# The Problem of Stability in Mechanical Systems Using the Example of Mine Hoist Installations
## 3 Parametric Vibrations of the Mining Skip
The complete elasticity coefficient for the whole system of guides and steel shaft (if we assume that rolling guide elements are connected in series at a fixed distance), could be expressed as:  
\[\frac{1}{k_{\rm c}}=\frac{k+k_{z}}{k\cdot k_{z}} \tag{8}\]  
After introducing all substitutions:  
\[k_{\rm c}=\frac{E\cdot C\cdot\left(1-\frac{D}{\rm C}\cos\omega t\right)}{(E+C) \cdot\left(1-\frac{D}{\rm C+C}\cos\omega t\right)} \tag{9}\]  
From Equation (9) we obtain:  
\[k_{\rm p}=\frac{E\cdot C}{E+C};\ \varepsilon_{1}=\frac{D}{E\cdot E};\ \varepsilon_{2}=\frac{D}{E+E},\]  
The values of parameters \(\varepsilon_{1}\) and \(\varepsilon_{2}\) are much smaller than zero, therefore Equation (10) could be written as:  
\[k_{\rm c}=k_{\rm p}\frac{1-\varepsilon_{1}\cos\omega t}{1-\varepsilon_{2}\cos \omega t} \tag{10}\]  
where  
\[k_{\rm p}\frac{1-\varepsilon_{1}\cos\omega t}{1-\varepsilon_{2}\cos\omega t} \approx k_{\rm p}(1-\varepsilon\cdot\cos\omega t) \tag{11}\]  
After introducing the resulting substitutions from Equation (9), we obtain:  
\[\varepsilon=\varepsilon_{1}-\varepsilon_{2}.\]  
Alternatively:  
\[\varepsilon=\frac{\rm D}{\rm C}-\frac{\rm D}{\rm E+C}=\frac{\rm ED}{\rm C(E+C )}.\]  
Finally, the equation of the complete elasticity coefficient of the steel shaft system with guides could be expressed as:  
\[k_{\rm c}=k_{\rm p}\left(1-\cos\left(2\pi\frac{V_{0}}{l_{0}}t\right)\right) \tag{12}\]  
where:  
\(V_{0}\left[\frac{\rm m}{\rm s}\right]\)--velocity of the hoist installation while moving at a constant velocity  
\(l_{0}\left[\rm m\right]\)--the distance between two supports. The system of Equations (1)-(4) is required to assess the influence of variable steel shaft stiffness on the stability of horizontal displacements of the mining skip in the course of moving at the constant velocity \(V_{0}=\rm const\). The elasticity in the system Equations (1)-(4) is constant and expressed as \(k\) and replaces the \(k_{\rm c}\) parameter defined in (12). The assessment procedure is limited to cases that meet the condition that \(I=me_{1}\varepsilon_{2}\) and vibrations of the head and the bottom frame are independent and enable the separation of the system of Equations (1)-(4) into two independent systems of equations (like 3.3 and 3.4 from [1]):  
\[\left\{ \begin{array}{l}\frac{I+me_{1}^{2}}{f^{2}}\cdot\ddot{x}_{\rm A}+k_{ \rm g}\left(x_{\rm A}-x_{\rm g}\right)+2h_{\rm g}\left(\ddot{x}_{\rm A}-\ddot {x}_{\rm g}\right)=0\\ m_{\rm g}\ddot{x}_{\rm g}+2k\left(x_{\rm g}-x_{1}(t)\right)-k_{\rm g}\left(x_{ \rm A}-x_{\rm g}\right)-2h_{\rm g}\left(\dot{x}_{\rm A}-\dot{x}_{\rm g}\right) +2h\left(\ddot{x}_{\rm g}-\dot{x}_{\rm i}(t)\right)=0\end{array} \right. \tag{13}\]\[\left\{ \begin{array}{c}\frac{l+m_{\rm g}^{2}}{l^{2}}\ddot{x}_{\rm B}+k_{\rm d}(x _{\rm B}-x_{\rm d})+2h_{\rm d}\big{(}\ddot{x}_{\rm B}-\dot{x}_{\rm d}\big{)}=0\\ m_{\rm d}\ddot{x}_{\rm d}+2k(x_{\rm d}-x_{1}(t+\tau))-k_{\rm d}(x_{\rm B}-x_{\rm d })-2h_{\rm d}\big{(}\ddot{x}_{\rm B}-\dot{x}_{\rm d}\big{)}+2h\big{(}\dot{x}_{ \rm d}-\dot{x}_{1}(t+\tau)\big{)}=0\end{array} \right. \tag{14}\]  
For free vibrations, the system of equations (3.3 from [1]) becomes:  
\[ \begin{array}{c}m_{\rm A}\ddot{x}_{\rm A}+k_{\rm g}\big{(}x_{\rm A}-x_{\rm g }\big{)}+2h_{\rm g}\big{(}\ddot{x}_{\rm A}-\ddot{x}_{\rm g}\big{)}=0\\ m_{\rm g}\ddot{x}_{\rm g}+2k_{\rm p}\big{(}1-\varepsilon\cos\!\left(2\pi\frac{ V_{0}}{l_{0}}l\right)\!\big{)}x_{\rm g}-k_{\rm g}\big{(}x_{\rm A}-x_{\rm g}\big{)}-2h_{ \rm g}\big{(}\ddot{x}_{\rm A}-\ddot{x}_{\rm g}\big{)}+2h\dot{x}_{\rm g}=0 \end{array}  \tag{15}\]  
Expressing time by the dimensionless parameter \(\tau=p_{10}\cdot t\) in the system of Equation (15), where \(p_{10}\) is the natural frequency of free vibrations for mass \(m_{\rm A}\), \(m_{\rm g}=0\), allows transformation after the individual substitutions:  
\[X_{\rm A}(t)=u(\tau) \tag{16}\]  
\[X_{\rm g}(t)=v(\tau) \tag{17}\]  
We then obtain:  
\[