## 2 Multibody Model of the Valvetrain System
* Natural frequency of the system: \[\omega_{n}=\sqrt{\frac{k(m_{1}+m_{2})}{m_{1}m_{2}}},\] (3)
* Damping factor: \[\zeta=\frac{c}{2}\sqrt{\frac{m_{1}+m_{2}}{km_{1}m_{2}}},\] (4)
* Critical damping coefficient, considering \(\zeta=1\): \[c_{r}=2\sqrt{k\frac{m_{1}m_{2}}{m_{1}+m_{2}}},\] (5)  
The coefficients used to define the contacts between the components of the valvetrain system are presented in Table 2.  
The next important element of the direct drive valvetrain mechanism is represented by the tappet. It is defined by the HLA (hydraulic lash adjuster), a hydraulic element, which allows the hydraulic system to model the properties of the oil, the oil pressure and the amount of air present in it [34]. The operation of this element is based on the force given by the pressure of the high-pressure oil chamber and the depreciation resulting from the leakage of the oil in the high-pressure chamber by the interstitis between the piston and the lower housing [34; 40]. Modelling the spring as a flexible body is absolutely necessary because it has a very important role in the dynamics of the valvetrain mechanism. Thus, the first step in Modelling the spring was to determine its constructive characteristics and its elastic characteristic. The valve springs are cylindrical in shape, its constructive characteristics being presented in Table 3 (Figure 6). \begin{table}
\begin{tabular}{c c c c} \hline
**Case No.** & **Bodies in Liaison** & **Rigidity [N/m]** & **Damping [Ns/m]** \\ \hline
1 & Camshaft–Outer tappet housing & \(7\times 10^{7}\) & 420 \\ \hline
2 & Tappet Housing–Valve & \(4\times 10^{7}\) & 250 \\ \hline
3 & Valve–Valve seat & \(10\times 10^{7}\) & 520 \\ \hline
4 & Valve–Valve retainer & \(3\times 10^{7}\) & 100 \\ \hline
5 & Valve retainer–Spring retainer & \(3\times 10^{7}\) & 100 \\ \hline
6 & Spring washer–Valve guide & \(4\times 10^{7}\) & 250 \\ \hline \end{tabular}
\end{table} 
Table 2: Coefficients of rigidity and damping. \begin{table}
\begin{tabular}{c c c} \hline Free height & \(H_{o}\) [mm] & 44.5 \\ \hline Diameter & \(d\) [mm] & 4 \\ \hline External diameter & \(D\) [mm] & 28 \\ \hline Internal diameter & \(D_{i}\) [mm] & 20 \\ \hline Average diameter & \(D_{m}\) [mm] & 24 \\ \hline The slope of the arch & \(\alpha_{0}\) [\({}^{\circ}\)] & 6 \\ \hline Number of spirals & \(n_{t}\) & 6.75 \\ \hline \end{tabular}
\end{table} 
Table 3: Constructive characteristics of the valve spring. The elastic characteristic of the spring was determined using a spring test stand, Figure 7. The operation of the stand was based on the rotation of a crank through which a worm-driven gear was driven which caused the spring to act on a spring with a certain force corresponding to a weight whose value could be read on the graduated dial. Simultaneously, the value of the spring deformation corresponding to the applied weight could be read on the graduated ruler. Thus, the springs corresponding to one cylinder were removed, two from the inlet valves and two from the exhaust valves, and their elastic characteristics were determined, entering in the table the values corresponding to the applied weights and their deformation (see Table 4). Figure 6: The spring of the valve. Figure 7: Test stand. The elastic characteristic of the spring is determined by the relationship [42]:  
\[c=\frac{F}{f}, \tag{6}\]  
Based on the calculations, an average elasticity resulted: c = 42,000 N/m (Figure 8). The definition of the spring as a flexible body was achieved by using the PDS module (powertrain dynamic simulator) of the Virtual Lab program, which allowed Modelling of the engine or its components, models that could then be used accurately to determine their mechanical and dynamic performance [29; 43; 44].