# Reducing Oscillations in Suspension of Mine Monorail Track
## 4 Discussion
The round-link chain which supports the monorail track is connected to the eyelets of the suspension by means of a split pin. The vertical oscillations of the monorail which occur during the operation of the mine suspended monorail track are taken up by the spring washers and damping inserts which ensure the energy absorbing of shocks and periodic oscillations. Figure 6: Suspension for monorail track: (**a**) general view; (**b**) schematic diagram, where: 1—case; 2—elastic damping device; 3—fastening bolt; 4—damping insert; 5—taper washer; 6—nut. Consider the movement of rolling stock on a monorail track with joints and equipped with the proposed device. We take into account the viscous resistance of the device and introduce the following notation: \(b_{1}\)--viscous drag coefficient of a monorail suspension; \(b_{2}\)--coefficient of viscous resistance of the rolling stock suspension. In this case, the emerging perturbations affecting the considered mass \(m_{2}\) can be presented as a single irregularity  
\[\eta(t)=\eta_{5}/2(1-\cos(2\pi x/l_{5}), \tag{16}\]  
where \(\eta_{s}\), \(l_{s}\) are, respectively, the height and length of the unevenness formed when the undercarriage moves along the junction of the monorail, \(x=V_{n}\)\(t\) is the distance traveled function; \(V_{n}\) is the rolling stock speed. Rolling stock speed \(V_{n}\) was adopted permanently. Under the influence of these perturbations on mass \(m_{2}\), due to deformation of suspension, oscillatory processes and dynamic forces arise, which are transmitted to suspension of monorail and support of excavation. These oscillations are described by the following equations:  
\[\left\{ \begin{array}{l}m_{1}\ddot{z}_{1}+b_{1}\dot{z}_{1}+b_{2}(\dot{z}_{1}- \dot{z}_{2})+C_{1}z_{1}+C_{2}(z_{1}-z_{2})=0;\\ m_{2}\ddot{z}_{2}+b_{2}(\dot{z}_{2}-\dot{z}_{1})+C_{2}(z_{2}-z_{1})=b_{2}\, \dot{\eta}(t)+C_{2}\,\eta(t)\end{array} \right.. \tag{17}\]  
Graphics of change in dependencies \(z_{1}\) and \(z_{2}\) in time obtained by solving Equation (17) using Runge-Kutta method are shown in Figure 7. The following values of the input parameters are accepted: \(m_{1}=0.2\) t; \(m_{2}=4\) t; \(\eta_{s}=0.005\) m, \(l_{s}=0.3\) m, \(V_{n}=3\) m/s. This figure also shows the dependencies of the forces in suspension of monorail track in time--\(F_{1}\), and in suspension of rolling stock--\(F_{2}\). These efforts were determined according to the formulas:  
\[ \begin{array}{l}F_{1}=g(m_{1}+m_{2})+C_{1}z_{1}+b_{1}\dot{z}_{1};\\ F_{2}=gm_{2}+C_{2}(z_{2}-z_{1})+b_{2}(\dot{z}_{2}-\dot{z}_{1}).\end{array}  \tag{18}\]  
Figure 7a shows that for different values of suspension parameters, oscillations occur with different amplitudes, frequencies, and damping rates. Moreover, if \(C_{1}=C_{2}=7000\) kN/m and \(b_{1}=b_{2}=15\) kN\(\cdot\)s/m, then, in absolute terms, \(z_{1}\) reaches up to 6.1 mm, and \(z_{2}\) reaches up to 12.0 mm. As follows from Figure 7c, the greatest force in suspension \(F_{1}=80.2\) kN, and between rolling stock and monorail it is \(F_{2}=77.3\) kN. As shown in Figure 7b,d, a significant reduction in suspension forces can be obtained by \(C_{1}=2600\) kN/m and \(b_{1}=50\) kN\(\cdot\)s/m, leaving values of other parameters the same. In these figures, it can be seen that \(z_{1}=11\) mm, \(z_{2}=15\) mm, \(F_{1}=59.6\) kN and \(F_{2}=57.0\) kN. If we compare the highest loads \(F_{1}\) arising in the first and last case, the difference is more than 1.35-fold. Therefore, by changing the values \(C_{1}\), \(C_{2}\) and \(b_{1}\), \(b_{2}\), it is possible to control dynamic loads acting on the suspension of the monorail track. As analysis of calculations for the I-155 monorail shows, the use of the suspension makes it possible to reduce the amplitude of the maximum suspension oscillations by 2-3 times and reduce the dynamic loads on the rolling stock and support by 30-40%. At the same time, the proposed suspension design can be used to create new mine suspended monorail roads, as well as to modernize existing ones.