate the numerical results when shaft and

bearing stiffness are both equally signi®cant numerically,

the values of the linear bearing stiffness parameters are

taken as

lb1� 0:5 lb2

� 1:0

The nonlinear parameters are chosen as

lNb1� 0:1 lN

b2� 0:1

Damping is chosen as j11 � j22 � j33 � 0:01 in all the


Fig. 5. Typical response samples. (a) Response x1h�t� at station 1. (b) Response x2h�t� at station 2. (c) Response x3h�t� at station 3.

cases. The excitation forces are simulated through random

number generating subroutines and are normalised with

respect to the maximum value of f1�t�: A typical sample

of the excitation is shown in Fig. 4a. The power spectrum

of the input averaged over an ensemble of 2000 force

samples is shown in Fig. 4b. Responses x1h; x2h and x3h

are generated numerically for 4096 number of instances in

the non-dimensional time (t) range 0±2048, by solving the

governing equations through fourth-order Runge±Kutta

subroutine. The responses are computed for 2000 number

of samples of the simulated random force and fed as inputs

to the parameter estimation algorithm. The various ®rst- and


Fig. 6. Power-spectra of the response (averaged over 2000 samples) of: (a) x1h�t�; (b) x2h�t�; and (c) x3h�t�:

higher order kernels are extracted from the responses and

consequently parameter estimation is carried out. The

output consists of the linear stiffness parameters lb1; lb2

;

the damping ratios j 11, j 22, j 33, the mass ratios m 1, m 2 and

the nonlinear stiffness parameters lNb1

and lNb2: The esti-

mated parameters are compared with those originally used

for the simulation of response and the accuracy of estimates

and errors involved are discussed.x1h; x2h and x3h numerically resulting from the force of

Fig. 4a and b, applied at station are shown in Fig. 5a±c (no

force is applied at the second bearing.) The corresponding

power-spectra are shown in Fig. 6a±c. The fundamental


Fig. 7. Estimates of the ®rst-order kernel transforms of: (a) x1 H�1�1 �v�; (b) x2 H�1�1 �v�; (c) x3 H�1�1 �v�; (d) x1 H�2�1 �v�; (e) x2 H�2�1 �v�; and (f) x3 H�2�1 �v�:

frequencies of the system can be noticed from these ®gures

to exist at 0.097, 0.175 and 0.219 cycles/t . The frequencies

correspond to those obtained from the eigenvalue solution

of the linear non-dimensional stiffness matrix described in

Eqs. (37) and (38) and can be called v 1, v 2, v 3. It can also

be noticed from the power-spectra of Fig. 6a±c that the

system nonlinearity is not apparently, equally displayed

by responses x1h; x2h and x3h: The plot of x2h (Fig. 6b)

displays additional peaks at frequencies 0.263, 0.291,

0.369, 0.490, 0.525 and 0.560 cycles/t , which can be iden-

ti®ed as �2v3 2 v2�; (3v 1), �2v1 1 v2�; �v1 1 v2 1 v3�;(3v 2) and �2v2 1 v3� harmonics, respectively. These


Fig. 7. (continued)

harmonics are not distinctly visible in the plots of x1h (Fig.

6a) and x3h (Fig. 6c). This is due to the fact that stations 1

and 3, being closer to the station of force application (station

1), the linear response levels are higher at these stations in

comparison to that at station 2 and overlap the nonlinear

contributions. Similar response plots can be obtained by

application of the force at station 2, while keeping the

force at station 1 equal to zero.

The applied force and resultant response at the three

stations are employed in Eqs. (27) and (32) to extract the

®rst-order Volterra kernels, kH�i�1 �v� shown in Fig. 7a±f.

These kernels show the three fundamental frequencies of


Fig. 8. Normalised error in the ®rst-order estimates of: (a) x1 H�1�1 �v�; (b) x2 H�1�1 �v�; (c) x3 H�1�1 �v�; (d) x1 H�2�1 �v�; (e) x2 H�2�1 �v�; and (f) x3 H�2�1 �v�:

the system, mentioned above. The errors in these estimated

kernels, is given in Fig. 8a±f. The errors are computed by

comparing the estimated kernels with those obtained from

the exact analytical expressions (Appendix A). The errors,

as expected [36], are high in the vicinity of the fundamental

frequencies. However, they are in the 10% zone up to a non-

dimensional frequency of 0.06 cycles/t . Parameter estima-

tion is carried out in this frequency range.

The third-order kernel factors x1x1C 1±1;1;13 �v;v;v�;

x1x2C 2±1;1;13 �v;v;v�; x2x1C 1±1;1;1

3 �v;v;v�; x2x2C 2±1;1;13

�v;v;v�, x3x1C 1±1;1;13 �v;v;v� and x3x2C 2±1;1;1

3 �v;v;v�synthesised from the ®rst-order kernels are shown in Fig.


Fig. 8. (continued)

9a±f. The third-order kernels x1 H�1�3 �v;v;v�; x2 H�1�3 �v;v;v�and x3 H�1�3 �v;v;v�; extracted from the measurements of the

force and response, in accordance with Eqs. (32) and (33)

are shown in Fig. 10a±c. While the ®rst-order kernels are

estimated in the entire available frequency range 0.0±1.0,

the third-order kernels involving a 3v factor, have to be

restricted to one-third of this frequency zone (i.e. 0.0±

0.33). It can be observed from the ®gures that while the

measured third-order kernels are reasonably accurate in

showing the harmonic at v1;2;3=3 (at non-dimensional


Fig. 9. Estimates of the third-order kernel factors of: (a) x1x1C 121;1;13 �v;v;v�; (b) x1x2C 221;1;1

3 �v;v;v�; (c) x2x1C 121;1;13 �v;v;v�; (d) x2x2C 221;1;1

3 �v;v;v�; (e)x3x1C 121;1;1

3 �v;v;v�; and (f) x3x2C 221;1;13 �v;v;v�:

frequencies� 0.032, 0.058 and 0.073 cycles/t ), identi®ca-

tion of the harmonics at v 1,2,3 (at non-dimensional

frequencies� 0.097, 0.175, 0.219 cycles/t) is weak, due

to higher statistical errors. The estimation of nonlinear para-

meters l 11N and l 22

N from these kernels is therefore restricted

to the frequency zone of 0.0±0.07.

The estimates of the nonlinear parameter l 11N and l 22

N ,

obtained in accordance with the relationship (36), are

shown in Fig. 11a and b. A fourth-order polynomial

curve regressed through the estimates of these nonlinear

parameters over the frequency range is also shown in

these ®gures. The mean values of the estimates are


Fig. 9. (continued)

found to be

lNb1� 0:09 lN

b2� 0:12

(The exact values of the above nonlinear parameters

are those chosen for response simulation, that is,

lNb1� lN

b2� 0:10�: The linear parameters have been

estimated from the ®rst-order kernels of Fig. 7a±f,

using a complex curve ®tting routine [35]. The estimates

are

m1 � 0:956 m2 � 0:897


Fig. 10. Estimates of third-order kernel transforms of: (a) x1 H�1�3 �v;v;v�; (b) x2 H�1�3 �v;v;v�; and (c) x3 H13 �v;v;v�:

lb1� 0:487 lb2

� 0:898

j11 � j22 � j33 � 0:011

7. Remarks

Reasonably good estimates have been obtained through

numerical illustration of the proposed procedure. The accu-

racy lies within the 10% zone for all the parameters. It is to

be noted that the response representation has been restricted

to third-order kernels only, in this study. Also, the ensemble

size has been restricted to 2000.

Acknowledgements

The authors wish to express their thanks to the ®nancial

aid being provided by the Propulsion Panel of Aeronautical

Research and Development Board, Ministry of Defence,

Government of India, in carrying out the study.

Appendix A. Synthesis of kernel transforms

Equating the like powers of c in Eq. (11)

c1 terms:

x1h 001x2h 001x3h 001

8>><>>:9>>=>>;1 2

j11 0 0

0 j22 0

0 0 j33

26643775

x1h 01x2h 01x3h 01

8>><>>:9>>=>>;

1

l11 1 lLb1

l12 l13

l21 l22 1 lLb2

l23

l31 l32 l33

2666437775

x1h1

x2h1

x3h1

8>><>>:9>>=>>; �

�f 1�t��f 2�t�

0

8>><>>:9>>=>>;�A1�

c2 terms:

x1h 002x2h 002x3h 002

8>><>>:9>>=>>;1 2

j11 0 0

0 j22 0

0 0 j33

26643775

x1h 02x2h 02x3h 02

8>><>>:9>>=>>;

1

l11 1 lLb1

l12 l13

l21 l22 1 lLb2

l23

l31 l32 l33

2666437775

x1h2

x2h2

x3h2

8>><>>:9>>=>>; �

0

0

0

8>><>>:9>>=>>; �A2�


Fig. 11. Estimates of the nonlinear parameters of: (a) lNb1; and (b) lN

b2:

c3 terms:

x1h 003x2h 003x3h 003

8>><>>:9>>=>>; 1 2

j11 0 0

0 j22 0

0 0 j33

26643775

x1h 03x2h 03x3h 03

8>><>>:9>>=>>;

1

l11 1 lLb1

l12 l13

l21 l22 1 lLb2

l23

l31 l32 l33

2666437775

x1h3

x2h3

x3h3

8>><>>:9>>=>>;

1

lNb1

0 0

0 lNb2

0

0 0 0

2666437775

x1h31

x2h31

x3h31

8>><>>:9>>=>>; �

0

0

0

8>><>>:9>>=>>; �A3�

Eqs. (A1)±(A3) can be solved sequentially. Taking Laplace

transforms of Eq. (A1), for zero initial conditions, one

obtains

�M�{h1�s�} � {F�s�}

with

�M� �

s2 1 2sj11 1 l11 1 lLb1

l12 l13

l21 s2 1 2sj22 1 l22 1 lLb2

l23

l31 l32 s2 1 2sj33 1 l33

2666437775

�A4�

{h1�s�} � { x1h1�s� x2h1�s� x3h1�s�}T

and {F�s�} � { F1�s� F2�s� 0}T

The solution for x1h1�s�; x2h1�s� and x3h1�s� from the above

is

{h1�s�} � �H�{F�s�} �A5�

where the kernel matrix

�H� � �M�21 �A6�

whose individual elements can be worked out to be

�H� �x1 H1�s� x1 H2�s� x1 H3�s�x2 H1�s� x2 H2�s� x2 H3�s�x3 H1�s� x3 H2�s� x3 H3�s�

2666437775

x1 H1�s� � �2l23l32 1 �s2 1 2sj22 1 l22 1 lLb2�

� �s2 1 2sj33 1 l33��=Dx1 H2�s� � �l13l32 2 l12�s2 1 2sj33 1 l33��=Dx1 H3�s� � �2l13�s2 1 2sj22 1 l22 1 lL

b2�1 l12l33�=D

x2 H1�s� � �l23l31 2 l21�s2 1 2sj33 1 l33��=Dx2 H2�s� � �2l13l31 1 �s2 1 2sj11 1 l11 1 lL

b1�

� �s2 1 2sj33 1 l33��=Dx2 H3�s� � �l13l21 2 �s2 1 2sj11 1 l11 1 lL

b1�l23�=D

x3 H1�s� � �2�s2 1 2sj22 1 l22 1 lLb2�l31 1 l21l32�=D

x3 H2�s� � �l12l31 2 �s2 1 2sj11 1 l11 1 lLb1�l32�=D

x3 H3�s� � �2l12l21 1 �s2 1 2sj11 1 l11 1 lLb1�

� �s2 1 2sj22 1 l22 1 lLb2��=D (A7)

with

D � �s2 1 2sj11 1 l11 1 lLb1�

� �s2 1 2sj22 1 l22 1 lLb2��s2 1 2sj33 1 l33�

2 �s2 1 2sj11 1 l11 1 lLb1�l23l32

2 l13�s2 1 2sj22 1 l22 1 lLb2�l31

2 l12l21�s2 1 2sj33 1 l33�1 l12l23l31 1 l13l21l32

�A8�Taking the Laplace transform of Eq. (A2) similarly gives

s2 1 2sj11 1 l11 1 lLb1

l12 l13

l21 s2 1 2sj22 1 l22 1 lLb2

l23

l31 l32 s2 1 2sj33 1 l33

2666437775

�x1h2�s�x2h2�s�x3h2�s�

8>><>>:9>>=>>; �

0

0

0

8>><>>:9>>=>>; (A9)

which yields

x1h2�s� � 0; x2h2�s� � 0; x3h2�s� � 0 �A10�which shows that the second-order kernel is identically zero,

i.e.

x1 k2�t1; t2� � x2 k2�t1; t2� � x3 k2�t1; t2� � 0 �A11�In order to synthesise expressions for third-order kernels,


Eq. (A3) can be written as

x1h 003x2h 003x3h 003

8>><>>:9>>=>>;1 2

j11 0 0

0 j22 0

0 0 j33

26643775

x1h 03x2h 03x3h 03

8>><>>:9>>=>>;

1

l11 1 lLb1

l12 l13

l21 l22 1 lLb2

l23

l31 l32 l33

2666437775

x1h3

x2h3

x3h3

8>><>>:9>>=>>;

�q1�t�q2�t�

0

8>><>>:9>>=>>; �A12�

where the following abbreviations have been used:

q1�t� � 2lN11

x1h31�t�; q2�t� � 2lN

22x2h3

1�t� �A13�Eq. (A12) is a set of linear equations, similar to Eq. (A3)

and therefore the solution in terms of Volterra operators can

be written as

x1h3�t� � x1 H1�q1; q2�;x2h3�t� � x2 H1�q1; q2�;x3h3�t� � x3 H1�q1; q2�

�A14�

that is

x1 H3� �f 1�t�; �f 2�t�� x1 H1�q1; q2�x2 H3� �f 1�t�; �f 2�t�� x2 H1�q1; q2�x3 H3� �f 1�t�; �f 2�t�� x3 H1�q1; q2�:

�A15�

Appendix B. Third-order kernel factors

x1x1C �i±j;k;l�3 �s1; s2; s3�

� 2x1 H�i�1 �s1 1 s2 1 s3�x1 H�j�1 �s1�x1 H�k�1 �s2�x1 H�l�1 �s3�

x2x2C �i±j;k;l�3 �s1; s2; s3�


x2x1C �l±i;j;k�3 �s1; s2; s3�

� 2x2 H�l�1 �s1 1 s2 1 s3�x1 H�i�1 �s1�x1 H�j�1 �s2�x1 H�k�1 �s3�

x3x1C �i±j;k;l�3 �s1; s2; s3�


x3x2C �l±i;j;k�3 �s1; s2; s3�

� 2x3 H�l�1 �s1 1 s2 1 s3�x2 H�i�1 �s1�x2 H�j�1 �s2�x2 H�k�1 �s3�

i � 1; 2; j � 1; 2; k � 1; 2; l � 1; 2:

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