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.