PARAMETER FREEZING PRECISE EXPONENTIAL INTEGRATOR AND ITS APPLICATION IN NONLINEAR VEHICLE-BRIDGE COUPLED VIBRATION
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摘要:描述车桥耦合作用的基本问题是一个时变系统问题, 且很多工况下需考虑非线性特性, 使得该问题难以得到解析解, 甚至数值解也可能很复杂. 针对于该问题的求解, 本文提出了一种参数冻结精细指数积分法, 将其应用于车桥耦合动力学模型的数值分析中. 该方法结合了精细积分和指数积分特点, 并将时变系数矩阵在每一积分步参数冻结, 用于获得系统振动响应的数值解. 考虑汽车轮胎与桥面的力和位移耦合关系、桥面沥青铺装层、桥梁材料粘弹性和几何非线性特性, 建立了车桥耦合动力学模型, 并应用参数冻结精细指数积分法对该模型进行了求解. 通过与近似解析解、辛Runge-Kutta算法以及经典的Newmark-β数值积分法计算结果进行对比, 验证了所提出方法计算结果的有效性和准确性. 在此基础上, 制作了缩尺车桥耦合系统模型, 测试了跨中挠度响应, 进一步验证了理论建模和所提算法的有效性和实用性. 通过数值计算分析了所提算法的数值特性, 结果表明: 本文提出的参数冻结精细指数积分法不仅可以处理时变、非线性问题, 且具有良好的数值计算精度和长时间数值稳定性; 由于精细积分的特点, 参数冻结精细指数积分法的计算时间步长可以取的较大, 可有效提高计算效率. 因此, 本文所提出的参数冻结精细指数积分法预期可成为求解车桥耦合动力学问题的一种新的高效算法.
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关键词:
- 精细指数积分法/
- 车桥耦合振动/
- 参数冻结/
- 非线性时变系统/
- 辛Runge-Kutta算法
Abstract:The basic problem describing vehicle-bridge interaction is a time-varying system problem, which the nonlinear characteristics need to be considered in many working conditions. It is difficult to get the analytical solution to this problem, and even the numerical solution may be complex. In this paper, focus on the solution to this problem, a parameter freezing precise exponential integrator is proposed and its application to vehicle-bridge coupled problem is analyzed. This algorithm combines the characteristics of precise integration and exponential integration, and freezes the parameters of time-varying coefficient matrix at each integration step to obtain the numerical solutions of vibration responses of the system. Considering the coupling relationship of the force and displacement between the vehicle tire and bridge deck, the asphalt pavement of the bridge deck, the viscoelastic and geometric nonlinear characteristics of the bridge, the vehicle-bridge coupled dynamic model is established. The parameter freezing precise exponential integrator is used to solve this dynamic model. By comparing with the approximate analytical solution, the solution calculated by the symplectic Runge-Kutta algorithm and the classical Newmark-β numerical integration method, the validity and accuracy of the proposed method are verified. On this basis, a scaled vehicle-bridge coupling system model is made. The mid-span deflection response is tested, which further verifies the validity and practicability of the theoretical model and proposed algorithm. The numerical characteristics of the proposed algorithm are analyzed by numerical calculation. Numerical results show that the proposed parameter freezing precise exponential integrator can not only deal with time-varying and nonlinear problems, but also has good numerical calculation accuracy and long-time numerical stability. Due to the characteristics of the precise integration, the time step can be set to be larger for the parameter freezing precise exponential integrator, which can effectively improve the calculation efficiency. Therefore, the proposed parameter freezing precise exponential integrator is expected to be a new and efficient algorithm for solving vehicle-bridge coupling dynamics problem. -
图 5近似解析解、时间步长为$ \tau = {10^{ - 2}}{\text{ s}} $的FPEI4-4格式和时间步长为$ \tau = {10^{ - 4}}{\text{ s}} $的SRK2-4格式计算跨中挠度结果
Figure 5.The results of mid-span deflection calculated by analytical approximate solution, FPEI4-4 scheme with $ \tau = {10^{ - 2}}{\text{ s}} $ and SRK2-4 scheme with $ \tau = {10^{ - 4}}{\text{ s}} $
图 6时间步长为$ \tau = {10^{ - 4}}{\text{ s}} $和$ \tau = {10^{ - 5}}{\text{ s}} $时SRK2-4格式计算跨中挠度结果之差
Figure 6.The difference between the results of mid-span deflection calculated by SRK2-4 scheme under time step $ \tau = {10^{ - 4}}{\text{ s}} $and $ \tau = {10^{ - 5}}{\text{ s}} $
图 7FPEI4-4格式和Newmark-β算法求解跨中挠度计算误差
Figure 7.The numerical errors of mid-span deflection calculated by FPEI4-4 scheme and Newmark-β algorithm
图 8FPEI4-4格式与Newmark-β算法得到的桥梁跨中挠度结果
Figure 8.The results of mid-span deflection calculated by FPEI4-4 scheme and Newmark-β algorithm
图 11$ \tau = {10^{ - 2}}{\text{ s}} $, 不同车速时FPEI4-4格式求解跨中挠度计算误差
Figure 11.The numerical errors of mid-span deflection calculated by FPEI4-4 scheme under different vehicle speeds with time step $ \tau = {10^{ - 2}}{\text{ s}} $
图 12$ \tau = {10^{ - 2}}{\text{ s}} $, FPEI4-4格式选取不同时变矩阵冻结方式的计算结果
Figure 12.The numerical results calculated by FPEI4-4 scheme under different parameter freezing forms with time step $ \tau = {10^{ - 2}}{\text{ s}} $
图 13$ \tau = {10^{ - 2}}{\text{s}} $, FPEI4-4格式不同时变矩阵冻结方式求解跨中挠度的计算误差
Figure 13.The numerical errors of mid-span deflection calculated by FPEI4-4 scheme under different parameter freezing forms with time step $ \tau = {10^{ - 2}}{\text{s}} $
表 1汽车参数
Table 1.Vehicle parameters
Symbol Value m2/kg 18000 m1/kg 2000 k2/N/m 9000000 c2/Ns/m 80000 k1/N/m 36000000 c1/Ns/m 700 
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表 2桥面沥青铺装层和桥体参数
Table 2.Pavement and bridge parameters
Symbol Value L/m 40 b/m 8 h1/m 0.1 h2/m 2 E1/GPa 8 E2/GPa 34.5 $ {\rho _1} $/kg/m3 2000 $ {\rho _2} $/kg/m3 2500 $ {\eta _1} $ 0.1 $ {\eta _2} $ 0.01 下载:
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表 3Newmark-β算法和FPEI4-4格式计算跨中挠度对比
Table 3.Comparison of mid-span deflection calculated by Newmark-β algorithm and FPEI4-4 scheme
Time step/s Computation time/s Maximum error/m Newmark-β algorithm $ {10^{ - 4}} $ 0.869262 $ - 2.222306 \times {10^{ - 7}} $ $ {10^{ - 3}} $ 0.078846 $ 2.269753 \times {10^{ - 6}} $ $ {10^{ - 2}} $ — — FPEI4-4 scheme $ {10^{ - 4}} $ 2.849518 $ 4.373681 \times {10^{ - 13}} $ $ {10^{ - 3}} $ 0.319489 $ 4.373730 \times {10^{ - 11}} $ $ {10^{ - 2}} $ 0.031098 $ 4.409290 \times {10^{ - 9}} $ $ {10^{ - 1}} $ 0.003465 $ - 1.741726 \times {10^{ - 6}} $ 下载:
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表 4不同冰冻近似方式的对比
Table 4.Comparison of different parameter freezing forms
Time step/s Form Maximum error/m
$ {10^{ - 2}} $LC1 $ 4.409290 \times {10^{ - 9}} $ LC2 $ 8.127175 \times {10^{ - 8}} $ LC3 $ 8.418344 \times {10^{ - 8}} $
$ {10^{ - 3}} $LC1 $ 4.373730 \times {10^{ - 11}} $ LC2 $ 8.131892 \times {10^{ - 9}} $ LC3 $ 8.139901 \times {10^{ - 9}} $ 下载:
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