## 2 Materials and Methods
### Equations of Motion and Their Solutions
 \tag{24}\]  
Replacing the mechanical characteristic of the motor with Expression (7), introducing dimensionless time \(\tau=\omega_{0u}t,\tau=\omega_{0d}t\) and after small transformations, we obtain the equation of motion of the device in the following form:  
For the piston lifting time,  
\[\varphi^{\prime\prime}+\frac{1}{\omega_{0u}}\bigg{(}\mu_{1}\sin^{2}\varphi+ \frac{b}{J_{mr}}\bigg{)}\varphi^{\prime}+\mu_{2}\sin^{3}\varphi{{}^{\prime}} ^{2}+\sin\varphi=\frac{a}{J_{mr}\omega_{0u}^{2}}, \tag{25}\]  
where \(\varphi\subseteq[0,\pi]\cup[2\pi,3\pi]\cup[4\pi,5\pi]\ldots[2n\pi,(2n+1)\pi], n=0,1,2,\ldots,\) and for the piston lowering time:  
\[\varphi^{\prime\prime}+\frac{1}{\omega_{0d}}\bigg{(}\mu_{1}\sin^{2}\varphi+ \frac{b}{J_{mr}}\bigg{)}\varphi^{\prime}+\mu_{2}\sin^{3}\varphi{{}^{\prime}} ^{2}+\sin\varphi=\frac{a}{J_{mr}\omega_{0d}^{2}}, \tag{26}\]  
where \(\varphi\subseteq[\pi,2\pi]\cup[3\pi,4\pi]\cup[5\pi,6\pi]\ldots[(2n-1)\pi,2n \pi],n=1,2,3,\ldots\)  
Equations (25) and (26), the dash sign denotes the dimensionless time derivative \(\tau\). The differential Equations (25) and (18) differ from each other by coefficients \(\omega_{0u}^{2}\) and \(\omega_{0d}^{2}\), but have the same form; therefore, it is sufficient to solve one of them. Equations (25) and (26) are significantly nonlinear since the nonlinear term enters the equations without a small parameter. The essentially nonlinear model of the differential equations of motion (25) and (26) allows for an in-depth study of the dynamics of the mixing-whipping device in both rotational and oscillatory or aperiodically damped modes of the driving member's motion. In contrast, vibrational systems with nonideal power sources and flexible supports (which are mostly weakly nonlinear) lack this capability and are generally limited to analyzing dynamics only in the rotational mode of the driving element. A comprehensive study of the influence of system parameters on the device's dynamics is possible only within essentially nonlinear systems. At \(a/\left(J_{mr}\omega_{0}^{2}\right)>1\), there is a rotational mode, and at \(a/\left(J_{mr}\omega_{0}^{2}\right)<1\), there are oscillatory and aperiodically damped crank motion modes. Depending on the value \(b/(\omega_{0}J_{mr})\), the crank movement is divided into oscillatory-damping and aperiodic-damping modes. The mixing and whipping processes are carried out mainly under the influence of the piston pressure force, and therefore the mixing and whipping force and its power are determined by the expressions\[F=\frac{1}{2}\rho\pi\Big{(}R_{1}^{2}-n_{2}R_{2}^{2}\Big{)}\Bigg{[}\frac{1}{n_{2}^{2} }\left(\frac{R_{c}}{R_{2}}\right)^{2}-1\Bigg{]}r_{0}^{2}\sin^{2}\varphi\dot{ \varphi}^{2},P=\frac{1}{2}\rho\pi\Big{(}R_{1}^{2}-n_{2}R_{2}^{2}\Big{)}\Bigg{[} \frac{1}{n_{2}^{2}}\left(\frac{R_{c}}{R_{2}}\right)^{2}-1\Bigg{]}r_{0}^{3}\sin ^{3}\varphi\dot{\varphi}^{3}. \tag{27}\]  
The equations of motion of the device (25) and (26) are solved using the MATLAB R2023b software package ([[https://www.mathworks.com/products/new_products/release2023b.html](https://www.mathworks.com/products/new_products/release2023b.html)]([https://www.mathworks.com/products/new_products/release2023b.html](https://www.mathworks.com/products/new_products/release2023b.html)) (accessed on 22 June 2025)) using the fourth-order Runge-Kutta method for oscillating and aperiodic damped modes of motion of the crank. This resulted in graphical dependences of the rotation angle and rotational speed of the actuating engine shaft on time, mixing and whipping force power from the angle of rotation and rotational speed of the excitation source shaft, and the parameters of the nonideal mechanical system. The obtained graphical dependences of the engine shaft rotation speed on time indicate the superiority of the main harmonic of the solutions of the equations of motion in them and, consequently, the inherent oscillatory characteristics for them.