Analytical solutions to VBI system (simply supported boundary condition) considering both vehicle and bridge damping effects and multiple bridge vibration modes

Matlab codes developed to calculate vehicle and bridge responses (displacement, velocity, and acceleration) for a theoretical vehicle bridge interaction (VBI) system (simply supported boundary condition) reestablished to consider both the vehicle and bridge damping effects and multiple bridge vibration modes. Hypothesis: Multiple bridge dynamic information may be extracted from the vehicle that travels on it. Assumption: 1. The magnitude of the vehicle acceleration signal (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, say >20%; 2. Uniformly distributed bridge property (mass, damping, section stiffness); 3. Vehicle traveling speed is constant. Limit: 1. The bridge is considered as the Bernoulli-Euler beam, 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. Note: Two forms of solutions are presented for cross-checking purposes, choose either SSB_dvb.m or SSB_dvb_check.m, the results for vehicle and bridge should be the same, but allow for minor differences due to truncations of numerical calculation.本研究开发的Matlab代码，用于计算理论车辆-桥梁耦合（Vehicle Bridge Interaction, VBI）系统的车辆与桥梁响应（位移、速度及加速度）；该系统采用简支边界条件，且已重构以同时考虑车辆与桥梁阻尼效应及多阶桥梁振动模态。 核心假设：可从行驶于桥梁之上的车辆中提取多维度桥梁动力学信息。 预设条件如下： 1. 车辆加速度信号沿重力方向的幅值，相较于重力加速度常数（g）可忽略不计，即幅值占比小于10%；若该条件未合理满足（占比大于20%），解析解将不再成立； 2. 桥梁的质量、阻尼及截面刚度均一分布； 3. 车辆行驶速度保持恒定。 适用限制如下： 1. 桥梁被建模为欧拉-伯努利梁（Bernoulli-Euler beam），未考虑剪力、转动惯量及轴力引发的弯曲效应； 2. 仅适用于桥梁与车辆均为零初始条件的场景； 3. 本代码仍可能存在其他未列明的局限性。 备注：本文提供两种形式的解用于交叉验证，可任选SSB_dvb.m或SSB_dvb_check.m进行计算；二者的车辆与桥梁响应理论上应完全一致，但因数值计算的截断误差，可能存在微小差异。