## 2 Materials and Methods
### Kinematic Design  
The cam profile plays a critical role in determining valve train performance by governing valve motion, acceleration, and contact forces, which in turn influence stress levels within the components. To minimize stress-inducing accelerations, it is essential to reduce abrupt changes in the curvature of the cam profile. The schematic of the kinematic model used in this study is shown in Figure 1. Within the software, cam profile designs were generated to achieve the desired lift values for both intake and exhaust models. Separate cam profiles were created for the intake and exhaust systems of the right and left banks. These profiles were subsequently positioned on the camshaft in accordance with the engine's firing order. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Part** & **Kinematic Model** & **Dynamic Model** & **Mass Percentage** \\ \hline Cam lobe & rigid & rigid & 20.16 \\ Cam follower & rigid & flex & 31.63 \\ Roller & rigid & rigid & 2.27 \\ Pushrod & rigid & multi-mass & 8.19 \\ Rocker arm & rigid & flex & 18.44 \\ Bridge & rigid & flex & 4.02 \\ Valve & rigid & rigid & 10.08 \\ Spring & not defined & multi-mass & 4.04 \\ Elephant foot & not defined & rigid & 1.16 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Part definitions in kinematic and dynamic model. Optimizing valve motion is crucial to achieving the desired engine performance characteristics while mitigating stress concentrations. This requires careful consideration of the dynamic interaction between contact forces and elastic deformation. To address these multifaceted requirements, the design process incorporates acceleration values in the development of the lift profile. In the kinematic model, the cold lash is specified as a constant value when creating the cam profile. In the dynamic model, this clearance is defined between the elephant foot and the bridge. The valve opening ramp zone is for compensating for the specified lash values, as shown by the lift profile acceleration values in Figure 2 (\(0^{\circ}\) crank angle indicates the beginning of the power stroke here and after). Figure 1: Kinematic MBD model of valvetrain. Figure 2: Lift profile acceleration values. The valve acceleration profile also significantly impacts the gas exchange process in the engine. Rapid acceleration and deceleration of the valve can improve airflow into and out of the cylinder, potentially improving engine power and efficiency. However, these aggressive acceleration profiles must be carefully balanced against the adverse effects of excessive valvetrain oscillations, which can compromise the stability and durability of the system [13]. Optimizing the valve acceleration profile involves striking a balance between maximizing engine performance and minimizing negative dynamic effects. This often requires tuning the acceleration curve to suppress oscillations, particularly during the valve deceleration phase. In addition, the valve closing velocity is a key factor affecting the noise, vibration and harshness (NVH) characteristics, as well as the durability of the valve, seat, and spring retainer. The vertical displacement of the cam follower is determined by the cam profile. For a simple harmonic motion cam, the displacement can be expressed as follows:  
\[y_{c}(\theta)=R_{b}(\theta)+R_{f}+rcos(\theta)-\sqrt{(R_{b}(\theta)+R_{f})^{2}- (rsin(\theta))^{2}}, \tag{1}\]  
where \(y_{c}\) is the displacement of the cam follower, \(R_{b}\) is the radius of the cam base circle, \(R_{f}\) is the radius of the roller, \(r\) is the lift of the cam lobe, and \(\theta\) is the angular position of the cam shaft. The pushrod is assumed to be rigid, so its length remains constant. The vertical displacement of the pushrod at the rocker arm end is moving with the ratio \(R_{cp}\) is as follows:  
\[R_{cp}=\frac{L_{pp}}{L_{rp}}, \tag{2}\]  
where \(L_{pp}\) is the horizontal distance from the end of the pushrod to the pivot point and \(L_{rp}\) is the horizontal distance from the roller to the pivot point.