## 4 Multi-Objective Optimal Structural Design of the Mechanical Amplifier  
Parametric studies of analytical models have been carried out to investigate the influences of key design parameters on the mechanical amplifier performances, respectively [33]. On the other hand, considering this micro-manipulator may be used in some special applications, it is necessary to coordinate and compromise these major design objectives such as displacement ratio, \(R\), input stiffness, \(K_{\text{in}}\), natural frequency, \(f\), the mass, \(T\) and the max stress, \(\sigma\), due to their complicated, variable and nonlinear relationships. When the output displacement produced by the PZT is 0.05 mm, the changing scopes of the cross section height, \(t\), and the radius of the flexure hinge, \(r\), are 0.3-0.9 mm and 2.2-4.5 mm respectively, the relationships of displacement ratio, input stiffness max stress and the first frequency are shown in Figure 3. From Figure 3a, one can find that the input stiffness, \(K_{\text{in}}\), and the max stress, \(\sigma\), is a nonlinear relationship along with the ratio, \(R\), changes. For small values of the ratio, \(R\), the \(\sigma\) increases slowly when \(K_{\text{in}}\) increases, but when \(R\) changes from 0.55 to 0.65 mm, it is noted that the \(\sigma\) increases first and then decreases. In Figure 3b, one can see that the frequency \(f\) sharp increases along with the stiffness, \(K_{\text{in}}\), increases, but it has an inconspicuous change with the ratio, \(R\). That means the objective, \(f\), is more sensitive to the changes in the stiffness, \(K_{\text{in}}\). The similar analysis processes can be carried out with another group of design parameters. When \(l_{x}\) changes from 6 to 16 mm and \(l_{y}\) changes from 2.2 to 2.9 mm, the relationships among these design objectives are different compared with the first group. From Figure 4a, it can be observed that the most significant change is the relationship between \(\sigma\) and ratio \(R\), and the frequency, \(f\), is less sensitive to the changes in other objectives from Figure 4b. Besides, the \(K_{\text{in}}\) decreases in 0.25-0.6 N/\(\upmu\)m along with \(l_{x}\) and \(l_{y}\) changes, this means that the ratio, \(R\), and \(K_{\text{in}}\) are more sensitive to the second parameter group. Figure 3: The relationships among the design objectives with \(r\) and \(t\)**(a)** relationships among \(R\)-\(K_{\text{in}}\)-\(\sigma\)**(b)** relationship among \(R\)-\(K_{\text{in}}\)-\(f\). It can be observed that the multi-objective optimization of the mechanical amplifier is complicated. In this case, the greater the displacement ratio, \(R\), the better. For a given PZT, the output displacement is 0.01 mm and its stiffness, \(K_{\mathrm{PZT}}\), has been determined; therefore the input stiffness, \(K_{\mathrm{in}}\) needs to be smaller than \(K_{\mathrm{PZT}}\). One also desires that the fundamental frequency, \(f\), is greater and the mass, \(T\), is smaller. Consequently this model can be described as:  
**Max:**: \(R\), \(K_{\mathrm{in}}\),\(f\)
**Min:**: \(T=4\rho B\{(\frac{1}{2}l_{{}_{ED}}+l_{{}_{EA}})B_{1}+(l_{x}+l_{{}_{EA}})(l_{y}+2 r+t)+2[l_{x}(2r+t)-\pi r^{2}]\}\)
**Subject to:**  
\( \begin{cases}15\mathrm{mm}\leq l_{x}\leq 20\mathrm{mm}\\ 2\mathrm{mm}\leq l_{y}\leq 3.2\mathrm{mm}\\ 2.0\mathrm{mm}\leq r\leq 4.0\mathrm{mm}\\ 0.6\leq t\leq 1.2\mathrm{mm}\\ 13\leq R\leq 17\\ 80\mathrm{Hz}\leq f\leq 125\mathrm{Hz}\\ K_{\mathrm{in}}\leq K_{\mathrm{PZT}}\end{cases} \)  
There are other kinds of optimization algorithms with respect to the multi-objective optimization problem. In this paper, non-dominated sorting genetic algorithm II (NSGA-II) is employed to solve this problem.