## 2 Structure Design
### Bending Internal Force and Deflection  
To better clarify the simplified model, the structure of the two-axis four-gimbal electro-optical pod structure is divided into five key components for analysis. As shown in Figures 1 and 2, S1 is the spherical cover, S2 is the outer pitch gimbal that is the core component of the simplified model of the cantilever beam, S3 is the fine-stage components (think of it as the load q in the middle of S2), S4 is the voice coil motor that outputs constant torque F, S5 is the end cover that is on the left side and the fixed end of the cantilever beam. What is more, because the rotation angle between the gimbals is relatively small, the torque change is ignored, and its maximum value is taken for analysis. First, the bending internal force of plane bending under external force is analyzed. Moreover, the internal force diagram of bending moment and shear force is drawn by force analysis. In additions, the core problem is to check the deflection error of the simplified model of the cantilever beam. In Figures 2 and 3, suppose that the connection between S2 and S5 is the origin O. Then, establish the Cartesian coordinate system Oxy. The distance of \(l1\), \(l2\), and \(l3\) are shown in Figure 2. \(l1\) is the distance between the fixed end of the left end cover and the fine-stage components. \(l2\) is the length of fine-stage components. \(l3\) is the distance between the fine-stage components and the voice coil motor (VCM). Span H is the sum of \(l1\), \(l2\), and \(l3\). F is the VCM output constant torque. q is the load that is enforced by the fine-stage components S2 in the middle of the cantilever beam S2 (outer pitch gimbal). In Figure 3, the \(x\)-axis is the length of the outer pitch gimbal, which is the simplified model of the cantilever beam. \(F_{x}\)-x and M-x represent the changing states of shear force \(F_{x}\) and bending moment M at different positions, X, of in the cantilever beam. What is more, the internal force diagram of bending moment and shear force is drawn. Figure 3 shows the change of bending moment \(M\) and shear force \(F_{x}\) with \(x\). Figure 1: Ultralight two-axis four-gimbal electro-optical pod and coarse–fine composite system. BLDC: brushless direct current motor; VCM: voice coil motor. Figure 2: The clarifications of the model as simplified to a cantilever beam. Because the deformation of the cantilever beam is very small, the change of the beam's span length after deformation is ignored. Since the example reports a cantilever beam, at the O-point (the fixed end), the bending moment is max. Figure 4a,b presents the changing states of the shear force \(F_{x}\) and bending moment M. Moreover, the material of the beam works within the elastic range of the beam, so the deflection and angle of the beam are linear with the load acting on the beam. Using static equilibrium analysis of material mechanics, the deflection of cantilever beam is calculated by the superposition principle. Because of its complicated force, it is divided into three force forms to solve the equations, respectively, which are finally superimposed together to obtain the deflection curve equation of the coarse-fine composite structure. In Figure 5, due its complexity, the force of the system is divided into three force forms in order to use the superposition principle to solve the equation. What is more, the rigid displacement of the free end A of the cantilever beam is selective analysis. The key is to decompose the load q in the middle into two loads q starting at the origin A. The first is down, the second is up. Figure 4: The internal force diagram. (**a**) \(F_{x}-x\); (**b**) \(M-x\). Figure 5: Superposition principle method static equilibrium analysis. (**a**) Form 1; (**b**) Form 2; (**c**) Form 3. Figure 3: Force analysis of coarse–fine composite structure. As shown in Table 1, the structure distance data obtained after Solidworks software simulation is analyzed.