# Modeling and NVH Analysis of a Full Engine Dynamic Model with Valve Train System
## 2 Theory  
The dynamic model is based on the floating frame of reference (FFoR) formulation [10]. For large-scale displacement, the Lagrangian formulation is employed. The small elastic deformations are considered relative to a local frame of reference, while this local frame of reference undergoes large motions with respect to a fixed global frame of reference [11]. Each discrete mass of the model is governed by Newton-Euler equation of motion. The analysis is performed by the implicit integration method (Newmark) in the time domain [12]. Considering the complexity of the engine model, it needs to be greatly simplified, as shown in Figure 1. Besides the parametric and lumped-mass models, the finite element method (FEM) is commonly used for simplifications. An accurate NVH analysis requires that the element size should not be too small. The lower limit of the wavelength is calculated according to the upper limit of the analysis frequency, and then the density of the element is determined. Due to the complexity of the engine structure of the components, there are hundreds of thousands of elements in the engine FEM model. In order to reduce the DoFs and modes of the matrices to an acceptable level for calculation, component mode synthesis (CMS) is applied in the condensation of the FEM models [13]. Only a few main modes of substructures are retained in the condensed FEM models and participate in the calculations of the equations of motion. The schematic representation of the FFoR is shown in Figure 2. \(e^{T}=(e_{1},e_{2},e_{3})\) and \(b^{T}=(b_{1},b_{2},b_{3})\) denote the global coordinate system and the local coordinate system, respectively. \(\stackrel{{\rightarrow}}{{R}}\) denotes the global body position vector, \(\stackrel{{\rightarrow}}{{\rho}}\) denotes the global coordinate system initial position vector, and \(\stackrel{{\rightarrow}}{{\mu}}\) denotes the elastic displacement of the node \(dm\). The reference position vector can be written as:  
\[\stackrel{{\rightarrow}}{{R}}_{m}=\stackrel{{ \rightarrow}}{{R}}+\stackrel{{\rightarrow}}{{\rho}}+\stackrel{{ \rightarrow}}{{\mu}}=e^{T}\hat{R}+b^{T}\hat{\rho}+b^{T}\hat{\mu}, \tag{1}\]  
where \(\hat{R}\), \(\hat{\rho}\), \(\hat{\mu}\) are the component matrices of \(\stackrel{{\rightarrow}}{{R}}\), \(\stackrel{{\rightarrow}}{{\rho}}\), \(\stackrel{{\rightarrow}}{{\mu}}\). The elastic displacement expanded in modal coordinates can be written as:  
\[\stackrel{{\rightarrow}}{{\mu}}=b^{T}\Phi\varepsilon, \tag{2}\]  
where \(\Phi\) is the eigenvector; \(\varepsilon\) is the modal participation factor. The Lagrange equation is defined as:  
\[L=T-U, \tag{3}\]  
where \(T\) is the kinetic energy; \(U\) is the potential energy. \[T=\frac{1}{2}\int\left|\stackrel{{\rightarrow}}{{v}}_{m}\right| \left|\stackrel{{\rightarrow}}{{v}}_{m}\right|dm=\frac{1}{2}\int \left|\frac{d\stackrel{{\rightarrow}}{{R}}_{m}}{dt}\right|\left| \frac{d\stackrel{{\rightarrow}}{{R}}_{m}}{dt}\right|dm, \tag{4}\]  
\[U=\frac{1}{2}\varepsilon^{T}\Lambda\varepsilon, \tag{5}\]  
Figure 1: Simplification process of the engine model. Figure 2: Schematic representation of the floating frame of reference (FFoR) approach. where \(\overrightarrow{v_{m}}\) is the velocity vector of \(dm\); \(\Lambda\) is the diagonal matrix of the natural frequencies, defined as \(\Lambda=diag\Big{[} \begin{array}{cc}\lambda_{1}^{2}&\lambda_{2}^{2}&\cdots \end{array} \Big{]}\). The basic dynamic equations in the FFoR can be obtained by substituting Equations (4) and (5) into the Lagrange equation:  
\[\frac{d}{dt}\frac{\partial L}{\partial\dot{q}}-\frac{\partial L}{\partial q}= Q_{q}, \tag{6}\]  
where \(Q_{q}\) denotes generalized force, and \(q\) denotes generalized coordinate.