vibratory diagnosis by finite element model updating

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Colloque “Analyse vibratoire Expérimentale”

Blois, 20 & 21 Novembre 2012

VIBRATORY DIAGNOSIS BY FINITE ELEMENT MODEL UPDATING AND OPERATIONAL MODAL ANALYSIS DIAGNOSTIC VIBRATOIRE PAR RECALAGE DE MODELE ELEMENTS FINIS ET ANALYSE MODALE OPERATIONNELLE G. Gautier, R. Serra and J. -M. Mencik ENI Val de Loire, Université François Rabelais de Tours, LMR Laboratory, Rue de la Chocolaterie, BP 3410, F-41034 Blois Cedex, France, e-mail: [email protected]

Keywords: Subspace fitting, Operational Modal Analysis, Finite Elements Model Updating, Vibratory Diagnosis, Rotating machine

Resume In this paper, a subspace fitting method is proposed to update the finite element model of a rotating machine. The procedure is achieved by considering an experimental observability matrix which is obtained through a Subspace Identification technique. The procedure is applied to determine the foundation stiffness of a rotating machine subject to a random noise.

1

Introduction

Evaluating damages occurring in mechanical systems constitutes a tough task. Their emergence and evolution are characterized by variations (those can be small) of the dynamic properties of structures [1]. Many damage diagnosis methods have been proposed to carry out this issue. The methods based on Finite Element (FE) model updating [2] perform the comparison between the modal parameters obtained experimentally with those of a numerical model. For industrial processes, Operational Modal Analysis (OMA) [3] approaches aim at extracting the structural parameters in operating conditions. Among these approaches, Subspace Identification (SI) techniques [4] appear highly efficient to determine the modal parameters of structures in the time domain. The framework of SI techniques is summarized as follows. From the consideration of input-output data, a so-called experimental observability matrix is obtained by projecting the output signal onto some appropriate subspaces. The observability matrix contains the modal parameters of the structure considered, which are extracted using a subspace fitting method [5, 6]. The motivation behind this work is to improve further on the accuracy of SI techniques to predict the modal parameters of any mechanical system. To this aim, a FE model of the structure is considered to carry out the subspace fitting procedure in a Least Squares (LS) sense. The modal parameters of the system are then updated by minimizing an error norm which depends on some unknown parameters of the structure (e.g., stiffnesses…). The proposed approach is applied to a rotating machine excited by a random noise. The experimental observability matrix, as obtained using the MOESP SI technique, is used to update the parameters of the structure. The accuracy of the method is highlighted.

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Deterministic-Stochastic Modal Analysis

The purpose of SI techniques [4] is to consider a discrete modal state-space representation of the form

 q  xk  Bmoduk  wk k1  obsqk  vk  yk 

where

uk and yk

(eqs. 1)

are the vectors of input and output data, respectively;

matrix of eigenvalues, Bmod and  obs shapes of the structure; also, wk and

 is a diagonal

are matrices expressed in terms of the mode

vk

are vectors of noises while

qk

is a vector of

generalized coordinates. In eqs. 1, the subscripts k and k 1 refer to vectors of data measured at two different times tk and tk1 . From the linear state space model, the inputoutput matrix equation is derived as [11]

Y   expQ H dU  H sM  N where

 t

k

Y

(eq. 2)

is a matrix of output data that are measured over different time intervals



tk1 ... tk 1  . Q , M and k

coordinates, noises and input data; also, d

N

are the related matrices of generalized

 exp

represents the experimental observability

s

matrix; otherwise, H and H are Hankel matrices. The basic idea behind SI techniques is to identify the experimental observability matrix  exp from the knowledge of Y . This is done by eliminating the terms H U  H M  N in Eq.3 by means of projection and weighting procedures. Clearly, a projection of the row space of Y onto the orthogonal d

s



complement U of the row space of U enables one to remove the influence of inputs. In addition, the facts to left and right multiply eq. 3 with some matrices W1 and W2 having some specific properties regarding noise uncorrelation enables those noise terms to be removed as well. Considering such procedures yields

O  W1Y /U W2  W1 expQ /U W2

(eq. 3)

More specifically, SI techniques deal with the matrix

W1 exp  U11S11/2

[12], where the matrix

S1 results from a SVD of O, i.e.  S 0 O   U1 U 2  1  0  0

 V T  1  V2T

   

(eq. 4)

 and mode shapes obs are determined from the experimental observability matrix  exp in different ways [11]. All the methods make use of the invariance property of the matrix  exp . The matrices of eigenvalues

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Subspace Fitting for Finite Element Model updating

3.1 Objective function The subspace fitting procedure [5, 6] is a concept that aims at correlating a theoretical matrix ( ) with the experimental observability  exp matrix as

 exp  ( )T where

T

(eq. 5)

is a similarity matrix. Here, the matrix

( )

is supposed to be dependent from a

set of parameters (denoted as  ), which are to be identified. The subspace fitting procedure can be formulated through the following LS problem:

,T  arg min  exp  ( )T where

.

F

the matrix of

2

(eq. 6)

F

denotes the Frobenius norm. This LS problem can be simplified by determining

T

in a preprocessing step as

( ) ), which yields

T  ( )  exp ( ( ) being the pseudo inverse

  argmin r( ) 2 2

where

(eq. 7)

r( )  vec(I  ( )( ) ) exp  .

The key idea behind the present work is to express the theoretical observability matrix ( ( ) ) by means of a FE model of the considered mechanical system. In doing so, the spatial dynamics of the system is taken into account to carry out the minimization procedure of r( ) with a view to identifying the parameters  in an accurate and unique way. Regarding rotating machines, a related FE based eigenproblem is considered as

 M   (  G)  K   2 j

where

j

M, K

and



j

0

(eq. 8)

refer to the mass, stiffness and damping matrices; also,

matrix that reflects the gyroscopic effects, while

 j ,  j 

the eigenproblem are

G is the

 is the angular velocity. The solutions of

which stand for complex eigenvalues and right

eigenvectors, respectively. Thus a theoretical FE-based observability matrix can be expressed as

  obs    obs ( )       obs 1 where

tdiag(  j )

e

     

(eq. 9)

and

 obs is the matrix of eigenvectors  j at the observation points

of output signals.

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3.2 Optimization algorithm The Gauss-Newton algorithm [10] is used to solve the minimization problem for This algorithm is based on the following relation

 f 1   f   f H 1g where  f is a step size. g matrices of

4

r( )

2 2

r( ) .

(eq. 10)

and

H are,

respectively, the Gradient and the Hessian

defined as

 r  gi  2 r H   i 

(eq. 11)

 r H r  H ij  2     i  j 

(eq. 12)

Experiments

4.1 Description of the structure The proposed method is applied to update the FE model of the rotating machine depicted in Fig. 1. The structure is composed of a shaft attached to one rigid disk and supported by two flexible bearings. The properties of the structure are reported in Tab. 1. The output signal of the structure is measured by means of one accelerometer which is attached to the first bearing. On the second bearing, a shaker generates a random noise. The experimental observability matrix is obtained using the MOESP SI technique [11]. Some of identified experimental eigenfrequencies are reported in Tab. 2.

Fig. 1: Experimental Rotor

Shaft

Disk

Parameter Density Young Modulus Length Diameter Density Young Modulus Thickness Diameter

Value 7850 kg.m-3 193 GPa 0.5 m 0.0254 m 7850 kg.m-3 193 GPa 0.01 m 0.13 m

Tab. 1: Rotor Parameters

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4.2 FE of the structure A FE model is considered which is composed of six Timoshenko beam elements for the shaft, with two translations ( u and v) and two rotations (  and  ), along the x - and y axis, per node.

Fig. 2: FE model of the structure

The nodal displacement vector can be written as





u v  

The mass matrix



T

(eq. 13)

M S and M Sr of a beam element, related to translational and rotational

displacements, are respectively expressed as

 156 0 0 22L 54 0 0  156 22L 0 0 54 13L  2  4L 0 0 13L 3L2  4L2 13L 0 0  SL  MS  156 0 0 420   sym 156 22L   4L2    36 0 0 3L  36 3L 0   4L2 0  4L2 I  M rS  30L   sym    

13L 0 0 3L2 22L 0 0 4L2

36 0 0 3L   0 36 3L 0  0 3L L2 0   3L 0 0 L2  36 0 0 3L  36 3L 0   4L2 0   4L2 

           

(eq. 14)

(eq. 15)

where L , S,  and I are the length, cross-sectional area, density and inertia moment of the shaft cross-section, respectively. Also, the related matrix of gyroscopic effects is given

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by



 0 36 3L 0 0 36 3L 0    0 0 3L 36 0 0 3L    0 4L2 3L 0 0 L2    I  0 0 3L L2 0  Gs  0 36L 3L 0  15L   0 0 3L    skew  sym 0 4L2    0  

(eq. 16)

The element stiffness matrix is given by

 12 0 0 6L 12 0 0 6L  12 6L 0 0 12 6L 0  2 2  4L  a 0 0 6L 2L  a 0  2 2 4L  a 6L 0 0 2L  a EI  KS  3 12 0 0 6L2 (1 a)L  sym 12 6L 0  2  0 4L  a  2 4L  a 

           

(eq. 17)

where

a

12EI is introduced to take account the shear deformation effects. E and G GSL2

being the Young and Shear moduli of the shaft. Finally, the whole equation of motion for the shaft element is expressed as

  (M S  M )   r S

     GS  2  

1 

    1 0   KS   2   2 

1 

(eq. 18)

where structural damping is neglected. Otherwise, the disk is modeled by means of concentrated mass and gyroscopic effects using the following matrix term:

   M D  GD      where

mD

and

I

Dx

mD

0

0

0

mD

0

0

0

I Dx

0

0

0

0  u   0   v   0    I Dx   

            

0 0 0 0 0 0

0 0 0

0 0 I Dz

0 0 I Dz 0

     

      

 I Dy , I Dz  are the mass and the inertia moments of the disk. 123

u v 

(eq. 19)

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Finally, the foundation of the bearings is modeled by means of the following stiffness matrix

  KB     

k 0 0 0

0 k 0 0

0 0 0 0

0 0 0 0

     

(eq. 20)

4.3 Updating procedure The updating procedure of the FE model is carried out considering the bearing stiffness as an unknown parameter. This parameter is updated through the subspace fitting procedure described previously, whose flowchart is postponed in Fig. 3.

k

The procedure is initialized with a value of 10 10 N/m for k . The eigenfrequencies of the system (rotating machine - foundation), obtained for this value, are reported in Table 2 and compared with the experimental eigenfrequencies. Thus the updating procedure is 6

6

carried out which yields a value of 3310 N/m for k . In that case, the errors between FE-based and experimental eigenfrequencies appear quite small, as expected. In a more general view, the updating procedure can be used to determine the variation of the stiffness parameter k over the time domain to carry out structural health monitoring. This yields an efficient way to detect the occurrence of defects as well as machine breakdowns.

Fig. 3: Flowchart of the Subspace Fitting procedure

1 2 3 4 5

Experimental Frequency (Hz) 631 884 1551 2399 3001

Before Updating Frequency (Hz) Error (%) 394.7 37 759.5 14.1 1294.3 16.6 2064.6 13.9 2606.1 13.2

After Updating Frequency (Hz) Error (%) 635.8 0.8 878.9 0.6 1552.1 0.1 2284.9 4.8 2962.9 1.3

Tab.2: Eigenfrequencies of the system (rotating machine - foundation)

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Conclusion

A finite element model updating procedure has been proposed. The procedure used a subspace fitting approach to adjust a FE-based observability matrix with an experimental observability matrix obtained from a subspace identification technique. The method has been successfully applied to determine the bearing stiffness of a rotating machine. In a more general view, this method seems to constitute an efficient tool to carry out the structural health monitoring of mechanical systems.

Acknowledgements The authors express their thanks for the financial support provided by European Union (FEDER Centre) and ”Conseil Régional du Centre”.

References [1] S. W. Doebling, C. R. Farrar, M. B. Prime, « A summary review of vibration-based damage identification methods », Engineering Analysis Group Los Alamos National Laboratory, Los Alamos, NM. [2] M.I. Friswell, J.E. Mottershead, « Finite Element Model Updating in structural dynamics », Springer, 1995. [3] L. Zhang, R. Brincker, P. Andersen « An overview of operational Modal Analysis: Major Development and issues », Issue: 1, Publisher: Kluwer Academic Publishers, Pages: 179-190, 2004. [4] W. Favoreel, B. De Moor, P. Van Overschee « Subspace state space system identification for industrial processes », Journal of Process Control 10 (2000), pp. 149155. [5] R. Serra, C. Gontier, M. Raffy, « A subspace fitting method for structural modal identification in time domain », In ISMA25: International Conference of Noise and Vibration Engineering, Leuven, Belgium, 2000. [6] A. Lee Swindlehurst, B. Ottersten, R. Roy, T. Kailath, « A subspace fitting method for identification of linear state-space models », In: IEEE Transaction on Automatic Control, Vol. 40(2), pp. 315-316, 1995. [7] G.H. Golub and V. Pereyra, « The di erentiation of pseudo-inverses and nonlinear least squares problems whose variables separate », SIAM J. Numer Anal 10:413-432 (1973). [8] M. Lalanne, G. Ferraris « Rotordynamics Prediction in Engineering», Wiley, 1990. [9] M.I. Friswell, J.E.T. Penny, « Crack Modelling for Structural Health Monitoring », Structural Health Monitoring, Vol 1(2): 0139–148, 2002. [10] A. Lee Swindlehurst, B. Ottersten, R. Roy, T. Kailath, « Multiple Invariance ESPRIT », IEEE Transactions on Signal Processing, vol. 40, 1992. [11] M. Viberg, B. Wahlberg and B. Ottersten, « Analysis of State Space System Identification Methods Based on Instrumental Variables and Subspace Fitting », Automatica, vol. 33(9), 1997, 1603-1616. [12] P. VanOverschee, B. DeMoor, « Subspace Identifcation of Linear Systems: Theory, Implementation Applications Kluwer Academic Publishers », Dordrecht, 1996.

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