The quantum mechanics approach to uncertainty modeling in structural dynamics
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1 p. 1/3 The quantum mechanics approach to uncertainty modeling in structural dynamics Andreas Kyprianou Department of Mechanical and Manufacturing Engineering, University of Cyprus
2 Outline Introduction Notions of uncertainty and probability Uncertainty in structural dynamics Theoretical Development Quantum mechanics motivation Density matrix Model of uncertain structures Updating of means and covariances Example Undamped three degree of freedom system p. 2/3
3 p. 3/3 Introduction Uncertainty: State of mind of an observer of an experiment whose outcome is an event out of many possible alternatives Probability Theory: Models experiments of the previous kind and facilitates an objective measure of uncertainty: that of entropy Jaynes Interpretation: Probability distribution is the knowledge of the observer about the experiment Entropy is measure of ignorance
4 p. 4/3 Introduction Uncertainty in Structural Dynamics Aleatoric Lack of knowledge about the exact values of the inertia, damping and elasticity Epistemic inability to model all the intricacies of a complicated structure Density Matrix to model uncertainty: Captures both the underlying system dynamics and statistics of uncertain structures Covariance properties easy to obtain Facilitates uncertain model updating
5 Quantum Mechanics Motivation Motivating Quantum Mechanics Principles: State of a quantum mechanical particle is expressed as wavefunction Φ(x) Φ(x) 2 is interpreted as probability density function Measurable physical quantities are described by Hermitian linear operators O Once the system in state Φ then the average value of the observed quantity corresponds to O = Φ H OΦ p. 5/3
6 p. 6/3 Quantum Mechanics Motivation Orthogonality relations for n-degree of freedom system described by n n symmetric M, K and C : 2ω r ζ r = c r m r = ΨH r CΨ r Ψ H r MΨ r r = 1..n ω 2 r = k r m r = ΨH r KΨ r Ψ H r MΨ r r = 1..n Damped Undamped Observables M, C,K M, K Observed ω r, 2ω r ζ r ωr 2
7 p. 7/3 Density Matrix Consider a damped n-degree of freedom system: Normalized-to-unity complex mode shape Ψ r C n Ψ r = n i α i u i where {u i },i = 1..n orthonormal basis of C n n i α i 2 = 1 Substitute Ψ r in the orthogonality relations
8 p. 8/3 Density Matrix 2ω r ζ r = c r m r = ni,j α iα j C ij ni,j α iα j M ij ω 2 r = k r m r = ni,j α iα j K ij ni,j α iα j M ij m r, c r and k r the weighted averages of the elements of the mass, damping and stiffness matrices
9 p. 9/3 Density Matrix Definition: The n n matrix P r that the coefficients α iα j create, P r = Ψ r Ψ H r since Ψ r is normalized to unity, Tr(P r ) = 1 Orthogonality relationships can be written as, 2ω r ζ r = Tr(P rc) Tr(P r M) ω 2 r = Tr(P rk) Tr(P r M)
10 p. 10/3 Uncertain Structures: Modeling How are uncertain structures modeled? Model Nominal Dynamics Nominal mass, M, stiffness, K and damping C matrices Model for Generating Statistics Average (nominal) density matrix for each mode Set of density matrices {P r 1 r n} together with M, stiffness, K and damping C matrices constitute the uncertain structure
11 p. 11/3 Example Three Degree of Freedom System
12 p. 12/3 Undamped System: Non-random mass Parameter Values of Nominal System M 1 = 2,M 2 = 3,M 3 = 4Kg K 3 = 10,K 4 = 10,K 6 = 30,K 1 = K 2 = K 5 = 20 N m Uncertain System K i = ( 20+ 5N (0,1) ) N m i = 1,2, N/m
13 p. 13/3 Undamped System: Non-random mass Sample: realizations Average density matrices P r 1 r 3 were computed Observer s state of knowledge about the uncertain system ω 2 r = Tr(P rk) Tr(P r M),( rad s (5.2082) ( ) ( ) ) 2 ω = Tr (P r K ) Tr(P r M ), rad s (2.2813) (4.5719) (5.9654)
14 Undamped System: Non-random mass Analysis of Covariance Take Cholesky decomposition of mass matrix M = R T R Substitute in orthogonality relation to get This is equivalent to ω 2 r = ΨT KΨ Ψ T R T RΨ ω 2 r = ΨT R T R T KR 1 RΨ Ψ T R T RΨ p. 14/3
15 p. 15/3 Undamped System: Non-random mass Analysis of Covariance Matrix A = R T KR 1 is symmetric Same eigenvalues as of the original system but the associated eigenvectors are given by RΨ New density matrices from the previous ones P A i = 1 Tr(RP i R T ) ( RPi R T)
16 p. 16/3 Uncertain System: Non-random mass Analysis of Covariance Hypothesis: Variability in the sample of natural frequencies is due to system covariance matrices n 2 n 2 covariance matrices of vectorized A Extending the theory of density matrices to tensor product R n R n, new density matrices are defined, P A ij = P A i P A j, 1 i,j n
17 p. 17/3 Uncertain System: Non-random mass Analysis of Covariance Cov ( ω 2 i,ω 2 j ) = Tr ( PA ij Cov(vec(A)) ) Since the mass matrix is non-random Cov ( ω 2 i,ω 2 j) depends on the Cov(vec(K)) Cov(vec(K)) is easily constructed from the variances and covariances of the individual stiffness elements
18 Undamped System: Non-random mass Analysis of Covariance veck = K 1 +K 4 +K 6 K 4 K 6 K 4 K 2 +K 4 +K 5 K 5 K 6 K 5 K 3 +K 5 +K 6 p. 18/3
19 Undamped System: Non-random mass Analysis of Covariance To construct the 9 9, Cov(vecK), use Var(K i +K j +K k ) = Var(K i )+Var(K j )+Var(K k )+ 2(Cov(K i,k j )+Cov(K i,k k )+Cov(K j,k k ))) Cov(K i +K j +K k, K k ) = Var(K k ) Cov(K i,k k ) Cov(K j,k k ) Cov(K i +K j +K k,k i +K l +K m ) = Var(K i )+Cov(K i,k l )+Cov(K i,k m ) +Cov(K j,k m )+Cov(K k,k i )+Cov(K k,k l )+ Cov(K k,k m ) p. 19/3
20 p. 20/3 Undamped System: Non-random mass Analysis of Covariance Since A = R T KR 1 then vec(a) = ( R T R T) vec(k) Covariance of A is given by Cov(vec(A)) = ( R T R T) T ( Cov(K) R T R T)
21 Undamped System: Non-random mass Analysis of Variance: Results, Cov ( ωi,ω 2 j 2 Using covariance expressions for Cov(vec(K)) (0.0767) (0.1325) (0.1677) (0.1325) (1.7709) (0.1349) (0.1677) (0.1349) (0.8023) ) Using sample covariance for Cov(vec(K)) (0.0767) (0.1325) (0.1677) (0.1325) (1.7709) (0.1349) (0.1677) (0.1349) (0.8023) p. 21/3
22 p. 22/3 Undamped system: Non-random mass Model Updating: Mean Problem formulation: K nom is unknown and an initial guess K in is given. Density matrices and nominal natural frequencies ω 2 i are known. K in = K nom +δk Objective: Find δk Solution ω 2 i = Tr(ρ ik nom ) Tr(ρ i M) = Tr(ρ i(k in δk)) Tr(ρ i M)
23 Undamped system: Non-random mass System of Updating Equations Tr(ρ i δk) = Tr(ρ i K in ) ω 2 itr(ρ i M) 1 i n Initial Stiffness Matrix, K in, and δk K 1 +K 4 +K 6 K 4 K 6 K 4 K 2 +K 4 +K 5 K 5 K 6 K 5 K 3 +K 5 +K 6 δk δk 2 +δk 5 δk 5 0 δk 5 δk 5 p. 23/3
24 p. 24/3 Undamped System: Non-random mass Numerical example Initial Nominal Values K i = 10 N m i = 1,2,5 Substituting the known ω 2 n and ρ i s in the system of updating equations 0.083δK δK δK 5 = δK δK δK 5 = δK δK δK 5 = 4.38
25 p. 25/3 Undamped System: Non-random mass Solution δk 1 δk 2 δk 5 = N m Nominal and updated nominal stiffness matrices, where K nomup = K in δk 60(60.12) 10( 10) 30( 30) 10( 10) 50(49.69) 20( 20.05) 30( 30) 20( 20.05) 60( 60.05)
26 p. 26/3 Undamped System: Non Random Mass Model Updating: Variance Problem Formulation: Variances of K i i = 1,2,5 are not known. No initial variances are required. Density matrices and variances of ω 2 i are known. Objective: Find the variances of individual stiffness parameters Solution var ( ω 2 i ) = Tr ( PA ii cov(veca) ) 1 i n
27 p. 27/3 Undamped System: Non Random Mass Built up cov(vec(k)) using the individual variances and covariances Set Cov(vec(A)) = ( R T R T) T Cov(K) ( R T R T) Substitute the observed var ( ωi 2 ) and PA ii in the last expression of the previous slide to get three equations in the three unknown variances
28 p. 28/3 Undamped System: Non Random Mass varK varK varK 5 = varK varK varK 5 = varK varK varK 5 =
29 p. 29/3 Undamped System: Non Random Mass Solution vark 1 vark 2 vark 5 =
30 p. 30/3 Conclusions Uncertainty model based on the concept of density matrix Reformulation of orthogonality relationships using the trace operator In uncertainty context these give the estimated mean values Theory extended through tensor products to account for covariances Updating of means and computation of unknown covariances
31 p. 31/3 Future Work Results not presented Damped systems Uncertain random mass. Distribution in natural frequencies and decay rates are non-gaussian Future Work Frequency domain formulation and structural modification of uncertain structures The uncertain system could be treated as if it is certain
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