# Stiffness-Oriented Structure Topology Optimization for Hinge-Free Compliant Mechanisms Design
## 4 Numerical Results
### Gripper Mechanism  
The first example demonstrated here is one of the gripper mechanisms design. The mechanism is to achieve the functionality that two opposite points at the output side move in the vertical direction to grip a workpiece or a cell when a horizontal force or displacement is applied at the input point. The gripper design domain \(\Omega\) as shown in Figure 2a is a 400 mm \(\times\) 400 mm rectangle with a 120 mm \(\times\) 120 mm gap at the output side. It is discretized into 3600 4-node quadrilateral elements. Partial fixed support is defined at both top and bottom corners on the left edge. The input load \(\mathrm{f}_{\mathrm{in}}=100\) N is exerted in the left direction at the input point. The objective is to synthesize the design so that the mechanism can produce the expected gripping force \(\mathrm{f}_{\mathrm{out}}\) or output displacement \(\Delta_{\mathrm{out}}\) to grip the workpiece which is simulated by a spring with constant stiffness \(\mathrm{k}_{\mathrm{out}}=500\) N/mm, furthermore, the design should be stiff enough to sustain the applied load and reaction force from workpieces as well. The material considered here is Aluminum with [PERSON]'s modulus E = 73 GPa, [PERSON]'s ratio \(\mathrm{v}=0.3\), and density \(\rho=2750\) kg/m\({}^{3}\).  
Figure 2b,c provides different optimized topology solutions which are well-distributed compliant designs with different weighting coefficients of the 1 st eigen-frequency and the volume fraction constraint restricted to be 30%. The optimized topology design shows in Figure 2b a continuous, monolithic, flexible, and free of single-node hinge feature. When further increase the weighting coefficient of the 1 st eigen-frequency, as Figure 2c shows the optimization process will switch to enhance the structural stiffness. Compared with the topology optimization result of References [19; 29], the optimization result of Figure 2b,c basically shares a similar topology layout, but the topology layouts of Figure 2b,c are simpler and the load transmission paths are more concise and clear, which means these two designs will be easier to fabricate. The set of design results shown in Table 1 demonstrate that the weighting coefficient has a significant effect on the optimized topology design. The benefit of this is that the designer can conveniently obtain the expected optimized topology layout based on actual design requirements by weighting interested eigen-frequency.  
Figure 3 illustrates the convergence history of output displacement and interested eigen-frequency of the gripper mechanism of case 1. Notice that the maximum displacement of 10.5 \(\upmu\)m is achieved after iteration 35 with a table iteration process. Both 1 st and 2 nd eigen-frequency gradually increased until iteration 55 to achieve their maximum value. It shows that the optimization procedure focuses on the maximum interested eigen-frequency  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Case**} & **Weighting Coefficient** & **Weighting** & **Output Stroke** & **The 1 st** & **The 2 nd** \\  & **of the 1 st** & **Coefficient of the** & **(\(\upmu\)m)** & **Eigen-Frequency** & **Eigen-Frequency** \\  & **Eigen-Frequency** & **MPE** & **MPE** & **(Hz)** & **(Hz)** \\ \hline \(1\) & 10 & 1 & 10.6 & 733 & 2310 \\ \(2\) & 100 & \(1\) & 7.5 & 856 & 2581 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Two types of optimized result with different weighting coefficient of the 1 st eigen-frequency.  
Figure 2: (**a**) Design domain and boundary of the gripper mechanism; (**b**) optimized topology solutions obtained by minimizing synthesis weighting index (SWI) with weighting coefficient 10 of the 1 st eigen-frequency; (**c**) optimized topology solutions obtained by minimizing synthesis weighting index (SWI) with weighting coefficient 100 of the 1 st eigen-frequency.  
after achieving the maximum MPE. The optimization results indicate that the proposed method works correctly.