# Thermo-Mechanical Coupled Analysis of Electric Vehicle Drive Shafts
## 2 Finite Element Method for CV Joint
### Solve Process 1 with Ls-Dyna  
As previously noted, Explicit analysis using LS-DYNA often entails extensive computational time. Therefore, this study focuses on extracting essential thermal characteristics to optimize computational efficiency. Specifically, the dynamic behavior was analyzed for a duration of 0.36 s, corresponding to three rotations under a 500 RPM condition. The thermo-mechanical coupling equations employed in the analysis integrate the fundamental structural equations derived from [PERSON]'s law with the heat transfer equations. In this  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Property** & **Value** & **Unit** \\ \hline Density & 7.85 & g/cm\({}^{3}\) \\ Young’s Modulus & 205 & GPa \\ Yield Stress & 785 & MPa \\ Poisson’s Ratio & 0.29 & \\ Thermal Expansion Coefficient (0-100 \({}^{\circ}\)C) & 12.2 & \(\upmu\)m/m\({}^{\circ}\)C \\ Thermal Conductivity (100 \({}^{\circ}\)C) & 42.6 & W/mK \\ \hline \hline \end{tabular}
\end{table}
Table 1: Mechanical properties for AISI 4140.  
Figure 3: Angle of joint.  
process, the thermal expansion of the material is modeled using the coefficient of thermal expansion (\(\alpha\)), as shown in Equation (1):  
\[\epsilon^{\text{th}}=\alpha(\text{T})(\text{T}-\text{T}_{\text{ref}}) \tag{1}\]  
The deformation due to thermal expansion calculated from Equation (1) is subsequently applied to the total strain calculation using Equation (2):  
\[\{\epsilon\}=\left\{\epsilon^{\text{th}}\right\}+\left[\text{D}\right]^{-1}\{ \sigma\} \tag{2}\]  
Here, [D] represents the elasticity matrix of the material. At each element level, the strain is iteratively adjusted to maintain stress equilibrium with adjacent elements. This iterative process determines the stress and strain distribution within each component. Finally, the degrees of freedom associated with the dynamics and thermal properties of each component are computed and linked through the coupled governing equation shown below. The stress-strain relationship from Equation (2) is applied as the element load (F), ensuring convergence by accounting for material stiffness, thermal expansion, and heat transfer effects.  
\[\begin{bmatrix}\text{M}&0\\ 0&0\end{bmatrix}\begin{Bmatrix}\ddot{\text{u}}\\ \ddot{\text{T}}\end{Bmatrix}+\begin{bmatrix}\text{C}&0\\ 0&\text{C}^{\text{I}}\end{bmatrix}\begin{Bmatrix}\dot{\text{u}}\\ \dot{\text{T}}\end{Bmatrix}+\begin{bmatrix}\text{K}&\text{G}^{\text{T}}\\ \text{G}&\text{K}^{\text{I}}\end{bmatrix}\begin{Bmatrix}\text{u}\\ \text{T}\end{Bmatrix}=\begin{Bmatrix}\text{F}\\ \text{Q}\end{Bmatrix} \tag{3}\]  
[M] = Element mass matrix;  
[C] = Element structural damping matrix;  
[K] = Element stiffness matrix;  
[G] = Thermal expansion matrix;  
[u] = Displacement vector;  
[F] = Combined element nodal force and pressure vectors;  
[C\({}^{\text{I}}\)] = Element specific heat matrix;  
[K\({}^{\text{I}}\)] = Element diffusion conductivity matrix;  
[T] = Temperature vector;  
[Q] = Combined element heat generation rate and convection surface heat flow vector.  
This framework ensures the accurate simulation of the thermo-mechanical behavior of components by coupling the effects of displacement, stress, heat, and thermal expansion in a unified analysis.  
Using these equations, it is possible to calculate structural deformation, heat generation, and deformation caused by thermal effects, allowing the precise simulation of the dynamic behavior of the system under external loads and temperature variations. The computation time for Explicit analysis using Ls-Dyna varies depending on factors such as geometry, mesh density, boundary conditions, and the computer specifications utilized. In this study, the analysis was conducted using Ls-Dyna in combination with Ls-PrePost 2024 R1 (4.11.0) software. The computations were performed on a system equipped with the following specifications: CPU: AMD Ryzen 9 5950X (16-Core), RAM: 128 GB, and GPU: NVIDIA GeForce GTX 1660 Super. The total computation time for the analysis amounted to 17 h, 26 min, and 40 s.