## 2 Multi-Body Dynamics Analysis Method of Disc Brake under Thermo-Solid Coupling  
To more accurately study the vibration characteristics of disc brakes, a multi-flexible body dynamics model of high-speed train disc brakes taking thermo-solid coupling into account is established. This model allows for the measurement of the temperature and stress distribution of the brake disc as well as the analysis of the impact of thermo-solid coupling on the brake disc's vibration characteristics; hence, the influence of braking parameters on the vibration response of the brake disc can be studied. Based on this research, this paper proposes a multi-body dynamics analysis method for disc brakes under the action of thermo-solid coupling. The establishment of the disc brake multi-physical field coupling model takes the initial pressure, initial speed, and initial temperature of the high-speed train as the input conditions, and using multi-body dynamics to analyze its contact changes, the contact pressure and friction force changes between the brake disc surface and the brake pad can be obtained. In addition, the dissipation rate of friction during braking is analyzed from the perspective of energy transfer. Based on the dissipation rate of friction, the frictional heat flux is gained. Additionally, the changing rules of temperature and thermal expansion of the brake disc surface and brake pad are obtained from the perspective of structural heat transfer, heat radiation, heat convection, and heat conduction. Furthermore, the change of the contact between the brake disc surface and the brake pad after the expansion deformation is analyzed, and the characteristics of vibration during the braking process are obtained. The coupling relationship between physical fields in this method is shown in Figure 2.  
Figure 2: Coupling relationship of disc brake multi-body dynamics analysis method under thermo-solid coupling. When the brake is working, the friction work causes the kinetic energy of the brake disc to be lost and converted into heat energy. The dissipation rate of friction is obtained through the calculation of multi-body dynamics. The dissipation rate of friction can be expressed as [22]:  
\[Q_{b}=\frac{p_{b}}{A} \tag{1}\]  
where \(P_{b}\) is the braking power of the friction force acting on the brake disc, and \(A\) is the contact area between the brake disc surface and the brake pad. Loading the dissipation rate of friction on the contact surface of the brake disc and the brake pad as a frictional heat source, the generated heat flux can be expressed as [23]:  
\[q=\mu p\omega\tau \tag{2}\]  
where \(q\) is the heat flux generated at the radius \(r\) of the brake disc, \(\mu\) is the friction coefficient, \(p\) is the load on the contact surface of the brake disc, and \(\omega\) is the rotation speed of the brake disc. The brake disc surface and brake pads generate significant heat due to friction. When the heat generated by the brake is the same as the output heat, the energy equation can be expressed as:  
\[Q=k\Delta\Delta T \tag{3}\]  
where \(Q\) is the heat transferred during the coupling process, \(k\) is the heat transfer coefficient, \(\Delta T\) is the average temperature difference during the heat transfer process, and A is the area that transfers heat. When the temperature of the brake changes, its material responds through a volume change, which can be expressed as the thermal strain of \(\varepsilon_{th}\), and its calculation formula can be expressed as:  
\[\varepsilon_{th}=\alpha(T-T_{\text{ref}}) \tag{4}\]  
where \(T\) is the working temperature of the brake; \(T_{\text{ref}}\) is the volume reference temperature of the brake, that is, the temperature when no thermal expansion occurs; and \(\alpha\) is the secant thermal expansion coefficient of the brake material, which can be expressed as:  
\[\alpha=\frac{e^{\frac{L}{L_{0}}}-1}{T-T_{\text{ref}}} \tag{5}\]  
where \(L\) is the length of the material after expansion, and \(L_{0}\) is the reference length of the material.