## 3 Engine Modifications and Mechanism Design
### Mechanism Design
[PERSON] et al. [34] synthesized a novel balancing cam mechanism for a single-piston engine to passively compensate resistive torque fluctuations of an engine camshaft at all positions and speeds. The camshaft was comprised of inlet and exhausted cams, each separately connected to a set of rocker, valve, and spring (see Figure 15). Three major assumptions were made in analysis of the camshaft torque: (i) the inlet and exhaust springs had identical geometry, stiffness (\(k\)), and preload (\(P\)); (ii) the engine was well-lubricated and frictionless; and (iii) the rollers were treated as two massless two-force members, with the contact force applying to their centers. Considering the camshaft rotates at a constant speed, the force and moment equations for inlet and exhaust components of the camshaft were derived using the free-body diagram (FBD) of each component:  
Figure 8:4  
Figure 8:5  
Figure 8:6  
Figure 8:7  
Figure 8:8  
Figure 8:9 Schematic of the balancing mechanism [34]. Figure is reprinted with permission from publisher. where subscript \(X\) could be replaced with either inlet or exhaust notations in Figure 16 (subscripts \(i\) and \(e\)). The description of each notation is represented in Figure 10:0  
By analyzing the inlet and exhaust valve displacements for one full revolution of the cam, they showed that large spring force variations were responsible for the fluctuation of the resistive camshaft torque. These dynamic speed fluctuations were recognized as the main vibration source of the camshaft component. To resolve this issue, a balancing mechanism was proposed, consisting of a cam, rocker, spring, and spring guide, similar to the valve structure (see Figure 17). From the FBD of the balancing mechanism and applying the same equations as Equation (7) to (11), the equation of the resistive torque generated by this mechanism was developed. The main purpose of this balancing mechanism was to cancel the resistive torque generated by the valve mechanism, at any rotation angle of \(\theta\), and gives a zero-output torque. To validate and explore the capability of the proposed mechanism, numerical simulations were conducted, and results showed that this methodology could be extended to multi-piston engines. In resemblance to methods of using flywheel, this mechanism was able to operate at all speeds, with no input power, and thus zero energy loss. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Notation** & **Description** & **Notation** & **Description** \\ \hline \(m_{vX}\) & Mass of inlet/exhaust valve & \(\mathbf{F}_{SX}\) & Inlet/exhaust spring force \\ \(\mathbf{F}_{nX}\) & Force from adjustment screw to inlet/exhaust valve & \(\mathbf{F}_{cX}\) & Force from inlet/exhaust cam to rocker \\ \(\mathbf{r}_{aX}\) & Position vector from rocker pivot to roller center & \(\mathbf{r}_{bX}\) & Position vector from rocker pivot to adjustment screw \\ \(I_{X}\) & Moment of inertia of inlet/exhaust rocker & \(\mathbf{r}_{cX}\) & Position vector from inlet/exhaust cam axis \\ \(T_{cX}\) & Required inlet/exhaust torque on camshaft to balance with force \(\mathbf{F}_{cX}\) & \(T\) & Total camshaft torque \\ \(\delta_{X}\) & Inlet/exhaust valve linear displacement & \(\phi_{X}\) & Inlet/exhaust rocker angular displacement \\ \(\mathbf{e}_{X}\) & Unit vector along inlet/exhaust valve & \(\mathbf{k}\) & Unit vector along rotation vector \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Notations used in derivation of the force and moment equations for the camshaft analysis. Figure 16: Free-body diagram (FBD) of the valve, rocker, cam, and roller of the engine camshaft at: (**a**) inlet; (**b**) exhaust [34]. Figures are reprinted with permission from publisher.