## 3 Model Description  
The analysis of the balancing of the OP engine has been conducted by developing an analytical model based on the mechanics of the slider-crank mechanism. The full derivation of crank mechanism dynamics is provided in numerous textbooks, for example [20] or [24]. The initial model does not account for the number of cylinders or the engine architecture but focuses on determining the loads of the individual slider-crank mechanism. The developed model enables the evaluation, for each angular position, of the forces due to moving masses and those arising from gas pressure, thereby obtaining the resulting forces at the crank supports and the output torque. The inertial forces were derived analytically by studying the kinematics of the crank mechanism, while the gas forces were obtained based on a pressure profile derived by preliminary thermo-fluid-dynamic calculations. Specifically, both analytical and empirical formulas were employed, and the operating cycle has been divided into different phases: intake/exhaust, compression, combustion, and expansion (in a two-stroke engine intake and exhaust phases are treated as a single phase). The intake/exhaust phase has been simply approximated using a constant pressure value equal to the intake/boost pressure. The compression and expansion phases were modeled considering generic polytropic transformations:  
\[p(\theta)V(\theta)^{1.35} =p_{sc}V_{sc}^{1.35}, \tag{10}\] \[p(\theta)V(\theta)^{1.25} =p_{se}V_{se}^{1.25}, \tag{11}\]  
where the subscript \(sc\) refers to the pressure \(p\) and volume \(V\) at the beginning of compression phase, while the subscript \(se\) refers to the pressure \(p\) and volume \(V\) at the beginning of expansion phase. Finally, the combustion phase has been modeled using the differential form of the first law of thermodynamics with the adoption of the Wiebe heat release model [20]. Figure 6 depicts the pressure profile considered. Figure 5: The resultant moment \(\mathbf{M}^{l}_{rec,r,tot}\) generated by the rotating portion of the first-order alternating forces \(\mathbf{F}^{l}_{rec,r,i}\) of each crank mechanism in the 3-cylinder co-rotating configuration, with and without offset. By considering the offset angle between the two shafts, the number of cylinders, and the specific firing order, it has been possible to allocate these factors in space and time, thereby deriving the global action of the forces on the entire engine system and obtaining the complete model. However, when working with a design concept that significantly diverges from conventional approaches, it is essential to exercise caution in scenarios where standard assumptions may no longer hold. In a conventional single-piston engine, the volume change is straightforward, as it is simply calculated as the cylinder area multiplied by the piston stroke. However, in an OP design, the two pistons may move at different speeds and in opposite directions while sharing a single working volume and pressure. As a result, the pressure profile can no longer be directly correlated with the crank angle, as in conventional approaches, since the volume may vary due to the offset angle. [PERSON] [25] analyzed this aspect in his article. He assumed the pressure profile to be constant with respect to the exhaust crankshaft, meaning that one piston always experiences the same pressure profile regardless of a possible offset between the two crankshafts, while only the intake one is affected by the offset. The authors of this paper believe that this approach could be potentially misleading and therefore adopt a different methodology, in which the pressure profile has been instead described as a function of the instantaneous volume within the cylinder, which varies according to the offset. The pressure profile does not account for the fluid dynamic advantage of the offset in enhancing trapping efficiency. This simplification is intentional, as the primary objective of this study is to isolate and analyze the impact of crankshafts offset on the engine structural behavior.