## 3 DASH Interferometer Mount
### Optimizing the Flexible Structure
#### 3.3.3 Optimal Parameters of Flexible Structure  
The dimensions of the flexible structure and its relative phase position can be optimized with the objective that the mechanical stresses of the flexible structure should be less than the tensile strength of the material, and the shear stress at the bonding surface should be less than the shear strength of the epoxy adhesive. The process of optimal design is as follows.  
The spring constant \(K\) at the wing position is inversely proportional to the length \(l\) of the wing, as expressed in Equation (11). However, the length \(l\) of the wing should be increased, according to Equations (12) and (13). Similarly, the spring constant \(K\) at the wing position is proportional to the thickness \(h_{1}\), the number \(n\) and the width \(b\) of the wings. Therefore, the parameters' design is a comprehensive topology optimization process.  
The optimal design of thickness \(h_{2}\) and the angle of the bonding surface \(\theta\) are related to the deformation of the bonding surface \(\omega\). Assuming that the acceleration response of the Mass is 30 g (root mean square value), as per the Pauta criterion (3 sigma guidelines), the Mass will undergo an acceleration (\(a=90\) g). The deformation of the bonding surface (upper and lower) \(\omega\) should be less than 1 \(\upmu\)m.  
Therefore, the optimal values selected for the lower flexible structure are \(h_{1}=1.5\) mm, \(h_{2}=1.75\) mm, \(b=1.6\) mm, \(\alpha=\pi/10\), \(l=2.43\) mm, \(n=2\), and \(\theta=\pi/4\). The optimal values for the upper flexible structure are \(h_{1}=1\) mm, \(h_{2}=1.5\) mm, \(b=1.3\) mm, \(\alpha=\pi/13\), \(l=2.43\) mm, \(n=2\), and \(\theta=\pi/4\).  
The positions (unit is mm) of the four groups of flexible structures and the coordinate position (unit is mm) for the center of mass are given as follows:  
\[\begin{bmatrix}X_{1}&Y_{1}\\ X_{2}&Y_{2}\\ X_{3}&Y_{3}\\ X_{4}&Y_{4}\\ X_{0}&Y_{0}\end{bmatrix}=\begin{bmatrix}-17&17\\ 17&17\\ 17&-17\\ -17&17\\ 20.2&11.5\end{bmatrix} \tag{14}\]  
The spring constants of the lower and upper flexible structures are \(K_{lower}=52,000\) N/mm and \(K_{upper}=12,400\) N/mm, respectively. The spring constant \(K_{lower}\) is permitted to be much stiffer than the spring constant \(K_{lower}\) in the Z direction because the lower spring plays a position-supporting role. The spring constant \(K_{T}\) and the corresponding resonant frequency \(f_{ref}\) are expressed as shown:  
\[f_{ref}=\frac{1}{2\pi}\sqrt{\frac{K_{T}}{m}}, \tag{15}\]  
Figure 10: Phase of flexible structures (90\({}^{\circ}\) phase).  
where \(K_{T}=K_{lower}*K_{upper}/\left(K_{lower}+K_{upper}\right)\) is the spring constant of the flexible structure.  
Table 3 lists the spring constants and frequencies of the DASH interferometer assembly. Therefore, the spring constants of flexible structures are optimized to adjust the natural frequency of the assembly.  
The Mass undergoes an acceleration (\(a=90\) g), which is substituted in Equations (12) and (13). According to the effective quality factor of the vibration mode, the maximum force on the flexible structure is \(F_{max}=62.7\) N. The maximum stress of a single flexible structure \(\tau_{max}\) and the shear stress \(\sigma_{max}\) at the bonding surface between the structure and optical elements can be given below, as follows [24]:  
\[\left\{\begin{array}{ll}\tau_{max}=\frac{3}{2}\frac{F_{max}}{bh_{1}}\\ \sigma_{max}=\frac{m}{5}=\frac{1}{8\pi R^{2}}\end{array}\right., \tag{16}\]  
where \(S\) is the area of bonding surface.  
The maximum shear stress of the flexible structure is 72 MPa, which is significantly lower than the tensile strength of Invar. The maximum shear stress at the bonding surface is 1.4 MPa, which is lower than the shear strength of the epoxy adhesive (8217).  
Additionally, the bending deformations of the bonding surface are 0.4 \(\upmu\)m and 0.5 \(\upmu\)m, respectively.