# Design General Cam Profiles Based on Finite Element Method
## 7 Case Study
### Computation of Cam Flat-Face Follower  
With the flat-face follower, as shown in Figure 4, the function of the follower displacement \(s_{0}\) can be written as follows:  
\[s_{0}=f(\varphi)\, \tag{66}\]  
where \(\varphi\) is the angle of the camshaft, and the displacement function \(f(\varphi)\) is shown in Appendix A. For cam plate, we can describe the tangent vector and others with \(\alpha=1\) and \(\beta=1\). \(\xi^{1}\) is denoted by \(\xi\). From the variation of the stabilized equation (see Equation (26)), \(\mathbf{f}_{\text{stab}}^{\text{e}}\) can be computed as follows:  
\[\mathbf{f}_{\text{stab}}^{\text{e}}=\int_{\Omega_{0}^{\text{e}}}\mu\Big{(} \text{A}^{11}-\ \text{a}^{11}\Big{)}\ \mathbf{N}_{,\xi}^{\text{T}}\ \mathbf{a}_{1}\ \text{dA}_{ \text{e}}. \tag{67}\] \begin{table}
\begin{tabular}{l l} \hline \multicolumn{2}{l}{1. Defined the initial configuration} \\ \hline \multicolumn{2}{l}{2. Step loop} \\ \hline \(\bullet\) & Given parameters: tolerances for the reduced penalty parameter \(\mu\), and \\  & augmentedLagrangian; penalty parameter \(\varepsilon\); parameter \(\lambda\) \\ \hline \multicolumn{2}{l}{2.1. Newton iteration:} \\ \hline \(\bullet\) & Compute \(\mathbf{k}_{\varphi},\mathbf{k}_{\text{stab}},\ \ \mathbf{f}_{\varphi},\ \mathbf{f}_{\text{stab}}\), as shown in Equations (48), (52), and (54) respectively \\ \(\bullet\) & Apply boundary conditions \\ \(\bullet\) & Step 1: Load step for penalty method \\ \(\bullet\) & Step 2: Reduce parameter \(\mu\) \\ \(\bullet\) & Step 3: Update new parameter \(\lambda\) of Augmented Lagrangian method following as equation \\ \(\lambda_{\text{new}}=\epsilon\ \xi+\lambda_{\text{old}}\) & Solve \(\ \mathbf{K}\ \Delta\mathbf{x}^{\text{i}}=-\mathbf{f}(\mathbf{x}^{\text{i}-1})\) and update \(\mathbf{x}^{\text{i}}=\mathbf{x}^{\text{i}-1}+\Delta\mathbf{x}^{\text{i}}\) \\ \hline \(\bullet\) & Update reference configuration \\ \hline \end{tabular}
\end{table} 
Table 1: The computational algorithm. Figure 4: Calculation parameters of cam flat-face follower. Likewise, the tangent matrix is calculated by  
\[\mathbf{k}^{\mathbf{e}}_{\mathrm{stab}}=(\mathbf{k}^{\mathbf{e}}_{\mathrm{stab1}}+ \mathbf{k}^{\mathbf{e}}_{\mathrm{stab2}})\, \tag{68}\]  
where  
\[\mathbf{k}^{\mathbf{e}}_{\mathrm{stab1}}=\int_{\Omega^{\mathbf{e}}_{0}}2\mu\; \mathrm{a}^{11}\;\mathbf{N}^{\mathrm{T}}\;\mathbf{a}_{1}\;\mathbf{a}_{1} \mathbf{N}^{\mathrm{T}}_{,\xi}\;\mathrm{d}A_{\mathrm{e}}\, \tag{69}\]  
As shown in Figure 4, the displacement of the follower can be computed as  
\[\mathbf{s}:=\;\boldsymbol{x}\cdot\mathbf{n}-\;\tau_{0}. \tag{70}\]  
Taking the variation of the current displacement in Equation (70) is presented by  
\[\delta\mathbf{s}=\;\delta\boldsymbol{x}\cdot\mathbf{n}+\boldsymbol{x}\cdot \delta\mathbf{n}. \tag{71}\]  
The angle of the camshaft can be written as (see Figure 4)  
\[\varphi=\arccos(\mathbf{n}\cdot\mathbf{e}_{1}). \tag{72}\]  
The variation of \(\mathbf{s}_{0}\) can be also computed by  
\[\delta\mathbf{s}_{0}=-\mathrm{d}_{\mathrm{c}}\;\mathbf{e}_{1}\cdot\delta \mathbf{n}. \tag{73}\]  
In Equation (73), \(\mathrm{d}_{\mathrm{c}}\) is calculated in Appendix A. From the calculation above, \(\mathbf{f}^{\mathbf{e}}_{\mathbf{p}}\) can be simply obtained by  
\[\mathbf{f}^{\mathbf{e}}_{\mathbf{p}}=\int_{\Omega^{\mathbf{e}}_{0}}\Bigl{(} \mathbf{N}^{\mathrm{T}}\mathbf{n}+\mathbf{N}^{\mathrm{T}}_{,\xi}\;\mathbf{R}^{ \mathrm{T}}_{\mathrm{A}}\,\mathbf{b}\Bigr{)}\;\mathrm{d}A_{\mathrm{e}}\, \tag{74}\]  
where, \(\mathbf{R}_{\mathrm{A}}:=\;\mathbf{R}^{\mathrm{I}}_{\mathrm{A}}\). The tangent matrix is  
\[\mathbf{k}^{\mathbf{e}}_{\mathbf{p}}=\mathbf{k}^{\mathbf{e}}_{\mathbf{p}1}+ \mathbf{k}^{\mathbf{e}}_{\mathbf{p}2}. \tag{75}\]  
Here  
Figure 4:0  
and  
Figure 4:1  
Tensor \(\mathbf{b}\) in Equation (76) and tensors \(\mathbf{G}_{1},\mathbf{G}_{2}\) in Equation (77) are defined in Appendix A. In the example, the cycloid function for the displacement of the follower as shown in Equations (A1) and (A2) in Appendix A is used for the design of the cam profile.