Influence of bridge geometric nonlinearity on dynamic response of vehicle-bridge coupling system

Based on the theory of vehicle-bridge coupling vibration, the differential equation of vehicle-bridge coupling system is set up according to different conditions. The differential equation of the system is converted into matrix form using mode decomposition method and is solved using MATLAB. The system equation has a non-linear matrix term when the geometric nonlinearity of the bridge is considered. The influence of wheel acceleration on the dynamic response of the bridge is analyzed without simplification under four speeds. The results show that it is acceptable to neglect the influence of wheel acceleration at low speeds, but it has a significant influence which must be considered at high speeds.


Introduction
With the remarkable and increasing span and vehicle load of bridges, as well as the gradually decreasing mass and stiffness of bridge structures, much attention has been focused on the complexity and diversity of vehicle-bridge coupling vibration.
There are numerous methods of analyzing vehicle-bridge coupling vibration. Xia [1] presented a method of simplifying the bridge into a modal model to analyze the vehicle-bridge-pier system. Based on the finite element method, Chen [2] analyzed the resonance of a rigid-frame bridge under vehicle-bridge coupling vibration. Li [3] conducted in-depth analyses of the vehicle-bridge interaction under stochastic vibration. Shen [4] used an ODE function based on the Runge-Kutta method to compile a secondary development function which can solve the differential equation of the vehicle-bridge coupling system.
The influence of the geometric nonlinearity of the structures on the vehicle-bridge coupling vibration is important, especially for long-span and low-mass bridges. Considering the geometric nonlinearity of the bridge, it can be more realistic to analyze the dynamic response of the structure, which is of great theoretical and practical significance in the design and construction of the bridge structure. Fig. 1 shows the simply supported beam subjected to sprung mass, the vehicle system passes through the beam at a uniform speed ,

Establishment of system equation considering the geometric nonlinearity of the bridge
, and denotes the dynamic deflection of the beam and the mass , respectively. Meanwhile, the displacement of the mass is consistent with the deflection of the beam at its position expressed as , . The main derivation process of the differential equation is as follows. First, the dynamic equilibrium equation of a simply-supported beam can be written as Eq. (1): (1)

Fig. 1. Simply supported beam subjected to sprung mass model
The dynamic equilibrium equation of can be written as: The external load acting on the beam can be expressed as Eq. (3): According to the mode decomposition method, the right end of Eq. (3) can be transformed into: The left end of the Eq. (1) can be transformed into: Therefore, Eq. (3) can be expressed as Eq. (6): ( ) + 2 ( ) sin sin Similarly, using the mode decomposition method, the Eq. (2) can be written as: By combining Eq. (6) and Eq. (7), the system dynamic equilibrium equations of the vehicle coupling with the simply-supported beam are obtained. The th order matrix expression of the system motion equation is: , ⋯ , 0 represents the generalized force vector, represents the generalized mass matrix, represents the generalized damping matrix and represents the generalized stiffness matrix, is used to describe the nonlinear matrix. This can be described by the following matrices: It can be seen from the above derivations that in cases where the geometric nonlinearity of the bridge is taken into consideration, the system equation has a non-linear matrix term. It can be found that the non-linear matrix is related only to the nature of the bridge itself, and not to the parameters of the vehicle.

Establishment of system equation without simplifying wheel acceleration
The external load acting on the bridge is deduced by Eq. (6). In fact, the overall analysis of the vehicle shows that the external loads on the bridge can be expressed as Eq. (9): According to the mode decomposition method, the right end of Eq. (6) can be obtained and converted into the following equations: The system dynamic equilibrium equations of a simply-supported beam, wheel and spring (damper) mass system can be obtained by combining Eq. (5), Eq. (13), Eq. (14) and Eq. (15).
The th-order matrix expression can be expressed as follows:

Effect of velocity on dynamic response of bridge without simplifying wheel acceleration
For the dynamic response of the wheel mass coupled acceleration of a high-speed vehicle on the bridge, it is necessary to analyze the influence of the last two terms of Eq. (12) ( ) and ( ) on the dynamic response of the vehicle-bridge coupling system [1,5].
In this section, the influence of wheel acceleration on the bridge's dynamic response is analyzed without simplification under four speeds: 20 m/s, 25 m/s, 30 m/s and 35 m/s. The calculating model of Eq. (12) is defined as Model A, and the model neglecting ( ) and ( ) is defined as Model B. Fig. 2 shows the vertical displacement response of the two models at different speeds. The maximum displacement of the structure and the maximum displacement ratio of the two models are shown in Table 1.  When the speed is ≤ 20 m/s, Model A and Model B exhibit little difference in their mid-span displacement response. However, when the speed is ≥ 30 m/s, the ratio / is 1.3557, and model A should be adopted. When the speed is 20 m/s < < 30 m/s, the appropriate model can be selected according to the calculation accuracy requirements.

Conclusions
In this paper, the effect of the geometric nonlinearity of bridges on the dynamic response of the vehicle-bridge coupling system is taken into account, and the following conclusions are obtained: (1) The system equation has a non-linear matrix term when the geometric nonlinearity of the bridge is considered. The non-linear matrix is related only to the nature of the bridge structures, and the parameters of the vehicles have no influence on the matrix.
(2) The value of the non-linear matrix will increase as the inertia moment decreases when the other conditions remain unchanged.
(3) The influence of wheel acceleration on the bridge dynamic response is analyzed without simplification under four speeds. It is acceptable to neglect the influence of wheel acceleration at ≤ 20 m/s, while it is necessary to consider the significant influence of the wheel acceleration at ≥ 30 m/s.