## 3 Derivation of a Mathematical Model
Hence, the kinetic energy (Equation (3)) takes the following form:  
\[E_{K_{tot}}=\frac{1}{2}\cdot I_{1}\cdot\dot{\varphi}_{1}^{2}+\frac{1}{2}\cdot m _{1}\cdot\dot{x}_{1}^{2}+\frac{1}{2}\cdot I_{2}\cdot\dot{\varphi}_{1}^{2}+ \frac{1}{2}\cdot I_{3}\cdot\dot{\varphi}_{1}^{2}+\frac{1}{2}\cdot I_{4}\cdot \left(2\cdot\dot{\varphi}_{1}\right)^{2}+\frac{1}{2}\cdot I_{5}\cdot\dot{ \varphi}_{3}^{2}+\frac{1}{2}\cdot I_{6}\cdot\dot{\varphi}_{4}^{2}. \tag{5}\]  
The final form of the total kinetic energy \(E_{K_{tot}}\) is as follows:  
\[E_{K_{tot}}=\frac{1}{2}\cdot\left[m_{1}\cdot\dot{x}_{1}^{2}+(I_{1}+I_{2}+I_{3} +4\cdot I_{4})\cdot\dot{\varphi}_{1}^{2}+I_{5}\cdot\dot{\varphi}_{3}^{2}+ \frac{1}{2}\cdot I_{6}\cdot\dot{\varphi}_{4}^{2}\right], \tag{6}\]  
where the relation in round brackets represents the reduced moment of inertia \(I_{1 red}\) for the mutual shaft--in this case, shaft 1. This moment can be written in the following form:  
Figure 5:0  
Furthermore, the dissipative energy represents the sum of the energies, which include energies wasted by the system itself. In our case, we consider the viscous friction in the system, namely, the friction of the bodies moving rectilinearly (with the generalized coordinate \(x_{1}\)) as well as the bodies performing rotating movements described by the generalized coordinate \(\varphi_{1}\). Friction-related phenomena are included in coefficients \(b_{1}\), \(b_{2}\), \(b_{3}\), and \(b_{4}\). Hence, the dissipative energy assumes the following form:  
Figure 5:1  
Considering a torsion stiffness of the individual sections of shafts and a connecting rod, we can derive the potential energy of the system. For this purpose, we use the following equation:  
Figure 5:2  
where \(k_{1}\) is the stiffness of the connecting rod, and \(k_{2}\) and \(k_{3}\) are the torsion stiffness values of individual parts of the shafts (Figure 7a). Next, we can proceed to derive the individual energies written in Equations (6), (8) and (9). Derivations are performed based on the general Equation (1), whereas the derivations themselves are not included. The final system of equations of motion for the solved mechanical system can then be written as follows:  
\[ \begin{gathered} m_{1}\cdot\ddot{x}_{1}+b_{1}\cdot\dot{x}_{1}+k_{ 1}\cdot x_{1}=F(t)\\ I_{s}\cdot\ddot{\varphi}_{1}+2\cdot b_{2}\cdot\dot{\varphi}_{1}+(4 \cdot k_{2}+k_{3})\cdot\dot{\varphi}_{1}-2\cdot k_{3}\cdot\dot{\varphi}_{3}-k _{3}\cdot\varphi_{4}=0\\ I_{5}\cdot\ddot{\varphi}_{3}+b_{3}\cdot\dot{\varphi}_{3}-2\cdot k_{2} \cdot\varphi_{1}+k_{2}\cdot\dot{\varphi}_{3}=0\\ I_{6}\cdot\ddot{\varphi}_{4}+b_{4}\cdot\dot{\varphi}_{4}-k_{3} \cdot\dot{\varphi}_{1}+k_{3}\cdot\dot{\varphi}_{4}=0\end{gathered} , \tag{10}\]  
Additionally, the related matrix form is \[\left[ \begin{array}{cccc}m_{1}&0&0&0\\ 0&I_{s}&0&0\\ 0&0&I_{5}&0\\ 0&0&0&I_{6}\end{array} \right].\left[ \begin{array}{c}\ddot{x}_{1}\\ \ddot{\varphi}_{1}\\ \ddot{\varphi}_{3}\\ \ddot{\varphi}_{4}\end{array} \right]+\left[ \begin{array}{cccc}b_{1}&0&0&0\\ 0&2\cdot b_{2}&0&0\\ 0&0&b_{3}&0\\ 0&0&0&b_{4}\end{array} \right].\left[ \begin{array}{c}\dot{x}_{1}\\ \dot{\varphi}_{1}\\ \dot{\varphi}_{3}\\ \dot{\varphi}_{4}\end{array} \right]+\left[ \begin{array}{cccc}k_{1}&0&0&0\\ 0&4\cdot k_{2}+k_{3}&-2\cdot k_{2}&-k_{3}\\ 0&-2\cdot k_{2}&k_{2}&0\\ 0&-k_{3}&0&k_{3}\end{array} \right].\left[ \begin{array}{c}x_{1}\\ \varphi_{1}\\ \varphi_{3}\\ \dot{\varphi}_{4}\end{array} \right]=\left[ \begin{array}{c}F(t)\\ 0\\ 0\\ 0\end{array} \right]. \tag{11}\]  
Adopting the known terminology,  
\[\mathbf{M}=\left[ \begin{array}{cccc}m_{1}&0&0&0\\ 0&I_{s}&0&0\\ 0&0&I_{5}&0\\ 0&0&0&I_{6}\end{array} \right] \tag{12}\]  
is the mass matrix,  
\[\mathbf{B}=\left[ \begin{array}{cccc}b_{1}&0&0&0\\ 0&2\cdot b_{2}&0&0\\ 0&0&b_{3}&0\\ 0&0&0&b_{4}\end{array} \right] \tag{13}\]  
is the damping matrix,  
\[\mathbf{K}=\left[ \begin{array}{cccc}k_{1}&0&0&0\\ 0&4\cdot k_{2}+k_{3}&-2\cdot k_{2}&-k_{3}\\ 0&-2\cdot k_{2}&k_{2}&0\\ 0&-k_{3}&0&k_{3}\end{array}