# Modeling and NVH Analysis of a Full Engine Dynamic Model with Valve Train System
## 3 Modeling and Analysis
### Valve Train System Model  
In the uncoupled approach, the dynamic calculation of the valve train system and the dynamic calculation of the engine are carried out separately. The valve train dynamic model is built and calculated independently from the engine; the force results of the joints between the valve train and the engine are loaded onto the engine dynamic model. The disadvantages of the approach are that the impact of the cylinder head deformation to the valve train and the support and constraints of the valve train on the cylinder head are not taken into consideration. The crank speed fluctuation is also neglected or pre-set. In fact, there are important interactions between the valve train and the cylinder head which have significant impacts on the precision of vibration calculation and noise prediction. In the proposed coupled approach, the dynamic models of the valve train and the engine are combined into one model, forming a full engine dynamic model with the valve train. In this model, the dynamic process between the valve train and the engine is fully analyzed, and the engine dynamic results including the valve train are obtained. The challenge of the coupled approach is the reduced efficiency and solver stability due to the increased complexity of the model. The schematic diagram of differences between the uncoupled and coupled approach is shown in Figure 11.  
Figure 10: The load of cylinder 1# piston-liner contact force. Figure 9: The force transmitting through the conrod. The valve train system model consists of two camshafts with variable valve timing (VVT) phasers, eight single intake valve trains, and eight single exhaust valve trains. The valve train type of the engine is the end-pivot rocker arm overhead camshaft (OHC), as shown in Figure 12. The topology of the valve train system is shown in Figure 13.  
Each single valve train (SVT) system model consists of a rocker, a hydraulic lash adjuster (HLA), a valve, and a valve spring, as shown in Figure 14. In each SVT system model, in addition to the joint connected to the camshaft, there are three joints connected to the cylinder head: valve-valve seat joint; valve spring-valve spring seat joint; and HLA-HLA seat joint. Each camshaft is connected to the cylinder head by six radial bearings and two thrust bearings (omitted in Figure 13). The front ends of the two camshafts are connected to the front end of the c  
Figure 11: Differences between the uncoupled and coupled approach. Figure 12: The structure of the intake and exhaust single valve train (SVT) system. Figure 13: The topology of the valve train system. The governing equations of motion are obtained for each element of the lumped masses through the Newton-Euler method and solved together in a time marching scheme [16]. In order to improve the accuracy of the model, the structure of the valve and the valve spring can be split into several discrete mass elements respectively, and the elements are connected in series by springs and dampers. Both the spring and the valve are limited to one-dimensional motion along the valve axis. The mass and stiffness of each discrete element of the spring are defined as [17]:  
\[ \begin{array}{l}m_{i}=2\pi\rho A\Delta N_{i}r_{i}\\ \kappa_{i}=\frac{I_{i}G_{i}\cos(\theta)}{2\pi r_{i}^{3}\Delta N_{i}}\end{array}  \tag{13}\]  
where \(m\) is the mass, \(\rho\) is the density, \(A\) is the cross section area, \(\Delta N\) is the discretized coil number, \(r\) is the radius of the discretized coil, \(\kappa\) is the stiffness, \(I\) is the torsional moment of inertia, \(G\) is the shear modulus, and \(\theta\) is the pitch angle. The rocker of the valve train is the roller finger follower type, which is mostly used in modern passenger cars, and has both rotational and translational DoFs. The overhead cam pushes the roller in the middle of the rocker, with the HLA as the pivot point, driving the valve to open and close. The deformation of the rocker has a great effect on the valve dynamic results.