The quantum mechanics approach to uncertainty modeling in structural dynamics Andreas Kyprianou Department of Mechanical and Manufacturing Engineering, University of Cyprus
– p. 1/3
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
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
– p. 3/3
Introduction Uncertainty in Structural Dynamics
Lack of knowledge about the exact values of the inertia, damping and elasticity Epistemic inability to model all the intricacies of a complicated structure Aleatoric
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
– p. 4/3
Quantum Mechanics Motivation Motivating Quantum Mechanics Principles:
State of a quantum mechanical particle is expressed as wavefunction Φ (x) 2
Φ (x) 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 H
hOi = Φ OΦ
– p. 5/3
Quantum Mechanics Motivation for n-degree of freedom system described by n × n symmetric M, K and C: ΨH cr r CΨr r = 1..n = H 2ωr ζr = mr Ψr MΨr Orthogonality relations
H k Ψ r r KΨr 2 r = 1..n ωr = = H mr Ψr MΨr
Observables Observed
Damped
Undamped
M, C,K ωr , 2ωr ζr
M, K ωr2
– p. 6/3
Density Matrix Consider a damped n-degree of freedom system: Normalized-to-unity complex mode shape Ψr ∈ C n Ψr =
n X
αi ui
i
where {ui }, i = 1..n orthonormal basis of C n n X i
2
|αi | = 1
Substitute Ψr in the orthogonality relations
– p. 7/3
Density Matrix Pn ∗ α cr i,j i αj Cij 2ωr ζr = = Pn ∗ mr i,j αi αj Mij Pn ∗ α k r i,j i αj Kij 2 = Pn ∗ ωr = mr i,j αi αj Mij
mr , cr and kr the weighted averages of the elements of the mass, damping and stiffness matrices
– p. 8/3
Density Matrix The n × n matrix Pr that the coefficients αi∗ αj create, Definition:
P r = Ψr ΨH r since Ψr is normalized to unity, T r (Pr ) = 1 Orthogonality relationships can be written as, 2ωr ζr =
T r(Pr C) T r(Pr M)
ωr2 =
T r(Pr K) T r(Pr M)
– p. 9/3
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 {Pr 1 ≤ r ≤ n} together with M, stiffness, K and damping C matrices constitute the uncertain structure
– p. 10/3
Example Three Degree of Freedom System
– p. 11/3
Undamped System: Non-random mass Parameter Values of Nominal System
M1 = 2, M2 = 3, M3 = 4Kg K3 = 10, K4 = 10, K6 = 30, K1 = K2 = K5 = 20 N m Uncertain System ³
Ki = 20 +
√
5N (0, 1)
´
N m
i = 1, 2, 5
3000
3000
3000
2500
2500
2500
2000
2000
2000
1500
1500
1500
1000
1000
1000
500
500
500
0 10
20
30
0 10
20 N/m
30
0 10
20
30
– p. 12/3
Undamped System: Non-random mass Sample: 10000 realizations ¯ r 1 ≤ r ≤ 3 were Average density matrices P computed Observer’s state of knowledge about the uncertain system ωr2
=
T r(Pr K) rad , T r(Pr M) s ³
´2
5.2284 (5.2082) 20.9078 (20.9241) 35.5304 (35.5912)
ω=
√
T r(Pr K) rad √ , T r ( Pr M ) s
2.2964 (2.2813) 4.5723 (4.5719) 5.9354 (5.9654)
– p. 13/3
Undamped System: Non-random mass Analysis of Covariance
Take Cholesky decomposition of mass matrix M = RT R Substitute in orthogonality relation to get T Ψ KΨ 2 ωr = T T Ψ R RΨ
This is equivalent to T T −T −1 Ψ R R KR RΨ 2 ωr = ΨT RT RΨ
– p. 14/3
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 ³ ´ 1 T RP R = PA i i T T r (RPi R )
– p. 15/3
Uncertain System: Non-random mass Analysis of Covariance Hypothesis:
Variability in the sample of natural frequencies is due to system covariance matrices n2 × n2 covariance matrices of vectorized A Extending the theory of density matrices to tensor product Rn ⊗ Rn , new density matrices are defined, A A ⊗ P = P PA j , 1 ≤ i, j ≤ n i ij
– p. 16/3
Uncertain System: Non-random mass Analysis of Covariance
Cov
³
ωi2 , ωj2
´
= Tr
³
¯ A Cov (vec (A)) P ij
´
Since matrix is non-random ´ ³ the mass Cov ωi2 , ωj2 depends on the Cov (vec (K)) Cov (vec (K)) is easily constructed from the variances and covariances of the individual stiffness elements
– p. 17/3
Undamped System: Non-random mass Analysis of Covariance
vecK =
K 1 + K 4 + K6 −K4 −K6 −K4 K 2 + K 4 + K5 −K5 −K6 −K5 K 3 + K 5 + K6
– p. 18/3
Undamped System: Non-random mass Analysis of Covariance
To construct the 9 × 9, Cov (vecK), use Var (Ki + Kj + Kk ) = V ar (Ki ) + V ar (Kj ) + V ar (Kk ) + 2 (Cov (Ki , Kj ) + Cov (Ki , Kk ) + Cov (Kj , Kk ))) Cov (Ki + Kj + Kk , −Kk ) = −V ar (Kk ) − Cov (Ki , Kk ) − Cov (Kj , Kk ) Cov (Ki + Kj + Kk , Ki + Kl + Km ) = V ar (Ki ) + Cov (Ki , Kl ) + Cov (Ki , Km ) +Cov (Kj , Km ) + Cov (Kk , Ki ) + Cov (Kk , Kl ) + Cov (Kk , Km )
– p. 19/3
Undamped System: Non-random mass Analysis of Covariance
Since A = R−T KR−1 then ³
vec (A) = R
−T
⊗R
−T
´
vec (K)
Covariance of A is given by Cov (vec (A)) = ³
R
−T
⊗R
´ −T T
³
Cov (K) R
−T
⊗R
−T
´
– p. 20/3
Undamped System: Non-random mass Cov ωi2 , ωj2 Using covariance expressions for Cov (vec (K)) Analysis of Variance: Results,
³
´
0.0758 (0.0767) 0.1327 (0.1325) 0.1663 (0.1677) 0.1327 (0.1325) 1.7547 (1.7709) 0.1385 (0.1349) 0.1663 (0.1677) 0.1385 (0.1349) 0.8012 (0.8023)
Using sample covariance for Cov (vec (K))
0.0764 (0.0767) 0.1323 (0.1325) 0.1667 (0.1677) 0.1323 (0.1325) 1.7702 (1.7709) 0.1404 (0.1349) 0.1667 (0.1677) 0.1404 (0.1349) 0.8019 (0.8023)
– p. 21/3
Undamped system: Non-random mass Model Updating: Mean
Knom is unknown and an initial guess Kin is given. Density matrices and nominal natural frequencies ωi2 are known. Kin = Knom + δK
Problem formulation:
Objective:
Find δK
Solution
ωi2
T r (ρi Knom ) T r (ρi (Kin − δK)) = = T r (ρi M) T r (ρi M)
– p. 22/3
Undamped system: Non-random mass System of Updating Equations T r (ρi δK) = T r (ρi Kin )−ωi2 T r (ρi M) 1 ≤ i ≤ n Initial Stiffness Matrix, Kin , and δK
K1 + K4 + K6 −K4 −K6 −K4 K2 + K4 + K5 −K5 −K6 −K5 K3 + K 5 + K6
δK1 0 0 0 δK2 + δK5 −δK5 0 −δK5 δK5
– p. 23/3
Undamped System: Non-random mass Numerical example
Initial Nominal Values N i = 1, 2, 5 Ki = 10 m Substituting the known ωn2 and ρi s in the system of updating equations 0.083δK1 + 0.091δK2 + 0.006δK5 = −1.77 0.19δK1 + 0.24δK2 + 0.54δK5 = −7.98
0.398δK1 + 0.001δK2 + 0.037δK5 = −4.38
– p. 24/3
Undamped System: Non-random mass Solution
δK1 δK2 δK5
=
−10.11 −9.64 −10.5
N m
Nominal and updated nominal stiffness matrices, where KnomU P = Kin − δK
60 (60.12) −10 (−10) −30 (−30) −10 (−10) 50 (49.69) −20 (−20.05) −30 (−30) −20 (−20.05) −60 (−60.05)
– p. 25/3
Undamped System: Non Random Mass Model Updating: Variance
Variances of Ki i = 1, 2, 5 are not known. No initial variances are required. Density matrices and variances of ωi2 are known.
Problem Formulation:
Objective:
Find the variances of individual stiffness parameters Solution
var ωi2 = T r P¯iiA cov (vecA) 1 ≤ i ≤ n ³
´
³
´
– p. 26/3
Undamped System: Non Random Mass Built up cov (vec (K)) using the individual variances and covariances Set Cov (vec (A)) = ³
R−T ⊗ R−T
´T
Cov (K) R−T ⊗ R−T ³
´
Substitute the observed var ωi2 and P¯iiA in the last expression of the previous slide to get three equations in the three unknown variances ³
´
– p. 27/3
Undamped System: Non Random Mass 0.006838varK1 +0.008299varK2 +0.00003137varK5 = 0.0767 0.00035varK1 + 0.05813varK2 + 0.2925varK5 = 1.7710 0.1589varK1 +0.000001294varK2 +0.001363varK5 = 0.8023
– p. 28/3
Undamped System: Non Random Mass Solution
varK1 varK2 varK5
=
5.0360 5.0982 5.0067
– p. 29/3
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
– p. 30/3
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
– p. 31/3
