Evolution of modal properties in the non-proportionally damped coupled vehicle–bridge system

The modal properties of bridges are crucial parameters in engineering applications, and their variation caused by moving vehicles has been increasingly recognized in recent years. However, existing closed-form analytical expressions for the varying modal properties during vehicle passage exist only for simple structural configurations. This paper proposes an innovative method for analyzing the instantaneous modal properties of the system, considering general boundary conditions and the damping effect. First, only general boundary conditions are considered, and the results show that the location of the maximum system frequency shift depends on the mode shape. Then the damping effect is considered, observing that the system is non-proportionally damped in most cases, making it difficult to obtain a closed-form solution using existing methods. To solve this, we transform the system of second-order ordinary differential equations into a set of fourth-order ordinary differential equations and derive the closed-form solution for system's modal properties. Based on this solution, we introduce the concept of Critical Coupling Damping (CCD), which defines the transition between the coupled description and the moving mass case. This work deepens understanding of changing modal properties during the traverse of sprung masses, and so has numerous potential engineering applications.