# Reducing Oscillations in Suspension of Mine Monorail Track
## 3 Results
1. According to the design of the suspended monozial road, we select the masses \(m_{1}\), and \(m_{2}\) calculate the coefficient \(\mu_{e}\). 2. We calculate the vibration amplitude of the monozial suspension without additional elastic elements. 3. We determine the natural frequency of oscillation of the suspension \(k_{e}\). 4. We determine the ratio \(\xi_{e}\) between the natural vibration frequency of the suspension \(k_{e}\) and the frequency of the perturbing influences \(\omega\) acting on the suspension. 5. We assign the allowable oscillation amplitude of the monorail track, which should not be more than 15 mm. 6. Taking into account clauses 1 and 5, we set the depreciation factor. 7. If \(\xi_{e}<1\) and \(k_{v}>1\), then, based on Expression (7), we calculate the coefficient using the formula \[C_{e}=\frac{\mu_{e}(\xi_{e}^{2}-1+\mu_{e})}{(W_{kp}-1)\xi_{e}^{2}(1-\xi_{e}^{2 })}.\] (10) If \(\xi_{e}>1\), then \[C_{e}=\frac{\mu_{e}(\xi_{e}^{2}-1+\mu_{e})}{(|W_{kp}|+1)\xi_{e}^{2}(\xi_{e}^{2 }-1)}.\] (11)
8. Based on the obtained value \(C_{e}\) and the suspension stiffness factor \(C_{1}\) determined by the design of the monorail suspension, we determine the stiffness factor \(C_{2}\) related to the elastic elements (poppet washers) as \(C_{2}=C_{1}\times C_{e}\). 9. According to the obtained value \(C_{2}\) and the maximum allowable load on the monorail track, we select poppet washers, set their parameters, specify the stiffness coefficient and the total compression stroke. We select the material of the shock-absorbing insert and determine its dimensions. 10. For the values of the stiffness coefficient and the total compression stroke of the disc springs, taken into account in Clause 9, we repeat the calculation and specify \(C_{e}\), \(A_{1}\), \(A_{2}\) and \(W_{kp}\). It should be noted that during the movement of heterogeneous rolling stock along a monorail track, the frequency of disturbing forces can vary within certain limits from \(\omega_{1}\) to \(\omega_{2}\), and therefore the method proposed above, which considers the action of disturbing forces of constant frequency \(\omega\), has limitations. Let us set these restrictions and consider the general case when \(\omega_{1}\leq\omega\leq\omega_{2}\). In this case, the natural frequencies \(k_{1}^{2}=C_{1}/m_{1}\) and \(k_{2}^{2}=C_{2}/m_{2}\) in relation to the natural frequency of the suspension without poppet washers \(k_{0}\) should be equal to \(k_{2}<k_{0}<k_{1}\), and the calculated amplitude of the suspension oscillations \(A_{1\omega}\) should not exceed the permissible oscillation amplitudes \(A_{1d}\). If \(\omega_{1}\leq k_{1}\leq\omega_{2}\) and \(\omega_{1}\leq k_{2}\leq\omega_{2}\), then the use of poppet washers is inappropriate, since resonance is possible if the natural frequencies \(k_{1}\) or \(k_{2}\) with the frequencies \(\omega_{1}\) of \(\omega_{2}\) disturb or coincide. Effective damping requires \(k_{1}>\omega_{2}\) or \(k_{2}<\omega_{1}\). Imagine the natural frequency \(k_{2}^{2}=C_{1}C_{e}/m_{2}\). Then, \(C_{1}=k_{2}^{2}m_{2}/C_{e}\). Considering that \(k_{e}^{2}=C_{1}/(m_{1}+m_{2})\), we obtain  
\[k_{e}^{2}=k_{2}^{2}\frac{\mu_{e}}{C_{e}}. \tag{12}\]  
Based on (6), (12) and transformations, we obtain  
\[A_{1}=\frac{m_{e}r_{e}\xi_{e2}^{2}\xi_{\omega\omega}^{2}}{(m_{1}+m_{2})\big{(} (1-\mu_{e})\xi_{\omega}^{4}-\xi_{\omega}^{2}\big{(}\xi_{v}^{2}+\xi_{2}^{2} \big{)}+\xi_{v}^{2}\xi_{2}^{2}\big{)}}, \tag{13}\]  
where \(\xi_{2}=k_{2}/\omega_{i}\); \(\xi_{v}=k_{e}/\omega_{i}\); \(\xi_{\omega}=\omega/\omega_{i}\); \(\omega_{i}\)--effective disturbance frequency from the range from \(\omega_{1}\) to \(\omega_{2}\). Similarly to (7), in the absence of poppet washers, the suspension oscillation amplitude is  
\[A_{0}=\lim_{\xi_{2}\rightarrow\infty}A_{1}=\frac{m_{e}r_{e}\xi_{\omega\omega} ^{2}}{(m_{1}+m_{2})(\xi_{\varphi}^{2}-\xi_{\omega}^{2})}. \tag{14}\]