Uncertainty quantification of critical speed for railway vehicle - PowerPoint PPT Presentation

Uncertainty quantification of critical speed for railway vehicle dynamics PhD student: D. Bigoni 1 Supervisors: A. P. Engsig-Karup 1 , H. True 1 , J.S. Hesthaven 2 1 The Technical University of Denmark, 2 Brown University

Railway vehicle dynamics - Euler’s formulation 1 m ¨ x 1 + 2 D 2 ˙ � � x 1 + 2 k 4 � x 1 + 2 F X ( ξ x 1 , ξ y 1 ) + 2 F X ( ξ x 2 , ξ y 2 ) = 0 I ¨ � x 2 + k 6 � x 2 + 2 ha [ F X ( ξ x 1 , ξ y 1 ) − F X ( ξ x 2 , ξ y 2 )]+ + a [ F Y ( ξ x 1 , ξ y 1 ) + F Y ( ξ x 2 , ξ y 2 )] = 0 where F X and F Y are the creep forces, and determine a non-linear coupling of � x 1 and � x 2 . Among other components, these forces involve also the running velocity v of the vehicle, the conicity of the wheels and the wheel-rail friction. 1 H.True and C.Kaas-Petersen 1983 2 DTU Informatics, Technical University of Denmark Uncertainty quantification of critical speed for railway vehicle dynamics

Railway vehicle dynamics - Hunting Speed 50.0m/s 1e 4 1e 5 1.0 1.0 0.8 0.5 0.6 0.4 0.0 0.2 x1(t) x2(t) 0.0 0.5 0.2 0.4 1.0 0.6 0.8 1.5 0 2 4 6 8 10 0 2 4 6 8 10 Time Time Speed 105.0m/s 1e 2 1e 2 4 1.0 Speed = 110m/s 20 3 15 2 0.5 10 1 5 x1(t) x2(t) Im( λ ) 0 0.0 0 5 1 10 2 0.5 15 3 20 80 60 40 20 0 Re( λ ) 4 1.0 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Time Time 3 DTU Informatics, Technical University of Denmark Uncertainty quantification of critical speed for railway vehicle dynamics

Railway vehicle dynamics - Stochastic Model Let’s now assume that the suspension components k 6 , k 4 and D 2 are known within a certain level of accuracy and model this by: k 6 ∼ N (3 . 44 · 10 6 , 2 . 96 · 10 10 ) , (std. of approx. 5%) k 4 ∼ N (9 . 12 · 10 4 , 4 . 15 · 10 7 ) , (std. of approx. 7%) D 2 ∼ N (1 . 46 · 10 4 , 1 . 07 · 10 6 ) , (std. of approx. 7%) 1e 6 4.5 1e 5 3.0 1e 4 3.0 4.0 2.5 2.5 3.5 2.0 3.0 2.0 2.5 PDF 1.5 1.5 2.0 1.0 1.5 1.0 1.0 0.5 0.5 0.5 0.0 0.0 0.0 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 K6 1e6 K4 1e5 D2 1e4 What are the dynamics of the system under these conditions? 4 DTU Informatics, Technical University of Denmark Uncertainty quantification of critical speed for railway vehicle dynamics

Uncertainty Quantification - Traditional Approaches 1e6 2.2 Analytical Methods 2.4 2.6 • Moment Equation K6 2.8 3.0 • Perturbation Method 1.5 1.6 1.7 1.8 1.9 2.0 2.1 Pros. : recover the exact solution K4 1e5 1e6 1e5 2.2 1.5 Cons. : problem-dependent, cumbersome 1.6 2.4 1.7 2.6 1.8 K6 K4 1.9 2.8 Sampling Methods 2.0 3.0 2.1 � N − 1 / 2 � 2.4 2.6 2.8 3.0 3.2 3.4 2.4 2.6 2.8 3.0 3.2 3.4 • (MC) Monte Carlo – O D2 1e4 D2 1e4 1e 1 1.8 1.6 √ (log N ) d / 1.4 � � • (QMC) Quasi Monte Carlo – O N 1.2 1.0 PDF 0.8 0.6 • (MCMC) Markov Chain Monte Carlo 0.4 Pros. : general applicability, MC convergence indepen- 0.2 0.0 0.85 0.90 0.95 1.00 1.05 1.10 dent from dimensionality d Linear Critical Speed 1e2 Figure: Linear critical speed Cons. : very slow convergence distribution using 10 4 realizations for MC method. 5 DTU Informatics, Technical University of Denmark Uncertainty quantification of critical speed for railway vehicle dynamics

UQ - Generalized Polynomial Chaos (gPC) 2 Let Y be a r.v. with CDF F Y ( y ) . Use the N th-degree gPC expansion of the random parameters and the solution N a k = 1 � � F − 1 Y N = a k Φ k ( Z ) , ˆ ˆ ( F Z ( z ))Φ k ( z ) dF Z ( z ) Y γ k I Z k =0 N � u N ( t, Z ) = u k ( t )Φ k ( Z ) ˆ k =0 � E [ ∂ t u N ( t, Z )Φ k ( Z )] = E [ f ( u N )Φ k ( Z )] , D × (0 , T ] 1 ˆ u k (0) = γ k E [ u (0 , Z )Φ k ( Z )] , D × { t = 0 } µ u ( t ) ≈ E [ u N ( t, Z )] = ˆ u 0 ( t ) Var [ u ( t, Z )] ≈ Var [ u N ( t, Z )] = � N u 2 k =1 γ k ˆ k ( x, t ) I Z f ( z ) dF Z ( z ) and { φ i ( Z ) } N � where E [ f ( Z )] = i =0 are proper orthonormal basis. 2 D.Xiu and G.Karniadakis 2004 6 DTU Informatics, Technical University of Denmark Uncertainty quantification of critical speed for railway vehicle dynamics

UQ - gPC on Railway Vehicle Dynamics The N -th order gPC expansion of the problem is given by  � � � � E ∂ t u 1 ,N φ k = E u 2 ,N φ k    � � � � � � ∂ t u 2 ,N φ k = − 2 E D 2 ,N u 2 ,N φ k − 2 E k 4 ,N u 1 ,N φ k  E     − 2 E [( F X ( ξ x 1 , ξ y 1 ) + F X ( ξ x 2 , ξ y 2 )) φ k ]  � � � � E ∂ t u 3 ,N φ k = E u 4 ,N φ k    � � � � ∂ t u 4 ,N φ k = − E k 6 ,N u 3 ,N φ k − 2 ha E [( F X ( ξ x 1 , ξ y 1 ) − F X ( ξ x 2 , ξ y 2 )) φ k ]  E     − a E [( F Y ( ξ x 1 , ξ y 1 ) + F Y ( ξ x 2 , ξ y 2 )) φ k ]  where k is a multi index such that � u i,N ( t, Z ) = u k ( t )Φ k ( Z ) , ˆ i = 1 , . . . , 4 | k |≤ N � i + ( d − 1) � We obtain a system of K = � N coupled equations that can be treated i =0 ( d − 1) using standard ODE solvers. The following table shows how this number scales: 1 2 3 4 5 6 7 N d = 1 2 3 4 5 6 7 8 d = 2 3 6 10 15 21 28 36 d = 3 4 10 20 35 56 84 120 d = 4 5 15 35 70 126 210 330 d = 5 6 21 56 126 252 462 792 7 DTU Informatics, Technical University of Denmark Uncertainty quantification of critical speed for railway vehicle dynamics

UQ - gPC on Railway Vehicle Dynamics MC vs gPC(N=5) - v=70m/s 1e 7 MC vs gPC(N=5) - v=70m/s 1e 2 1.2 1.5 Monte Carlo Monte Carlo Lateral Displacement - Variance 1.0 gPC gPC 1.0 Pros : Elegant fomulation, one Lateral Displacement 0.8 0.5 single solution of the system, 0.6 0.0 optimal accuracy 0.4 0.5 0.2 Cons : Intrusive and cumbersome 1.0 0.0 0 1 2 3 4 5 0 1 2 3 4 5 Time Time to implement, non-linearities must (a) Mean and variance (b) Variance be treated carefully, weak on -4 -7 10 10 time-dependent problems (but there exist improvements). k 2 k 2 k Var [ u ] − σ max k E [ u ] − µ max u u 10 -8 -5 -9 10 10 0 50 100 150 200 250 0 50 100 150 200 250 N N 8 DTU Informatics, Technical University of Denmark Uncertainty quantification of critical speed for railway vehicle dynamics

UQ - Probabilistic Collocation Methods (PCM) Z ( j ) � M � Solve the deterministic ODE on a ”proper” set Θ M = j =1 of nodes in the random space: � ∂ t u ( t, Z ( j ) ) = f ( u ) , D × (0 , T ] u (0) = u 0 , D × { t = 0 } This will give u ( j ) = u ( t, Z ( j ) ) solutions on which we can apply interpolation rules or projection rules. Let’s consider the discrete projection: � u N ( Z ) = ˆ u k ( t )Φ k ( Z ) | k |≤ N u k ( t ) = 1 E [ u ( t, Z ) φ k ( Z )] = 1 � ˆ u ( z ) φ k ( z ) dF Z ( z ) γ k γ k where the integral can be computed by cubature rules using the ”properly” selected set of nodes Θ M . Then statistics can be easily obtained: µ u ( t ) ≈ E [ u N ( t, Z )] = ˆ u 0 ( t ) � u 2 Var [ u ( t, Z )] ≈ Var [ u N ( t, Z )] = γ k ˆ k ( x, t ) | k |≤ N Target: obtain the ”best” statistics out of the smallest number of simulation! 9 DTU Informatics, Technical University of Denmark Uncertainty quantification of critical speed for railway vehicle dynamics

UQ - PCM on Railway Vehicle Dynamics 1e5 2.05 Hermite polynomials are chosen as 2.00 1.95 1.90 basis for the projection/cubature. 1.85 K6 1.80 1.75 Projection with these polynomials can 1.70 1.65 be highly accurate, using proper Gauss 1.60 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 K4 1e6 1e4 1e4 3.3 3.3 quadrature nodes and weights, for 3.2 3.2 3.1 3.1 3.0 3.0 which analytical formulas exist. 2.9 2.9 K6 K4 2.8 2.8 2.7 2.7 10 -4 10 -6 2.6 2.6 2.5 2.5 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 1.60 1.65 1.70 1.75 1.80 1.85 1.90 1.95 2.00 2.05 D2 1e6 D2 1e5 -7 10 (c) Collocation points, N = 3 10 -5 k 2 k 2 k Var [ u ] − σ max k E [ u ] − µ max u u -8 10 2 MC vs PCM(4th-ord) - v=70m/s 7 MC vs PCM(4th-ord) - v=70m/s 1e 1e 1.5 1.2 Monte Carlo Monte Carlo -6 10 Lateral Displacement - Variance Collocation 1.0 Collocation 10 -9 1.0 Lateral Displacement 0.8 0.5 10 -7 10 -10 0.6 0 50 100 150 200 250 300 350 0 50 100 150 200 250 300 350 N N 0.0 0.4 Figure: PCM convergence to highest 0.5 0.2 accuracy (mean and variance). 1.0 0.0 0 1 2 3 4 5 0 1 2 3 4 5 Time Time (d) Mean and variance (e) Variance Figure: PCM vs. Monte Carlo 10 DTU Informatics, Technical University of Denmark Uncertainty quantification of critical speed for railway vehicle dynamics