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Theoretical vehicle bridge interaction (VBI) models have been widely studied for decades for the simply supported boundary condition but not for the other boundary conditions. This paper presents the mathematical models for several non-simply supported boundary conditions including both ends fixed, fixed simply supported, and one end fixed the other end free (cantilever) boundary condition. The closed-form solutions can be found under the assumption that the vehicle acceleration magnitude is far lower than the gravitational acceleration constant. The analytical solutions are then illustrated on a specific bridge example to compare the responses due to different bridge boundary conditions, and to study different vehicle parameter effects on extracting multiple bridge frequencies (five) from the vehicle responses. A signal drift phenomenon can be observed on the acceleration response of both the bridge and the vehicle, while a camel hump phenomenon can be observed on the Fast Fourier analysis of the vehicle acceleration signal. The parameter study shows that the vehicle frequency is preferred to be high due to the attenuation effect on the bridge frequencies that are higher than the vehicle frequency. The vehicle speed parameter is preferred to be low to reduce both the camel hump phenomenon and the vehicle acceleration magnitude, while both the vehicle mass and damping parameter have little effect on the multiple bridge frequencies extraction from the vehicle. Besides presenting the explicit solutions for calibrating other numerical models, this study also demonstrates the feasibility of the vehicle-based bridge health monitoring approach, as any bridge anomaly due to deterioration may be sensitively reflected on the bridge frequency list extracted from the vehicle response. 

The coupled differential equation group for the vehicle bridge interaction system is reestablished to include both the vehicle and bridge damping effects. The equation group can be uncoupled and closed-form solutions for both the bridge and vehicle can be obtained under the assumption that the vehicle acceleration magnitude is much lower than the gravitational acceleration constant. Then based on a simply supported boundary condition scenario, several critical parameters including bridge damping, vehicle frequency, vehicle speed, vehicle mass, and vehicle damping are studied to investigate their effects on extracting multiple bridge frequencies from the vehicle. The results show that the bridge damping plays a significant role in the vibration behaviour of both the vehicle and the bridge compared to the vehicle damping. The vehicle is preferred to be designed with a high frequency beyond the interested bridge frequencies to be extracted since low vehicle frequency tends to attenuate bridge frequencies that are higher than the vehicle frequency. A camel hump phenomenon can be observed on the extracted bridge frequencies from the vehicle, especially for scenarios that involve high bridge vibration mode and high vehicle speed. Vehicle speed is preferred to be maintained low to meet the theoretical assumption and to reduce the camel hump phenomenon. Although vehicle mass is not necessarily limited in this study, there is a magnitude balance among vehicle mass, vehicle speed, and damping to meet the theoretical assumption. This theoretical work may give some indications for designing a special field test vehicle to monitor bridge in a more comprehensive way.