# Assessment of Uneven Wear of Freight Wagon Brake Pads
## 2 Materials and Methods  
This research was conducted using the 2 TP-11 brake pad (Tribo, Bila Tserkva, Ukraine) as an example, since it is the most common type found in freight wagons with a gauge of 1520 mm. The 2 TP-11 brake pad is an important element of the wagon brake system due to its high resistance to wear and stability at elevated temperatures. However, the uneven distribution of the pressure force caused by the structural features of the brake system leads to different wear rates of the pad, which reduces its service life and increases maintenance costs. The distribution of the pressure force between the brake pads in the wagon is a complex process, affected by structural, dynamic, operational, technological and external factors. Their combination leads to uneven force distribution, increased wear of pads, lower braking efficiency, and higher operating costs [16, 17, 18]. In order to eliminate the causes of unevenness, an integrated approach is needed; it will improve structural solutions, enhance the material properties, optimize operating conditions and introduce innovative regulation technologies. Taking into account the above, a methodology based on a modern mathematical approach has been developed. It is based on the analysis of functional dependencies and integral calculations for accurate modeling of processes in the brake system of the wagon. This technique is presented as a tool to minimize uneven wear by optimizing the distribution of the pressure force between the pads. The optimal distribution of the pressure force between the brake pads is one of the most important tasks in ensuring the efficiency and durability of the wagon brake systems. The distribution of the pressure force p on the i-th pad is defined as follows:  
\[p_{i}=k_{i}\cdot K, \tag{1}\]  
where \(k_{i}\) is the weight coefficient of the pressure force distribution for i-th pad and \(K\) is the total pressure force on the pad, kN. This equation takes into account that each pad receives a portion of the pressure force proportional to the weight coefficient \(k_{i}\). To ensure physical correctness, the sum of all coefficients should be equal to one:  
\[\sum\limits_{i=1}^{n}k_{i}=1. \tag{2}\]  
where \(n\) is the number of brake pads involved in optimizing the distribution of the pressing force. The wear rate \(J_{i}\) is determined by the nonlinear dependence on the pressure force \(p_{i}\):  
\[J_{i}=\gamma\cdot\varphi_{k}\cdot p_{i}^{n}, \tag{3}\]  
where \(\gamma\) is a coefficient including the physical and mechanical properties of the pad material; \(\varphi_{k}\) is the friction coefficient; and \(\alpha\) is the exponent characterizing the wear sensitivity to the pressure force of the pad, \(\alpha>1\). This formula can be used to model the actual dependence of wear on the pressure force and properties of the material of the pad. The total wear of the brake pad is determined by integrating the wear rate:  
\[H=\sum\limits_{i=1}^{n}H_{i}=\sum\limits_{i=1}^{n}\int\limits_{0}^{t}J_{i}dt, \tag{4}\]  
where \(t\) is the operating time. Substituting (3) into (4), the following can be obtained:  
\[H=\sum\limits_{i=1}^{n}\gamma\varphi_{k}\int\limits_{0}^{t}(k_{i}K)^{\alpha}dt. \tag{5}\]  
With constant pressure, the integral is simplified to the following:  
\[H=\gamma\varphi_{k}t\sum\limits_{i=1}^{n}(k_{i}K)^{\alpha}. \tag{6}\]  
This formula is the basis for estimating the total pad wear according to the known parameters of the pressure force and weight factors. _Optimization of pressure distribution._  
Minimization of uneven wear is based on the deviation function, as follows:  
\[f=\sum\limits_{i=1}^{n}\left|H_{i}-\frac{H}{n}\right|, \tag{7}\]  
where \(H_{i}\) is the wear of i-th pad and \(H\)/\(n\) is the average wear of pads. This function can be used to evaluate the uneven wear of the pads and is the basis for optimization. _Lagrange function._  
The Lagrange method was proposed in this research to find the optimal distribution of the pressure force between the brake pads of a freight wagon.