## 4 Results  
A multibody model (MBS model) of the engine with rotating cylinders was created in the Simpack software package (Dassault Systemes, Velizy-Villacoublay, France; Figure 8a). Figure 8b shows the detail of joints defined in the Simpack software package. The model includes both mechanical and kinematic joints. Simpack is a widely used MBS software package that can render models of various mechanical systems, including simplified subsystems of vehicles, such as engines [39] and gearboxes [40,41], parts of chassis and suspension systems, entire vehicles [42,43], and sets of vehicles or other mechanisms [44,45]. Simpack works using numerical methods in which a mathematical model describing the dynamic behavior of a mechanical system can be derived automatically. Firstly, markers for placing individual components and defining mechanical and kinematical couplings were defined. Individual bodies of the solved mechanism were then connected to each other by the corresponding joints and constraints. In the created MBS model, all bodies were considered rigid. This means that the dynamic properties of the bodies were fully described by their masses, moments of inertia, and centers of gravity. For better visualization, the geometries of the bodies were defined by models imported from the Autodesk Inventor software (Figure 4). However, as the definitions of materials of individual bodies require one to perform strength analyses, the motion properties of the engine are analyzed in terms of the engine's kinematics, and the mass and inertia properties are defined as units. Based on the created computational models and their parameters, we selected waveforms for some kinematic quantities of the engine. The engine waveforms were compared, achieving results for the two approaches:  
* The mathematical model derived in Section 3 and solved in MATLAB (MathWorks, Natick, MA, USA);  
Figure 8: A multibody model of the engine in the Simpack software package (**a**), the detail of joints (mechanical and kinematic joints) (**b**): 1—a plate, 2—a connecting rod, 3—an auxiliary stone, 4—a stone, 5—a first output, 6—a second output, 8—pistons, 9—heads of cylinders, 10—a stand. * The numerical model is represented by the MBS model created in the Simpack software (Figure 8). The MBS model consists only of rigid bodies in which all the linking elements, such as shafts, are also considered rigid. Therefore, the compared numerical model developed in MATLAB was based on equations of motion in compliance with Equation (23). It should be noted that calculations were carried out for unit masses and inertia properties because current calculations are focused on analyses of the kinematic properties of the engine mechanism. The resulting waveforms and graphs were processed in MATLAB to compare them within individual figures. From all the results, we selected waveforms for some kinematic quantities, including for a functional block (Figure 9), a piston mechanism (Figures 10 and 11), the first output, the second output, and a switchboard (Figures 12 and 13). Here, the angular velocity of the functional block is \(\varphi_{1}\) = 20 rad\(\cdot\)s\({}^{-1}\), the angular velocity of the first output is \(\varphi_{2}\) = 10 rad\(\cdot\)s\({}^{-1}\), the angular velocity of the second output due to the technical solution is \(\varphi_{1}\) = 20 rad\(\cdot\)s\({}^{-1}\), and the gear ratio is \(i_{g}\) = 0.5. Figure 9 shows the waveforms for the angular velocity of the functional block of the engine mechanism obtained from Simpack and MATLAB. Both results correspond to each other with only small deviations. Figure 9 shows the waveform of the angular velocity of the functional block. Here, the velocity oscillates despite the chosen constant value. This oscillation is due to the compound motion of the pistons, which perform both rotational and translational movement at the same time. This compound movement changes the position of the center of gravity in the mechanical system. Figure 11: Translational velocity of pistons—results from MATLAB.