# Influence of Non-Uniform Rail Loads on the Rotation of Railway Sleepers
## 2 Movement Equations and Their Solutions
### Solution for Time-Invariable Forces
The series factors, in a general case, i.e., with constant \(a_{0}\), cosine components \(a_{i}\), and sine components \(b_{i}\), have the following form:  
\[\begin{split}& a_{0}=\frac{NQ}{\lambda};\\ & a_{i}=\frac{2Q}{\lambda}\cdot e^{-\frac{2\pi^{2}\sigma^{2}}{ \lambda^{2}}}\cdot(\sum\limits_{k=1}^{N}\cos\frac{2\cdot\pi\cdot l_{k}\cdot i} {\lambda});\\ & b_{i}=\frac{2Q}{\lambda}\cdot e^{-\frac{2\pi^{2}\sigma^{2}}{ \lambda^{2}}}\cdot(\sum\limits_{k=1}^{N}\sin\frac{Substituting the relationships (9) in the equation system (4), the algebraic equation system for unknown factors \(\text{A}_{\text{i}}\), \(\text{B}_{\text{i}}\), \(\text{C}_{\text{i}}\), and \(\text{D}_{\text{i}}\) is obtained:  
\[ \begin{split}& A_{i}\Big{[}EI\Omega_{i}^{4}-\big{(}N_{t}+m_{r}v^{2} \big{)}\Omega_{i}^{2}+k_{r}\Big{]}+B_{i}[-c_{r}v\Omega_{i}]+C_{i}[-k_{r}]+D_{i}[ c_{r}v\Omega_{i}]=a_{i}\\ & A_{i}[c_{r}v\Omega_{i}]+B_{i}\Big{[}EI\Omega_{i}^{4}-(N_{t}+m_{ r}v^{2})\Omega_{i}^{2}+k_{r}\Big{]}+C_{i}[-c_{r}v\Omega_{i}]+D_{i}[-k_{r}]=b_{i}\\ & A_{i}[-k_{r}]+B_{i}[c_{r}v\Omega_{i}]+C_{i}\Big{[}k_{r}+k_{p}-m_ {p}v^{2}\Omega_{i}^{2}\Big{]}+D_{i}\big{[}-(c_{r}+c_{p})v\Omega_{i}\big{]}=0\\ & A_{i}[-c_{r}v\Omega_{i}]+B_{i}[-k_{r}]+C_{i}\big{[}(c_{r}+c_{p} )v\Omega_{i}\big{]}+D_{i}\Big{[}k_{r}+k_{p}-m_{p}v^{2}\Omega_{i}^{2}\Big{]}=0 \end{split}  \tag{10}\]  
whereas the constants in Equation (9) are determined by the following formulas:  
\[ \begin{split}& y_{ra0}=a_{0}\Big{(}\frac{1}{k_{r}}+\frac{1}{k_{p}} \Big{)}\\ & y_{pa0}=\frac{a_{0}}{k_{p}}\end{split}  \tag{11}\]  
The same method is used in the antisymmetric form of rail and sleeper vibrations with a time-invariable load. If we express the load in Equation (5) as a Fourier series, in the same interval \([0,\lambda]\) as before (factors: \(\circ\)--constant, \(\text{c}_{\text{i}}\)--the amplitude of the cosine components, and \(\text{d}_{\text{i}}\)--the amplitude of the sine components):  
\[ \begin{split}&\Delta^{n}q(\xi)=\frac{\alpha_{0}}{2}+\sum\limits_{i= 1}^{\infty}\,(c_{i}\cdot\cos\Omega_{i}\xi+d_{i}\cdot\sin\Omega_{i}\xi);\\ &\xi\in[0,\lambda];\Omega_{i}=\frac{2\pi\cdot i}{\lambda}\end{split}  \tag{12}\]  
and also expressing the searched-for functions of rail vibrations in the form of a Fourier series in the same interval \([0,\lambda]\), i.e.,  
\[ \begin{split}& y_{r\Delta}(\xi)=\frac{y_{r\Delta 0}}{2}+\sum\limits_{i=1}^{\infty}\,(E_{i}\cdot\cos\Omega_{i}\xi+F_{i}\cdot \sin\Omega_{i}\xi);\\ & y_{p\Delta}(\xi)=\frac{y_{p\Delta 0}}{2}+\sum\limits_{i=1}^{ \infty}\,(G_{i}\cdot\cos\Omega_{i}\xi+H_{i}\sin\Omega_{i}\xi);\\ &\xi\in[0,\lambda];\Omega_{i}=\frac{2\pi\cdot i}{\lambda}\end{split}  \tag{13}\]  
the formulas necessary to solve the problem are obtained. Differentiating Equation (13) and substituting the obtained relationships with Equation (12) in Equation (5), the algebraic equations system with unknowns \(\text{E}_{\text{i}}\), \(\text{F}_{\text{i}}\), \(\text{G}_{\text{i}}\), and \(\text{H}_{\text{i}}\) is obtained:  
\[ \begin{split}& E_{i}\Big{[}EI\Omega_{i}^{4}-(N_{t}+m_{r}v^{2}) \Omega_{i}^{2}+k_{r}\Big{]}+F_{i}[-c_{r}v\Omega_{i}]+G_{i}[-k_{r}]+H_{i}[c_{r}v \Omega_{i}]=c_{i}\\ & E_{i}[c_{r}v\Omega_{i}]+F_{i}\Big{[}EI\Omega_{i}^{4}-(N_{t}+m_{ r}v^{2})\Omega_{i}^{2}+k_{r}\Big{]}+G_{i}[-c_{r}v\Omega_{i}]+H_{i}[-k_{r}]=d_{i}\\ & E_{i}[-k_{r}]+F_{i}[c_{r}v\Omega_{i}]+G_{i}\Big{[}(k_{r}+k_{p}) -I_{p}\frac{v^{2}}{b^{2}}\Omega_{i}^{2}\Big{]}+H_{i}\big{[}-(c_{r}+c_{p})v \Omega_{i}\big{]}=0\\ & E_{i}[-c_{r}v\Omega_{i}]+F_{i}[-k_{r}]+G_{i}\big{[}(c_{r}+c_{p} )v\Omega_{i}\big{]}+H_{i}\Big{[}(k_{r}+k_{p})-I_{p}\frac{v^{2}}{b^{2}}\Omega_{ i}^{2}\Big{]}=0\end{split}  \tag{14}\]  
and the constants in Equation (13) are determined by the following formulas:  
\[ \begin{split}& y_{r\Delta 0}=c_{0}\Big{(}\frac{1}{k_{r}}+\frac{1}{k_{p}} \Big{)}\\ & y_{p\Delta 0}=\frac{c_{0}}{k_{p}}\end{split}  \tag{15}\]