# Multi-Objective Parameter Optimization of Flexible Support System of Optical Mirror
## 2 Modeling of Motion Resistance
### Internal Resistance Modeling of Single Cylinder  
There is no friction resistance in the support cylinder. Its internal resistance is only related to the elastic deformation of the BBR and RD. As shown in Figure 2, the deformation directions of the BBR and RD on the upper and lower chambers are different when the support cylinder moves.  
\[F=\left[\begin{array}{c}H_{1}(d,w,L,h,p_{r},p_{s})\\ H_{2}(d,w,L,h,p_{r},p_{s})\\ P_{1}(R_{p},K,D_{p1},W_{h1},W_{sw},H_{sd})\\ P_{2}(R_{p},K,D_{p1},W_{h1},W_{sw},H_{sd})\\ P_{3}(R_{p},K,D_{p2},W_{h2},W_{sw},H_{sd})\end{array}\right] \tag{3}\]  
where \(d\) is the elastic beam thickness of the BBR, \(w\) is the elastic beam width of the BBR, \(L\) is the effective length of the elastic beam, \(h\) is the bending height, \(p_{r}\) is the bending angle, \(p_{s}\) is the bending position. \(R_{p}\) is the radius of the piston chamfer, \(K\) is the forming height of the RD, \(D_{p1}\) and \(D_{p2}\) are the installation diameters of the small and intermediate RDs, respectively; \(W_{h1}\) and \(W_{h2}\) are the thicknesses of the small and intermediate RDs, respectively; \(W_{sw}\) is the side wall thickness; and \(H_{sd}\) is the RD hardness. The structures of the BBR and RD are shown in Figures 3 and 4, respectively.  
Figure 2: Schematic diagram of support cylinder.  
The movement resistance of the BBR exhibits non-linear relationships with \(d\), \(w\), \(L\), \(h\), \(p_{r}\), and \(p_{s}\). \(w\), \(L\), and \(h\) are determined according to the design of the support cylinder, whereas \(d\), \(p_{r}\), and \(p_{s}\) need to be optimized according to the performance parameters of the support cylinder.  
The movement resistance of the RD exhibits non-linear relationships with \(R_{p}\), \(K\), \(D_{p}\), \(W_{h}\), \(W_{sw}\), and \(H_{sd}\). \(R_{p}\), \(D_{p}\), \(W_{h}\), and \(W_{sw}\) are determined according to the design of the support cylinder, and \(H_{sd}\) need to be optimized according to the performance parameters of the support cylinder.  
\[K=\frac{1}{2}(S_{M}+C+2R_{P}+2W_{h}) \tag{4}\]  
where \(C\) is the distance between the cylinder wall and piston. \(S_{M}\) is the maximum half-stroke:  
\[S_{M}=\text{Max}(S_{A},S_{B}) \tag{5}\]  
where \(S_{A}\) and \(S_{B}\) are the upward and downward strokes, respectively.  
During the operation of the support cylinder, the hydraulic oil in the upper chamber would exert a force on the small RD in the upper cavity and the middle RD, and that in the lower chamber would exert a force on the small RD in the lower chamber and the middle RD. The RD deformations in the upper and lower chambers of the support cylinder differ owing to the pressure difference between these.  
(1) The deformation of the small RD of the upper chamber is:  
\[\Delta L_{su}=\frac{C_{1}K_{1}-C_{1}/2}{2 EW_{h1}}P_{u}+\frac{C_{1}^{2}P_{u}}{4 EW_{h1}} \tag{6}\]  
where \(C_{1}\) is the distance between the cylinder wall and external piston, \(P_{u}\) is the upper chamber pressure, \(K_{1}\) is the height of the small RD, and \(E\) is the elastic modulus of the RD.  
(2) The deformation of the small RD of the lower chamber is:  
\[\Delta L_{sd}=\frac{C_{1}K_{1}-C_{1}/2}{2 EW_{h1}}P_{d}+\frac{C_{1}^{2}P_{d}}{4 EW_{h1}} \tag{7}\]  
where \(P_{d}\) is the lower chamber pressure.  
(3) The deformation of the middle RD is:  
\[\Delta L_{b}=\frac{C_{2}K_{2}-C_{2}/2}{2 EW_{h2}}(P_{u}-P_{d})+\frac{C_{2}^{2}( P_{u}-P_{d})}{4 EW_{h2}} \tag{8}\]  
where \(C_{2}\) is the distance between the cylinder wall and internal piston. \(K_{2}\) is the height of the large RD.  
Figure 4: Structural parameters of rolling diaphragm (RD).  
Figure 3: Structural parameters of beryllium bronze reed (BBR).