Smooth Modes as a Tool for Operational Modal Analysis: New Developments Rubens Sampaio PUC-Rio rsampaio@puc-rio.br PUC-Rio Rubens Sampaio (PUC-Rio rsampaio@puc-rio.br) Smooth Orthogonal Decomposition 1 / 39
Outline Review of Karhunen-Lo` eve History of SOD in modal analysis SOD in operational modal analysis Noise sensitivity of standard SOD New developments Conclusions PUC-Rio Rubens Sampaio (PUC-Rio rsampaio@puc-rio.br) Smooth Orthogonal Decomposition 2 / 39
First part-KLD: Naive look Some history-main applications Main idea of KL decomposition The mathematical problem Construction of the KL basis How to combine KL with Galerkin Reduced model given by KL Practical question: how to compute the KLD An example Different guises of KL; basic ingredients Karhunen-Lo` eve expansion: main hypothesis Karhunen-Lo` eve Theorem Basic properties Applications to Random Mechanics Examples PUC-Rio Rubens Sampaio (PUC-Rio rsampaio@puc-rio.br) Smooth Orthogonal Decomposition 3 / 39
Some history Beginning: works in Statistics and Probability and Spectral Theory in Hilbert Spaces. Some contributions: Kosambi (1943) Lo` eve (1945) Karhunen (1946) Pougachev (1953) Obukhov (1954) Applications: Lumley (1967): method applied to Turbulence (coherent structures, 1970) Sirovich (1987): snapshot method An important book appeared in 1996: Holmes, Lumley, Berkooz. In Solid Mechanics the applications started around 1993. In finite dimension it appears under different guises. PUC-Rio Rubens Sampaio (PUC-Rio rsampaio@puc-rio.br) Smooth Orthogonal Decomposition 4 / 39
Main Applications Principal Component Analysis (PCA): Statistics and image processing Empirical orthogonal functions: Oceanography and Metereology (Lorenz 1956) Factor analysis: Psychology and Economics Data analysis: Principal Component Analysis (PCA) Reduced models, through Galerkin approximations Dynamical Systems: to understand the dynamics Signal Analysis PUC-Rio Rubens Sampaio (PUC-Rio rsampaio@puc-rio.br) Smooth Orthogonal Decomposition 5 / 39
Two main purposes of KLD order reduction by projecting high-dimensional data in lower-dimensional space feature extraction by revealing relevant, but unexpected, structure hidden in the data PUC-Rio Rubens Sampaio (PUC-Rio rsampaio@puc-rio.br) Smooth Orthogonal Decomposition 6 / 39
Main idea of KLD In plain words: The key idea of KL is to reduce a large number of interdependent variables to a much smaller number of uncorrelated variables while retaining as much as possible the variation in the original data. PUC-Rio Rubens Sampaio (PUC-Rio rsampaio@puc-rio.br) Smooth Orthogonal Decomposition 7 / 39
Main idea of KL decomposition more precisely Suppose we have an ensemble { u k } of scalar fields, each being a function defined in ( a , b ) ⊂ R . We work in a Hilbert space L 2 (( a , b )). n =1 of L 2 that We want to find a (orthonormal) basis { ψ n } ∞ is optimal for the given data set in the sense that a finite dimensional representation of the form N � u ( x ) = ˆ a k ψ k ( x ) , k =1 for N given, describes a typical member of the ensemble better than representations of the same dimension in any other basis. The notion of typical implies the use of an average over the ensemble { u k } and optimality means maximazing the average normalized projection of u into { ψ n } N n =1 . PUC-Rio Rubens Sampaio (PUC-Rio rsampaio@puc-rio.br) Smooth Orthogonal Decomposition 8 / 39
The mathematical problem Suppose, for simplicity, we have just one function ψ max ψ ∈ L 2 E ( | < u , ψ > | 2 ) � ψ � 2 This implies J ( ψ ) = E ( | < u , ψ > | 2 ) − λ ( � ψ � 2 − 1) d d ε J ( ψ + εφ ) | ε =0 = 0 � b R ( x , y ) ψ ( y ) dy = λψ ( x ) a with R ( x , y ) = E ( u ( x ) u ( y )) PUC-Rio Rubens Sampaio (PUC-Rio rsampaio@puc-rio.br) Smooth Orthogonal Decomposition 9 / 39
Construction of the KL basis Construct R(x,y) from the data Solve the eigenvalue problem: � R ( x , y ) ψ ( y ) dy = λψ ( x ) D to get the pair ( λ i , ψ i ) If u is the field then the N-order approximation of it is u N ( t , x ) = E ( u ( t , x )) + Σ N ˆ i =1 a i ( t ) ψ ( x ) To make predictions use the Galerkin method taking the ψ ’s as trial functions PUC-Rio Rubens Sampaio (PUC-Rio rsampaio@puc-rio.br) Smooth Orthogonal Decomposition 10 / 39
Galerkin projections Suppose we have a dynamical system governed by ∂ v v ∈ { ( a , b ) × D → R n } = A ( v ) ∂ t v (0 , x ) = v 0 ( x ) initial condition B ( v ) = 0 boundary condition The Galerkin method is a discretization scheme for PDE based on separation of variables. One searches solutions in the form: N � v ( x , t ) = ˆ a k ( t ) ψ k ( x ) k =1 PUC-Rio Rubens Sampaio (PUC-Rio rsampaio@puc-rio.br) Smooth Orthogonal Decomposition 11 / 39
Reduced equations The reduced equation is obtained making the error of the approximation orthogonal to the first N KL elements of the basis. ∂ ˆ v errorequation ( t , x ) = ∂ t − A (ˆ v ) errorinicond ( x ) = v (0 , x ) − v 0 ( x ) ˆ < errors , ψ i ( x ) > = 0 i = 1 , ..., N . for � da i D A (Σ N dt ( t ) = n =1 a n ( t ) ψ n ( x )) ψ i ( x ) dx for i = 1 , ..., N � a i (0) = D v 0 ( x ) ψ i ( x ) dx i = 1 , ..., N for PUC-Rio Rubens Sampaio (PUC-Rio rsampaio@puc-rio.br) Smooth Orthogonal Decomposition 12 / 39
History of SOD in modal analysis PUC-Rio Rubens Sampaio (PUC-Rio rsampaio@puc-rio.br) Smooth Orthogonal Decomposition 13 / 39
Smooth Orthogonal Decomposition - Definition The Smooth Orthogonal Decomposition (SOD) consists in an extension of the Proper Orthogonal Decomposition (Karhunen-Lo` eve decomposition) with a constraint on the smoothness. When applied to a vector field, the SOD consists in: Find the basis for an orthogonal projection that maintains the maxi- mum variance of the original field and also performs it as smooth as possible in relation to time . PUC-Rio Rubens Sampaio (PUC-Rio rsampaio@puc-rio.br) Smooth Orthogonal Decomposition 14 / 39
Smooth Orthogonal Decomposition - Definition Let x ( t ) = [ x 1 ( t ) , . . . , x n ( t )] T be a real-valued stochastic field with respect to time t ∈ R . It is assumed to be stationary and have zero mean. The orthogonal projection of x ( t ) with respect to the vector ψ is defined as v ( t ) = proj ψ x ( t ) = � x ( t ) , ψ � ψ � ψ � Assuming for simplicity that ψ is an unit vector, the norm of v(t) is � v ( t ) � = � x ( t ) , ψ � PUC-Rio Rubens Sampaio (PUC-Rio rsampaio@puc-rio.br) Smooth Orthogonal Decomposition 15 / 39
Smooth Orthogonal Decomposition - Definition maintains the maximum variance of the original field � � � v ( t ) � 2 �� max ψ E = max ψ { E [ � ψ , x ( t ) � . � x ( t ) , ψ � ] } � � � � ψ T E x ( t ) x T ( t ) = max ψ . ψ performs it as smooth as possible in relation to time � � v i ( t ) � 2 �� min ψ i E � ˙ = min ψ i { E [ � ψ i , ˙ x ( t ) � . � ˙ x ( t ) , ψ i � ] } � � � � ψ T x T ( t ) = min ψ i x ( t ) ˙ . i E ˙ ψ i PUC-Rio Rubens Sampaio (PUC-Rio rsampaio@puc-rio.br) Smooth Orthogonal Decomposition 16 / 39
Smooth Orthogonal Decomposition - Definition This two optimization problems can be brought together to only one maximization problem by creating the function α ( ψ ) as following: � � � � v ( t ) � 2 � E max ψ { α ( ψ ) } = max ψ v ( t ) � 2 ] E [ � ˙ � � ψ T R xx (0) ψ = max ψ . ψ T R ˙ x (0) ψ x ˙ Where R xx (0) and R ˙ x (0) are correlation function matrices defined as: x ˙ � � x ( t + τ ) x T ( t ) R xx ( τ ) = E � � x T ( t ) R ˙ x ( τ ) = E x ( t + τ ) ˙ ˙ x ˙ PUC-Rio Rubens Sampaio (PUC-Rio rsampaio@puc-rio.br) Smooth Orthogonal Decomposition 17 / 39
Smooth Orthogonal Decomposition - Definition The solution of this maximization problem is obtained by finding its stationary point, which leads to the following eigenvalue problem: R xx (0) ψ = λ R ˙ x (0) ψ x ˙ Since R xx (0) and R ˙ x (0) are n × n matrices, this eigenvalue problem leads x ˙ to the set { ψ i } n i =1 , called smooth modes the set { λ i } n i =1 , called smooth values PUC-Rio Rubens Sampaio (PUC-Rio rsampaio@puc-rio.br) Smooth Orthogonal Decomposition 18 / 39
SOD in operational modal analysis Operational modal analysis consists in identify the modal parameters (natural frequencies and vibration modes) of a system under ambient excitation monitoring only its responses (displacements, velocity or acceleration). ? x (t) System To identify the modal parameters using SOD, one has to establish the following relationship: smooth modes ⇔ vibration modes smooth values ⇔ natural frequencies PUC-Rio Rubens Sampaio (PUC-Rio rsampaio@puc-rio.br) Smooth Orthogonal Decomposition 19 / 39
SOD in operational modal analysis Vibration of an undamped MDOF system: The dynamics of a discretized linear conservative mechanical system under ambient excitation can be modeled by the equation M¨ x ( t ) + Kx ( t ) = f ( t ) , where f ( t ) is assumed to be a zero-mean white noise process and the mass matrix M is assumed to be invertible. The acceleration can be expressed as x ( t ) = − M − 1 Kx ( t ) + M − 1 f ( t ) . ¨ Also, the natural frequencies and the normal modes can be extracted by the eigenvalue problem M − 1 K φ = ω 2 φ , where { ω } are the natural frequencies and { φ } the normal modes. PUC-Rio Rubens Sampaio (PUC-Rio rsampaio@puc-rio.br) Smooth Orthogonal Decomposition 20 / 39
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