Analytical solutions to a theoretical double-axle VBI system

Matlab codes were developed to calculate vehicle and bridge responses (displacement, velocity, and acceleration) for a theoretical double-axle vehicle bridge interaction (VBI) system. Both the vehicle and bridge damping effects and multiple bridge vibration modes are considered. Contact patch length and vehicle external excitation are considered. The time step is determined by contact patch length and vehicle speed.
Assumption: 1. The magnitude of the vehicle acceleration (gravitational direction) is negligible compared to the gravitational acceleration constant (g), say <10%. Analytical solutions will be invalid if this condition is not reasonably met; 2. Uniformly distributed bridge property (mass, damping, section stiffness); 3. Vehicle speed is constant.
Limit: 1. Based on Bernoulli-Euler beam theory, flexure effects caused by shear forces, rotary inertial forces, and axial forces are not considered; 2. For zero initial conditions of bridge and vehicle only; 3. Other flaws may also apply, use with caution.
Note: Two forms of solutions (alpha and benchmark) are presented for cross-checking purposes, the results are the same. Simply run the initial.m file then the plot.m file. Edit as needed.

本研究开发了一套 Matlab 编程代码，旨在计算理论上的双轴车辆与桥梁交互系统（VBI）的车辆和桥梁响应（位移、速度和加速度）。该计算模型同时考虑了车辆和桥梁的阻尼效应以及桥梁的多重振动模式。接触片长度和车辆外部激励也被纳入考量。时间步长由接触片长度和车辆速度决定。

假设：1. 车辆加速度（重力方向）的幅度相对于重力加速度常数（g）而言可忽略不计，例如小于10%。若此条件无法得到合理满足，则解析解将无效；2. 桥梁属性（质量、阻尼、截面刚度）均匀分布；3. 车辆速度保持恒定。

限制：1. 基于伯努利-欧拉梁理论，不考虑由剪切力、旋转惯性力和轴向力引起的弯曲效应；2. 仅适用于桥梁和车辆的初始条件为零的情况；3. 可能存在其他缺陷，使用时请谨慎。

备注：为交叉验证目的，提供了两种形式的解决方案（alpha 和基准），结果一致。只需依次运行 initial.m 文件和 plot.m 文件，如有需要可进行编辑。