## 3 Derivation of a Mathematical Model
\right] \tag{14}\]  
is the stiffness matrix, and  
\[\ddot{\mathbf{x}}=\left[ \begin{array}{cccc}\ddot{x}_{1}&\ddot{\varphi}_{1}& \ddot{\varphi}_{3}&\ddot{\varphi}_{4}\end{array} \right]^{T}, \tag{15}\]  
\[\dot{\mathbf{x}}=\left[ \begin{array}{cccc}\ddot{x}_{1}&\dot{\varphi}_{1}&\dot{ \varphi}_{3}&\dot{\varphi}_{4}\end{array} \right]^{T}, \tag{16}\]  
\[\mathbf{x}=\left[ \begin{array}{cccc}x_{1}&\varphi_{1}&\varphi_{3}&\varphi_{4} \end{array} \right]^{T} \tag{17}\]  
are vectors of generalized accelerations and velocities, respectively. Furthermore,  
\[\mathbf{F}=\left[ \begin{array}{cccc}F(t)&0&0&0\end{array} \right]^{T} \tag{18}\]  
is the vector of external loads or the excitation of the system. As this engine is assumed to be powered by compressed air, the force \(F(t)\) in vector \(\mathbf{F}\) represents the vector's time function. However, we do not yet know how to mathematically describe this force. If we take into account the dimensions and functions of the engine mechanism, we can consider certain simplifications. First, the plate, switchboard, and cylinders are considered as one functional unit with a mutual moment of inertia \(I_{s}\) (i.e., the reduced moment of inertia in Equation (7)). Due to the very short lengths of the shafts, the torsion stiffness is very high. Consequently, the relative rotation of the shaft ends, i.e., torsion deformation, can be neglected. We also consider the very high stiffness of the connecting rod. As a result of this high stiffness, the differences in the deformations of individual bodies described by Equation (9) are zero; therefore, the potential energy of this system is also considered zero. These simplifications do not have significant effects on the output variables (Figure 7b). Hence, the initial generalized coordinates \(\varphi_{1}\), \(\varphi_{2}\), \(\varphi_{3}\), and \(\varphi_{4}\) are identical to the generalized coordinate \(\varphi_{1}\) (i.e., \(\varphi_{1}=\varphi_{2}=\varphi_{3}=\varphi_{4}\)). Therefore, these simplifications lead to modified kinetic energy \(E_{K_{\text{mod}}}\) as follows:  
\[E_{K_{\text{mod}}}=\frac{1}{2}\cdot m_{1}\cdot\dot{x}_{1}^{2}+\frac{1}{2}\cdot I _{p}\cdot\dot{\varphi}_{1}^{2}+\frac{1}{2}\cdot I_{s}\cdot\dot{\varphi}_{1}^{ 2}+\frac{1}{2}\cdot I_{i}\cdot\left(2\cdot\dot{\varphi}_{1}\right)^{2}+\frac{1} {2}\cdot I_{v1}\cdot\dot{\varphi}_{1}^{2}+\frac{1}{2}\cdot I_{v2}\cdot\left(2 \cdot\dot{\varphi}_{1}\right)^{2}. \tag{19}\]We can then modify Equation (19) as follows:  
\[E_{K_{\rm mod}}=\frac{1}{2}\cdot\Big{[}m_{1}\cdot\dot{x}_{1}^{2}+\big{(}I_{p}+I_{s }+4\cdot I_{l}+I_{v1}+4\cdot I_{v2}\big{)}\cdot\dot{\varphi}_{1}^{2}\Big{]}. \tag{20}\]  
The expression in round brackets is called the reduced moment of inertia, which we mark as \(I_{red}\). The final form of the modified kinetic energy \(E_{K_{\rm mod}}\) is then written as  
\[E_{K_{\rm mod}}=\frac{1}{2}\cdot\Big{[}m_{1}\cdot\dot{x}_{1}^{2}+I_{red}\cdot \dot{\varphi}_{1}^{2}\Big{]}. \tag{21}\]  
As mentioned above, the potential energy equals zero. Considering only the modified kinetic energy and the dissipative energy and performing derivations according to Equation (1), we can obtain simplified equations of motion:  
\[ \begin{array}{c}m_{1}\cdot\ddot{x}_{1}+b_{1}\cdot\dot{x}_{1}=F(t)\\ I_{red}\cdot\ddot{\varphi}_{1}+(b_{1}+2\cdot b_{2})\cdot\dot{\varphi}_{1}=0 \end{array} . \tag{22}\]  
The matrix form of the equations of motion written in Equation (22) is  
\[\left[ \begin{array}{cc}m_{1}&0\\ 0&I_{red}\end{array} \right]\cdot\left[ \begin{array}{c}\ddot{x}_{1}\\ \ddot{\varphi}_{1}\end{array} \right]+\left[ \begin{array}{cc}b_{1}&0\\ 0&b_{1}+2\cdot b_{2}\end{array} \right]\cdot\left[ \begin{array}{c}\dot{x}_{1 }\\ \dot{\varphi}_{1}\end{array} \right]=\left[ \begin{array}{c}F(t)\\ 0\end{array} \right], \tag{23}\]  
where  
\[M_{\rm mod}=\left[ \begin{array}{cc}m_{1}&0\\ 0&I_{red}\end{array} \right] \tag{24}\]  
is the modified mass matrix;  
\[B_{\rm mod}=\left[ \begin{array}{cc}b_{1}&0\\ 0&b_{1}+2\cdot b_{2}\end{array}