Identification of Damping Using Proper Orthogonal Decomposition

Transcription

1 Identification of Damping Using Proper Orthogonal Decomposition M Khalil, S Adhikari and A Sarkar Department of Aerospace Engineering, University of Bristol, Bristol, U.K. S.Adhikari@bristol.ac.uk Identification of Damping p.1/26

2 Outline Motivation Brief overview of damping identification Independent Component Analysis Numerical Validation Conclusions Identification of Damping p.2/26

3 Damping Identification 1 Unlike the inertia and stiffness forcers, in general damping cannot be obtained using first principle. Two briad approaches are: damping identification from modal testing and analysis direct damping identification from the forced response measurements in the frequency or time domain Identification of Damping p.3/26

4 Some References 1. Adhikari, S. and Woodhouse, J., Identification of damping: part 1, viscous damping, Journal of Sound and Vibration, Vol. 243, No. 1, May 2001, pp Adhikari, S. and Woodhouse, J., Identification of damping: part 2, non-viscous damping, Journal of Sound and Vibration, Vol. 243, No. 1, May 2001, pp Adhikari, S. and Woodhouse, J., Identification of damping: part 3, symmetry-preserving method, Journal of Sound and Vibration, Vol. 251, No. 3, March 2002, pp Adhikari, S. and Woodhouse, J., Identification of damping: part 4, error analysis, Journal of Sound and Vibration, Vol. 251, No. 3, March 2002, pp Adhikari, S., Lancaster s method of damping identification revisited, Transactions of ASME, Journal of Vibration and Acoustics, Vol. 124, No. 4, October 2002, pp Adhikari, S., Damping modelling using generalized proportional damping, Journal of Sound and Vibration, Vol. 293, No. 1-2, May 2006, pp Adhikari, S. and Phani, A., Experimental identification of generalized proportional damping, Transactions of ASME, Journal of Vibration and Acoustics, 2006, submitted. Identification of Damping p.4/26

5 Damping Identification 2 Some shortcomings of the modal analysis based methodologies are: Difficult to extend in the mid-frequency range: Relies on the presence of FRF distinct peaks Computationally expensive and time-consuming for large systems Non-proportional damping leading to complex modes adds to the computational burden Identification of Damping p.5/26

6 Mid-Frequency Range Low-Frequency Range: A uniform low modal density High-Frequency Rage: A uniform high modal density Mid-Frequency Range: Intermediate band in which modal density varies greatly Identification of Damping p.6/26

7 Discrete Linear Systems The equations describing the forced vibration of a viscously damped linear discrete system with n dof: M n ü n (t) + C n u n (t) + K n u n (t) = f n (t) M n is the mass matrix, C n is the damping matrix and K n is the stiffness matrix u n (t) is the displacement vector, and f n (t) is the forcing vector at time t In the frequency domain, one has [ ω 2 M n + iωc n + K n ] Un (ω) = F n (ω) Identification of Damping p.7/26

8 Damping Matrix Identification Applying Kronecker Algebra and taking the vec operator to the frequency domain representation ( Un (ω i ) T iω i I n ) veccn = F n (ω i )+ω 2 i M n U n (ω i ) K n U n (ω i ), For many frequencies, we have ¾ U n (ω 1 ) T iω 1 I n U n (ω 2 ) T iω 2 I n. U n (ω J ) T iω J I n vecc n = F n (ω 1 ) + ω 2 1 M nu n (ω 1 ) K n U n (ω 1 ) F n (ω 2 ) + ω2 2M nu n (ω 2 ) K n U n (ω 2 ). F n (ω J ) + ω 2 J M nu n (ω J ) K n U n (ω J ). The above equation can be written as Ax = y Identification of Damping p.8/26

9 Least-Square Approach In case the system of equations being overdetermined, x can be solved in the least-square sense using the least-square inverse of the matrix A, as follows x = [ A T A ] 1 A T y. x is the least-square estimate of x and [ A T A ] 1 A T is the Moore-Penrose inverse of A Identification of Damping p.9/26

10 Physics-Based Tikhonov Regularisation In order to satisfy symmetry, for instance, in the damping matrix C m, we need to have C n = C T n The symmetry condition in the mass matrix gives rise to the constraint equation: L C x = 0 n 2 0 m 2 is the zero vector of order m 2 and the subscript in L C indicates that the constraint is on the damping matrix Identification of Damping p.10/26

11 Tikhonov Regularisation Applying Tikhonov Regularisation to estimate x, we obtain the following solution x = ( ) A T A + λ 2 CL T 1 ( CL C A T y ). The above solution depends on the values chosen for the regularisation parameter λ C If λ C is very large, the constraint enforcing the symmetry condition predominates in the solution of x On the other hand, if it is chosen to be small, the symmetry constraint is less satisfied and the solution depends more heavily on the observed data y Identification of Damping p.11/26

12 The Need for Model Order Reduction In the proposed recursive least squares method, we are required to obtain the inverse of a square matrix of order n 2 If we are trying to estimate the damping matrix of a complex system with large n, this is not feasible, even on high performance computers There is a need to reduce the order of the model prior to the system identification step Identification of Damping p.12/26

13 Proper Orthogonal Decomposition Entails the extraction of the dominant eigenspace of the response correlation matrix over a given frequency band These dominant eigenvectors span the system response optimally on the prescribed frequency range of interest POD is essentially the following eigenvalue problem R uu ϕ = λϕ R uu is the response correlation matrix given by R uu = u n (t)u n (t) T 1 T T u n (t)u T n (t) t=1 Identification of Damping p.13/26

14 Spectral Decomposition of R uu Using the spectral decomposition of R uu, one obtains R uu = n i=1 λ i ϕ i ϕ T i The eigenvalues are arranged: λ 1 λ 2... λ n The first few modes capture most of the systems energy, i.e. R uu can be approximated by R uu m i=1 λ iϕ i ϕ T i m is the number of dominant POD modes, generally much smaller than n Identification of Damping p.14/26

15 Model Reduction using POD The output vector can be approximated by a linear representation involving the first m POD modes using u n (t) = m i=1 a i (t)ϕ i = Σa (t) Σ is the matrix containing the first m dominant POD eigenvectors: Σ = [ϕ 1,...,ϕ m ] R n m Identification of Damping p.15/26

16 Model Reduction using POD Using Σ as a transformation matrix, our reduced order model becomes Σ T M n Σä (t) + Σ T C n Σȧ (t) + Σ T K n Σa (t) = Σ T f n (t) The system of equations can now be rewritten in the reduced-order dimension as M m ü m (t) + C m u m (t) + K m u m (t) = f m (t) Identification of Damping p.16/26

17 Model Reduction using POD The reduced order mass, damping, and stiffness matrices as well as the reduced order displacement and forcing vectors are M m C m K m = Σ T M n Σ R m m = Σ T C n Σ R m m = Σ T K n Σ R m m u m (t) = Σ T u n (t) = a (t) f m (t) = Σ T f f (t) Identification of Damping p.17/26

18 Reduced-Order Model Identification Once either the POD transformation is applied, there will be m 2 unknowns to be identified, as opposed to n 2 for our original model, where m is much smaller than n The aforementioned least square estimation method can now be used to estimate the reduced order damping matrix Once the reduced order damping matrix is estimated, we can carry out system simulations at the lower order dimension m, and project the displacement results back into the original n-dimensional space Identification of Damping p.18/26

19 Numerical Validation A coupled linear array of mass-spring oscillators is considered to be the original system A lighter system is coupled with a heavier system The lighter system posses higher modal densities compared to the heavier system Identification of Damping p.19/26

20 System Description The mass and stiffness matrices have the form ¾ M n = Kn = k u 0 n/2 m 2 I n/2 ¾ m 1I n/2 0 n/ n, m 1, and m 2 are chosen to be 100 DOFs, 0.1kg, and 1kg and k u = N/m The system is assumed to have Rayleigh damping by C n = α 0 M n + α 1 K n, where α 0 = 0.5 and α 1 = Identification of Damping p.20/26

21 Typical System Response Original FRF Frequency band of interest Log amplitude (db) of H (60,40) (ω) Frequency (rad/s) The frequency range considered for the construction of the POD is shown Identification of Damping p.21/26

22 POD Eigenvalues Log (db) of normalized POD eigenvalues ( λ / λ max ) Eigenvalue number Normalized eigenvalues of the correlation matrix Identification of Damping p.22/26

23 POD-Reduced Model A typical FRF of the POD reduced model is compared with the original FRF below The reduced order model FRF match reasonably well with the original FRF in the frequency band of interest 105 Original TF, n= 100 POD Simulated TF, m= 13 Log amplitude (db) of H (60,40) (ω) Frequency (rad/s) Identification of Damping p.23/26

24 Noise-Free Identification In the noise-free case, we obtain the identified POD reduced order damping matrix The identified matrix is used to obtain a typical FRF of the system below 105 Original TF, n= 100 POD Identified TF, m= 13 Log amplitude (db) of H (60,40) (ω) Frequency (rad/s) Identification of Damping p.24/26

25 Effect of Noise The system response is contaminated with noise The variance of the noise is ten times smaller than that of the response We obtain the identified FRF shown below 105 Original TF, n= 100 POD Identified TF, m= 13 Log amplitude (db) of H (60,40) (ω) Frequency (rad/s) Identification of Damping p.25/26

26 Conclusion The salient features that emerged from the current investigation are: POD can be successfully applied for reduced-order modelling Kronecker Algebra in conjunction with Tikhonov Regularisation provide an elegant theoretical formulation involving identification of the damping matrix Using a noise-sensitivity study, the identification method is demonstrated to be robust in a noisy environment Identification of Damping p.26/26