## 4 Results
### Engine Balancing
#### 4.1.1 Difference Between Co-Rotating and Counter-Rotating Configurations  
Before analyzing the effects of offset, it is crucial to consider the differences between co-rotating and counter-rotating shaft configurations. In the co-rotating shaft configuration, since the two shafts are identical and exhibit a phase difference of 180\({}^{\circ}\) between them, each moment component of one shaft is balanced by the corresponding component of the other shaft. Conversely, for counter-rotating shafts, where the two shafts are symmetric with respect to the plane orthogonal to the cylinder axes, the rotating component of the first shaft balances the counter-rotating component of the second shaft, and vice versa. Table 3 depicts the direction of forces and how the moments were evaluated. Only the 3-cylinder configuration is reported, as the derivation of the 4-cylinder one is straightforward.  
\begin{table}
\begin{tabular}{c c} \hline \hline \multicolumn{2}{c}{**Engine Main Parameters**} \\ \hline Masses & \\ \hline Piston pin & 137.50 g \\ Total piston rings & 18.50 g \\ Total reciprocating masses \((m_{rec})\) & 915.5 g \\ Crankpin \((m_{cp})\) & 330 g \\ \hline \hline \end{tabular}
\end{table}
Table 2: _Cont._Considering the two scenarios presented above, an important difference emerges between co-rotating and counter-rotating configurations.  
On one side, in the case of co-rotating shafts, the engine is perfectly balanced in both cases. In fact, if the moments generated by the rotating portion of the first-order reciprocating forces are left unbalanced (Scenario 2), they will be counteracted by the moments induced by the corresponding components from the other shaft. Therefore, in this configuration of co-rotating shafts, either balancing or not balancing the rotating component of the first-order moments generated by the reciprocating forces is a valid option. However,  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & Co-Rotating Shafts & Counter-Rotating Shafts \\ \hline \(M_{rec,r}^{I}\  
adopting co-rotating shafts with the first-order rotating forces balanced by counterweights on both shafts (Scenario 1) offers the advantage of reducing the load on the crankshaft journals and bearings, increasing their reliability.  
In contrast, for counter-rotating shafts, the two scenarios lead to entirely different balancing outcomes. Given that the moments generated by the rotating components of the first-order reciprocating forces of the two shafts rotate in opposite directions due to the counter-rotating configuration, they cannot cancel each other out. The same applies to the moments induced by the counter-rotating components of the first-order reciprocating forces. As a result, in Scenario 1, where the moments of the rotating component of the first-order reciprocating forces are balanced by counterweights, the counter-rotating counterparts remain unbalanced, generating a maximum imbalance moment. The only way to achieve a perfectly balanced engine in the counter-rotating configuration is to not balance the moments generated by the rotating components of the first-order reciprocating forces using counterweights (Scenario 2).  
Figures 7 and 8 present the resultant moment divided into x-component, y-component, and magnitude for 3-cylinder and 4-cylinder configurations, respectively.