# Modeling and NVH Analysis of a Full Engine Dynamic Model with Valve Train System
## 3 Modeling and Analysis
### Multi-Body Dynamic Model  
Based on the actual structures and connections of the engine, a simplified multi-body dynamic model is established, as shown in Figure 5. The excitation of the model comes from the pressure in each cylinder. The pressure forces act downward on the pistons and upward on the cylinder head and valves (omitted in Figure 5). The moving parts of the engine include pistons, conrods, crankshaft, valve train, and balance shafts. The forces from the moving parts act on the structures through joints, including liners, bearings, and seats. The structure bodies of the engine include a cylinder block, cylinder head, cylinder head cover, intake manifold, exhaust system, timing cover, and oil pan. The conrod is simplified as a beam element, and the mass is distributed to the big end and the small end as lumped mass points. The piston-liner interaction relationship is simplified as a joint with a spring and a damper, as shown in Figure 6. The piston with a piston ring and piston pin is simplified to a mass point which is connected to the points on the liner thrust side (TS) and anti-thrust side (ATS). Each point-to-point connection consists of a linear spring and a damper. Figure 5: Schematic diagram of the engine model structure. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Components** & **Order** & **Simulation/Hz** & **Measurement/Hz** & **Error** \\ \hline \multirow{6}{*}{Cylinder head cover} & 1 & 128.5 & 133.2 & \(-\)3.5\% \\  & 2 & 204.1 & 201.7 & 1.2\% \\  & 3 & 348.6 & 346.6 & 0.6\% \\  & 4 & 386.6 & 401.1 & 3.6\% \\  & 5 &  
Since the piston's second-order motion will cause a shift in the piston's center of mass, the force in the y direction of the piston can be calculated by the following formula:  
\[F_{p}=-k\Delta u_{y}-c\Delta\dot{u}_{y}, \tag{9}\]  
where \(k\) is the equivalent spring stiffness, \(c\) is the equivalent damping, \(\Delta u_{y}\) is the piston's displacement in the y direction, and \(\Delta\dot{u}_{y}\) is the component of the piston velocity projected in the y direction. The interaction relationship between the conrod big end and the crank pin is simplified as a point-to-point joint with a non-linear spring and a non-linear damper. In local coordinates, the bearing force can be calculated as:  
\[F_{b}=k_{0}\Big{(}\frac{k_{b}}{k_{0}}\Big{)}^{\frac{|\Delta u_{ij}|}{\tau_{0}}} \Delta u_{ij}+c_{0}\Big{(}\frac{c_{b}}{c_{0}}\Big{)}^{\frac{|\Delta u_{ij}|}{ \tau_{0}}}\Delta\dot{u}_{ij}, \tag{10}\]  
where \(\Delta u_{ij}\) is the displacement vector from the center of the conrod big end to the center of the crank pin, \(k_{0}\) and \(c_{0}\) are the equivalent stiffness and damping when \(\left|\Delta u_{ij}\right|=0\), \(k_{b}\) and \(c_{b}\) are the equivalent stiffness and damping corresponding to \(\Delta u_{ij}\), and \(u_{0}\) is the bearing radial clearance. The joint connecting the main bearing and the main journal is very important in transmitting the load from the crankshaft to the engine block, which is significant to the vibration and acoustic calculation. Therefore, the nonlinear spring and damper (NONL) model is used for the main joint, which is more precise and detailed. Considering that the oil film only transmits pressure but not tension, only a part of the oil film is compressed when offset at the center of the main journal. As shown in Figure 7, the compressed oil film accounts for approximately 1/3 of the entire bearing (120 degrees). Figure 6: Simplified piston–liner interaction model. Figure 7: Simplified main journal–main bearing interaction model. Each line in Figure 7 represents a set of spring and damper. Considering the oil leakage at both ends of the main bearing, the oil film pressure should be parabolic in the axial direction.