RANDOM VIBRATION ANALYSIS OF MAGLEV VEHICLE-BRIDGE COUPLED SYSTEMS WITH THE EXPLICIT TIME-DOMAIN METHOD
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摘要: 轨道不平顺所引起的磁浮车辆-桥梁耦合系统随机振动问题是磁浮列车行驶过程中需要关注的重点问题之一，传统的随机振动方法在解决这类问题时存在工作量大、计算效率低等问题。分别从磁浮车辆系统和桥梁系统出发，建立磁浮车辆系统响应和桥梁系统响应关于车桥相互作用电磁力的时域显式表达式；利用车轨间的电磁力方程及几何相容条件，构建电磁力关于轨道不平顺的时域显式表达式；进一步推导得到轨道不平顺作用下磁浮车辆系统和桥梁系统关键动力响应显式表达式。在此基础上，利用统计矩运算法则，直接获得磁浮车辆系统和桥梁系统关键响应的演变统计矩；也可以利用轨道不平顺的数字生成技术，结合随机模拟法，获得车桥耦合系统关键响应的统计信息。在上述过程中，由于车桥耦合系统关键动力响应的显式表达式已先行构建完毕，因此大幅提升了随机振动分析的计算效率。以二自由度磁浮车辆与桥梁耦合模型为例，阐明了方法列式过程。磁浮列车过多跨简支梁桥的数值算例结果表明，时域显式法具有理想的计算精度和效率。Abstract: The vibration of maglev vehicle-bridge coupled system induced by random guideway irregularities is one of the major concerns for running maglev trains. Traditional methods for random vibration analysis are tedious and time-consuming. The explicit expressions for the dynamic responses of the maglev vehicle and the bridge subsystems with respect to the interacting electromagnetic force were first formulated. By applying the electromagnetic force formula and the displacement compatibility condition between the vehicle and the guideway, the explicit expression for the electromagnetic force in terms of guideway irregularities was further derived. Subsequently, the explicit expressions for the critical responses of the maglev vehicle and the bridge subsystems subjected to guideway irregularities were elicited. Based on the explicit formulations, the evolutionary statistical moments of critical responses of the maglev vehicle and the bridge subsystems were calculated by direct application of the statistical moment operation rules. In addition, a random simulation method was put forward to obtain the statistical metrics of critical responses of the maglev vehicle-bridge coupled system with generated guideway irregularity samples. Owing to the merits of explicit formulations, a more efficient stochastic analysis of the maglev vehicle-bridge coupled system was achieved. A coupled model involving a 2-DOF maglev vehicle and a simply-supported beam was used to demonstrate the formulation procedure of the proposed method. The numerical example with a maglev train traversing a multi-span bridge indicates that the explicit time-domain method has high accuracy and superior computational efficiency.
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图 1 二自由度磁浮车辆-桥梁耦合系统力学模型
Figure 1. Mechanical model for a 2-DOF maglev vehicle-bridge coupled system
图 3 桥梁跨中竖向位移
${y_{{\text{bm}}}}$ 时程Figure 3. Time history of vertical displacement
${y_{{\text{bm}}}}$ at mid-span of bridge
图 4 车体竖向加速度
${a_1}$ 时程Figure 4. Time history of vertical acceleration
${a_1}$ of car body
图 7 桥梁跨中竖向位移
${y_{{\text{bm}}}}$ 均值时程Figure 7. Time history of mean value of vertical displacement
${y_{{\text{bm}}}}$ at mid-span of bridge
图 8 桥梁跨中竖向位移
${y_{{\text{bm}}}}$ 标准差时程Figure 8. Time history of standard deviation of vertical displacement
${y_{{\text{bm}}}}$ at mid-span of bridge
图 9 车体竖向加速度
${a_1}$ 均值时程Figure 9. Time history of mean value of vertical acceleration
${a_1}$ of car body
图 10 车体竖向加速度
${a_1}$ 标准差时程Figure 10. Time history of standard deviation of vertical acceleration
${a_1}$ of car body
图 11 电流变化量
$\Delta I$ 均值时程Figure 11. Time history of mean value of electric current change
$\Delta I$ 
图 12 电流变化量
$\Delta I$ 标准差时程Figure 12. Time history of standard deviation of electric current change
$\Delta I$ 
图 13 悬浮间隙变化量
$\Delta c$ 均值时程Figure 13. Time history of mean value of air gap change
$\Delta c$ 
图 14 悬浮间隙变化量
$\Delta c$ 标准差时程Figure 14. Time history of standard deviation of air gap change
$\Delta c$ 
图 15 磁浮列车过多跨简支梁桥模型
Figure 15. Mechanical model for a maglev train traversing a multi-span simply-supported bridge
图 16 某跨简支梁跨中竖向位移
${y_{{\text{bm}}}}$ 均值时程Figure 16. Time history of mean value of vertical displacement
${y_{{\text{bm}}}}$ at mid-span of a simply-supported beam
图 17 某跨简支梁跨中竖向位移
${y_{{\text{bm}}}}$ 标准差时程Figure 17. Time history of standard deviation of vertical displacement
${y_{{\text{bm}}}}$ at mid-span of a simply-supported beam
图 18 第2节车辆车体竖向加速度
${a_1}$ 均值时程Figure 18. Time history of mean value of vertical acceleration
${a_1}$ for car body of the 2nd maglev vehicle
图 19 第2节车辆车体竖向加速度
${a_1}$ 标准差时程Figure 19. Time history of standard deviation of vertical acceleration
${a_1}$ for car body of the 2nd maglev vehicle表 1 单样本时程分析耗时对比
Table 1. Comparison of computation time for sample analysis
方法 计算耗时/s 时域显式法 0.739 + 0.013 = 0.752 Newmark-β逐步迭代法 1.413 
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表 2 算例1响应统计矩分析的计算耗时
Table 2. Computation time for statistical moment analysis of responses for the 1st numerical example
方法 样本数 计算耗时/s 显式表达式建立 响应统计矩分析 总耗时 时域显式直接法 − 0.739 0.100 0.839 时域显式随机模拟法 200 0.072 0.811 600 0.162 0.901 1000 0.211 0.950 3000 0.502 1.241 5000 0.693 1.432
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表 3 各响应平均峰值
Table 3. Mean peak values of responses
样本数 响应平均峰值 ${y_{{\text{bm,peak}}}}{\text{/m}}$ ${a_{ {{1} },{\text{peak} } } }/({\text{m/} }{ {\text{s} }^2})$ $\Delta {I_{{\text{peak}}}}/{\text{A}}$ $\Delta {c_{{\text{peak}}}}/{\text{m}}$ 200 1.157×10−2 0.409 2.841 6.677×10−4 600 1.157×10−2 0.411 2.865 6.740×10−4 1000 1.157×10−2 0.411 2.869 6.751×10−4 3000 1.157×10−2 0.414 2.880 6.770×10−4 5000 1.157×10−2 0.415 2.883 6.781×10−4
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表 4 算例2响应统计矩分析的计算耗时
Table 4. Computation time for statistical moment analysis of responses for the 2nd numerical example
方法 计算耗时/s 显式表达式
建立响应统计矩
分析总耗时 时域显式
直接法5.894 11.850 17.744 时域显式
随机模拟法4.582
(1000个样本)10.476
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