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.