## 2 Problem Description
### Design Parameters
#### 2.2.4 Offset Angle Between the Two Crankshafts  
The OP engine combines the advantage of piston-controlled ports with the benefit of unidirectional scavenging [7]. The optimization of the scavenging process depends on the height of the ports and the angular offset between the intake and exhaust crankshafts (Figure 4) [23]. The offset angle, \(\alpha\), is defined as the phase shift of the exhaust crankshaft relative to the intake crankshaft. A positive angle indicates that the exhaust piston is in advance, and it reaches the dead centers before the intake piston. Since the ports are opened and closed by the piston motion, the opening profile is symmetric with respect to both top and bottom dead centers. For this reason, the exhaust ports must be greater than the intake ports to ensure they open earlier and allow the exhaust gases to exit. However, with zero angular offset, the intake and exhaust ports reach their maximum opening simultaneously, when the pistons are at the BDC. This not only causes the exhaust ports to open earlier but also delays their closure, resulting in fresh charge losses through the exhaust. Disregarding port height, a positive offset delays the opening, and consequently the closing of the intake ports relative to the exhaust ports. This reduces the amount of fresh charge lost through the exhaust and increases the trapping efficiency [23]. Although increasing the offset angle improves the scavenging process and enhances performance, it also introduces imbalance in the engine. This imbalance arises because the resulting forces and moments on each crankshaft become misaligned. Just as a preliminary example, consider only the rotating portion of the first-order reciprocating forces in the 3-cylinder co-rotating configuration, without offset. The two resultant moments are aligned in opposite directions, effectively canceling each other out. However, with a non-zero offset angle, a resultant moment arises in the \(\theta+\frac{n-\pi}{2}\) direction. Figure 5 illustrates the \begin{table}
\begin{tabular}{c|c|c|c|c|c} \hline  & \(\left|\mathbf{F}^{I}_{rec,r}\right|=\) & \(\left|\sum\mathbf{F}^{I}_{rec,r,i}\right|\) & 0 & \(\left|\mathbf{M}^{I}_{rec,r}\right|=\) & \(\left|\sum M^{I}_{rec,r,i}\right|\) & \(\frac{\sqrt{3}}{2}m_{rec}\omega^{2}\tau I_{cyl}\) \\ \hline \(\left|\mathbf{F}^{I}_{rec,r}\right|=\) & \(\left|\sum\mathbf{F}^{I}_{rec,r,i}\right|\) & 0 & \(\left|\mathbf{M}^{I}_{rec,r}\right|=\) & \(\left|\sum M^{I}_{rec,r,i}\right|\) & \(\frac{\sqrt{3}}{2}m_{rec}\omega^{2}\tau I_{cyl}\) \\ \hline \(\left|\mathbf{F}^{II}_{rec,r}\right|=\) & \(\left|\sum\mathbf{F}^{II}_{rec,r,i}\right|\) & 0 & \(\left|\mathbf{M}^{II}_{rec,r}\right|=\) & \(\left|\sum M^{II}_{rec,r,i}\right|\) & \(\frac{\sqrt{3}}{2}m_{rec}\omega^{2}\tau I_{cyl}\) \\ \hline \(\left|\mathbf{F}^{II}_{rec,r}\right|=\) & \(\left|\sum\mathbf{F}^{II}_{rec,r,i}\right|\) & 0 & \(\left|\mathbf{M}^{II}_{rec,r}\right|=\) & \(\left|\sum M^{II}_{rec,r,i}\right|\) & \(\frac{\sqrt{3}}{2}m_{rec}\omega^{2}\tau I_{cyl}\) \\ \hline \(\left|\mathbf{F}^{I}_{rec,r}\right|=\) & \(\left|\sum\mathbf{F}^{I}_{rec,r,i}\right|\) & 0 & \(\left|\mathbf{M}^{I}_{rec,r}\right|=\) & \(\left|\sum M^{II}_{rec,r,i}\right|\) & \(\sqrt{2}m_{rec}\omega^{2}\tau I_{cyl}\) \\ \hline \(\left|\mathbf{F}^{I}_{rec,r}\right|=\) & \(\left|\sum\mathbf{F}^{I}_{rec,r,i}\right|\) & 0 & \(\left|\mathbf{M}^{I}_{rec,r}\right|=\) & \(\left|\sum M^{I}_{rec,r,i}\right|\) & \(\sqrt{2}m_{rec}\omega^{2}\tau I_{cyl}\) \\ \hline \(\left|\mathbf{F}^{I}_{rec,r}\right|=\) & \(\left|\sum\mathbf{F}^{I}_{rec,r,i}\right|\) & 0 & \(\left|\mathbf{M}^{I}_{rec,r}\right|=\) & \(\left|\sum M^{II}_{rec,r,r,i}\right|\) & \(4m_{rec}\omega^{2}\tau I_{cyl}\) \\ \hline \(\left|\mathbf{F}^{II}_{rec,r}\right|=\) & \(\left|\sum\mathbf{F}^{II}_{rec,r,i}\right|\) & 0 & \(\left|\mathbf{M}^{II}_{rec,r}\right|=\) & \(\left|\sum M^{II}_{rec,r,r,i}\right|\) & \(4m_{rec}\omega^{2}\tau I_{cyl}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Resultant forces and moments for the 3-cylinder and 4-cylinder crankshafts.