2.1.2. The Vehicle Model

In this section, a vehicle model is established, which involves mass, spring, and damper [23]. The influence of road roughness on vehicle-bridge interaction vibration are considered, as shown in Figure 1. These assumptions are used in the VBI dynamic model: (i) the vehicle is traveling on the bridge with a constant speed v; (ii) the wheel is always in contact with the beam by point contact; (iii) the displacement of the wheel and the beam at the contact point is consistent.

**Figure 1.** The VBI system.

According to the Newton's second law, the vibration equation of the vehicle can be written as:

$$m\_v \ddot{y}\_v = F\_{vb} - F\_G \tag{3}$$

where *mv* and *yv* are the vehicle weight and vehicle displacement, respectively; *FG* and *Fvb* are the vehicle gravity and the interaction force, respectively. *Fvb* can be calculated by:

$$F\_{\upsilon b} = -k\_{\upsilon}(y\_{\upsilon} - y\_{\text{bc}} - r) - c\_{\upsilon}(\dot{y}\_{\upsilon} - \dot{y}\_{\text{bc}} - \dot{r})\tag{4}$$

where *kv*, *cv* and *ybc* are spring stiffness, the damping, the bridge deflection, respectively. *ybc* can be calculated by:

$$y\_{bc} = \mathbf{N}\_b \cdot y\_b \tag{5}$$

where *y<sup>b</sup>* denotes the global displacement vector of the bridge, *N<sup>b</sup>* denotes the bridge shape function.

Combining the Equations (3)–(5), we can obtain:

$$m\_{\upsilon}\ddot{y}\_{\upsilon} - \mathcal{C}\_{\upsilon b}\cdot\dot{y}\_{b} + c\_{\upsilon}\dot{y}\_{\upsilon} - \mathcal{K}\_{\upsilon b}\cdot y\_{b} + k\_{\upsilon}y\_{\upsilon} = F\_{\upsilon r} - F\_{\mathsf{G}}\tag{6}$$

where *Cvb* = *cv*·*Nb*, *Kvb* = *cv*· . *N<sup>b</sup>* + *kv*·*Nb*, *Fvr* = *cv* . *r* + *kvr* are the vehicle additional damping, stiffness, and load terms, respectively.

2.1.3. The Bridge Model

Under vehicle load, bridge dynamic equation can be expressed as:

$$\mathcal{M}\_b \ddot{y}\_b + \mathcal{C}\_b \dot{y}\_b + \mathcal{K}\_b y\_b = -F\_{bv} \tag{7}$$

where *M<sup>b</sup>* is bridge mass matrices, *C<sup>b</sup>* is bridge damping matrices, and *K<sup>b</sup>* is bridge stiffness matrices; *Fbv* is the equivalent nodal force of *Fbv*. It has the following relationship:

$$F\_{b\upsilon} = \mathbf{N}\_b^T \cdot F\_{l\upsilon} = \mathbf{N}\_b^T \cdot F\_{\upsilon b} \tag{8}$$

Substituting Equations (8) and (4) into Equation (7), we can obtain:

$$\mathbf{M}\_{b}\ddot{\mathbf{y}}\_{b} + (\mathbf{C}\_{b} + \mathbf{C}\_{bb})\dot{\mathbf{y}}\_{b} - \mathbf{C}\_{bv}\cdot\dot{\mathbf{y}}\_{v} + (\mathbf{K}\_{b} + \mathbf{K}\_{bb} + \mathbf{K}\_{bc})\mathbf{y}\_{b} - \mathbf{K}\_{bv}\cdot\mathbf{y}\_{v} = -\mathbf{K}\_{b}^{T}\cdot\mathbf{F}\_{vr} \tag{9}$$

where *Cbb* = *N<sup>T</sup> <sup>b</sup>* ·*cv*·*Nb*, *<sup>C</sup>bv* <sup>=</sup> *<sup>N</sup><sup>T</sup> <sup>b</sup>* ·*cv*, *<sup>K</sup>bb* <sup>=</sup> *<sup>N</sup><sup>T</sup> <sup>b</sup>* ·*kv*·*Nb*, *<sup>K</sup>bc* <sup>=</sup> *<sup>N</sup><sup>T</sup> <sup>b</sup>* ·*cv*· . *N<sup>b</sup>* and *Kbv* = *N<sup>T</sup> <sup>b</sup>* ·*kv* are the bridge additional damping and stiffness, respectively.

2.1.4. The VBI Dynamic Model

The VBI dynamic equation can be obtained by combining Equations (6) and (9) in the matrix form:

$$
\begin{bmatrix} \mathbf{M}\_{b} & \mathbf{0} \\ \mathbf{0} & m\_{v} \end{bmatrix} \begin{bmatrix} \ddot{\mathbf{y}}\_{b} \\ \ddot{\mathbf{y}}\_{v} \end{bmatrix} + \begin{bmatrix} \mathbf{C}\_{b} + \mathbf{C}\_{bb} & -\mathbf{C}\_{bv} \\ -\mathbf{C}\_{vb} & c\_{v} \end{bmatrix} \begin{bmatrix} \dot{\mathbf{y}}\_{b} \\ \dot{\mathbf{y}}\_{v} \end{bmatrix} + \begin{bmatrix} \mathbf{K}\_{b} + \mathbf{K}\_{bb} + \mathbf{K}\_{bc} & -\mathbf{K}\_{bv} \\ -\mathbf{K}\_{vb} & k\_{v} \end{bmatrix} \begin{bmatrix} \mathbf{y}\_{b} \\ \mathbf{y}\_{v} \end{bmatrix} = \begin{bmatrix} -\mathbf{N}\_{b}^{T} \cdot \mathbf{F}\_{\mathrm{vr}} \\ \mathbf{F}\_{\mathrm{vr}} - \mathbf{F}\_{\mathrm{G}} \end{bmatrix} \tag{10}
$$

To improve the computational efficiency, the model synthesis method is used to reduce the computational degrees of freedom of the bridge, and the vibration equations of the VBI dynamic model are rewritten as follows:

$$
\begin{bmatrix} I & 0 \\ 0 & m\_{\upsilon} \end{bmatrix} \begin{bmatrix} \ddot{q}\_{b} \\ \ddot{y}\_{\upsilon} \end{bmatrix} + \begin{bmatrix} \mathbf{C}\_{b} + \mathbf{\Phi}\_{b}^{T} \mathbf{C}\_{bb} \Phi\_{b} & -\mathbf{\Phi}\_{b}^{T} \mathbf{C}\_{b\upsilon} \\ -\mathbf{C}\_{\upsilon b} \Phi\_{b} & c\_{\upsilon} \end{bmatrix} \begin{bmatrix} \dot{q}\_{b} \\ \dot{y}\_{\upsilon} \end{bmatrix} + \begin{bmatrix} \mathbf{K}\_{b} + \mathbf{\Phi}\_{b}^{T} (\mathbf{K}\_{bb} + \mathbf{K}\_{bc}) \Phi\_{b} & -\mathbf{\Phi}\_{b}^{T} \mathbf{K}\_{bc} \\ -\mathbf{K}\_{b} \Phi\_{b} & k\_{\upsilon} \end{bmatrix} \begin{bmatrix} q\_{b} \\ y\_{\upsilon} \end{bmatrix} = F\_{R} + F\_{G} \tag{11}
$$

where **Φ**<sup>b</sup> is the modal shape matrix of the bridge.

$$F\_R = \begin{bmatrix} -\Phi\_b^T \mathbf{N}\_b^T \cdot F\_{vr} \\ F\_{vr} \end{bmatrix}, \; F\_G = \begin{bmatrix} 0 \\ F\_G \end{bmatrix}$$

$$\begin{aligned} \text{Let } \mathcal{M} = \begin{bmatrix} \mathcal{M}\_b & 0 \\ 0 & m\_v \end{bmatrix}, \; \mathcal{C} = \begin{bmatrix} \mathcal{C}\_b + \mathcal{C}\_{bb} & -\mathcal{C}\_{lv} \\ -\mathcal{C}\_{vb} & \mathcal{C}\_v \end{bmatrix}, \mathcal{K} = \begin{bmatrix} \mathcal{K}\_b + \mathcal{K}\_{bb} + \mathcal{K}\_{bv} & -\mathcal{K}\_{bv} \\ -\mathcal{K}\_{vb} & k\_v \end{bmatrix}, \; F\_{eq} = \begin{bmatrix} \mathcal{K}\_b + \mathcal{K}\_{vb} & -\mathcal{K}\_{vb} \\ -\mathcal{K}\_{vb} & k\_v \end{bmatrix}, \; \mathcal{C}\_{eq} = \begin{bmatrix} \mathcal{K}\_b + \mathcal{K}\_{vb} & -\mathcal{K}\_{vb} \\ -\mathcal{K}\_{vb} & k\_v \end{bmatrix}, \; \mathcal{C}\_{eq} = \begin{bmatrix} \mathcal{K}\_b + \mathcal{K}\_{vb} & -\mathcal{K}\_{vb} \\ -\mathcal{K}\_{vb} & \mathcal{K}\_{vb} \end{bmatrix}. \end{aligned}$$

According to the Newmark-β method, one can obtain:

$$\mathbf{K}\_{eq} y\_i = \mathbf{F}\_{eq} \tag{12}$$

where

$$F\_{eq} = F + \mathcal{M}[\frac{1}{\beta \Delta t^2} y\_i + \frac{1}{\beta \Delta t} \dot{y}\_i + (\frac{1}{2\beta} - 1)\ddot{y}\_i] + \mathcal{C}[\frac{\gamma}{\beta \Delta t} y\_i + \frac{\gamma}{\beta} - 1)\dot{y}\_i + (\frac{\gamma}{\beta} - 2)\ddot{y}\_i]$$

$$\mathcal{K}\_{eq} = \mathcal{K} + \frac{1}{\beta \Delta t^2} \mathcal{M} + \frac{\gamma}{\beta \Delta t} \mathcal{C}$$

According to the Newmark-β method, when the initial displacement, velocity, and acceleration at the initial time are given, the response of the system at any time can be determined based on Equation (12), and the complete time series data for dynamic response can be obtained.
