# Stiffness-Oriented Structure Topology Optimization for Hinge-Free Compliant Mechanisms Design
## 4 Numerical Results
### Three-Dimensional Steering Mechanism
Furthermore, to focus the optimization iteration process on the stiffness in the first eigenmode in the vertical direction to make full use of material, different weighting coefficients of the 1 st eigen-frequency are employed as shown in Table 3.  
Figure 7 shows four kinds of topology optimized results using different weighting coefficients of the 1 st eigen-frequency from 1 to 2550. All of the optimized materials are well distributed and the topology layouts are continuous, monolithic, and free of single-node hinge features. The rotation pivots are close to the bottom non-design block and far away from the output point to achieve flexibility requirements. Based on Table 3, as the increasing weighting coefficient of the 1 st eigen-frequency, the optimization iteration process will focus more on enhancing the 1 st eigen-frequency, from 1424 Hz to 2404 Hz, by adding more material on rotation pivot while the output stroke reduced from 83 \(\upmu\)m to 51 \(\upmu\)m. Clearly, the topology optimized results vary with the weighting coefficient and the topology process will focus on the interested eigen-frequency with the help of the weighting coefficient. Case 4 is chosen for further study of this steering mechanism, as Figure 8 shows, the optimal topology of this mechanism was obtained after 37 iterations using the proposed formulation. Figure 7g,h shows two different views of the optimized layout. The optimized design is a well-distributed compliant design and free of single-point hinge features. It clearly shows that the input force from PZT directly transmits to the output port to generate desired vertical stroke via the rotation pivot. There is a slanted bar connected between the output port and the main body to achieve the mentioned stiffness requirement. Figure 8 shows the convergence history of input/output strokes and the interested eigenfrequencies. The initial output stroke is 11 \(\upmu\)m downwards and the final output stroke is 51 \(\upmu\)m downwards which is larger than the required 50 \(\upmu\)m requirement with the final GA of 4.6. The desired stiffness related to the first eigenmode swing in the vertical direction is enhanced and its corresponding eigen-frequency increased from 1252 Hz to 2404 Hz, which is also larger than the required 1500 Hz. Both of them indicate that the proposed method works correctly. Compared with the other three cases, case 4 optimized design added more material to pivot to enhance 1 st eigen-frequency with the help of weighting efficiency. Finally, the 1 st eigen-frequency 2404 HZ is achieved and increased the control bandwidth a lot. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Case** & **1** & **2** & **3** & **4** \\ \hline Weighting coefficient of the 1 st eigen-frequency & 1 & 100 & 1000 & 2250 \\ Weighting coefficient of the MPE & 1 & 1 & 1 & 1 \\ Output Stroke (\(\upmu\)m) & 83 & 74 & 58 & 51 \\ The 1 st eigen-frequency (Hz) & 1424 & 1721 & 2136 & 2404 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Four types of optimized result with different weighting coefficient of the 1 st eigen-frequency. Figure 7: **(a,b)** continuous optimized layout different views with weighting coefficient 1 of the 1 st eigen-frequency; **(c,d)** continuous optimized layout different views with weighting coefficient 100 of the 1 st eigen-frequency; **(e,f)** continuous optimized layout different views with weighting coefficient 1000 of the 1 st eigen-frequency; **(g,h)** continuous optimized layout different views with weighting coefficient 2250 of the 1 st eigen-frequency.