Bigoni, Daniele

Engsig-Karup, Allan Peter

True, Hans

Online Research Database In Technology

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Downloaded from orbit.dtu.dk on: Dec 20, 2017Uncertainty Quantification on High-speed Railway DynamicsBigoni, Daniele; Engsig-Karup, Allan Peter; True, HansPublication date:2012Document VersionPublisher's PDF, also known as Version of recordLink back to DTU OrbitCitation (APA):Bigoni, D., Engsig-Karup, A. P., & True, H. (2012). Uncertainty Quantification on High-speed Railway Dynamics.Poster session presented at Uncertainty Quantification for High-Performance Computing Workshop, Oak Ridge,TN, United States.Department of Informatics and Mathematical ModelingDTU InformaticsUncertainty Quantificationon High-speed Railway DynamicsDaniele Bigoni1, Allan P. Engsig-Karup, Hans True1Corresponding author: dabi@imm.dtu.dkIntroductionModern developments in railway vehicle dynamics are boosting the idea of running faster trains with improved safety. Important contributions to thesedevelopments are being given by precise numerical approximations. The Newton-Euler formulation of vehicle models always contain nonliearities due to thewheel-rail contact forces and components in the suspension system. These make the solution of these systems computationally challenging [4].On the other hand, the nonlinear terms in these systems have a critical influence on the dynamics and cannot be neglected. Model parameters of thevehicle as well as track irregularities are often known within a certain degree of accuracy due to measurement errors. It is then fundamental to try and detectthe parameters that can be critical for the riding safety. The goal of this work is to employ modern uncertainty quantification techniques (e.g. generalizedPolynomial Chaos) in order to obtain estimates of the sensitivity of the dynamics to such parameters which can then be taken into account in vehicle designs.The modelThe study focuses first on a simplified model[3] (see fig. 1) formed by a wheel set con-nected to the car body by a spring, withstiffnessKs, and a dry friction damper, withstatic and dynamic friction coefficients υand µ. The model runs on an irregularstraight track, where the irregularities aredescribed by a sinusoidal function. The in-vestigation is restricted to the lateral (x1)and the yaw (x3) dynamics, that can be ex-pressed by the nonlinear system in (1).A complete realistic railway wagon model[1] will be used for more complex investi-gations. The model considers 35 degreesof freedom (66 nonlinear ODEs).x˙1 = x˜2x˙2 =Fin+Fdmx˙3 = x4x˙4 =−2AFyI(1)x˜1 = x1 + a sin(2piVlt), x˜2 = x2 + a2piVlcos(2piVlt)Fin = −2Fx − 2Ksx˜1, Fsl = −sign (x˜2)µdNFst ={−sign (Fin)µsN |Fin| > µsN−Fin |Fin| ≤ µsNFd = Fstsech (αx˜2) + Fsl(1− sech (αx˜2))Fx =ξxFrΨξr, Fy =ξyFrΦξr, Fr =ξrC(1− Cξr3µt +C2ξ2r27µ2t)if Cξr < 3µtµt otherwiseξx =x˜2V− x3, ξy =Ax4V+λx˜1r0, ξr =√(ξxΨ)2+(ξyΦ)2Figure 1: Simplified model.Nonlinear DynamicsRailway vehicles are known to develop lateral oscillations, even on astraight smooth track, when running at high speeds [2]. This is due tothe loss of stability of the centered solution. This can result in the start-ing of periodic motion through a super/subcritical Hopf Bifurcation (seefig. 2) and the development of chaotic motion.Figure 3a shows that it is not sufficient to identify the speed at whichthe Hopf bifurcation takes place, but also to determine the shape of thehysteresis that can be present. The linear critical speed can be iden-tified as the velocity at which the centered solution loses stability. Thenonlinear critical speed is instead identified as the speed above whichthe system can have multiple solutions.The identification of bifurcation points and hysteresis is a computation-ally expensive task, due to the presence of non-smoothnesses in thesystem that slow down the convergence of conventional ODE solvers.The dynamics of the system can change drastically depending on thevalues of its parameters. Figure 3b shows an example of such aspectwhen the conicity of the wheel set is modified.Also, the presence of dry friction dampers can have destabilizing ef-fects on the dynamics as shown in figure 3c.0 2 4 6 8 10Time0.00200.00150.00100.00050.00000.00050.00100.0015x1(t)Speed V = 80 m/s(a) Stable solution0 2 4 6 8 10Time0.80.60.40.20.00.20.40.60.8x1(t)Speed V = 130 m/s(b) Hunting300 250 200 150 100 50 0 50<(λ)40302010010203040=(λ)Loss of stabilityV = 80 m/sV = 130 m/s(c) Migration of the eigenvalues of the Jaco-bian matrixFigure 2: Loss of stability of the centered solution above the linear critical speed (Hopf bifurcation) on the simplifiedmodel. Here, the dry friction damper and the sinusoidal irregularities of the track are disregarded.0 20 40 60 80 100 120 1400123456789x 10−3AmplitudeLeadingwheelset(L.B)Speed (m/s)Hopf Bifurcation − No precession forces(a) Subcritical Hopf Bifurcation of a com-plete vehicle on straight track.0 20 40 60 80 100 120 140V0.00.10.20.30.40.5λHopf bifurcation w.r.t. conicity and speed(b) Linear critical speed with respect to theconicity of the wheel set.0 2 4 6 8 10t0.0060.0040.0020.0000.0020.0040.006x1(t)Effect of dry friction damper - V = 50 m/s(c) Destabilizing effect due to the dry frictiondamper.Figure 3: Nonlinear dynamics effectsUncertainty QuantificationIn the vehicle industry all the parameters of the components ofthe suspension system are known within certain tolerances, dueto measurement errors as well as wear. In the railway industrythe irregularities of the rails should be considered as well. Inthe simple model described by (1) these deformations were de-scribed by sinusoidal functions for simplicity. In reality track ir-regularities have a more complex pattern that can be describedby stochastic processes.The figures in 4 show the Monte Carlo approach in the detectionof the linear critical speed with respect to the stiffness value ofthe lateral spring. The detection of linear critical speed is notcomputationally expensive and can be done observing the mi-gration of the eigenvalues of the Jacobian as shown in fig. 2c,so the slow convergence of the Monte Carlo approach is accept-able. On the contrary, the nonlinear critical speed can only beobtained calculating all the dynamics with respect to the speed,thus alternative techniques are required for the computation ofthe statistics.0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20K 1e60100200300400500600700n of samples(a) Realizations of stiffness Ks.105 110 115 120 125 130critical speed0100200300400500600700n of outcomes(b) Outcomes of linear critical speed.Figure 4: Detection of the linear critical speed of the simplified model using Monte Carlo method.References[1] BIGONI, D. Curving Dynamics in High Speed Trains. Master’s thesis, IMM, The Technical University of Denmark, KongensLyngby, Denmark, 2011.[2] TRUE, H. On the theory of nonlinear dynamics and its applications in vehicle systems dynamics. Vehicle System Dynamics31, 5-6 (1999), 393–421.[3] TRUE, H., AND ASMUND, R. The dynamics of a railway freight wagon wheelset with dry friction damping. Vehicle SystemDynamics 38 (2002), 149–163.[4] TRUE, H., ENGSIG-KARUP, A. P., AND BIGONI, D. On the numerical and computational aspects of non-smoothnesses thatoccur in railway vehicle dynamics. Mathematics and Computers in Simulation (2011). Submitted.

Uncertainty Quantification on High-speed Railway Dynamics

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Uncertainty Quantification on High-speed Railway Dynamics

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