## 5 Conclusions  
This paper presented an analytical methodology for evaluating the balance of two-stroke OP engines. The proposed approach took four key design parameters into account: two that also influence the balancing of conventional four-stroke reciprocating engines (inline, V, and boxer configurations), which are the number of cylinders and the angle \begin{table}
\begin{tabular}{c c c} \hline \hline
**Architecture** & **Offset Angle** & **Averaged Output Torque** \\ \hline \multirow{3}{*}{3-cylinder} & 0\({}^{\circ}\) & 318 Nm \\  & 5\({}^{\circ}\) & 294 Nm \\  & 10\({}^{\circ}\) & 269 Nm \\  & 15\({}^{\circ}\) & 240 Nm \\ \hline \multirow{3}{*}{4-cylinder} & 0\({}^{\circ}\) & 424 Nm \\  & 5\({}^{\circ}\) & 392 Nm \\  & 10\({}^{\circ}\) & 359 Nm \\  & 15\({}^{\circ}\) & 320 Nm \\ \hline \hline \end{tabular}
\end{table} 
Table 4: OP engine output torque. Figure 15: Output torque profile of a 4-cylinder OP engine as the offset angle varies: **(a)** 0\({}^{\circ}\) offset; **(b)** 5\({}^{\circ}\) offset; **(c)** 10\({}^{\circ}\) offset; **(d)** 15\({}^{\circ}\) offset. between the crank throws, and two parameters specific to OP engines, which are the rotational direction of the two crankshafts (co-rotating and counter-rotating) and the offset angle. The offset angle was also analyzed concerning its effect on the engine output torque. An analytical model was developed to assess the resulting imbalance moment and its variation over a full engine cycle. This model was then applied to investigate two configurations: a 3-cylinder and a 4-cylinder OP engine. For both configurations, co-rotating and counter-rotating crankshafts were examined under four offset angle conditions: \(0^{\circ},5^{\circ},10^{\circ}\), and \(15^{\circ}\). Two different balancing scenarios were considered: one in which the rotating component of first-order reciprocating forces was fully balanced by counterweights in each individual crankshaft, and another in which this component was left unbalanced. The results provided valuable insights. Counter-rotating crankshafts could not reach a global balancing when the rotating component of first-order reciprocating forces were balanced with counterweights, as the counter-rotating counterpart remained entirely unbalanced. Conversely, co-rotating crankshafts could be utilized in both scenarios. However, leaving the rotating component of first-order reciprocating forces locally unbalanced increased the loads on the main bearings, potentially leading to higher wear. Introducing an offset angle resulted in increased imbalance in both configurations. Nevertheless, the co-rotating shafts configuration emerged as the preferable choice, as it led to a lower overall imbalance and offered greater design flexibility. Finally, the impact of the offset angle on engine output torque was also analyzed. The offset causes an uneven torque distribution between the two crankshafts. Specifically, the exhaust-side crankshaft is positioned more favorably, generating higher torque, while the intake-side crankshaft produces lower torque, and in extreme cases, even zero mean torque. In conclusion, the proposed methodology represents a powerful tool in the early stages of OP engine design. Thanks to a spreadsheet-based model, this approach enables the rapid, effective, and straightforward generation of key design guidelines, reducing the time and complexity typically associated with early-stage development. The model does not account for dynamic effects; therefore, for a more detailed analysis, particularly in the final design stages, more comprehensive and complex dynamic models (such as using FEA or multibody analysis) could be employed. In addition, from a fluid dynamics point of view, more studies must be conducted to consider the transient characteristics of hydrogen combustion and evaluate a more precise pressure profile.