NON-STATIONARY MECHANICAL VIBRATION MODELING AND ANALYSIS

Transcription

1 NON-STATIONARY MECHANICAL VIBRATION MODELING AND ANALYSIS VIA FUNCTIONAL SERIES TARMA MODELS A.G. Poulimenos and S.D. Fassois DEPARTMENT OF MECHANICAL &AERONAUTICAL ENGINEERING GR PATRAS, GREECE sms 13 th IFAC Symposium on System Identification (SYSID 2003) Rotterdam, The Netherlands, August 2003 Research supported by the VolkswagenStiftung.

2 NON-STATIONARY MECHANICAL VIBRATION AND ANALYSIS VIA FUNCTIONAL SERIES TARMA MODELS 1 TALK OUTLINE 1. Introduction & Aims of the Work 2. The Problem 3. Functional Series TARMA Modeling 4. Non-Stationary Random Vibration Modeling and Analysis 5. Concluding Remarks

3 NON-STATIONARY MECHANICAL VIBRATION AND ANALYSIS VIA FUNCTIONAL SERIES TARMA MODELS 2 1. INTRODUCTION &AIMS OF THE WORK The General Problem Modeling and analysis of non-stationary mechanical vibration. Non-stationary random vibration is characterized by time-varying statistics. requires time-frequency methods for its analysis. Problem Characteristics Problem Significance Time (sec) Non-stationary random vibration is commonly encountered in: Mechanical systems Automotive & aircraft systems Rotating machinery Seismic & structural systems etc.

4 NON-STATIONARY MECHANICAL VIBRATION AND ANALYSIS VIA FUNCTIONAL SERIES TARMA MODELS 3 Literature Survey Non-stationary random vibration modeling and analysis may be based upon: Wigner-Ville distributions and their extensions (Oehlmann et al., 1997; Lee et al, 2001; Xu et al., 2002) Locally stationary modeling (Gersh and Brotherton, 1982; Owen et al., 2001) Wavelet-based methods (Ghanem and Romeo, 2000; Luo et el., 2002) Adaptive filtering (Cooper and Worden, 2000) FS-TAR/TARMA approaches (parametric) (Conforto and D Alessio, 1999; Petsounis and Fassois, 2000) Why Parametric Modeling? Offers a number of advantages over non-parametric methods: Physical significance and correspondence to underlying physical system Representation parsimony Improved accuracy and resolution

5 NON-STATIONARY MECHANICAL VIBRATION AND ANALYSIS VIA FUNCTIONAL SERIES TARMA MODELS 4 Why Functional Series TARMA Modeling? PARAMETRIC METHODS Parameter Evolution Statistical Evolution Accuracy Parsimony LOCALLY STATIONARY unstructured abrupt limited low ADAPTIVE unstructured slow limited low STOCHASTIC EVOLUTION stochastic slow & faster high medium FS-TARMA deterministic slow & fast high high Aims of the Work 1. Demonstration of the applicability of the FS-TARMA approach for modeling and predicting non-stationary mechanical vibration. 2. Assessment of the achievable accuracy and effectiveness of the FS-TARMA approach for recovering the underlying time-varying structural dynamics. 3. Comparison of the FS-TARMA approach with alternative approaches (STFT, RARMA RML)

6 NON-STATIONARY MECHANICAL VIBRATION AND ANALYSIS VIA FUNCTIONAL SERIES TARMA MODELS 5 2. THE PROBLEM 2.67 m Exciter DC motor conditioner DAQ PC Random excitation via electromechanical shaker. Vertical accelerations measured via piezoelectric accelerometers. Beam: 2670 (L) 50 (W) 70 (H) cm weight: 13.2 kgr Data acquisition (DAQ) [Siglab 20-42]. Band-pass filtering (focus on Hz frequency range). Cylindrical mass: 52.5 (R) 75.0 (H) cm weight: 5.4 kgr

7 NON-STATIONARY MECHANICAL VIBRATION AND ANALYSIS VIA FUNCTIONAL SERIES TARMA MODELS 6 Non-Stationary Vibration THE EXPERIMENT: In a single experiment the mass slides on the beam being pulled by a DC motor at a constant speed (u 3cm/sec) THE PROBLEM: Modeling, analysis and prediction of the non-stationary vibration Recovery of the underlying time-varying structural dynamics u constant velocity DC motor Exciter Acceleration m/sec 2 ( 0.7) Output Signal [Point (3)] Time (sec)

8 NON-STATIONARY MECHANICAL VIBRATION AND ANALYSIS VIA FUNCTIONAL SERIES TARMA MODELS 7 The Underlying Structural Dynamics The underlying structural dynamics corresponding to various mass positions are estimated via separate stationary experiments and subsequent analysis. d cm fixed position Exciter Acceleration m/sec 2 ( 0.7) Output Signal [Point (3)] Time (sec)

9 NON-STATIONARY MECHANICAL VIBRATION AND ANALYSIS VIA FUNCTIONAL SERIES TARMA MODELS 8 Model-based modal parameters 90 + : ARMA estimates Background: STFT Frequency (Hz) f 3 f 2 f Damping Ratio ζ 3 ζ ζ Time (sec)

10 NON-STATIONARY MECHANICAL VIBRATION AND ANALYSIS VIA FUNCTIONAL SERIES TARMA MODELS 9 3. FUNCTIONAL SERIES TARMA MODELING A FS-TARMA(na, nc) [pa,pc] model is of the form: x[t]+ n a i=1 a i [t] x[t i] =w[t]+ t : discrete time index x[t] : the non-stationary signal modeled w[t] : innovations sequence N (0,σw[t]) 2 n c i=1 c i [t] w[t i], t t 0 n a,n c : orders of the AR/MA polynomials a i [t],c i [t] : AR/MA time-varying parameters The time-varying model parameters (a i [t], c i [t], σ 2 w[t]) are expanded on functional spaces: a i [t] Δ = p a j=1 a i,j G ba (j)[t], c i [t] Δ = p c j=1 c i,j G bc (j)[t], σ 2 w[t] Δ = p s a i,j, c i,j, s j : AR, MA, σw[t] 2 coefficients of projection G r [t] : functional space basis functions b a (j), b c (j), b s (j) : AR, MA, σw[t] 2 basis function indices p a, p c, p s : AR, MA, σw[t] 2 functional space dimensionalities j=1 s j G bs (j)[t]

11 NON-STATIONARY MECHANICAL VIBRATION AND ANALYSIS VIA FUNCTIONAL SERIES TARMA MODELS 10 The Inverse Function Representation A FS-TARMA model may expressed in the inverse function form : A[B,t] {( }} ){ n a 1+ a i [t] B i x[t] = i=1 C[B,t] ({}} ){ n c 1+ c i [t] B i w[t] i=1 A[B,t] x[t] =C[B,t] w[t] I[B,t] =1+ C 1 [B,t] A[B,t] x[t] =w[t] }{{} I[B,t] i i [t] B i i=1 B : Backshift operator (B x[t] =x[t 1]) A[B,t]/ C[B,t]/ I[B,t] : AR/ MA/ Inverse function polynomial operators : B i B j = B i+j, B i g[t] =g[t i] B i ( skew multiplication)

12 NON-STATIONARY MECHANICAL VIBRATION AND ANALYSIS VIA FUNCTIONAL SERIES TARMA MODELS 11 Why we use FS-TARMA models? FS-TARMA models feature smooth, deterministic, parameter evolution. reflects a corresponding evolution in the underlying structural dynamics. FS-TARMA Model Estimation For given model orders (n a, n c ) and basis function indices (b a (j), b c (j), b s (j)), the determination of a FS-TARMA model consists of the estimation of the projection coefficient vector: Approach: θ Δ =[a T. c T. s T ] T A. Initial estimation via: The Two Stage Least Squares (2SLS) method or The Polynomial Algebraic (P A) method B. Final estimation via: The Prediction Error (PE) method

13 NON-STATIONARY MECHANICAL VIBRATION AND ANALYSIS VIA FUNCTIONAL SERIES TARMA MODELS 12 The Two-Stage Least Squares method (2SLS) STEP 1. Residual Series Estimation: The model residuals are evaluated via a truncated-order inverse function (estimated via linear regression): I[B,t,i] x[t] =e[t, i] STEP 2. AR/MA Projection Coefficient Estimation: The AR/MA coefficients are estimated using e[t, î] and linear regression: x[t]+ n a p a i=1 j=1 a i,j G ba (j) x[t i] = n c p c i=1 j=1 c i,j G bc (j) e[t i, î]+e[t, θ] STEP 3. Residual Variance Projection Coefficient Estimation: The residual variance is projected on the selected functional subspace.

14 NON-STATIONARY MECHANICAL VIBRATION AND ANALYSIS VIA FUNCTIONAL SERIES TARMA MODELS 13 The Polynomial Algebraic method (P A) STEP 1. Inverse Function Estimation: A truncated inverse function is estimated via linear regression: I[B,t,i] x[t] =e[t, i] STEP 2. Initial AR/MA Projection Coefficient Estimation: Based upon the obtained inverse function via deconvolution: A[B,t,a] =C[B,t,c] I[B,t,î] STEP 3. Signal Filtering: An auxilliary signal x[t] satisfying the FS-TAR part of the model is obtained as: C[B,t,ĉ] z[t] =A[B,t,a] x[t] A[B,t,â] x[t] =z[t]

15 NON-STATIONARY MECHANICAL VIBRATION AND ANALYSIS VIA FUNCTIONAL SERIES TARMA MODELS 14 STEP 4. Final AR/MA Projection Coefficient Estimation: The final AR coefficients are estimated using the obtained signal x[t] and liner regression: A[B,t,a] x[t] =w[t] The final MA coefficients are computed, based upon the new AR estimates (deconvolution): A[B,t,a] =C[B,t,c] I[B,t,î] STEP 5. Residual Variance Projection Coefficient Estimation: The residual variance is projected on the selected functional subspace.

16 NON-STATIONARY MECHANICAL VIBRATION AND ANALYSIS VIA FUNCTIONAL SERIES TARMA MODELS NON-STATIONARY RANDOM VIBRATION MODELING AND ANALYSIS FS-TARMA Modeling AR/MA orders obtained from preliminary stationary ARMA analysis. Functional space selection based upon a backward regression technique Functional Space Optimizaton (backward regression) RSS/SSS (%) Initial functional spaces: Type: continuous Dimension: p a = p c = Rejected Basis Functions

17 NON-STATIONARY MECHANICAL VIBRATION AND ANALYSIS VIA FUNCTIONAL SERIES TARMA MODELS 16 TARMA(6,6) [18,3] with Chebyshev II polynomial functional spaces F AR = {G 1 [t]..., G 20 [t]} except for G 4 [t] and G 19 [t] F MA = {G 1 [t],g 2 [t],g 7 [t]} RARMA modeling Recursive Maximum Likelihood (RML) method Common AR/MA orders with the FS-TARMA models Three consecutive passes (forward-backward-forward), Optimized forgetting factor (λ =0.992)

18 NON-STATIONARY MECHANICAL VIBRATION AND ANALYSIS VIA FUNCTIONAL SERIES TARMA MODELS 17 Model based Prediction Acceleration FS TARMA signal prediction RSS/SSS(%) Residual Series FS TARMA RARMA Residual Variance FS TARMA RARMA Time (sec)

19 NON-STATIONARY MECHANICAL VIBRATION AND ANALYSIS VIA FUNCTIONAL SERIES TARMA MODELS 18 Model-based time-dependent Power Spectral Density

20 NON-STATIONARY MECHANICAL VIBRATION AND ANALYSIS VIA FUNCTIONAL SERIES TARMA MODELS 19 Model-based time-dependent natural frequencies --: FS-TARMA estimates : RARMA estimates : ARMA estimates Background: Stationary vibration P.S.D Frequency (Hz) Time (sec)

21 NON-STATIONARY MECHANICAL VIBRATION AND ANALYSIS VIA FUNCTIONAL SERIES TARMA MODELS f 3 FS TARMA RARMA ARMA 80 Frequency (Hz) f f Time (sec)

22 NON-STATIONARY MECHANICAL VIBRATION AND ANALYSIS VIA FUNCTIONAL SERIES TARMA MODELS CONCLUDING REMARKS 1. The applicability of the FS-TARMA approach for modeling and predicting non-stationary mechanical vibration was demonstrated. 2. The effectiveness of the FS-TARMA approach for accurately recovering the underlying time-varying structural dynamics was demonstrated. 3. The FS-TARMA approach was shown to outperform alternative non-stationary analysis approaches by providing: Increased accuracy and resolution Smooth evolution of the underlying structural dynamics Increased prediction ability (RSS decreased by 42% compared to the RARMA-RML approach)