Parameter Estimation and Structural Model Updating Using Modal - PowerPoint PPT Presentation

SICON Final Conference Rome, September 21-25, 2009, SICON FC Parameter Estimation and Structural Model Updating Using Modal Methods in the Presence of Nonlinearity Jean-Claude Golinval University of Liege, Belgium Department of Aerospace and Mechanical Engineering Chemin des Chevreuils, 1 Bât. B 52 B-4000 Liège (Belgium) E-mail : JC.Golinval@ulg.ac.be

2 Outline 1. Introduction 2. Theoretical Modal Analysis of Nonlinear Systems 3. Nonlinear Experimental Modal Analysis 4. Model Parameter Estimation Techniques 5. Concluding Remarks

3 Introduction Design of engineering structures relies on • Numerical predictions � modal analysis (FEM) • Dynamic testing � experimental modal analysis (EMA) In the case of linear structures , the techniques available for EMA are mature e.g. • Eigensystem realization algorithm • Stochastic subspace identification • Polyreference least-squares complex exponentials frequency domain • etc

4 Introduction Nonlinearity in Engineering Applications backlash and friction in control surfaces and joints fluid-structure interaction composite materials hardening nonlinearities in engine-to-pylon connections Many works are reported in the literature on dynamic testing and identification of nonlinear systems but very few address nonlinear phenomena during modal survey tests.

5 Introduction Aim of this presentation • To extend experimental modal analysis to a practical analogue using the nonlinear normal mode (NNM) theory. • Validate mathematical models of non-linear structures against experimental data. Why? • NNMs offer a solid and rigorous mathematical tool. • They have a clear conceptual relation to the classical LNMs. • They are capable of handling strong structural nonlinearity.

6 Outline 1. Introduction 2. Theoretical Modal Analysis of Nonlinear Systems • Nonlinear Normal Modes (NNMs) • Numerical Computation of NNMs • Frequency-Energy Plot 3. Nonlinear Experimental Modal Analysis 4. Model Parameter Estimation Techniques 5. Concluding Remarks

7 Theoretical Modal Analysis � � � � M x C x K x f (x, x ) p MDOF system + + + = ( t ) NL Vector of nonlinear forces • Dynamic analysis Prediction of the responses using a numerical integration procedure (e.g. Newmark’s schema) • Modal analysis of the MDOF system (with no damping) In the nonlinear case In the linear case ( ) M � � � � � x K x M x K x f x, x + = + + = 0 0 NL Structural eigenproblem Use of the concept of nonlinear normal modes (NNMs) which is a j th eigenvector rigorous extension of the concept of 2 K Φ M Φ � eigenmodes to nonlinear systems. = ω = 1 , , j n j j j j th natural frequency

8 Nonlinear Normal Modes Definitions Two definitions of an NNM in the literature: 1. Targeting a straightforward nonlinear extension of the linear normal mode (LNM) concept, Rosenberg defined an NNM motion as a vibration in unison of the system (i.e., a synchronous periodic oscillation). 2. To provide an extension of the NNM concept to damped systems, Shaw and Pierre defined an NNM as a two-dimensional invariant manifold in phase space. Such a manifold is invariant under the flow (i.e., orbits that start out in the manifold remain in it for all time), which generalizes the invariance property of LNMs to nonlinear systems. In the present study , an NNM motion is defined as a (non-necessarily synchronous) periodic motion of the undamped mechanical system � this extended definition is particularly attractive when targeting a numerical computation of the NNMs.

9 Nonlinear Normal Modes Illustrative example: 2 DOF-system with a cubic stiffness ( ) 3 � � + − + = x 2 x x 0 . 5 x 0 1 1 2 1 ( ) � � + − = 2 0 x x x 2 2 1

10 Nonlinear Normal Modes In-Phase NNMs for Increasing Energy Moderate energy Low energy High energy Time-series Configuration space Power spectral density Phase space

11 Numerical Computation of NNMs General equation of the nonlinear system (with no damping) ( ) � � � M x K x f x, x + + = 0 NL Vector of nonlinear forces The numerical computation of NNMs relies on two main techniques, namely a shooting procedure and a method for the continuation of periodic solutions .

12 Numerical Computation of NNMs • Shooting method The shooting method consists in finding, in an iterative way, the ( ) ( ) � x x initial conditions and the period T inducing an isolated 0 , 0 periodic motion (i.e., an NNM motion) of the conservative system. = = t T t 0 Numerical ( ) ( ) ( ) ( ) � x x � x x T , T 0 , 0 integration Newton-Raphson

13 Numerical Computation of NNMs • Pseudo-arclength continuation method NNM branch Pseudo-arclength continuation method: Initial conditions predictor step tangent to the branch corrector step perpendicular to the predictor step (shooting) Period T

14 Frequency-Energy Plot (FEP) Backbone Modal curves of the FEP

15 Outline 1. Introduction 2. Theoretical Modal Analysis of Nonlinear Systems 3. Nonlinear Experimental Modal Analysis • Phase Separation Methods - Proper Orthogonal Decomposition • Phase Resonance Methods - Nonlinear Normal Mode Testing 4. Model Parameter Estimation Techniques 5. Concluding Remarks

16 Experimental Modal Analysis (EMA) Linear systems Theoretical Approach Experimental Approach Finite Element Model Response Measurements M � � x K x + = 0 Acc (m/s 2 ) Time series Time Eigenvalue problem 2 K Φ M Φ Identification methods = ω j j j Natural frequencies ( ω j 2 ) Mode shapes ( Φ j ) EMA for linear systems is now mature and widely used in structural engineering � well established techniques [1], [2].

17 Experimental Modal Analysis (EMA) Nonlinear systems Theoretical Approach Experimental Approach Finite Element Model Response Measurements ( ) � � � M x K x f x, x + + = 0 Acc (m/s 2 ) NL Time series Time Numerical NNM computation Experimental NNM extraction NNM frequencies NNM modal curves EMA for nonlinear systems is still a challenge.

18 Experimental Modal Analysis (EMA) There are two main techniques for EMA. 1. Phase separation methods Several modes are excited at once using either broadband excitation (e.g., hammer impact and random excitation) or swept-sine excitation in the frequency range of interest. in the nonlinear case, extraction of individual NNMs is not � possible generally, because modal superposition is no longer valid. use of the proper orthogonal decomposition (POD) method � to extract features from the time series . Remark • All structures encountered in practice are nonlinear to some degree. • If a nonlinear structure is excited with a broadband excitation signal (e.g. random force), then the results will appear linear � experimental modal analysis will lead to an updated linearized model !

19 Proper Orthogonal Decomposition (POD) Ω Instrumented structure x 1 x i x N snapshots M ⎡ ⎤ � x ( t ) x ( t ) 1 1 1 N M ⎢ ⎥ X � � � measurement = ⎢ ⎥ co-ordinates ⎢ ⎥ � ⎣ ⎦ x ( t ) x ( t ) M 1 M N [ ] X x … x is the observation matrix = ( 1 t ) ( t ) N

20 Proper Orthogonal Decomposition (POD) The M x M correlation matrix R is built 1 T R X X = M The eigenvalue problem is solved Eigenvalues (POVs) R u u = λ Eigenvectors of XX T (POMs)

21 Proper Orthogonal Decomposition (POD) Computation of the POMs using SVD M measurement co-ordinates Ω N time samples x 1 x ⎡ ⎤ � i x ( t ) x ( t ) 1 1 1 N ⎢ ⎥ � � � X = ⎢ ⎥ M x N x M ⎢ ⎥ � ⎣ ⎦ ( ) ( ) x t x t M 1 M N λ λ ≡ diag ( ) ( POV ) Using SVD i i T X U V = Σ × × × × M N M M M N N N Eigenvectors of XX T (POM)

22 Proper Orthogonal Decomposition (POD) Geometric Interpretation of the POMs Comparison of LNM, NNM and POM on the 2 DOF example POM First mode 2 The POM is the best linear x 2 representation of the LNM nonlinear normal mode. NNM -2 -1.5 1.5 x 1

23 Proper Orthogonal Decomposition (POD) Key idea: Application of the POD to Features Extraction Linear Systems Nonlinear Systems � � � M x C x K x p � � � � M x C x K x f (x, x ) p + + = + + + = ( t ) ( t ) NL Deterministic approach Statistical approach Eigenvalue problem : Proper Orthogonal Decomposition : T 2 K M Φ 0 X U V = Σ − ω = ( ) Response : Response : Spatial Spatial information information n ∑ n ∑ x Φ = η x u ( t ) i t ( ) = ( t ) a j t ( ) ( i ) ( j ) = 1 = j i 1 Time information Natural frequencies � Instantaneous η = ω + ω A cos( t ) B sin( t ) frequencies i i i i i