## 2 Materials and Methods
### Equations of Motion and Their Solutions
 \tag{15}\]  
Then, taking into account \(k\to 0:\frac{\ln\left[r/\left(kR_{2}\right)\right]}{\ln\left(1/k\right)} \to 1\), Equation (11) will take the form of  
\[u_{x}=v+\frac{1}{4\eta}\left(\frac{\partial p}{\partial x}-\rho g_{x}\right) \left(r^{2}-R_{2}^{2}\right). \tag{16}\]  
In Expression (16), we replace the value \(v\) with its expression from (4), substitute the gradient \(\partial p/\partial x\) with \(\Delta p/h\), and then replace the value \(\Delta P\) with its expression from (8). By substituting the resulting laminar flow velocity in the piston hole into [PERSON]'s law, we obtain the projections of the friction forces between the fluid and the surfaces of the holes:  
\[F_{3s}\approx-n_{2}\left\{-\frac{\rho}{2}\left[\frac{1}{n_{2}^{2}}\left( \frac{R_{c}}{R_{2}}\right)^{2}-1\right]\left(r_{0}\sin\varphi\dot{\varphi} \right)^{2}+\frac{mg_{x}}{\pi\left(R_{1}^{2}-n_{2}R_{2}^{2}\right)}\right\} \pi R_{2}^{2}, \tag{17}\]  
The expression for the virtual work of the projection of the resultant force \(F_{x}=\sum\limits_{i=1}^{3}F_{ix}\), exerted by the fluid on the piston and the engine torque \(M\) is given by the following form:  
\[\delta A=M\delta\varphi+F\delta x, \tag{18}\]  
where \(\delta x=r_{0}\sin\varphi\delta\varphi\) is the change in the coordinate \(x\), and substituting this value in (18), the following is derived:  
\[\delta A=Q_{\varphi}\delta\varphi=(M+F_{x}r_{0}\sin\varphi)\delta\varphi. \tag{19}\]  
Generalized forces of the system, according to (19), are as follows:  
\[Q_{\varphi}=M+F_{x}r_{0}\sin\varphi, \tag{20}\]  
where  
\[ \begin{array}{l}F_{x}=-\frac{2\pi\eta h}{\ln\left(R_{1}/R_{c}\right)}r_{0} \sin\varphi\dot{\varphi}+\varrho\left[\frac{1}{n_{2}^{2}}\left(\frac{R_{c}}{R_ {2}}\right)^{2}-1\right]\left[2\pi n_{2}R_{2}^{2}+\frac{\pi\left(R_{c}^{2}-R_{ 1}^{2}\right)}{2\ln\left(R_{1}/R_{c}\right)}\right]r_{0}^{2}\sin^{2}\varphi \dot{\varphi}^{2}+\\ \left\{\rho g_{x}h\pi\left(R_{1}^{2}-n_{2}R_{2}^{2}\right)-\frac{mg_{x}}{\pi \left(R_{1}^{2}-n_{2}R_{2}^{2}\right)}\left[2\pi n_{2}R_{2}^{2}+\frac{\pi \left(R_{c}^{2}-R_{1}^{2}\right)}{2\ln\left(R_{1}/R_{c}\right)}\right]\right\} \cdot\end{array}  \tag{21}\]  
Here, \(F_{x}=F_{u}\) or \(F_{d}\) depending on the value \(g_{x}\) or direction of the piston's movement (i.e., the direction of fluid flow through the piston holes and the gaps between the piston and cylinder). Substituting (6), (21) into (1), taking into account (13), (8)-(10), we obtain the following equations:  
For the upward movement of the piston:  
\[\ddot{\varphi}+\Big{(}f_{1}\sin^{2}\varphi+b/J_{mr}\Big{)}\dot{\varphi}+f_{2} \sin^{3}\varphi\dot{\varphi}^{2}+\omega_{0u}^{2}\sin\varphi=M/J_{mr}, \tag{22}\]  
and for downward movement of the piston:  
\[\ddot{\varphi}+\Big{(}f_{1}\sin^{2}\varphi+b/J_{mr}\Big{)}\dot{\varphi}+f_{2} \sin^{3}\varphi\dot{\varphi}^{2}+\omega_{0d}^{2}\sin\varphi=M/J_{mr}, \tag{23}\]  
where  
\[ \begin{array}{l}\mu_{1}=\frac{1}{I_{mr}}\frac{2\pi\eta h}{\ln(R_{1}/R_{c})}r_ {0}^{2},\mu_{2}=\frac{1}{I_{mr}}\bigg{\{}-\varepsilon\bigg{[}\frac{1}{n_{2}^{2 }}\left(\frac{R_{c}}{R_{2}}\right)^{2}-1\bigg{]}\bigg{[}2\pi n_{2}R_{2}^{2}+ \frac{\pi\left(\frac{R_{c}^{2}-R_{1}^{2}}{2}\right)}{2\ln(R_{1}/R_{c})}\bigg{]} r_{0}^{3}\bigg{\}},\\ \omega_{0u}^{2}=\frac{1}{I_{mr}}\bigg{\{}-\frac{m_{g}}{\pi\left(\frac{R_{1}^{ 2}-R_{2}R_{c}^{2}}{2}\right)}\bigg{[}2\pi n_{2}R_{2}^{2}+\frac{\pi\left(\frac{ R_{c}^{2}-R_{1}^{2}}{2}\right)}{2\ln(R_{1}/R_{c})}\bigg{]}+\rho gh\pi\left(R_{1}^{2}- n_{2}R_{2}^{2}\right)\bigg{\}}r_{0},\\ \omega_{0d}^{2}=\frac{1}{I_{mr}}\bigg{\{}\frac{m_{g}}{\pi\left(\frac{R_{1}^{ 2}-R_{2}R_{c}^{2}}{2}\right)}\bigg{[}2\pi n_{2}R_{2}^{2}+\frac{\pi\left(\frac{ R_{c}^{2}-R_{1}^{2}}{2}\right)}{2\ln(R_{1}/R_{c})}\bigg{]}-\rho gh\pi\left(R_{1}^{2}- n_{2}R_{2}^{2}\right)\bigg{\}}r_{0}\end{array}