## 2 Dynamic Model of a High-Speed Train  
The HST is subject to a variety of forces during operation. Each carriage is subject to traction and braking forces \(u\) from the traction unit, basic operating resistance \(f_{a}\), additional resistance \(f_{b}\), and interaction forces \(f_{d}\) between the carriages. The basic running resistance \(f_{a}\) is the resistance that the HST must be subjected to during operation, including the friction between the HST internal components, rolling resistance, and air resistance. Its calculation equation satisfies the Davis equation [31]:  
\[f_{a}(v)=c_{o}+c_{v}v+c_{a}v^{2} \tag{1}\]  
where \(c_{o}\), \(c_{o}\), and \(c_{a}\) are Davis coefficients, and their values are generally determined by empirical constants. \(c_{o}\) is the rolling mechanical resistance coefficient independent of train speed, \(c_{v}\) is the rolling friction resistance coefficient, and \(c_{a}\) is the air resistance coefficient.  
The additional running resistance \(f_{b}\) represents the running resistance generated when the HST passes through ramps, curves, and tunnels. Its calculation equation is as follows:  
\[\begin{array}{c}f_{1}=mg\theta\\ f_{2}=mg(600/R)\\ f_{3}=mg(0.00013L)\end{array} \tag{2}\]  
where \(g=9.8\) m/s\({}^{2}\) is the acceleration of gravity, \(m\) is the mass of a single carriage, \(f_{1}\) is the ramp additional resistance, \(\theta\) is the angle between the track and the horizontal road surface, \(f_{2}\) is the curve additional resistance, \(R\) is the curve radius, \(f_{3}\) is the tunnel additional resistance, and \(L\) is the tunnel length. The additional resistance during HST operation can be expressed as \(f_{b}=f_{1}+f_{2}+f_{3}\).  
The HST is a multi-body system consisting of multiple carriages connected by hooks. \(f_{d}\) is the interaction force between carriages during operation. Each carriage is subjected to the coupling force from other carriages, which can be expressed by an \"elastic-damping\" structure [32] as:  
\[f_{di(i+1)}=k(x_{i}-x_{i+1})+d(v_{i}-v_{i+1}) \tag{3}\]  
where \(f_{di(i+1)}\) is the interaction force between carriage \(i\) and carriage \(i+1\), \(k\) and \(d\) are the elastic coupling coefficient and damping coupling coefficient of the hook, respectively, and \(x_{i}\) and \(v_{i}\) are the displacement and speed of the carriage \(i\), respectively.  
In summary, the force analysis of each carriage according to [PERSON]'s laws of motion can describe the HST multi-mass model as:  
\[\left\{\begin{array}{l}m_{1}\dot{v}_{1}=u_{1}-f_{a1}-f_{d12}-f_{b1}\\ m_{i}\dot{v}_{i}=u_{i}-f_{ai}-f_{di(i+1)}+f_{d(i-1)i}-f_{bi}\\ \vdots\ \ \ (i=2,3,\cdots n-1)\\ m_{n}\dot{v}_{n}=u_{n}-f_{an}+f_{d(n-1)n}-f_{bn}\\ \dot{x}_{i}=v_{i},\ (i=1,2,\ldots,n)\end{array}\right. \tag{4}\]  
where \(n\) is the number of carriages and \(u_{i}\) stands for traction or braking force output.  
**Remark 1**.: _In a general simulation [33], the coefficients \(c_{o}\), \(c_{v}\), and \(c_{a}\) of the Davis equation and the mass \(m\) of a single carriage are generally fixed values determined by empirical constants. However, the value of these part coefficients will not be constant due to the change of a HST's running line environment, the difference of track adhesion, the consumption of the train itself, and the movement of carriage personnel. Therefore, the uncertain time-varying coefficients \(c_{o}\), \(c_{v}\), \(c_{a}\), and \(m\) are adopted in this paper, which is more consistent with the actual situation._