Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014 Porto, Portugal, 30 June - 2 July 2014 A. Cunha, E. Caetano, P. Ribeiro, G. Müller (eds.) ISSN: 2311-9020; ISBN: 978-972-752-165-4 An efficient numerical method for the vehicle-bridge nonlinear dynamic interaction 1 Zhibin Jin1, Xiaozhen Li1, Dejun Liu1, Shiling Pei2 Department of Bridge Eng., Southwest Jiaotong University, 610031, Chengdu, P.R.China 2 Department of Civil and Enviromental Eng., Colorado School of Mines, Golden, USA email: [email protected], [email protected] ABSTRACT: This study aims at providing an efficient and accurate method for the time history simulation of the vehicle-bridge dynamic interaction. With the geometric nonlinearity in wheel-rail contact fully considered, a simplified assumption was adopted that vertical wheel-load fluctuation has no influence on the lateral wheel-rail interaction. The wheel-rail creep forces are computed using Kalker’s theory, and the normal forces are solved from wheel-set vertical and torsional equilibrium equations. The method limited the number of nonlinear parameters for each wheel-set to five. In addition, mode superposition method is adopted for bridge responses to minimize the overall system DOFs. The virtual work principle method is used to derive the system matrix. Numerical examples show that the present method gives results comparable in terms of accuracy to that using a more complex model. But it is at least 50 times faster than the detailed model. The efficiency and accuracy of the proposed method will make it suitable in the reliability-based vehicle-bridge interaction analyses. KEY WORDS: Vehicle-bridge; Dynamic interaction; Wheel-rail contact; Mode superposition. 1 INTRODUCTION The rail way vehicles passing across a bridge arise vibrations both to the bridge and vehicles. These vibrations are of great concerns in bridge and vehicle design, and have been the topic of research for many years. In the earlier stage, the vehicle is simply treated as moving loads along the bridge[1], but significant difference to the effect of a real vehicle has been observed[2]. Another limitation of moving load model is that it cannot account for the lateral vibrations of the bridge. To take account the effect of vehicle vibration, various dynamic interaction models are developed later [3-12]. All these interaction models are solved numerically based on different wheel-rail interaction assumptions. For example, Yang and Yau et al.[3] assumed the wheel-set motion is identical to the rail beneath; Xia, Zhang and Xu et al. [4-5] took a set sine waves as the lateral gap between the wheel and rail; Li, Qiang et al.[6-7] allows the wheel has independent yaw and lateral motion; Zhai and Xia et al. [8-10] taken all 5 DOFs of the wheel-set are independent. The solution scheme for vehicle-bridge interaction varies in existing research works, including direct time history integration[4-5], iterative approach [6-7], the VBI element method [3,13], and explicit integration method[8-10]. Through combination of wheel-rail model assumption and integration scheme, a family of solution procedures was developed. But it is very challenging to increase the computational efficiency of these traditional methods. A simulation can often take a long time to complete even on state-of-the-art computers. This has become a major barrier for studying the vehicle-bridge system from a probabilistic perspective. The intent of this paper is to develop an efficient but still realistic model for vehicle-bridge random simulations. The wheel-rail interaction in this model is modified from [14-15]. The main modification here is the consideration of the rotational motion of rail, which is of significant for vehicles on the bridge. This model fully considers the wheel-rail geometric nonlinearity, which maybe the most significant nonlinear factor in vehicle bridge interaction. The model also adopts the mode superposition method for the bridge and results a hybrid model in which the vehicle are represented by physical DOFs, while the bridge by generalized DOFs in the frequency domain. The stiffness formulation was conducted through principle of virtual work using Matlab. The accuracy and efficiency of this method are proved in an example by comparing with that of detailed wheel-rail model. 2 THE VEHICLE AND BRIDGE AND EXCITATION MODELS The interaction model consists of two sub systems, namely, the vehicle and the bridge. There are a variety of vehicle models in literatures, here we use the model with 23 DOFs proposed by Li and Qiang[6] as depicted in Figure 1 and 2. A passage vehicle is composed of one car body, two trucks, and four wheel-sets. These components are all treated as rigid bodies. For each rigid body, 5 DOFs are considered, omitting the longitudinal motion. The 5 DOFs of the car-body is , thoses of the the i-th truck(i=1,2) is: . Symbols denotes the lateral, vertical, rolling, pitching and yawing motion respectively; suberscripts “c”and “t” stand for “car-body” and “truck”. ψ 1 Carbody z θ y 2lc ϕ x 3 Rear truch 12 4 11 7 Bridge deck 1 3 Front truch 7 9 10 Left rail 6 Wheel-set Right rail 8 2 2 5 6 5 2lt 4 er Section controid hr Figure 1. Vehicle-bridge interaction model. 1341 Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014 Carbody Secondary suspension 2b2 h1 Truck h2 h3 Primary suspension Wheel-set 2b1 Figure 2. Front view of the vehicle. For the wheel-set, since the wheels are assumed not to lift from the rail, the vertical and rolling DOFs are not free ones but dependent on the track motion and wheelset’s lateral motion. Thus, only yaw and lateral motions are free DOFs. Displacements of the j-th(j=1,2,3,4) wheel-sets is represented by . Totally, a passage vehicle’s motion are described by 23 DOFs as . To minimize the number of the bridge’s DOFs, we use the mode superposition method to represent the bridge. The mode superposition method[16] is a classical technique using generalized DOFs to decrease the physical DOFs of a system. The DOFs of bridge is the former N modal displacements , n=1~N. The n-th mode shapes is denoted as , which is mass normalized. Other parameters related to the bridge includes: the n-th natural frequency , n-th damping ratio . For a given bridge, all these parameters can be obtained from an eigenvalue analysis using any structure software. Rail irregularities are random excitations to the vehiclebridge system. Several nations have established rail irregularities spectrums based on field measurements [17]. We consider three components of rail irregularities, namely, the vertical, lateral and torsional irregularity. The rail irregularities to the j-th wheel-set are depicted below. In Figure 3, is the vertical rail irregularity at where the j-th wheel-set locates longitudinally, and are the lateral and torsional irregularities respectively. 3 VIRTUAL WORK FORMULATION When a systems is at equilibrium, the virtual work sum done by the forces (both internal and external, either linear or nonlinear) on any permissible virtual displacement δxi is zero, that is (1) If the inertial forces are treated as external forces, the formula above can be applied to dynamic problems. Knowing the virtual work of a system, the mass, stiffness, damping and external force matrix can be easily derived [16]. The virtual work considered in the study including virtual work done by vehicle suspension spring and damping forces and the work by bridge structure inertia and internal forces. We write these forces in terms of vehicle motions firstly, then calculate the virtual works done by these forces. We give each spring and damper set an index number (from 1 to 12 in the boxes of figure 1), also an index to each rigid body of the vehicle (from 1 to 7), as depicted in figure 1. For the i-th spring set, the two ends of the spring are connected to rigid body “ ” and “ ”, and the connection 1342 point in the local coordinate of body “ ” is and ]in “ ”. The stretches in x, y and z directions of the i-th spring are: (2) where (3) and are the displacements of the “s” and “e” body. Note that the longitudinal motion is not included. It’s easy to know, the virtual work done by this spring is (4) For damping forces, the virtual work can be expressed in a similar way. For bridge structure, take the n-th mode of bridge as a vibrator with mass as one, stiffness as , and damping coefficient as , the virtual work done by the stiffness and damping force of this vibrator can be written as , 4 (5) THE WHEEL-RAIL INTERACTION The contact geometry is the essential non-linear factor in the vehicle-bridge interaction. The key contact geometric parameters are show in figure 4. The nonlinear geometric parameters include: contact angles, and , at the left and right rail; rolling radius and ; and rotational angle of the wheel-set . For a real flanged wheel profile, these parameters are all non-linear functions of wheel-rail relative motion in lateral direction . In this study, the Chinese LM wheel tread and UIC60 rail profile are used. z Right Left y o rR r0 ϕw rL r0 Rail center yw δR δL Rail center Figure 4. Wheel-rail contact geometry. In the stationary track case in reference[15], the lateral movement of wheel in the track frame is identical that in the inertial frame . While in our moving track case (see Fig.5), the lateral movement at wheel in the inertial frame and the track frame are related by (6) LA z o wheelset z' rail NR ϕw δR ϕw ϕt y o' y' ϕw δL NL track bridge deck Figure 5. Movement of Wheel-set on the Bridge. Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014 All nonlinear parameters in wheel rail contact depend on the lateral relative displacement of the wheel-set to the track. Note that this relative displacement should be calculated in frame o’x’y’z’. There are two type forces between the wheel and the rail: contact normal forces and the creep forces. The normal forces are due to the normal compress between the wheel and rail materials. And the creep forces are due to the slip and adhesion between the wheel and the rail along their contact path. Figure 6 depicts the wheel-rail forces, where and are normal forces at the left and right wheel-rail contact point, and are the longitudinal creep forces at the left and right wheel-rail contact point respectively, and are the lateral creep forces, and are the creep moment along the vertical direction. Since the wheel-rail contact forces on stationary tracks haven been well established in [15], we derive the formulas of these forces on the moving track on the bridge in a parallel way. The total wheel-rail lateral force and yawing moment can be expressed in matrix form as In equation 9, =- (9) ′ is the lateral displacement between the wheel-set and the rail, and are the stiffness and damping matrix of wheel-rail contact respectively as (10.a) (10.b) In equation 10 , , , and are lateral, lateral/spin, spin and longitudinal creep coefficients, respectively; and are nonlinear terms whose formulas can be found in [15]. Knowing the wheel-rail forces, the virtual work by the wheel rail can be obtained as rL r0 rR r0 FLy FLz FRy NL NR FRz ML MR wheelset o MRz FRy rail FRx MLz FLy y ψw x FLx rail Figure.6. Wheel-rail forces. In derivation of the wheel-rail normal forces on the stationary track, the normal contact forces at left and right wheel are solved from two equilibrium equations. This is possible because the wheel and rail are assumed to be in constant contact, and neither vertical nor rolling motion is an independent freedom. Instead they are determined by lateral motion of the wheel-set. In our moving track case, component of normal forces along y direction is (7) where is the vertical wheel-set load, and the nonlinear term is used to calculate the gravitational stiffness can be found in reference [15]. Here an additional term appears in contrast to the stationary track case. The gravitational stiffness moment can be approximated as (8) where is the contact angle at wheel-set’s centered position, a is half of the wheel-set gauge and is the yaw angle of the wheel-set. The gravitational stiffness moment is the same as on the unrolled track. Creep forces are calculated from the creepages between wheel and rail using the Kalker’s linear theory. The creep forces on the moving track can be expressed in the similar way to that of the stationary track in [15]. The only difference lies in that Eq.6 is used to calculate the creepage. (11) where system equations and solving method The total virtual work of the vehicle-bridge system is the sum of all virtual work from vehicle suspensions, wheel rail forces, bridge modal forces and inertial forces. Then differentiate the total virtual work with first respect to the virtual displacements then to the displacements, velocities or rail irregularities. The differentiated results are members of the stiffness, damping and load distributing matrixes. The resulting vehicle-bridge system equation derived using the virtual work method is (12) where M is the diagonal mass matrix; and are the damping and stiffness matrix both dependent of time and the wheelrail lateral displacement ; X is the displacement vector; is the load distribute matrix, and the irregularity vector. The nonlinear equation 12 is solved iteratively at discretized time points using the Newmark’s integration method. Here only one loop of iteration for wheel-rail nonlinear geometry is required. 5 5.1 NUMERICAL EXAMPLES Bridge parameters The bridge in study is a continuous rigid frame with span of 88+168+88m, and two piers with height 120m. The bridge is made of reinforced concrete, whose modulus is 3.4 × 104 MPa, density 2550 kg/m3. Damping ratio is 0.05 for the first and second frequencies. The properties of the girder and pier sections are listed in Table 2. In table 2, A is the area of the sections, J the torsional moment of inertial and Iy, Iz the bending moment about horizontal and vertical directions respectively. The first lateral and vertical mode shapes of the bridge are plotted in figure 7. 1343 Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014 Table 2. Section properties of the bridge sections. 2 4 4 5.3 4 A/m J/m Iy/m Iz/m Girder 17.12 151.06 93.62 187.20 Pier 48.28 871.31 564.62 522.49 Lateral /mm Vertical /mm -3 Torsional /10 rad 10 Irregularities Secion Rail irregularities The rail irregularities are simulated from the USA spectrum, class 6. The wave length of the irregularities ranges from 1 to 100 m. The simulated results are plotted in figure 8. 5 0 -5 (a) First lateral mode(plane view, f=2.14 rad/sec) -10 0 100 200 300 400 500 Distance/m Figure 8. Simulated rail irregularities 5.4 5.2 Vehicle parameters A typical railway vehicle is selected in this analysis. The parameters of the vehicle are listed in table 3. The train is composed of 8 identical vehicles. The train runs at 100 km/h across the bridge. Table 3. Vehicle parameters. Parameter Mw Iwz Mt Itx Ity Itz Mc Icx Icy Icz k1x k1y k1z k2x k2y k2z c1x c1y Value 2100 kg 1030 kg.m2 2600 kg 2100 kg.m2 1420 kg.m2 2600 kg.m2 3.41×104 kg 1.10×105 kg.m2 1.70×106 kg.m2 1.70×106 kg.m2 1.57×107 N/m 7.48×106 N/m 1.18×106 N/m 1.78×105 N/m 1.78×105 N/m 2.21×105 N/m 0 N.s/m 0 N.s/m Parameter c1z c2x c2y c2z L lc lt b1 b2 h1 h2 h3 a δ0 f11 f12 f22 f33 Value 1.96×104 N.s/m 1.10×106 N.s/m 2.18×104 N.s/m 9.80×103 N.s/m 26.0 m 8.75 m 1.25 m 1.00 m 1.23 m 0.50 m 0.20 m 0.065 m 0.75 m 0.05 rad 8.28×106 N 2.22×104 N.m 11.3 N.m2 8.13×106 N Most parameters in table 3 have been defined early in this paper except for the inertial parameters. For the inertial parameters in table 3, M denotes mass, I denotes mass moment; the subscribers, ‘w’ stands for wheel-set, ‘c’ for carbody and t for truck; ‘x’, ‘y’ and ‘z’ represent directions. 1344 Detailed Model Present Model 0.10 0.05 0.00 -0.05 -0.10 0 5 10 15 Time / s 20 25 30 Figure 9. Lateral displacement at the middle main-span Deflection at midspan / mm Figure 7. Mode shapes of the bridge The dynamic responses of the bridge, the vehicle and their interaction are simulated using two methods. One method used the detailed wheel-rail model and iteration solving scheme, the other is the present method. Figure 9 and 10 are the lateral displacement and deflection at middle of the main-span of the bridge. It is clear, that the amplitude and phase of the bridge responses from the two methods are pretty close. Lateral displacement at midspan / mm (b) First vertical mode(front view, f=5.94 rad/sec) Numerical results 0 -5 -10 Detailed model Present model -15 0 5 10 15 Time / s 20 25 30 Figure 10. Deflection at the middle main-span The time histories of vehicle-bridge interaction forces are shown in figure 11. From the figure, the coincidence of the two methods can be observed, only small difference occurs in high frequencies components. This difference may be aroused from neglecting the effect of the vertical force fluctuations on the lateral forces in the present method. Wheel-rail lateral force / kN Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014 40 Detailed model Present model 20 0 -20 -40 4 6 8 10 Time / s 12 14 Figure 11. Wheel-rail lateral force The lateral and vertical accelerations at the centroid of the car-body are shown in figure 12 and 13 respectively. The results from the two method fit fairly well from the point view of engineering. Especially, the vertical acceleration fit much better than the lateral one. This can be understood since the vertical wheel-rail models are identical, and the lateral motion of the vehicle has minor effects on the vertical ones. Lateral acceleration of 2 the carbody / m/s 0.4 achieved by 2 approaches: (1) a simplified wheel-rail nonlinear interaction model which cancels two loops of iteration of the more detailed model; (2) The mode superposition method is used to decrease the bridge DOFs significantly. The accuracy of this model is still maintained by retain the most important non-linearity of vehicle-bridge system, the nonlinear wheel-rail contact geometry. Also, we introduced the virtual work method to derive the sophisticated motion equations of vehicle-bridge system to ease the task of manually operations. This approach is easy to expand to vehicles of other topology. Numerical results show that vehicle-bridge responses using present method are of adequate accuracy compared to those from a more detail model, while the present method is about 60 times faster than the detailed model. With these merits, the present model may find its future application in the random simulations of vehicle-bridge vibrations. ACKNOWLEDGMENTS This optional section contains acknowledgments. This study is supported by the National Natural Science Foundation of China (Grant No. 51008250), Southwest Jiaotong University Foundation for Young Researchers (Grant No. 2007Q110, SWJTU12BR037). Detailed model Present model 0.2 0.0 REFERENCES [1] -0.2 [2] -0.4 3 6 9 Time / s 12 15 [3] Figure 12. Lateral acceleration at the centroid of the car-body [4] [5] Detailed model Present model Vertical acceleration of 2 the carbody / m/s 0.2 [6] 0.0 [7] -0.2 [8] 3 6 9 Time / s 12 15 [9] Figure 13. Vertical acceleration at the centroid of the car-body Although the present method loses minor accuracy in the lateral behavior of the vehicle-bridge system, the tradeoff for efficiency is significant. For such a large span bridge interaction with 8 vehicles, the present method only token 317 second to simulate on a laptop using Matlab program. While the detailed method programmed in C++ language used 302 minutes. Or the present method is about 60 times efficient as the detailed model. [10] [11] [12] [13] 6 CONCLUSION This paper presented a new model for the vehicle-bridge dynamic interaction analysis. This model intends to get a balance between the efficiency and accuracy. The efficiency is [14] L. Fryba, Dynamics of vehicles on railway bridges, Vehicle System Dynamics, 1975, 4(2-3): 208-210. J. M. Goicolea, F. Gabaldón, F. Riquelme, Design issues for dynamics of high speed railway bridges, International Association for Bridge Maintenance and Safety, Porto, 2006. Y.B.Yang, J. 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