# An Indirect Identification Method for Train Basic Resistance Parameters
## 2 Train Dynamics Modeling  
The objective function of BRPI-SF in this paper is to minimize the deviation between the fitting curve and the actual operating curve. The fitting curve is calculated using the traction calculation method based on train operation data and the train dynamics model. The state of the train is affected by train weight, traction or braking force, basic resistance and additional resistance during operation. Considering the uneven additional resistance caused by different positions of each vehicle, this paper establishes the multi-mass train model, and its schematic diagram is shown in Figure 1.  
According to [PERSON]'s second law, the dynamic functions are described as:  
\[\left\{ \begin{array}{l}\frac{ds}{dt}=v\\ \frac{dp}{dt}=\frac{F-W_{0}(\mathbf{\theta},v)-G(s)}{(1+\gamma)\cdot M}\end{array} \right. \tag{3}\]  
where \(v\) is running speed (m/s) of train, \(t\) is running time (s), \(s\) is train position (m). \(M\) is train mass (t), \(\gamma\) is the rotating mass coefficient. \(F\) represents the train control force (kN), with positive and negative indicating traction and braking forces, respectively. \(W_{0}\)(\(\mathbf{\theta}\),\(v\)) is train basic resistance (kN). According to [PERSON] formula, \(W_{0}\)(\(\mathbf{\theta}\),\(v\)) can be expressed as a univariate quadratic equation with train running speed as a variable, as shown in Equation (4):  
\[W_{0}(\mathbf{\theta},v)=Mgw_{0}(\mathbf{\theta},v)/1000=Mg(a+bv+cv^{2})/1000 \tag{4}\]  
where \(w_{0}\)(\(\mathbf{\theta}\),\(v\)) represents unit basic resistance (N/kN). \(g\) is the gravity coefficient, taken as 9.81 (N/kg or m/s2). \(\mathbf{\theta}=[a\), \(b\), \(c]\), \(a\), \(b\), and \(c\) are basic resistance parameters. \(G(s)\) represents the additional resistance (kN), which is affected by the train operation position \(s\), and its calculation formula is written as follows:  
\[ \begin{array}{ll}G(s)&=Mg(w_{i}+w_{r})/1000\\ &=\sum\limits_{j=1}^{J}m_{iS}\Big{[}i(s_{j})+\frac{\alpha}{r(s_{j})}\Big{]}/1000 \end{array}  \tag{5}\]  
where \(w_{i}\) and \(w_{r}\) represent unit additional resistance (N/kN) of gradient and curve. \(m_{j}\) is the mass (t) of \(j\)-th car. \(s_{j}\) is the position of the \(j\)-th car's centroid. \(i(s_{j})\) means the gradient thousand fraction at position \(s_{j}\). \(r(s_{j})\) means the curve radius (m) at position \(s_{j}\). \(\alpha\) is usually 600. To facilitate analysis, train speed \(v\) and running time \(t\) are expressed as functions with train position \(s\) as independent variables, using Equation (3) to get:  
\[\left\{ \begin{array}{l}\frac{dv}{dt}(1+\gamma)Mv=F-W_{0}(\mathbf{\theta},v)-G(s) \\ \frac{dv}{dt}=\frac{1}{v}\end{array} \right. \tag{6}\]  
With the changes in speed and position, the combined force acting on the train will also change. Therefore, the train operation process is a complicated non-uniform motion, which cannot be solved directly by Equation (6). The traction calculation method divides the train operation process into some stages through discretization and assumes that combined force acting remains constant within each stage. Therefore, the train acceleration remains unchanged within a single stage and is considered constant acceleration operation. The train operation process is divided into \(I\) stages. In the \(i\)-th stage, the train start position is \(s_{i\text{-}1}\). End position is \(s_{i\text{-}1}\). Force is \(F_{i\text{-}1}\). Initial speed is \(v_{i\text{-}1}\), which is obtained  
Figure 1: Schematic diagram of the multi-mass train model. in (_i_-1)-th stage. We can obtain \(v_{i}\) by combining Equations (4)-(6), which is shown in Equation (7):  
\[v_{i}(\mathbf{\theta})=v_{i}(a,b,c)=\sqrt{v_{i-1}^{2}+(s_{i}-s_{i-1})\frac{2g}{1+ \gamma}\left[\frac{F_{i-1}}{Mg}-\frac{a+bv_{i-1}+cv_{i-1}^{2}+G(s_{i-1})}{1000} \right]} \tag{7}\]  
where \(v_{i}(\theta)\) is the calculation speed under parameter \(\theta\).