# Study on Passenger Comfort Based on Human-Bus-Road Coupled Vibration
## 2 Human-Bus-Road Coupled Vibration System
### Differential Equations of Human-Bus-Road Coupled Vibration  
The human-bus-road coupled system can be divided into three subsystems, i.e., the human, bus, and road. The interaction between the wheel and road surface and the interaction between the bus body and the human body are achieved through the contact points. That is, the displacement compatibility condition and the force equilibrium condition are used in the process of solving the differential equation of vibration [18]. It is noteworthy that only the vertical displacement and force are transmitted between the bus wheel and the road surface, which is the same for the interaction between the human body and the bus body. (1) Displacement compatibility condition. The vertical displacement of the human-bus-road model is shown in Figure 3. Assuming that the wheel always keeps close contact to the road surface and the human body is always attached to the bus body, the relative vertical displacements are as follows:  
\[\Delta_{bri}(t)=z_{wi}(t)-z_{ri}(t)-r_{i}(x) \tag{7}\]  
\[\Delta_{bhij}(t)=z_{b}(t)-z_{hj}(t) \tag{8}\]  
where \(Z_{wi}(t)\) is the vertical displacement of the ith-wheel; \(Z_{ri}(t)\) is the vertical displacement of the road at the point where the ith-wheel locates; \(Z_{b}(t)\) is the vertical displacement of the bus body; \(Z_{hj}(t)\) is the displacement of the jth-human body; \(r_{i}(x)\) is the road surface roughness at the contact point; \(\Delta_{bri}(t)\) is the relative vertical displacement between the road surface and the ith-wheel; \(\Delta_{bhij}(t)\) is the relative vertical displacement between the bus body and the jth-human body. (2) Force equilibrium condition: The interaction force follows [PERSON]'s third law, that is, the contact forces between wheel and road, between the bus body and the human body, are equal in magnitude and opposite in the direction:  
\[F_{bri}(t)=-F_{rbi}(t)=c_{wi}(\dot{\Delta}_{bri}(t))+k_{wi}(\Delta_{bri}(t)) \tag{9}\]  
\[F_{bhij}(t)=-F_{hbj}(t)=c_{hj}(\dot{\Delta}_{bhij}(t))+k_{hj}(\Delta_{bhij}(t)) \tag{10}\]  
where \(F_{bri}\) and \(F_{rbi}\) are the contact force between the road surface and the \(i\)th-wheel; \(F_{bhij}\) and \(F_{hbj}\) are the contact force between the bus body and the \(j\)th-human body; \(C_{Wi}\) is the \(i\)th-wheel damping; \(k_{Wi}\) is  
Figure 3: The vertical displacement of the human–bus–road model. the \(i\)th-wheel stiffness; C\({}_{hj}\) is the \(j\)th-human body damping; \(k_{hj}\) is the \(j\)th-human body stiffness; \(\Delta_{bri}(t)\) and \(\Delta_{bhj}(t)\) are the first-order derivative of \(\Delta_{bri}(t)\) and \(\Delta_{bhj}(t)\) to time. These above motion equations of the human, bus, and road are coupled through the displacement compatibility condition and force equilibrium condition. The combined equation is as follows:  
\[ \begin{array}{l}\left[\begin{array}{ccc}M_{r}&0&0\\ 0&M_{b}&0\\ 0&0&M_{h}\end{array}\right]\left(\begin{array}{c}\tilde{Z}_{r}(t)\\ \tilde{Z}_{b}(t)\\ \tilde{Z}_{h}(t)\end{array}\right)+\left[\begin{array}{ccc}C_{r}+C_{rr}&C_{ rb}&0\\ C_{br}&C_{b}&C_{bh}\\ 0&C_{hb}&C_{h}\end{array}\right]\left(\begin{array}{c}\dot{Z}_{r}(t)\\ \dot{Z}_{b}(t)\\ \dot{Z}_{h}(t)\end{array}\right)\\ +\left[\begin{array}{ccc}K_{r}+K_{rr}&K_{rb}&0\\ K_{br}&K_{b}&K_{bh}\end{array}\right]\left(\begin{array}{c}Z_{r}(t)\\ Z_{b}(t)\\ Z_{h}(t)\end{array}\right)=\left\{\begin{array}{c}F_{rb}(t)\\ F_{b}{}^{G}+F_{br}(t)+F_{bh}(t)\\ F_{h}{}^{G}+F_{br}(t)\end{array}\right\}\end{array}  \tag{11}\]  
In comparison with Equations (1), (5), and (6), \(c_{rr}\), \(c_{rb}\), \(c_{br}\), \(K_{rr}\), \(K_{rb}\), \(K_{br}\), \(F_{rb}\) and \(F_{br}\) are due to the wheel-road contact forces; \(c_{thr}\), \(c_{hb}\), \(K_{bh}\), \(K_{hb}\), \(F_{bh}\) and \(F_{hb}\) are due to the interaction forces between the bus body and the human body. When the bus is moving on the road, the positions of contact points, as well as the values of the forces, change with the bus location.