PROPER ORTHOGONAL DECOMPOSITION FOR MODEL UPDATING OF NON-LINEAR MECHANICAL SYSTEMS

Transcription

1 Mechanical Systems and Signal Processing (2001) 15(1), 31}43 doi: /mssp , available online at on PROPER ORTHOGONAL DECOMPOSITION FOR MODEL UPDATING OF NON-LINEAR MECHANICAL SYSTEMS V. LENAERTS, G. KERSCHEN AND J. C. GOLINVAL LTAS-Vibrations et Identixcation des Structures, University of Lie% ge, 1, Chemin des Chevreuils, B52, B-4000 Lie% ge, Belgium. (Accepted 1 September 2000) Proper orthogonal decomposition (POD), also known as Karhunen}Loeve (K}L) decomposition, is emerging as a useful experimental tool in dynamics and vibrations. The POD is a means of extracting spatial information from a set of time-series data available on a domain. The use of (K}L) transform is of great help in non-linear settings where traditional linear techniques such as modal-testing and power-spectrum analyses cannot be applied. These decomposition can be used as an orthogonal basis for e$cient representation of the ensemble. The POM have been interpreted mainly as empirical system modes and the application of POD to measured displacements of a discrete structure with a known mass matrix leads to an estimation of the normal modes. We investigate the use of the proper orthogonal modes of displacements for the identi"cation of parameters of non-linear dynamical structures with an optimisation procedure based on the di!erence between the experimental and simulated POM. A numerical example of a beam with a local non-linear component will illustrate the method Academic Press 1. INTRODUCTION The identi"cation of the dynamic characteristics of linear systems is now widely used and interest in non-linear systems has increased. Identi"cation of non-linear systems ranges from methods which simply detect the presence or type of a non-linearity to those which seek to quantify the dynamic behaviour through a mathematical model. Generally speaking, it requires the knowledge of the applied force and of the response of the system. Once the model parameters are identi"ed, the model may be used afterwards to predict the behaviour of the system. For dynamic analysis, most mechanical structures may be approximated by a linear model. However, when these structures are subjected to large displacement amplitudes, non-linear e!ects may become important and the linear model consequently fails. Even when the amplitudes remain restricted, some non-linear distortions may occur due to dry friction for instance. Both reasons demonstrate why interest in non-linear identi"cation is increasing. The importance of diagnosing and modelling non-linearity is recognised since a long time but it is only recently that non-linear theory is beginning to be exploited by structural dynamicists. A review can be found in reference [1]. Much of the work before the 1980s concentrated on the behaviour of non-linear systems with comparatively simple equations of motion. However, with the advent of a rapid expansion in computer-processor power, the simulation and identi"cation of multi-degree-of-freedom (mdof) structures became possible. One of the "rst approach to the identi"cation of sdof systems began with Masri and Caughey [2]. The procedure of the identi"cation is based on Newton's second law. The 0888}3270/01/010031#13 $35.00/ Academic Press

2 32 V. LENAERTS E¹ A. method analyses non-linear systems in terms of their internal restoring forces. It was generalised to mdof systems but su!ered from the necessary knowledge of both the displacement, velocity and acceleration data for each dof. Lot of researchers has worked on the method [3}5] in the following years and some experimental studies have been published on the identi"cation of non-linear automotive components [6, 7]. The use of Volterra's and Wiener's series gave information about the non-linear transfer of energy between frequencies. An identi"cation procedure was based on the higher-order FRF [8]. One serious problem of the method is the range of validity of the expansion of the series. The discontinuity of some non-linear systems cannot be represented by the series. Another technique which has been applied for modelling and prediction purposes is based on auto regressive moving average (ARMA) models. The non-linear auto regressive moving average with exogenous input model (NARMAX) [9, 10] also makes a useful link with Volterra's series. It allows the extraction of the higher-order FRFs directly from the NARMAX model. Recently, the conditioned reverse-path method which is a promising frequency-domain algorithm, has been developed for mdof systems [11]. New works have now started to take into account the spatial information. Adams [12] introduces a new principle of superposition for non-linear systems based on a spatial perspective of non-linearities as internal forces. Feeny [13] interprets the proper orthogonal modes (POM) as an approximation of the non-linear normal modes. The proper orthogonal decomposition (POD) [14] can be viewed as a time}space bi-orthogonal decomposition. The aim of this paper is to present a time domain method for the identi"cation of non-linear parameters of a model. The proposed method investigates the use of the spatial and time information contained in the POD to identify the linear as well as the non-linear parameters of a non-linear dynamical structure. The optimisation procedure is based on the di!erence between the experimental and simulated POM. 2. PROPER ORTHOGONAL DECOMPOSITION The POD, also employed to determine low-dimensional models, was "rst applied to turbulence problems by Lumley [15]. It identi"es a useful set of basis functions and the dimension of the subspace necessary to achieve a satisfactory approximation of the system. The POD also facilitates the resolution of the partial di!erential equations through their projection into a reduced-order model [16]. The POD has been applied successfully in the "eld of #uid dynamics [17] to identify coherent structures in turbulence. Many recent investigations have examined impacting systems [16, 18], thermics [19] and signal processing [20]. A recent work [13] has shown that the application of the POD to measured displacements of a discrete structure with a known mass matrix leads to an estimation of the normal modes MATHEMATICAL FORMULATION OF THE POD The POM are shown here to be the eigenfunctions of the space correlation tensor. The de"nitions and formulation presented here follow closely the ones used in Azeez [18]. Let u(x, t) be a random "eld on some domain Ω. Since the POD requires to deal with zero-mean signals, it is necessary to de"ne (v(x, t)) by subtracting the mean (;(x)" u ) from (v(x, t)): u(x, t)";(x)#v(x, t). (1) These "elds are sampled at "nite number of points in time. Then, at a "xed time t, the system displays a snapshot v (x), which is a continuous function on Ω. The aim of the POD

3 is to "nd the most representative structure (x) of the ensemble of N snapshots. This is accomplished by minimising the objective function (OF) λ: Minimise λ" ( (x)!v (x)) x3ω. (2) Equation (2) means that the sum of the di!erences squared between v (x) and (x) should be minimised and can also be written in term of a maximisation problem: Maximise PROPER ORTHOGONAL DECOMPOSITION λ"(1/n) ( Ω (x)v (x)dω) Ω (x) (x)dω x3ω. (3) In order to make the computation unique, the following orthonormality condition is imposed: (x) dx"1. (4) Ω With the following notations: 33 ( f, g), f (x)g(x)dω Ω inner product of f and g v, 1 v (x) average of snapshots N it turns out that the ensemble average of the inner products between v and must be maximised: Maximise λ" (, v ) (, ) The numerator of equation (3) can be expanded (, v ) " 1 (x)v (x)dx N (x )v (x )dx Ω Ω "( Ω Ω 1 N x3ω. (5) v (x)v (x ) (x)dx (x )dx. (6) The two-point correlation function (K) isde"ned as K(x, x )" 1 v (x)v (x ). (7) N Hence, equation (6) becomes (, v ) (K(x, x )) (x)dx (x )dx. (8) Ω Ω Inspection of equations (8) and (3) reveals that the optimisation problem can be reduced to the following integral eigenvalue problem [17, 18]: K(x, x ) (x )dx "λ (x ). (9) Ω

4 34 V. LENAERTS E¹ A. Equation (9) has a "nite number of orthogonal solutions (x), called the POM with corresponding real and positive eigenvalues λ. The eigenvalue with the largest magnitude is the maximum which is achieved in the maximisation problem (5). The second largest eigenvalue is the maximum of the same problem restricted to the space orthogonal to the "rst eigenfunction, and so forth. In order to make the computation unique, the eigenfunctions are normalised. Therefore, the POM can be used as a basis for the decomposition of the "eld v(x, t): v(x, t)" a (t) (x). (10) Moreover, by construction, the POM capture more energy than any other modes. It should also be noted that time-dependent coe$cients a (t) in equation (10) are uncorrelated [16]. Thus, the POD can be viewed as a bi-orthogonal decomposition because of the space}time symmetry of the decomposition. For an accurate approximation of the tensor (7) it is necessary to perform a long and expensive simulation. The computation of the eigenfunctions is even more expensive. Two methods exist to solve the problem, the direct [14] and the snapshot method [18] DISCRETE FORMULATION Suppose N linear snapshots v of size M obtained for instance by measurements of the acceleration at M locations. The M M covariance matrix C [20] is de"ne as Its eigensolutions (, λ ) which satisfy with C" 1 N v v. (11) C "λ, k"0,2, N!1 (12) λ *λ *2*λ '0 (13) characterise the proper orthogonal decomposition. Each eigenvector is associated with an eigenvalue λ. If the eigenvalues are normalised, they represent the relative energy captured by the corresponding POM. This decomposition is the best basis in terms of de-correlation. The brute computation of the eigensolutions of C is called the direct method. However, due to the space}time symmetry property, an alternative method, called the &&method of snapshots'', can be employed [18] COMPUTATION OF THE POD USING SVD The complete bi-orthogonal decomposition of the data may be obtained by the use of the SVD [21]. For instance, the SVD which is related to principal component analysis, is used in reference [22, 23] to compute modal metrics to solve model-updating problems in an optimisation procedure. Let x(t) denote a response time-history, where x is a vector containing the displacement, velocity or acceleration at M discrete locations. The discrete matrix X is formed: X" x (t ) 2 x (t ) x (t ) 2 x (t ). (14)

5 PROPER ORTHOGONAL DECOMPOSITION Thus, each row corresponds to a time history at one location and each column corresponds to a snapshot of the system at a speci"c time. Now, the singular-value decomposition of matrix X can be written as X"UΣV (15) with U an orthonormal matrix (size M M) of eigenvectors of XX and V an orthonormal matrix (size N N) of eigenvectors of X X. The size of the matrix Σ is M N but only the main diagonal has non-zero elements which are the singular values of X, sorted in descending order. If the matrix X is rank de"cient, i.e. some rows (or columns) can be generated by a linear superposition of the others, a few singular values will be zero. The SVD has a lot of applications, e.g. the estimation of the rank of a matrix, the "ltering of measurement noise and so forth. In this paper, the aim of the SVD is to compute the POMs and the normalised basic shapes including the response time histories [23] APPLICATION TO PARAMETER IDENTIFICATION The identi"cation of the non-linear parameters of a structure is based on the solution of an optimisation problem which consists in minimising the di!erence between the bi-orthogonal decompositions of the measured and simulated data, respectively. Let us de"ne the objective function F as F" ( ; ) # ( Σ ) # ( < ) (16) where ;, Σ and < are the di!erences between the matrices containing the bi-orthogonal decompositions, i.e. the di!erences between the SVD of the measured and simulated data [equation (15)]. It must be stated that the full decomposition is not retained in the (OF). Only the terms corresponding to the higher singular values are considered, which means that we take the proper orthogonal modes that contain the greatest amount of energy in the signal. Let be Σ 'Σ '2'Σ be the decreasing singular values of the experimental data. If we de"ne E" Σ the total energy in the data, only the p modes corresponding to a given percentage (e.g. 90%) of this energy are retained in the objective function, i.e. Σ '90%. (17) E Then the objective function F may be minimised using standard optimisation algorithms. 4. NUMERICAL EXAMPLE To illustrate the method, let us consider the clamped steel beam with a local non-linearity de"ned in Fig. 1. The steel beam is m long and has a square section of m. The beam is modelled with six "nite elements. Each node possesses 2-dof: one for the vertical de#ection and one for the rotation. The whole structure has 12- dof. The nonlinearity is a spring that exhibits a cubic sti!ness. For each iteration of the optimisation process, two sets of time histories have to be compared: The "rst one, which represents the reference case, has to be generated only one time. The second one, which corresponds to the model, has to be generated at each iteration of the optimisation process.

6 36 V. LENAERTS E¹ A. Figure 1. Non-linear beam with a local non-linearity. The ;, <, and Σ matrices are obtained from equation (15) for both cases and introduced in the objective function (16). Then the optimisation yields the solution of the parameter identi"cation IDENTIFICATION OF THE NON-LINEAR PARAMETER In this section, the linear part is supposed to be known so that the only parameter to identify is the non-linear cubic sti!ness. The free vibration of the beam is simulated with an initial displacement given by a static force applied at the end of the beam. The simulation is performed over a time period of 0.1 s, with a time step of 2 10 s. A Gaussian noise with an amplitude of 1% of the initial displacement has been added to test the robustness of the technique. In this example, the objective function is written in terms of the "rst POM only, i.e. with the highest singular value, which represents 70% of the energy. The normalised value of the non-linear parameter in the reference case is 50. This value is perturbed to start the optimisation process. Di!erent starting points are tested which lead to different minima. To obtain the correct optimum, the optimisation process needs to start with an initial value for the non-linear coe$cient not bigger than 80% of the correct value. The comparison between the original and the reconstructed signals at the starting point is given in Fig. 2a, while in Fig. 2b, the comparison is shown after the optimisation. The reconstructed signal (shown in dashed line) matches very closely the original one (in solid line) IMPROVEMENT OF THE OBJECTIVE FUNCTION To improve the optimisation process, the in#uence of di!erent parameters were tested, such as the total time of the simulation, the number of POD or the type of objective function. For each test the OF has been calculated for a large range of variation of the non-linear parameter Duration of the simulation The simulation period has a great in#uence on the picture of the OF because the time decomposition of the data is included in the OF (16). When the simulation duration is long, the time decomposition contains a lot of oscillations involving more oscillations in the plot of the objective function. In Fig. 3, two OFs are plotted with a simulation time of 0.02 s and

7 PROPER ORTHOGONAL DECOMPOSITION 37 Figure 2. Comparison of displacements before and after optimisation. Figure 3. Objective function for two simulation times. 0.2 s respectively. The "gure clearly shows that a lot of minima appear in the case of a long simulation time which can be bad for the research of the global minimum during the optimisation process. Of course a small simulation time (e.g. one oscillating period of the proper orthogonal mode) gives very little information on the system so that the proper orthogonal modes lose physical signi"cance. One solution consists into dropping the time decomposition in the OF (16). The plot of the objective function in Fig. 4 shows a more suited curve for the optimisation process. The drawback is again the loss of time information over the system.

8 38 V. LENAERTS E¹ A. Figure 4. Objective function without the time decomposition. In order to retain the time information but without the drawback of its oscillatory nature, the wavelet transform of the decomposition may be performed Wavelet transform of the time decomposition The wavelet transform =( f, t) of the time variation associated with each POM is computed, giving a time}frequency representation of the energy contained in the signal. The frequencies that correspond to the maximum of = are extracted. This information, which is time-dependent, is included into the objective function (4) instead of the right-singular vectors; it allows to transform the oscillatory nature of the time information into a more suited one. The comparison between the two formulations is shown in Fig. 5 for a simulation time of 1 s. The OF without the wavelet transform shows a quite &&horizontal'' line connected to a narrow valley that contains the global minimum (see Fig. 5). When the wavelet transform is applied, the objective function decreases from the start to the optimum, showing a large valley. The "rst part of each curve is zoomed in Fig. 6. In the case of the wavelet transform the function decreases very smoothly, which allows to reach easily the optimum of the function. Without the wavelet transform, the horizontal line is oscillating before the narrow valley; this may compromise the e$ciency of the optimisation procedure. As seen in Fig. 5, the non-linear phenomena are too important for high values of the non-linear parameter and the objective function presents large oscillations. Therefore, the optimisation process has to begin with small values of the non-linear parameter Number of POM In the previous sections only the "rst proper orthogonal mode was included in the objective function. Since no external excitation is applied, this mode contains more than 60% of the total amount of energy, which is su$cient for the identi"cation of the non-linear parameter. Some tests have been performed with more than one proper orthogonal mode [i.e. including more and more singular values into equation (16)] and the same behaviour as before is observed IDENTIFICATION OF THE WHOLE SYSTEM Let us consider now both the identi"cation of the linear and non-linear parts. In this case, two updating parameters are considered: the non-linear sti!ness and Young's modulus of the beam. Again the free vibration of the beam is simulated with an initial displacement given by a static force applied at the end of the beam.

9 PROPER ORTHOGONAL DECOMPOSITION 39 Figure 5. Objective function with and without wavelet transform. Figure 6. Zoom of the objective function with and without wavelet transform. The OF is written in terms of the "rst POM. The wavelet transform of the time decomposition is included into the function in place of the right-singular vectors. The simulation is performed over a time period of 0.4 s. A Gaussian noise with an amplitude of 1% of the initial displacement is added to the simulated displacement representing the measured data. The optimisation process starts with an initial value for the non-linear coe$cient of about 10% of the correct value, and an initial value for Young's modulus of about 50% of the correct value. The comparison between the original and the reconstructed signals at the starting point is given in Fig. 7(a), while in Fig. 7(b), the comparison is shown

10 40 V. LENAERTS E¹ A. Figure 7. Comparison of displacements before and after optimisation. Figure 8. Objective function during the optimisation process. after the optimisation. The reconstructed signal matches perfectly the original one. The error on the updated parameters is less than 0.1%. The plot of the objective function for each step of the optimisation is shown in Fig. 8. In Fig. 9 the OF is plotted for a large domain of the two variables. This plot shows that the OF is well suited for the optimisation process. The function presents only disturbances in the domain where the non-linear parameter is much higher than the linear one. For high values of the non-linear sti!ness, the non-linear phenomena are too strong so that any small change of one parameter can lead to signi"cant change of the OF. The comparison with Fig. 10 where the wavelet transform of the time decomposition is not performed into the OF, shows a lot of local minima which compromises the e$ciency of the optimisation procedure.

11 PROPER ORTHOGONAL DECOMPOSITION 41 Figure 9. Contour plot of the OF for the two updating parameters. Figure 10. Contour plot of the OF for the two updating parameters (no wavelet). 5. CONCLUSIONS In this paper, the proper orthogonal decomposition has been used to identify parameters of non-linear dynamical structures. An optimisation procedure based on the di!erence between the experimental and simulated POM has been used. A numerical example of a beam with a local non-linear component illustrates the method. The wavelet transform is then used to improve the optimisation procedure. The extension of the technique to the optimisation of several parameters is straightforward. The advantage of the method compared to the classical identi"cation techniques is that it can easily be applied to mdof systems. The systems considered up to now are submitted to free vibrations but it is planned to study the in#uence of random and sinusoidal excitations. The method has been tested on a simple simulated example and more tests are needed to

12 42 V. LENAERTS E¹ A. judge the importance of the WT and the number of POM included in the OF. A drawback of the method, is the requirement for generating time domain data using the predictive model at each iteration of the optimisation process, which is actually di$cult to realise for typical engineering FE model with several thousand dof. However, the method seems to be promising in the di$cult area of non-linear dynamic systems and will be veri"ed on experimental cases in the near future. ACKNOWLEDGEMENTS This work is supported by a grant from the Walloon government as a part of the research convention no &&Analyse inteh greh edereh sultats numeh riques et expeh rimentaux en dynamique des structures''. Part of this text also presents research results of the Belgian programme on Inter-University Poles of Attraction initiated by the Belgian state, Prime Minister's O$ce, Science Policy Programming. The scienti"c responsibility is assumed by its authors. REFERENCES 1. K. WORDEN 2000 Proceedings of the European COS¹ F3 Conference on System Identi,cation and Structural Health Monitoring. Nonlinearity in structural dynamics: the last ten years. 2. S.MASRI and T. CAUGHEY 1979 Journal of Applied Mechanics 433}447. A nonparametric identi- "cation technique for nonlinear dynamic problems. 3. M. AL-HADID and J. WRIGHT 1990 Mechanical Systems and Signal Processing 4, 463}482. Application of the force}state mapping approach to the identi"cation of nonlinear systems. 4. S. DUYM and P. GUILLAUME 1996 International Journal of Analytical and Experimental Modal Analysis 5. A local restoring force surface. 5. K. S. MOHAMMAD and G. TOMLINSON 1991 Journal of Sound and <ibration 152, 471}499. Direct parameter estimation for linear and nonlinear structures. 6. C. SURACE and G. TOMLINSON 1992 Proceedings of the I. Mech. E., Part D } Journal of Automobile Engineering 206, pp. 3}16. On the nonlinear characteristics of automotive shock absorbers. 7. S. DUYM and K. REYBROUCK Proceedings of the 21st International Seminar on Modal Analysis, euven. Fast parametric and nonparametric identi"cation of shock absorbers. 8. D. STORER and G. TOMLINSON 1993 Mechanical Systems and Signal Processing 7, 173}189. Recent developments in the measurement and interpretation of higher order transfer function from nonlinear structures. 9. S. BILLINGS and K. TSANG 1989 Mechanical Systems and Signal Processing 3, 319}339. Spectral analysis for nonlinear systems, part 1: parametric nonlinear spectral analysis. 10. S. BILLINGS and K. TSANG 1989 Mechanical Systems and Signal Processing 3, 341}359. Spectral analysis for nonlinear systems, part II: interpretation of nonlinear frequency response functions. 11. C. RICHARDS and R. SINGH 1998 Journal of Sound and <ibration 213, 673}708. Identi"cation of multi-degree-of-freedom nonlinear systems under random excitations by the reverse-path spectral method. 12. D. ADAMS and R. ALLEMANG 1999 ASME Journal of <ibration and Acoustics 121, 495}500. Characterisation of nonlinear vibrating systems using internal feedback and frequency response modulation. 13. B. F. FEENY 1997 Proceedings of DE¹197 ASME Design Engineering ¹echnical Conferences. Interpreting proper orthogonal modes in vibrations. 14. M. XIANGHONG, M. F. A. AZEEZ and A. F. VAKAKIS 1998 Modal Analysis and ¹esting, NA¹O- Advanced Study Institute at Sesimbra, Portugal. Nonlinear normal modes and nonparametric system identi"cation of nonlinear oscillators. 15. J. L. LUMLEY 1967 Atmospheric ¹urbulence and Radio =ave Propagation. The structure of inhomogeneous turbulent #ows. 16. M. F. A. AZEEZ and A. F. VAKAKIS 1997 Proceedings of DE¹C197, ASME Design Engineering ¹echnical Conferences, Sacramento, CA. Numerical and experimental analysis of the nonlinear dynamics due to impacts of a continuous overhung rotor.

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