# Stiffness-Oriented Structure Topology Optimization for Hinge-Free Compliant Mechanisms Design
## 4 Numerical Results
### Three-Dimensional Steering Mechanism  
In this section, we consider the design of a 3D steering complaint mechanism using topology optimization to illustrate how to focus the optimization progress on interested eigen-frequency and its corresponding eigenmodes by proposed formulation and how to balance the flexibility and stiffness requirement by using weighting coefficient as well. The mechanisms introduced here achieve the functionality that changes the input stroke in the horizontal direction, generated by PZT, to the output stroke in the vertical direction and amplify the input stroke simultaneously. The design domain \(\Omega\) as shown in Figure 6a, a 40 mm \(\times\) 25 mm \(\times\) 15 mm block and another three blocks with gray color are non-design domains. The upper two small gray blocks attached on the upper surface are the adaptors of PZT and a bottom gray block regarded as a support beam. The whole design domain is discretized into 15,300 eight-node cubic elements. Partially fixed supports are defined at left two non-design blocks on the left side surfaces. The PZT, simulated as a spring with constant stiffness 20,000 N/mm, generated force \(\text{f}_{\text{in}}=200\) N is applied in the horizontal direction at the center of two PZT mounting surfaces. Similar to a cantilever mechanism, the first two eigenmodes are swing modal in the vertical and horizontal direction, respectively, as shown in Figure 6b,c. The first eigen-frequency requires more than 1500 Hz to achieve the required bandwidth for system motion control. The output  
Figure 4: (**a**) Design domain and boundary of the inverter mechanism; (**b**) continuous optimized layout with weighting coefficient 10 of the 1 st eigen-frequency; (**c**) continuous optimized layout with weighting coefficient 100 of the 1 st 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** & **(\(\upmu\)Hz)** & **(Hz)** \\ \hline
1 & 10 & 1 & 49.8 & 927 & 2480 \\
2 & 100 & 1 & 32.4 & 1157 & 2613 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Two types of optimized result with different weighting coefficient of the 1 st eigen-frequency. stroke should not be less than 50 um to achieve the flexibility requirement. Accordingly, the objective is to balance these contradictory requirements, the output stroke, and the 1 st eigenfrequencies, simultaneously with a 20% volume constraint. For the elastic workpiece which is simulated as a spring, the constant stiffness of 500 N/mm is assigned. The material considered here is aluminum with Young's modulus **E** = 73 GPa, Poisson's ratio \(\mathbf{v}=0.3\), and density \(\rho=2750\) kg/m\({}^{3}\). Figure 5: Convergence history of output displacement and eigen-frequency for the invent mechanism. Figure 6: (**a**) Design domain and boundary of the steering mechanism; (**b**) the first eigenmode is a swing modal in vertical direction with eigen-frequency 1912 Hz; (**c**) the second eigenmode is a swing modal in horizontal direction with eigen-frequency 2186 Hz. Clearly, the direction of the first eigenmode of the steering mechanism, swinging in the vertical direction, is the same as the required output stroke direction, as shown in Figure 6b. To fulfill the required the 1 st eigen-frequency, not less than 1500 Hz, and output stroke, more than 50 \(\upmu\)m, requirement simultaneously, the desired optimal result should properly balance the stiffness and flexibility requirements.