# Deformation Analysis of the Multiplanar Conformal Off-Axis Four-Mirror in an Aerospace Environment

Authors: Haodong Huo; Shijun Ji; Ji Zhao; Rengquan Sun; Handa Dai
Journal: International Journal of Aerospace Engineering
 (2021)

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## Abstract

The effect of maneuver overload on a new type of the multiplanar conformal off-axis four-mirror is studied under the noninertial environment of aerospace. Two kinds of maneuvering flight, horizontal maneuvering and vertical maneuvering, are taken as examples, and the force analysis of the aircraft in the noninertial environment is considered. And the force analysis of the aircraft and the multiplanar conformal off-axis four-mirror is connected by the overload coefficient. Finally, the finite element simulation analysis of the multiplanar conformal off-axis four-mirror is carried out. The results show that the influence of maneuver overload on the multiplanar conformal off-axis four-mirror cannot be ignored. Large overload will result in large deformation of the multiplanar conformal off-axis four-mirror. The deformation is closely related to the structure of the multiplanar conformal off-axis four-mirror. Under the action of lateral force, nonuniform deformation is more likely to occur on the surface of the multiplanar conformal off-axis four-mirror.


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## Body

## 1. Introduction

The large field of view and high-resolution photoelectric sighting system is the “eye” of various physical space-time sensing equipment and the core point of technological breakthroughs in the fields of national defense security, space exploration, and high and new science and technology. Its manufacturing technology is highly regarded by all countries. The use of an off-axis free-form surface is a revolutionary technology to break through the short action distance of a multiband photoelectric sighting system and realize the common caliber, light, and small size. The existing off-axis four-mirror has the advantages of high imaging quality, wide field of view, and no shielding, but it is relatively difficult to manufacture and adjust and is easily affected by an external load. In order to solve these problems, a kind of large field of view multiplanar conformal off-axis four-mirror system is designed. At the same time, the system still has the advantages of an unobstructed center, good imaging quality, large field of view, and compact structure [1, 2].The effects of external loads on optical components have been studied extensively. Banyal and Ravindra [3] considered the structural response caused by heating due to a smooth and gradual increase of the solar flux during the daytime observations and cooling resulting from the exponentially decaying ambient temperature at night for a 2-meter solar telescope’s reflector. Segato et al. [4] performed thermal deformation analysis of the camera optics and their mountings in the expected operating temperature range for the Mercury mission. Lin et al. [5] studied the relationship between mirror optical aberration and gravity effect for different mirror mount designs. Banyal and Ravindra [6] used 3D finite element analysis to simulate the nonuniform temperature distribution and associated thermoelastic distortions of the mirror blank of a lightweight solar telescope and compared three commonly used substrate materials. Song et al. [7] found that in thermal analysis of an optical system, not only thermoelastic deformation should be considered but also refractive index heterogeneity caused by temperature heterogeneity should be considered. Gu et al. [8] proposed a thermal control model for parameter estimation, which can minimize the temperature difference between the primary mirror surface and the surrounding air. By optimizing the integrated flexible support, Zhang et al. [9] reduced the influence of lateral gravity on the optical performance of the mirror, as well as the influence of temperature variation and assembly tolerance. For the study of noninertial systems and maneuver overload, Reddy et al. [10] studied the structural safety of aircraft fuel tanks under different inertial loads. The research of Hou et al. [11] showed that the dynamic load has a significant influence on the ultraharmonic response of the rotor-ball bearing system at subcritical speed. Han and Ding [12] used the finite element method to study the dynamic characteristics of the system during maneuvering with linear damping supports and squeeze film damper supports at the shaft ends, respectively, and the research results showed that squeeze film damper could effectively suppress the amplitude of the transient response. Wei et al. [13] carried out dynamic modeling and analysis on planetary gear transmission under the noninertial system. Li and Xu [14] established the dynamic model of simulator of flight overload expression, and the attitude of the flight simulator under different overload conditions was simulated.The researchers used advanced optimization methods and means to analyze the target parts. Sentyakov et al. [15] selected the optimal parameters of the boring mandrel damping element through the mathematical model and algorithms for the numerical solution to the differential equation. Delyová et al. [16] used a genetic algorithm to optimize truss structure and obtained a more favorable state in terms of weight and stress ratios. Benderskiy and Frankovský [17] studied the change of topological features of the flow structure in the prenozzle volume of the combustion chamber of a power plant with a cylindrical slot channel at the power plant of the mass supply surface along with the change of geometric shape of mass passage. Sága and Jakubovičová [18] evolved the application of nonstationary random vibration theory and evaluated the reliability of the vehicle structure through the response of the nonstationary random function model established by the vehicle model, which proved the accuracy of the theory. The study of Kopas et al. [19] provided a theoretical and practical basis for multiaxial fatigue evaluation of weld joints. By using digital and classical holographic interferometry to analyze how the beam is deformed under the force and how much it can withstand, Černecký et al. [20] concluded that digital holographic interferometry is more convenient, fast, and accurate than classical holographic interferometry. These advanced optimization methods should be extended to the optimization design of mirrors, which will improve the quality of the optimization design of mirrors.Previous studies focused on the influence of temperature and gravity loads on the surface figure accuracy of the mirror, while there were few studies on the deformation of the mirror in the noninertial environment of aerospace during the orbit operation. In this paper, a new type of the multiplanar conformal off-axis four-mirror is taken as the research object. Considering the force state of the aircraft in two kinds of maneuvering flights, namely, horizontal maneuvering and vertical maneuvering, in a noninertial environment, the finite element software is used to calculate, and then, the deformation of each mirror in different states is analyzed. The research results provide a valuable basis for the optimization of multiplanar conformal off-axis four-mirror.


## 2. Finite Element Model

In the course of the flight, the aircraft often needs to make continuous curve flight in different planes, such as horizontal turn, somersault, and roll and dive. Such flight of the aircraft is called maneuvering flight. However, in the process of maneuvering flight, the aircraft is subject to additional inertial force, so it is necessary to introduce an imaginary force—inertial force—into the noninertial reference frame to conduct research using Newton’s law of motion in the noninertial reference frame.


### 2.1. The Establishment of a Noninertial Coordinate System

The noninertial frame of reference moves in a nonuniform straight line relative to the inertial frame of reference, so the airframe coordinate system of the aircraft is taken as the noninertial reference frame, as shown in Figure1. The two coordinate systems in the figure are included: (1) the geodetic coordinate system O−xyz and (2) the airframe coordinate system O0−x0y0z0. When the aircraft is flying, the geodetic coordinate system is used to describe the specific position of the aircraft. Compared with the geodetic coordinate system, the airframe coordinate system of the aircraft is noninertial, which is used to describe the attitude and various maneuverable flights of the aircraft, for example, the rolling motion of the aircraft around its X-axis, yaw motion around its Y-axis, and pitching motion around its Z-axis.Figure 1
Coordinate system.


### 2.2. Overload Introduction

As the core component of the photoelectric sighting system, the multiplanar conformal off-axis four-mirror is assumed to be installed at the position circled in red as shown in Figure1. Local magnification is shown in the upper right corner of Figure 1. The force of the aircraft in the noninertial frame will be transmitted to the multiplanar conformal off-axis four-mirror. In order to describe the actual external force on the aircraft, overload is usually introduced to measure.Overload is the ratio of the resultant surface force acting on a certain direction of the aircraft to the gravity of the aircraft. The overload components along the coordinate axis in theO0−x0y0z0 coordinate system are, respectively, nx=F−D/G, ny=L/G, and nz=Pz/G. F is the thrust of the engine. D, L, and Y are the aerodynamics of the aircraft in the X, Y, and Z directions, respectively. G is the gravity of the aircraft. When the aircraft is maneuvering, the overload in the Y direction is often large, which has the most serious impact on the structure of the aircraft. Therefore, the following studies focus on the analysis of the overload in Y direction while ignoring the overload in X direction and Z direction.


### 2.3. Finite Element Model of the Multiplanar Conformal Off-Axis Four-Mirror

In this paper, the multiplanar conformal off-axis four-mirror is taken as the research object. The plane structure diagram of the multiplanar conformal off-axis four-mirror and the number of four mirrors are shown in Figure2. The local finite element model of the multiplanar conformal off-axis four-mirror is shown in Figure 3. The region of A is the M2 mirror. In order to improve the calculation accuracy, local mesh refinement is carried out. The region of B is in a circumferential direction. It is the key position of mirror support and also the position where deformation is easy to occur, so the transition grid is adopted. The region of C is not the main location, and a larger grid is used. Fine mesh and transition mesh are used in D and E regions, respectively. The total number of elements finally divided into the whole model is 524,500, and the total number of nodes is 804,200.Figure 2
Planar structural graph.Figure 3
Local finite element model.Since SiC has excellent mechanical properties and thermophysical properties, after overcoming the processing difficulties, it has been widely used in the optical field. Therefore, in this paper, CVD SiC is used as the material of the multiplanar conformal off-axis four-mirror for analysis, and its material properties [21, 22] are listed in Table 1. At the same time, the bottom of the multiplanar conformal off-axis four-mirror is completely constrained. Under different maneuverable flight conditions, the load is applied to the model for calculation according to the analysis. By comparing and analyzing the RMS and PV values of each mirror separately deformed, the deformations at different positions of the same mirror, and the relative position changes between adjacent mirrors, the deformation rules of the multiplanar conformal off-axis four-mirror under a noninertial environment are obtained.Table 1
Material properties of the multiplanar conformal off-axis four-mirror.
MaterialDensity (g/cm3)Young’s modulus (GPa)Poisson’s ratioCVD SiC3.214660.21


### 2.4. Definition of Change in Relative Position

In order to better describe the deformation of the multiplanar conformal off-axis four-mirror under maneuver overload, the relative position change of any two adjacent mirrors before and after deformation is introduced. The distance between the vertices of two adjacent mirrors is denoted asdij, and the change of the relative position of the mirror is described by dij after deformation minus dij before deformation. The diagram of the distance between the vertices of adjacent mirrors is shown in Figure 4.Figure 4
The distance between adjacent mirror vertex.


## 2.1. The Establishment of a Noninertial Coordinate System

The noninertial frame of reference moves in a nonuniform straight line relative to the inertial frame of reference, so the airframe coordinate system of the aircraft is taken as the noninertial reference frame, as shown in Figure1. The two coordinate systems in the figure are included: (1) the geodetic coordinate system O−xyz and (2) the airframe coordinate system O0−x0y0z0. When the aircraft is flying, the geodetic coordinate system is used to describe the specific position of the aircraft. Compared with the geodetic coordinate system, the airframe coordinate system of the aircraft is noninertial, which is used to describe the attitude and various maneuverable flights of the aircraft, for example, the rolling motion of the aircraft around its X-axis, yaw motion around its Y-axis, and pitching motion around its Z-axis.Figure 1
Coordinate system.


## 2.2. Overload Introduction

As the core component of the photoelectric sighting system, the multiplanar conformal off-axis four-mirror is assumed to be installed at the position circled in red as shown in Figure1. Local magnification is shown in the upper right corner of Figure 1. The force of the aircraft in the noninertial frame will be transmitted to the multiplanar conformal off-axis four-mirror. In order to describe the actual external force on the aircraft, overload is usually introduced to measure.Overload is the ratio of the resultant surface force acting on a certain direction of the aircraft to the gravity of the aircraft. The overload components along the coordinate axis in theO0−x0y0z0 coordinate system are, respectively, nx=F−D/G, ny=L/G, and nz=Pz/G. F is the thrust of the engine. D, L, and Y are the aerodynamics of the aircraft in the X, Y, and Z directions, respectively. G is the gravity of the aircraft. When the aircraft is maneuvering, the overload in the Y direction is often large, which has the most serious impact on the structure of the aircraft. Therefore, the following studies focus on the analysis of the overload in Y direction while ignoring the overload in X direction and Z direction.


## 2.3. Finite Element Model of the Multiplanar Conformal Off-Axis Four-Mirror

In this paper, the multiplanar conformal off-axis four-mirror is taken as the research object. The plane structure diagram of the multiplanar conformal off-axis four-mirror and the number of four mirrors are shown in Figure2. The local finite element model of the multiplanar conformal off-axis four-mirror is shown in Figure 3. The region of A is the M2 mirror. In order to improve the calculation accuracy, local mesh refinement is carried out. The region of B is in a circumferential direction. It is the key position of mirror support and also the position where deformation is easy to occur, so the transition grid is adopted. The region of C is not the main location, and a larger grid is used. Fine mesh and transition mesh are used in D and E regions, respectively. The total number of elements finally divided into the whole model is 524,500, and the total number of nodes is 804,200.Figure 2
Planar structural graph.Figure 3
Local finite element model.Since SiC has excellent mechanical properties and thermophysical properties, after overcoming the processing difficulties, it has been widely used in the optical field. Therefore, in this paper, CVD SiC is used as the material of the multiplanar conformal off-axis four-mirror for analysis, and its material properties [21, 22] are listed in Table 1. At the same time, the bottom of the multiplanar conformal off-axis four-mirror is completely constrained. Under different maneuverable flight conditions, the load is applied to the model for calculation according to the analysis. By comparing and analyzing the RMS and PV values of each mirror separately deformed, the deformations at different positions of the same mirror, and the relative position changes between adjacent mirrors, the deformation rules of the multiplanar conformal off-axis four-mirror under a noninertial environment are obtained.Table 1
Material properties of the multiplanar conformal off-axis four-mirror.
MaterialDensity (g/cm3)Young’s modulus (GPa)Poisson’s ratioCVD SiC3.214660.21


## 2.4. Definition of Change in Relative Position

In order to better describe the deformation of the multiplanar conformal off-axis four-mirror under maneuver overload, the relative position change of any two adjacent mirrors before and after deformation is introduced. The distance between the vertices of two adjacent mirrors is denoted asdij, and the change of the relative position of the mirror is described by dij after deformation minus dij before deformation. The diagram of the distance between the vertices of adjacent mirrors is shown in Figure 4.Figure 4
The distance between adjacent mirror vertex.


## 3. Force Analysis of the Aircraft in Maneuvering Flight

The maneuvering flight of aircraft in the air mainly includes horizontal turn, somersault, and roll and dive. In this paper, the influence of overload on the deformation of the multiplanar conformal off-axis four-mirror due to horizontal maneuvering and vertical maneuvering of the aircraft is analyzed.


### 3.1. The Force Analysis of Horizontal Maneuvering of the Aircraft

The diagram of horizontal maneuvering of the aircraft is shown in Figure5. θr is the bank angle of the aircraft in a turn. The load acting on the aircraft at this time is also shown in Figure 5. According to D’Alembert’s principle, in the horizontal direction, the horizontal component of the lift force of the aircraft is balanced with the hypothetical inertial centrifugal force Nh=Gv2/gR, so the equilibrium equation of the aircraft in the horizontal direction is Lsinθr−Gv2/gR=0. In the vertical direction, the vertical component of the lift force of the aircraft is balanced with the gravity G of the aircraft, so the equilibrium equation of the aircraft in the vertical direction is Lcosθr−G=0. The two overload coefficients can be derived from the two equilibrium equations, ny=1/cosθr and ny=1+v2/gR1/2. The overload coefficient can be obtained in both formulas. But, for formula ny=1/cosθr, the overload coefficient can be obtained only by determining the bank angle of the aircraft. However, in the actual flight, the lift force that the aircraft can generate is limited by the thrust of the engine and the critical angle of the wing, so the bank angle of the aircraft is limited when it turns. At present, the maximum bank angle range of the aircraft is about 75°~80° when the craft is horizontal maneuvering, and the overload coefficient range is 4~6 by calculation. For the convenience of analysis, the overload coefficient is set as 4, 5, and 6, respectively, for analysis. At this time, the specific bank angle of the aircraft is 75.5°, 78.5°, and 80.4°, respectively.Figure 5
Sketch of horizontal maneuvering.At this time, the aircraft has no angular acceleration around its own axis and no additional overload occurs. It is considered that the overload coefficient of the aircraft is the same as the overload coefficient of the multiplanar conformal off-axis four-mirror. The force analysis of the multiplanar conformal off-axis four-mirror is shown in Figure6. F is the force generated by overload. The overload coefficients are 4, 5, and 6, and the corresponding inclination angles are 75.5°, 78.5°, and 80.4°, respectively, and the deformation of the multiplanar conformal off-axis four-mirror is calculated by finite element method.Figure 6
Force analysis diagram of multiplanar conformal off-axis four-mirror.


### 3.2. The Force Analysis of Vertical Maneuvering of the Aircraft

When the aircraft is maneuvering in the vertical plane, the overload suffered by the aircraft is much more complicated than that suffered by the aircraft in the horizontal maneuvering. The vertical dive of the aircraft can be divided into two stages: entering the dive and pulling up. The overload suffered by the aircraft when entering the dive is much less than that when the aircraft is pulling up. Therefore, in the following, the overload suffered by the aircraft when the aircraft is pulling up is analyzed.Figure7 shows the schematic diagram of the aircraft pulling up in the vertical plane. At this time, the lift force of the aircraft is greater than the radial component of gravity. The difference between the two forces is the centripetal force, which causes the aircraft to generate centripetal acceleration and the track to bend upward. According to D’Alembert’s principle, in the radial direction of the movement of the aircraft, in addition to the lift force L and the gravity component Gcosθ, it is also subjected to the hypothetical inertial centrifugal force Nn=L−Gcosθ, and the equilibrium equation is L−Gcosθ−mv2/r=0. In the tangential direction of the aircraft movement, in addition to the component forces of resistance, thrust, and gravity, there is also a hypothetical tangential inertial force Nτ. Because the tangential acceleration of the aircraft is much smaller than the centripetal acceleration, the tangential inertial force can often be ignored. According to the balance equation in the radial direction, the overload coefficient of the aircraft during pulling up can be deduced as follows: ny=cosθ+v2/gr.Figure 7
Force analysis diagram of aircraft.When the aircraft is pulling up, the anglesθ at each position are different. Assuming that the velocity of the aircraft and the curvature radius of the flight trajectory remain unchanged, it can be known from the overload coefficient equation deduced above that, when the aircraft reaches the lowest position of the flight trajectory, θ=0, cosθ=1; at this time, the corresponding overload coefficient reaches the maximum: ny=1+v2/gr. It is assumed that the maximum overload that the aircraft can bear is 9 g, corresponding to the lowest position of the flight trajectory. Then, calculate the overload coefficient when the angle θ is 10°, 20°, and 30°.It is also considered that the overload coefficient of the aircraft is the same as that of the multiplanar conformal off-axis four-mirror. The force analysis of the multiplanar conformal off-axis four-mirror is shown in Figure8. F is the force generated by overload. The finite element method is used to calculate the deformation of the multiplanar conformal off-axis four-mirror at different positions.Figure 8
Force analysis diagram of the multiplanar conformal off-axis four-mirror.


## 3.1. The Force Analysis of Horizontal Maneuvering of the Aircraft

The diagram of horizontal maneuvering of the aircraft is shown in Figure5. θr is the bank angle of the aircraft in a turn. The load acting on the aircraft at this time is also shown in Figure 5. According to D’Alembert’s principle, in the horizontal direction, the horizontal component of the lift force of the aircraft is balanced with the hypothetical inertial centrifugal force Nh=Gv2/gR, so the equilibrium equation of the aircraft in the horizontal direction is Lsinθr−Gv2/gR=0. In the vertical direction, the vertical component of the lift force of the aircraft is balanced with the gravity G of the aircraft, so the equilibrium equation of the aircraft in the vertical direction is Lcosθr−G=0. The two overload coefficients can be derived from the two equilibrium equations, ny=1/cosθr and ny=1+v2/gR1/2. The overload coefficient can be obtained in both formulas. But, for formula ny=1/cosθr, the overload coefficient can be obtained only by determining the bank angle of the aircraft. However, in the actual flight, the lift force that the aircraft can generate is limited by the thrust of the engine and the critical angle of the wing, so the bank angle of the aircraft is limited when it turns. At present, the maximum bank angle range of the aircraft is about 75°~80° when the craft is horizontal maneuvering, and the overload coefficient range is 4~6 by calculation. For the convenience of analysis, the overload coefficient is set as 4, 5, and 6, respectively, for analysis. At this time, the specific bank angle of the aircraft is 75.5°, 78.5°, and 80.4°, respectively.Figure 5
Sketch of horizontal maneuvering.At this time, the aircraft has no angular acceleration around its own axis and no additional overload occurs. It is considered that the overload coefficient of the aircraft is the same as the overload coefficient of the multiplanar conformal off-axis four-mirror. The force analysis of the multiplanar conformal off-axis four-mirror is shown in Figure6. F is the force generated by overload. The overload coefficients are 4, 5, and 6, and the corresponding inclination angles are 75.5°, 78.5°, and 80.4°, respectively, and the deformation of the multiplanar conformal off-axis four-mirror is calculated by finite element method.Figure 6
Force analysis diagram of multiplanar conformal off-axis four-mirror.


## 3.2. The Force Analysis of Vertical Maneuvering of the Aircraft

When the aircraft is maneuvering in the vertical plane, the overload suffered by the aircraft is much more complicated than that suffered by the aircraft in the horizontal maneuvering. The vertical dive of the aircraft can be divided into two stages: entering the dive and pulling up. The overload suffered by the aircraft when entering the dive is much less than that when the aircraft is pulling up. Therefore, in the following, the overload suffered by the aircraft when the aircraft is pulling up is analyzed.Figure7 shows the schematic diagram of the aircraft pulling up in the vertical plane. At this time, the lift force of the aircraft is greater than the radial component of gravity. The difference between the two forces is the centripetal force, which causes the aircraft to generate centripetal acceleration and the track to bend upward. According to D’Alembert’s principle, in the radial direction of the movement of the aircraft, in addition to the lift force L and the gravity component Gcosθ, it is also subjected to the hypothetical inertial centrifugal force Nn=L−Gcosθ, and the equilibrium equation is L−Gcosθ−mv2/r=0. In the tangential direction of the aircraft movement, in addition to the component forces of resistance, thrust, and gravity, there is also a hypothetical tangential inertial force Nτ. Because the tangential acceleration of the aircraft is much smaller than the centripetal acceleration, the tangential inertial force can often be ignored. According to the balance equation in the radial direction, the overload coefficient of the aircraft during pulling up can be deduced as follows: ny=cosθ+v2/gr.Figure 7
Force analysis diagram of aircraft.When the aircraft is pulling up, the anglesθ at each position are different. Assuming that the velocity of the aircraft and the curvature radius of the flight trajectory remain unchanged, it can be known from the overload coefficient equation deduced above that, when the aircraft reaches the lowest position of the flight trajectory, θ=0, cosθ=1; at this time, the corresponding overload coefficient reaches the maximum: ny=1+v2/gr. It is assumed that the maximum overload that the aircraft can bear is 9 g, corresponding to the lowest position of the flight trajectory. Then, calculate the overload coefficient when the angle θ is 10°, 20°, and 30°.It is also considered that the overload coefficient of the aircraft is the same as that of the multiplanar conformal off-axis four-mirror. The force analysis of the multiplanar conformal off-axis four-mirror is shown in Figure8. F is the force generated by overload. The finite element method is used to calculate the deformation of the multiplanar conformal off-axis four-mirror at different positions.Figure 8
Force analysis diagram of the multiplanar conformal off-axis four-mirror.


## 4. Result Analysis

### 4.1. Deformation Analysis of the Multiplanar Conformal Off-Axis Four-Mirror When the Aircraft Is Horizontal Maneuvering

Figure9 shows the RMS value and PV value of the mirror deformation under different overload conditions when the aircraft is horizontal maneuvering. As can be seen from Figure 9(a), the RMS value of mirror deformation increases with the increase of overload. At the same time, it can be found that the deformation degree of the M4 mirror is the largest, while the deformation degree of the M2 mirror is the least. In addition, the M4 mirror has the largest RMS change of 7 μm under different overloads, while the M1 mirror has the smallest RMS change of only 0.12 μm. It can be seen from Figure 9(b) that the PV value of mirror deformation also increases with the increase of overload. The PV value of the M1 mirror and M3 mirror is larger, while that of the M2 mirror and the M4 mirror is smaller. Under different overloads, the PV values of the M1 mirror and the M3 mirror changed greatly, while those of the M2 mirror and M4 mirror changed little.Figure 9
RMS value and PV value: (a) RMS value; (b) PV value.
(a)(b)As can be seen from Figure6, the component of force F in the vertical direction cancels out with gravity and the horizontal component along with the increasing of overload. The horizontal component of force F is also the resultant force of external forces on the multiplanar conformal off-axis four-mirror. The increase of external force resultant force leads to the increase of mirror deformation so that both RMS value and PV value increase with the increase of overload. The force F of the multiplanar conformal off-axis four-mirror in the y0 direction is much larger than the component of gravity in this direction, so the deformation of the M4 mirror is the largest. However, the deformation of the M2 mirror is the least because it is constrained by the bottom. At the same time, under different overload conditions, the change of the M4 mirror is the largest, while the change of the M2 mirror is the least. For the PV value, it reflects the degree of uneven deformation of the mirror. The external force on the multiplanar conformal off-axis four-mirror can be alleviated by the weak structure of its nonmirror, because the weak structure has a smaller moment of inertia. In addition, since the multiplanar conformal off-axis four-mirror is subjected to a gravitational component in a negative z0 direction and the volume of the M1 mirror is the largest, the degree of nonuniform deformation occurs to the largest extent. The M2 mirror and the M3 mirror are small in volume and close to the base, so the degree of uneven deformation is small. Although the M4 mirror is far from the base, its volume is small, so the degree of uneven deformation is also small.When the overload coefficient is 4, the deformation cloud diagram of each mirror of the multiplanar conformal off-axis four-mirror is shown in Figure10. In the center of the cloud image is the structure of the multiplanar conformal off-axis four-mirror. The arrows in the structure map indicate the viewing directions of the four mirrors and the number of vertices of the mirrors. The letters ABCD in Ai, Bi, Ci, and Di are the different vertices of the mirror, and the subscript i is the number of the mirror. At the same time, the outward deformation of the mirror is negative, and the inward deformation is positive. As can be seen from Figure 10, the deformation of the mirror is not uniform stripes but is uneven stripes. The maximum position of deformation of the four mirrors is A1, D2, A3, and A4, respectively. At the same time, as can be seen from the structure diagram in the center of the cloud, the points of the maximum deformation position are located in the side of the mirrors; this is because the mirrors are subjected to the component of gravity in the negative direction of z0.Figure 10
Deformation cloud diagram of different mirrors.Under different overload conditions, the relative position changes of adjacent mirrors when the aircraft is horizontal maneuvering are shown in Figure11. The change of relative position between the M1 and M2 mirrors is the least, but the change of relative position between M2 and M3 mirrors is the largest, and the value is negative. This indicates that when subjected to overload, M2 and M3 mirrors deform to the inside of the structure, which can be well explained by the force analysis in Figure 9. When the multiplanar conformal off-axis four-mirror is subjected to the force F generated by overload in the direction of y0, this force will stretch the structure and at the same time be constrained by the base, making the M2 mirror and M3 mirror close to each other. It is also the force F that makes the distance between the M3 mirror and the M4 mirror larger.Figure 11
Variation in relative position.


### 4.2. Deformation Analysis of the Multiplanar Conformal Off-Axis Four-Mirror When the Aircraft Is Vertical Maneuvering

Figure12 shows the RMS and PV values of the mirror deformation at three different positions when the aircraft is pulling up. As can be seen from Figure 12(a), although the change trends of the four mirrors are different, the deformation degree of the mirrors is still the largest in the M4 and the smallest in the M2. In addition, M4 has the largest variation of RMS at different angles or positions (0.63 μm), while M1 has the smallest variation of RMS (0.07 μm). As can be seen from Figure 12(b), the PV value of M3 is large, while that of M4 is small. PV values of M1 and M3 changed greatly in different positions, while PV values of M2 and M4 changed slightly. The M1 mirror, alone, has a minimum PV value at the 20° position. Most of these laws are consistent with those of the horizontal maneuvering of the aircraft.Figure 12
RMS value and PV value: (a) RMS value; (b) PV value.
(a)(b)Figure8 is the force analysis diagram of the multiplanar conformal off-axis four-mirror during pulling up. Due to the overload coefficient during pulling up being greater than that during the horizontal maneuvering, the overall deformation of the four mirrors is relatively large. However, it is more coincidental that the deformation degree of the four mirrors deforms to the same degree when the aircraft is pulling up and when the aircraft is horizontal maneuvering, all of which are M4>M1>M3>M2. Similarly, because the force F is much larger than the component of gravity along the y0 direction, the deformation of the M4 mirror is the largest, and it is greatly affected by the position of the aircraft. So the mirror RMS of M4 changes the most when the aircraft is in different positions. For the PV value, the influence of the M1 mirror and M3 mirror is greater than that of the M2 mirror and M4 mirror when it is affected by the component of gravity along the x0 direction. When θ=0, the overload of the aircraft is the maximum, but the RMS and PV values of the four mirrors are not the maximum at this time, but when θ is greater, the RMS and PV values of the M3 mirror are the maximum, which indicates that the lateral force of gravity along the x0 direction is more likely to cause the uneven deformation of the M3 mirror.When the loading position is atθ=30°, the deformation cloud diagram of each mirror of the multiplanar conformal off-axis four-mirror is shown in Figure 13. The definition of the physical quantity in the cloud image is consistent with the horizontal maneuvering of the aircraft. As can be seen from Figure 13, the deformation of the mirrors is not uniform streaks, but uneven streaks, and even rings appear in the M1 and M4 mirrors. In the center of the M4 mirror, there is a region with the largest deformation, while in the center of the M1 mirror, there is a region with relatively small deformation. At the same time, by comparing Figures 9 and 12, it can be found that the RMS value and PV value are larger when the aircraft is pulling up. This is because the overload is greater when the aircraft is pulling up, resulting in more serious deformation of the mirrors. The area with the largest deformation in the center of the mirror M4 is affected by the force F along the y0 direction. While the area with relatively small deformation in the center of the mirror M1 is affected by the component of gravity along the x0 direction. At the same time, the mirror deforms more at the edge and less at the center.Figure 13
Deformation cloud diagram of different mirrors.The relative position changes of adjacent mirrors at different positions when the aircraft is pulling up are shown in Figure14. As can be seen from Figure 14, the change of relative position between mirror 1 and mirror 2 is still the smallest, and the change of relative position between mirror 2 and mirror 3 is still negative. The difference is that the relative position of the adjacent mirrors changes significantly when the aircraft is at the position of θ=10° to 30°. This is because the M4 mirror is far away from the constraint of the base, so the relative change is the largest under the action of force.Figure 14
Variation in relative position.


## 4.1. Deformation Analysis of the Multiplanar Conformal Off-Axis Four-Mirror When the Aircraft Is Horizontal Maneuvering

Figure9 shows the RMS value and PV value of the mirror deformation under different overload conditions when the aircraft is horizontal maneuvering. As can be seen from Figure 9(a), the RMS value of mirror deformation increases with the increase of overload. At the same time, it can be found that the deformation degree of the M4 mirror is the largest, while the deformation degree of the M2 mirror is the least. In addition, the M4 mirror has the largest RMS change of 7 μm under different overloads, while the M1 mirror has the smallest RMS change of only 0.12 μm. It can be seen from Figure 9(b) that the PV value of mirror deformation also increases with the increase of overload. The PV value of the M1 mirror and M3 mirror is larger, while that of the M2 mirror and the M4 mirror is smaller. Under different overloads, the PV values of the M1 mirror and the M3 mirror changed greatly, while those of the M2 mirror and M4 mirror changed little.Figure 9
RMS value and PV value: (a) RMS value; (b) PV value.
(a)(b)As can be seen from Figure6, the component of force F in the vertical direction cancels out with gravity and the horizontal component along with the increasing of overload. The horizontal component of force F is also the resultant force of external forces on the multiplanar conformal off-axis four-mirror. The increase of external force resultant force leads to the increase of mirror deformation so that both RMS value and PV value increase with the increase of overload. The force F of the multiplanar conformal off-axis four-mirror in the y0 direction is much larger than the component of gravity in this direction, so the deformation of the M4 mirror is the largest. However, the deformation of the M2 mirror is the least because it is constrained by the bottom. At the same time, under different overload conditions, the change of the M4 mirror is the largest, while the change of the M2 mirror is the least. For the PV value, it reflects the degree of uneven deformation of the mirror. The external force on the multiplanar conformal off-axis four-mirror can be alleviated by the weak structure of its nonmirror, because the weak structure has a smaller moment of inertia. In addition, since the multiplanar conformal off-axis four-mirror is subjected to a gravitational component in a negative z0 direction and the volume of the M1 mirror is the largest, the degree of nonuniform deformation occurs to the largest extent. The M2 mirror and the M3 mirror are small in volume and close to the base, so the degree of uneven deformation is small. Although the M4 mirror is far from the base, its volume is small, so the degree of uneven deformation is also small.When the overload coefficient is 4, the deformation cloud diagram of each mirror of the multiplanar conformal off-axis four-mirror is shown in Figure10. In the center of the cloud image is the structure of the multiplanar conformal off-axis four-mirror. The arrows in the structure map indicate the viewing directions of the four mirrors and the number of vertices of the mirrors. The letters ABCD in Ai, Bi, Ci, and Di are the different vertices of the mirror, and the subscript i is the number of the mirror. At the same time, the outward deformation of the mirror is negative, and the inward deformation is positive. As can be seen from Figure 10, the deformation of the mirror is not uniform stripes but is uneven stripes. The maximum position of deformation of the four mirrors is A1, D2, A3, and A4, respectively. At the same time, as can be seen from the structure diagram in the center of the cloud, the points of the maximum deformation position are located in the side of the mirrors; this is because the mirrors are subjected to the component of gravity in the negative direction of z0.Figure 10
Deformation cloud diagram of different mirrors.Under different overload conditions, the relative position changes of adjacent mirrors when the aircraft is horizontal maneuvering are shown in Figure11. The change of relative position between the M1 and M2 mirrors is the least, but the change of relative position between M2 and M3 mirrors is the largest, and the value is negative. This indicates that when subjected to overload, M2 and M3 mirrors deform to the inside of the structure, which can be well explained by the force analysis in Figure 9. When the multiplanar conformal off-axis four-mirror is subjected to the force F generated by overload in the direction of y0, this force will stretch the structure and at the same time be constrained by the base, making the M2 mirror and M3 mirror close to each other. It is also the force F that makes the distance between the M3 mirror and the M4 mirror larger.Figure 11
Variation in relative position.


## 4.2. Deformation Analysis of the Multiplanar Conformal Off-Axis Four-Mirror When the Aircraft Is Vertical Maneuvering

Figure12 shows the RMS and PV values of the mirror deformation at three different positions when the aircraft is pulling up. As can be seen from Figure 12(a), although the change trends of the four mirrors are different, the deformation degree of the mirrors is still the largest in the M4 and the smallest in the M2. In addition, M4 has the largest variation of RMS at different angles or positions (0.63 μm), while M1 has the smallest variation of RMS (0.07 μm). As can be seen from Figure 12(b), the PV value of M3 is large, while that of M4 is small. PV values of M1 and M3 changed greatly in different positions, while PV values of M2 and M4 changed slightly. The M1 mirror, alone, has a minimum PV value at the 20° position. Most of these laws are consistent with those of the horizontal maneuvering of the aircraft.Figure 12
RMS value and PV value: (a) RMS value; (b) PV value.
(a)(b)Figure8 is the force analysis diagram of the multiplanar conformal off-axis four-mirror during pulling up. Due to the overload coefficient during pulling up being greater than that during the horizontal maneuvering, the overall deformation of the four mirrors is relatively large. However, it is more coincidental that the deformation degree of the four mirrors deforms to the same degree when the aircraft is pulling up and when the aircraft is horizontal maneuvering, all of which are M4>M1>M3>M2. Similarly, because the force F is much larger than the component of gravity along the y0 direction, the deformation of the M4 mirror is the largest, and it is greatly affected by the position of the aircraft. So the mirror RMS of M4 changes the most when the aircraft is in different positions. For the PV value, the influence of the M1 mirror and M3 mirror is greater than that of the M2 mirror and M4 mirror when it is affected by the component of gravity along the x0 direction. When θ=0, the overload of the aircraft is the maximum, but the RMS and PV values of the four mirrors are not the maximum at this time, but when θ is greater, the RMS and PV values of the M3 mirror are the maximum, which indicates that the lateral force of gravity along the x0 direction is more likely to cause the uneven deformation of the M3 mirror.When the loading position is atθ=30°, the deformation cloud diagram of each mirror of the multiplanar conformal off-axis four-mirror is shown in Figure 13. The definition of the physical quantity in the cloud image is consistent with the horizontal maneuvering of the aircraft. As can be seen from Figure 13, the deformation of the mirrors is not uniform streaks, but uneven streaks, and even rings appear in the M1 and M4 mirrors. In the center of the M4 mirror, there is a region with the largest deformation, while in the center of the M1 mirror, there is a region with relatively small deformation. At the same time, by comparing Figures 9 and 12, it can be found that the RMS value and PV value are larger when the aircraft is pulling up. This is because the overload is greater when the aircraft is pulling up, resulting in more serious deformation of the mirrors. The area with the largest deformation in the center of the mirror M4 is affected by the force F along the y0 direction. While the area with relatively small deformation in the center of the mirror M1 is affected by the component of gravity along the x0 direction. At the same time, the mirror deforms more at the edge and less at the center.Figure 13
Deformation cloud diagram of different mirrors.The relative position changes of adjacent mirrors at different positions when the aircraft is pulling up are shown in Figure14. As can be seen from Figure 14, the change of relative position between mirror 1 and mirror 2 is still the smallest, and the change of relative position between mirror 2 and mirror 3 is still negative. The difference is that the relative position of the adjacent mirrors changes significantly when the aircraft is at the position of θ=10° to 30°. This is because the M4 mirror is far away from the constraint of the base, so the relative change is the largest under the action of force.Figure 14
Variation in relative position.


## 5. Conclusion

In this paper, for a new type of the multiplanar conformal off-axis four-mirror, the deformation of the multiplanar conformal off-axis four-mirror is analyzed in the non-inertial environment when the aircraft is maneuvering in the horizontal plane and vertical plane, and the conclusions are obtained as follows:(1)
The increase of overload will lead to the increase of mirror deformation. A large overload will cause a large deformation of the mirror. When the aircraft maneuvers in the vertical plane, the deformation of the mirror is greater than that of the aircraft maneuvers in the horizontal plane(2)
The deformation is closely related to the structure of the multiplanar conformal off-axis four-mirror. In both cases, the deformation degree of the M4 mirror is the largest, while that of the M2 mirror and M3 mirror is smaller(3)
Lateral force is more likely to cause uneven deformation of the mirror. The uneven deformation of the mirror will be intensified when the volume of the mirror is large and the constraint is small(4)
The research results extend the analytical method for the optimal design of multiplanar conformal off-axis four-mirror. In the future, optimized multiplanar conformal off-axis four-mirror will be manufactured and tested under various loads in the laboratory. If the test is passed, the actual application can be carried out. At the same time, the research results provide valuable basis for the analysis of other components in the space environment



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*Source: 2970501-2021-10-11.xml*