# Influence of Non-Uniform Rail Loads on the Rotation of Railway Sleepers
## 2 Movement Equations and Their Solutions  
### Solution for Time-Invariable Forces  
With the adopted assumptions, two systems of movement equations are formed:  
* For the symmetric load: \[ \begin{split}& EI\frac{\partial^{4}y_{tra}}{\partial x^{2}}+N_{t}\frac{ \partial^{2}y_{tra}}{\partial x^{2}}+m_{r}\frac{\partial^{2}y_{tra}}{\partial t^ {2}}+c_{r}(\frac{\partial y_{tra}}{\partial t}-\frac{\partial y_{tra}}{ \partial t})+k_{r}(y_{ra}-y_{pa})=q_{av}(x,t)\\ & m_{p}\frac{\partial^{2}y_{pa}}{\partial t^{2}}+c_{p}\frac{ \partial y_{tra}}{\partial t}+k_{p}y_{pa}=c_{r}(\frac{\partial y_{tra}}{ \partial t}-\frac{\partial y_{tra}}{\partial t})+k_{r}(y_{ra}-y_{pa})\end{split} \] (2)
* For the antisymmetric load: \[ \begin{split}& EI\frac{\partial^{4}y_{rA}}{\partial x^{2}}+N_{t} \frac{\partial^{2}y_{rA}}{\partial x^{2}}+m_{r}\frac{\partial^{2}y_{rA}}{ \partial t^{2}}+c_{r}(\frac{\partial y_{rA}}{\partial t}-b\frac{\partial \phi}{\partial t})+k_{r}(y_{rA}-b\phi)=\Delta q(x,t)\\ & I_{p}\frac{\partial^{2}\phi}{\partial t^{2}}+c_{p}b^{2}\frac{ \partial\phi}{\partial t}+k_{p}b^{2}\phi=c_{r}b(\frac{\partial y_{rA}}{ \partial t}-b\frac{\partial\phi}{\partial t})+k_{r}b(y_{rA}-b\phi)\end{split} \] (3) where y_rA_--the rail displacement with an antisymmetric load (opposite signs for both rails) [m]; I_P--the moment of inertia for individual sleepers (I_P = I_s/I_s, where I_s--the moment of inertia of a sleeper [kgm\({}^{2}\)], and I_s--the sleeper axis spacing along the track [m]); --Other symbols: as in Figures 1 and 2. The solutions for both load cases will be the sums of the solutions for the forces that are constant (non-oscillating--load symbol qn) and time-variable (load symbol qo). In a mobile coordinate system (n = y, \(\xi\) = x - vt), where v--the constant train speed, the system of Equation (2) takes the following form for a system of non-time-variable forces:  
Figure 2: Track model, model parameters, and reference system [11]. \[ \begin{split}& EI\frac{d^{4}y_{ra}}{d^{6}}+(N_{t}+m_{r}v^{2})\frac{d^{2 }y_{ra}}{d^{2}}-c_{r}v\frac{dy_{ra}}{d^{5}}+k_{r}y_{ra}+c_{r}v\frac{dy_{ra}}{d^{5} }-k_{r}y_{pa}=q^{n}_{av}(\xi)\\ & m_{p}v^{2}\frac{d^{2}y_{pa}}{d^{2}}-c_{p}v\frac{dy_{ra}}{d^{5}}+k _{p}y_{pa}+c_{r}v\frac{dy_{ra}}{d^{5}}-c_{r}v\frac{dy_{ra}}{d^{5}}-k_{r}y_{ra}+k_ {r}y_{pa}=0\end{split}  \tag{4}\]  
When marking the vertical displacement of a sleeper under the rail with an antisymmetric load with \(y_{pA}=\varphi\text{b}\), the system of Equation (3)--in a mobile coordinate system with a non-oscillating load--assumes the following form:  
\[ \begin{split}& EI\frac{d^{4}y_{pA}}{d^{6}}+(N_{t}+m_{r}v^{2}) \frac{d^{2}y_{pA}}{d^{2}}-c_{r}v\frac{dy_{ra}}{d^{5}}+k_{r}y_{r\Delta}+c_{r}v \frac{dy_{pA}}{d^{5}}-k_{r}y_{pA}=\Delta q^{n}(\xi)\\ & I_{p}\frac{v^{2}}{b^{2}}\frac{d^{2}y_{pa}}{d^{2}}-c_{p}v\frac{dy _{pA}}{d^{5}}+k_{p}y_{pA}+c_{r}v\frac{dy_{pA}}{d^{5}}-c_{r}v\frac{dy_{pA}}{d^{ 5}}-k_{r}y_{r\Delta}+k_{r}y_{pA}=0\end{split}  \tag{5}\]  
It is assumed that the load in Equation (4) is distributed throughout the length of the rail. Another assumption is that the function of the load distribution for every wheel of a vehicle has the form of a Gaussian function, i.e.,  
\[\rho(\xi)=\frac{Q}{\sigma\cdot\sqrt{2\pi}}\cdot e^{-\frac{(\xi-k)^{2}}{2\sigma ^{2}}};\xi\in[0,\lambda];k=1,\ldots,N \tag{6}\]  
where  
Q--the static wheel load [N];  
\(\sigma\)--the distribution parameter (its value should allow for the distribution of the load of a single wheel in the rail axis to correspond to the real distribution) [m];  
\(\lambda\)--the position of the wheel in relation to coordinate \(\xi=0\) [m];  
\(\lambda\)--the assumed span of distribution (outside the [0, \(\lambda\)] interval, function \(\rho\) practically equals 0);  
N--the number of analysed wheels of a vehicle. Load functions are now transformed into Fourier series.