# Reducing Oscillations in Suspension of Mine Monorail Track
## 2 Materials and Methods  
To solve the issues which were set, the modeling of the process of movement of a suspended monorail was carried out using the Lagrange method, a methodology was developed and necessary parameters of the monorail suspension were determined. Based on this, a device was proposed that allows implementation of these parameters. The source of oscillations of suspension monorail components is the rolling stock moving along the monorail, which is fixed in the upper part of the mine excavation. On the one hand, dynamic influences--forces arising in the places of interaction of bogies with a monorail, are transferred to suspension of the rolling stock body. On the other hand, dynamic effects appear in attachment points of the monorail and are transmitted to the support of mine excavation. Such fluctuations do not contribute to the performance of useful work, and therefore increase energy costs, lead to additional wear of the monorail and reduce traffic safety. The decrease in the intensity of oscillations in attachment points of the monorail and the body can be achieved by reducing these dynamic effects, by changing the design of the suspension components and with the help of additional devices. A generalized model of overhead monorail can be represented as a set of interconnected systems (subsystems) consisting of many parts, which in certain tasks can be considered as independent objects of study. At vertical oscillations of the monorail and under the effect of harmonic disturbing force caused by the movement of the suspension, the system of the monorail suspension can be represented in the form of a dual-mass system shown in Figure 2.  
The notations of the calculation scheme: \(m_{1}\)--mass of suspension; \(m_{2}\)--mass of a part of the monorail track and rolling stock affecting the suspension; \(C_{1}\)--stiffness factor of the  
Figure 2: Calculation scheme of absorption of shock applied to the suspension. monorail suspension; \(C_{2}\)--stiffness factor of a part of the monorail track and rolling stock affecting the suspension; \(P_{v}\) sinu_t_--vertical disturbing force caused by the movement of the rolling stock carriages along the monorail; \(\omega\)--frequency of impact on rolling stock of forced force \(P_{v}\), \(t\)--the time of movement of rolling stock on monorail. The differential equations of forced vertical oscillations of the monorail suspension are as follows:  
\[\left\{ \begin{array}{l}m_{1}\tilde{z}_{1}+C_{1}z_{1}-C_{2}(z_{2}-z_{1})=0;\\ m_{2}\tilde{z}_{2}+C_{2}(z_{2}-z_{1})=P_{v}\sin\omega\;t,\end{array} \right. \tag{1}\]  
where \(z_{1}\), \(z_{2}\)--vertical shifts of gravity centers of masses \(m_{1}\) and \(m_{2}\), respectively. The solution to the set of equations is in the following form:  
\[\left\{ \begin{array}{l}z_{1}=A_{1}\sin\omega\;t;\\ z_{2}=A_{2}\sin\omega\;t,\end{array} \right. \tag{2}\]  
where \(A_{1}\), \(A_{2}\)--amplitudes of forced oscillations of the gravity centers of the respective masses \(m_{1}\) and \(m_{2}\). Therefore, set (1) can be formulated as follows:  
\[\left\{ \begin{array}{l}A_{1}(m_{1}\omega^{2}+C_{1}+C_{2})-A_{2}C_{2}=0;\\ -A_{1}C_{2}+A_{2}(C_{2}-m_{2}\omega^{2})=P_{v}.\end{array} \right. \tag{3}\]  
The solution to the set of equations is the following:  
\[ \begin{array}{l}A_{1}=\frac{P_{v}C_{2}}{m_{2}(m_{1}+m_{2})\left[\left(\frac{ C_{1}}{m_{1}+m_{2}}-\omega^{2}\right)\left(\frac{C_{2}}{m_{2}}-\omega^{2} \right)-\frac{\omega^{4}m_{2}}{m_{1}+m_{2}}\right]};\\ A_{2}=\frac{P_{v}C_{1}+C_{2}-m_{1}\omega^{2})}{m_{2}(m_{1}+m_{2})\left[\left( \frac{C_{1}}{m_{1}+m_{2}}-\omega^{2}\right)\left(\frac{C_{2}}{m_{2}}-\omega^{ 2}\right)-\frac{\omega^{4}m_{2}}{m_{1}+m_{2}}\right]}.\end{array} \right.\end{array} \tag{4}\]  
Taking into consideration that the amplitude of disturbing forces is proportional to squared frequency, we consider  
\[P_{v}=m_{e}r_{e}\omega^{2}, \tag{5}\]  
where \(m_{e}r_{e}\)--static moment of the reduced unbalanced mass \(m_{e}\) that is equivalent in action with the radius of reduction \(r_{e}\).