Flexible support device for airborne free-form surface reflector

By designing a flexible support device of the free curved mirror reflex mirrors with polygonal frame and flexible legs, the problem of poor support structure stability and large volume weight in existing technologies, realizing high -precision, high stability and compact types Optical systems improve the quality and reliability of imaging.

CN115308873BActive Publication Date: 2025-01-24SHAANXI ARTS & CRAFTS SPECTRUM TECH CO LTD
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Patent Information

Application Number
CN202210727085.8
Authority / Receiving Office
CN · China
Patent Type
Patents(China)
Current Assignee / Owner
Filing Date
2022-06-24
Publication Date
2025-01-24
Estimated Expiration
2042-06-24

AI Technical Summary

Technical Problem

The support structure of the existing free-curved mirror has problems of poor stability and low accuracy, which affects the imaging quality of optical remote sensing instruments. In compact optical systems, the mirror has a large volume and heavier weight, making it inconvenient to use.

Method used

A flexible support device of the free curved mirror of the vehicle is designed. It uses a polygonal frame and multiple flexible legs. The flexible legs are the same as the BIPOD structure. It can release stress during temperature changes and improve stability. The device also sets multiple flexible shields and small glue pores on the back of the frame. The reflector is fixed through epoxy adhesion to reduce the coherent curing stress.

Benefits of technology

The support of high accuracy and high stability has been achieved, the volume and weight of the optical system is reduced, and the space of phase adjustment and installation is provided, which improves the reliability of imaging quality and device, and reduces the launch cost.

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Abstract

The present invention belongs to the technical field of flexible support for airborne reflectors, and specifically relates to a flexible support device for airborne free-form surface reflectors, comprising a mirror frame and a plurality of flexible legs arranged on the back of the mirror frame; the mirror frame is a polygonal axisymmetric structure with a central square hole at the center; the flexible legs comprise a connector and two flexible legs respectively connected to the connector; the flexible legs comprise a first flexible structure, a leg, a second flexible structure and an external mounting interface of the flexible legs connected in sequence; the first flexible structure is connected to the back of the mirror frame through a connector; a flexible groove is also arranged on the leg; and a trimming pad is also arranged between the mirror frame and the connector. The present invention has high precision and high stability, releases assembly stress and temperature stress, and has the advantages of compact structure, easy installation and integration.
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Description

Technical Field

[0001] The invention belongs to the technical field of flexible support for airborne reflectors, and in particular relates to a flexible support device for an airborne free-form surface reflector. Background Art

[0002] With the continuous development of optical remote sensing technology, the technical indicators of airborne remote sensing instruments are constantly improving, and the design difficulty of optical systems is becoming increasingly higher. In order to achieve higher instrument design indicators and better optical imaging quality, the requirements for the mirror shape of each optical element in the optical system are becoming more and more complex. Therefore, optical free-form surface technology has gradually become a hot and cutting-edge issue in the development of optical instruments.

[0003] Optical free-form surface technology can achieve better aberration correction and improve imaging quality in optical systems by designing the mirror shape of optical elements into free-form surfaces; reduce the number and weight of optical elements in the system and increase the system's light flux; and achieve better optical path folding and more flexible optical design. However, free-form surface optical elements are different from common optical elements and are often core elements in optical systems. Existing free-form surface reflectors have the following problems: (1) The existing support structure has poor stability and low precision, which directly affects the imaging quality of optical remote sensing instruments; (2) In existing free-form surface reflectors, the effective aperture is mostly elliptical, which makes the reflector large and heavy in size and inconvenient to use in large-sized compact optical systems; (3) In existing free-form surface reflectors, the phase adjustment and installation space of optical elements are limited, which brings inconvenience to installation and adjustment. Summary of the invention

[0004] In view of the technical problems of poor precision and stability of existing support devices for reflectors, the present invention provides an airborne free-form surface reflector flexible support device with high precision and high stability, which can release assembly stress and temperature stress, and has the advantages of compact structure, easy installation and integration.

[0005] In order to achieve the above object, the technical solution adopted by the present invention is:

[0006] A flexible support device for an airborne free-form surface reflector comprises a mirror frame and a plurality of flexible legs arranged on the back of the mirror frame; the mirror frame is a polygonal axisymmetric structure with a central square hole at the center.

[0007] Furthermore, the flexible legs include a connecting piece and two flexible feet respectively connected to the connecting piece; the flexible feet include a first flexible structure, a foot, a second flexible structure and an external installation interface of the flexible legs connected in sequence; the first flexible structures are connected to the back of the frame through a connecting piece; a flexible groove is also provided on the foot; and a trimming pad is also provided between the frame and the connecting piece.

[0008] Furthermore, there are three flexible legs, which are distributed on the back of the frame and the connecting line of the three flexible legs forms an isosceles triangle structure.

[0009] Furthermore, the airborne free-form surface reflector flexible support device also includes a plurality of flexible baffles disposed on the back of the mirror frame; the plurality of flexible baffles are staggered with the plurality of flexible legs.

[0010] Furthermore, the flexible baffle includes a sheet body and a horizontal flexible groove and a vertical flexible groove respectively arranged on the sheet body; the horizontal flexible groove is axially perpendicular to the vertical flexible groove; the sheet body is connected to the back of the frame; the horizontal flexible groove and the vertical flexible groove are each one or more.

[0011] Furthermore, the horizontal flexible groove includes a first horizontal flexible groove, a second horizontal flexible groove, a third horizontal flexible groove and a fourth horizontal flexible groove; the vertical flexible groove includes a first vertical flexible groove; the first horizontal flexible groove is axially parallel to the second horizontal flexible groove, and the first vertical flexible groove extends axially from the edge of the flexible baffle to the center of the flexible baffle, passes through the second horizontal flexible groove, and is connected to the first horizontal flexible groove; the third horizontal flexible groove and the fourth horizontal flexible groove are symmetrically distributed along the axis of the first vertical flexible groove and extend to the edge of the flexible baffle, and the third horizontal flexible groove is axially parallel to the first horizontal flexible groove.

[0012] Furthermore, there are three flexible baffles, which are distributed on the back of the frame, and the center points of the three flexible baffles are connected to form an equilateral triangle structure; the vertices of the isosceles triangle structure are located below the base of the equilateral triangle structure, and the line connecting the vertices of the isosceles triangle structure and the vertices of the equilateral triangle structure is parallel to the vertical axis of the frame, and the base of the isosceles triangle structure is located below the base of the equilateral triangle structure.

[0013] Furthermore, the airborne free-form surface reflector flexible support device also includes a plurality of small glue injection holes arranged on the outer side wall of the mirror frame and connected to the central square hole.

[0014] Furthermore, a glue overflow groove is arranged on the hole wall of the central square hole, and the glue overflow groove is in a cross shape.

[0015] Furthermore, there are two glue overflow grooves, which are symmetrically distributed on the hole wall of the central square hole along the central axis of the frame.

[0016] The beneficial effects of the present invention are:

[0017] 1. The optical element of the present invention is processed into a polygonal mirror body, which can reduce the volume and weight of the optical system on the one hand, and provide more sufficient space for phase adjustment and installation of the optical element on the other hand, making adjustment and installation convenient.

[0018] 2. The airborne free-form surface reflector flexible support device proposed in the present invention is provided with multiple flexible legs as flexible supports, and the flexible legs have the same structure as the bipod legs, which is convenient for stress release of the entire device when the temperature changes greatly, thereby improving the stability of the device.

[0019] 3. The flexible support device for an airborne free-form surface reflector proposed in the present invention has multiple flexible baffles on the back of the mirror frame, and each flexible baffle is provided with multiple flexible grooves in different directions to assist in supporting the fixation of the free-form surface reflector. The structure is compact and the reliability of the device is improved. In addition, the weight of the device is reduced through the lightweight design of the reasonable weight-reducing grooves, thereby reducing the launch cost.

[0020] 4. The airborne free-form surface reflector flexible support device provided by the present invention adopts a small-point porous epoxy adhesive bonding method to fix the free-form surface reflector, which reduces the influence of the epoxy adhesive bonding and curing stress on the mirror group surface shape, and is provided with a reasonable glue overflow groove to better ensure the thickness uniformity of the glue layer and improve the mechanical stability of the device. BRIEF DESCRIPTION OF THE DRAWINGS

[0021] Figure 1 A front axonometric view of the flexible support device for an airborne free-form surface reflector provided by the present invention;

[0022] Figure 2 A rear axonometric view of the flexible support device for an airborne free-form surface reflector provided by the present invention;

[0023] Figure 3 A three-dimensional schematic diagram of the mirror frame provided by the present invention;

[0024] Figure 4 A three-dimensional schematic diagram of the flexible legs provided by the present invention;

[0025] Figure 5 A three-dimensional schematic diagram of the flexible baffle provided by the present invention;

[0026] in:

[0027] 1—mirror frame; 11—small glue injection hole; 12—vertical glue overflow groove; 13—horizontal glue overflow groove; 14—connection hole; 15—blocking piece connecting hole; 16—weight reduction groove; 17—center square hole; 2—free-form surface reflector; 3—flexible support leg; 31—mirror frame connecting hole; 32—first flexible structure; 33—flexible groove; 34—second flexible structure; 35—flexible support leg external mounting interface; 4—trimming pad; 5—flexible baffle; 51—mounting interface; 52—first horizontal flexible groove; 53—second horizontal flexible groove; 54—third horizontal flexible groove; 55—fourth horizontal flexible groove; 56—first vertical flexible groove. DETAILED DESCRIPTION

[0028] The present invention is described in detail below with reference to the accompanying drawings and embodiments.

[0029] Example 1

[0030] See also Figure 1 to Figure 3 The airborne free-form surface reflector flexible support device provided in this embodiment includes a mirror frame 1 and a flexible support leg 3 disposed on the back of the mirror frame 1 .

[0031] In this embodiment, the frame 1 is an octagonal structure with axial symmetry, and the vertical height of the frame 1 is greater than the horizontal width. A central square hole 17 is provided at the center of the frame 1; the free-form surface reflector 2 is embedded in the central square hole 17, and the mirror surface of the free-form surface reflector 2 is located on the front of the frame 1. An overflow glue groove is provided on the hole wall of the central square hole 17, and the overflow glue groove is in a cross shape. There are two overflow glue grooves, which are symmetrically distributed on the hole wall of the central square hole 17 along the central axis of the frame 1 in the vertical direction. A plurality of small glue injection holes 11 connected to the central square hole 17 are provided on the outer wall of the frame 1.

[0032] In this embodiment, the optical element is processed into a polygonal square mirror body, which can reduce the volume and weight of the optical system on the one hand, and provide more sufficient space for phase adjustment and installation of the optical element on the other hand, making adjustment and installation more convenient.

[0033] Specifically, the outer contour of the central square hole 17 is similar to that of the free-form surface reflector 2, but its size needs to ensure that after the free-form surface reflector 2 is installed, there is still a gap of 0.1mm±0.05mm on one side, and this gap is used to inject 2216 epoxy glue.

[0034] Specifically, a plurality of small glue injection holes 11 are symmetrically arranged on the left and right sides of the hole wall of the central square hole 17 in the frame 1, which are used to inject epoxy glue 2216 for bonding the mirror body, thereby bonding and fixing the frame 1 to the free-form surface reflector 2. The diameter of the small glue injection holes 11 does not exceed 3 mm, and the number of the small glue injection holes 11 needs to be calculated based on the weight and mechanical conditions of the free-form surface.

[0035] Specifically, the glue overflow groove is a cross shape, specifically including a vertical glue overflow groove 12 and a horizontal glue overflow groove 13 that intersect vertically. The axial direction of the vertical glue overflow groove 12 is parallel to the side of the frame 1, and the axial direction of the horizontal glue overflow groove 13 is the same as the central square hole 17. Its function is to overflow the excess epoxy glue injected from the small glue injection hole 11 to ensure that the thickness of the glue layer is more uniform, thereby improving the bonding strength.

[0036] Specifically, the frame 1 is provided with three connection holes 14 for mounting the three flexible legs 3 on the frame 1, and the frame 1 is provided with three baffle connection holes 15 for conveniently connecting and fixing the three flexible baffles 5 to the frame 1. A plurality of frame weight-reducing grooves 16 are provided on the outer wall of the frame 1 for reducing weight and lowering launch costs.

[0037] In this embodiment, the free-form surface reflector 2 is processed into a square mirror body, the back of which is a flat surface and the front surface is a free-form surface.

[0038] In this embodiment, there are three flexible legs 3, which are distributed on the back of the frame 1 and the connection line of the three flexible legs 3 forms an isosceles triangle structure. Specifically, the three flexible legs 3 are respectively arranged at the center of the top edge, the center of the lower left edge, and the center of the lower right edge of the frame 1, and the three flexible legs 3 form an isosceles triangle structure, the flexible leg 3 at the center of the top edge is the vertex of the isosceles triangle structure, and the connection line of the flexible leg 3 at the center of the lower left edge and the flexible leg 3 at the center of the lower right edge is the base of the isosceles triangle structure.

[0039] The airborne free-form surface reflector flexible support device provided in this embodiment is further provided with a flexible baffle 5 on the back of the mirror frame 1 .

[0040] In this embodiment, there are three flexible baffles 5, which are distributed on the back of the frame 1, and the connection line of the three flexible baffles 5 forms an equilateral triangle structure. Specifically, the three flexible baffles 5 are respectively arranged at the center of the bottom side, the center of the left side, and the center of the right side of the frame 1, and the three flexible baffles 5 form an equilateral triangle structure, the flexible baffle 5 at the center of the bottom side is the vertex of the equilateral triangle structure, and the connection line of the flexible baffle 5 at the center of the left side and the flexible baffle 5 at the center of the right side is the bottom side of the equilateral triangle structure.

[0041] The three flexible baffles 5 and the three flexible legs 3 are staggered. Specifically, the vertices of the isosceles triangle structure formed by the three flexible legs 3 are located below the base of the equilateral triangle structure formed by the three flexible baffles 5, and the line connecting the vertices of the isosceles triangle structure and the vertices of the equilateral triangle structure is parallel to the vertical axis of the frame 1, and the base of the isosceles triangle structure is located below the base of the equilateral triangle structure.

[0042] In this embodiment, the flexible legs 3 serve as external mounting connection devices for the flexible support device of the free-form surface reflector, and the three flexible baffles 5 are ground to ensure that their equal thickness tolerances are within 0.01 mm, thereby assisting in the fixation of the free-form surface reflector 2 and improving the stability and reliability of the free-form surface reflector 2. In this embodiment, a total of 6 flexible structures are installed on the back of the mirror frame 1, which can be used to release the assembly stress and temperature stress of the mirror group.

[0043] In the above implementation, three flexible baffles 5 and three flexible legs 3 are provided, and the three flexible legs 3 form a triangle, and the three flexible baffles 5 also form a triangle, and the triangular structure has good stability; however, in implementation, the number of flexible baffles 5 and flexible legs 3 is multiple, and can be designed according to the shape of the frame 1. While achieving stable support, the assembly stress and temperature stress of the assembly are released through the flexible groove.

[0044] Example 2

[0045] Based on Example 1, see Figure 4 The flexible legs 3 are bipod flexible legs, specifically including a connecting piece and two flexible feet respectively arranged on the connecting piece; the flexible feet include a first flexible structure 32, a foot, a second flexible structure 34 and a flexible leg external mounting interface 35 connected in sequence; the first flexible structure 32 is connected to the back of the frame 1 through a connecting piece; a flexible groove 33 is also provided on the foot; a trimming pad 4 is also provided between the frame 1 and the connecting piece.

[0046] Specifically, the frame 1 is provided with three connection holes 14 for the flexible legs 3 , and the connecting piece of the flexible legs 3 is provided with a frame connection hole 31 . The frame 1 and the flexible legs 3 are connected and fixed via the connection holes 14 and the frame connection holes 31 .

[0047] Two flexible legs with the same structure are connected to the connecting piece of the flexible leg 3. The angle formed by the two flexible legs is α, 0°<α<180°. The structure of the flexible leg is similar to that of a bipod leg, and a first flexible structure 32 and a second flexible structure 34 are respectively provided. A flexible groove 33 is provided on the leg. The first flexible structure 32 is connected to the connecting piece, and the second flexible structure 34 is connected to the flexible leg external mounting interface 35. The flexibility direction formed by the first flexible structure 32 and the second flexible structure 34 is inconsistent, that is, the thickness direction of the thinnest part of the first flexible structure 32 and the thickness direction of the thinnest part of the second flexible structure 34 are perpendicular, and the thickness directions of the thinnest parts of the two are inconsistent, and the flexibility is different. The function of the two flexible structures is to make the flexible leg 3 flexible in more directions, so as to better play the role of stress release. The main function of the flexible groove 33 is to make the entire device flexible in multiple directions, and to maintain high position accuracy, high posture accuracy and high surface accuracy in complex force and thermal environment. In addition, the flexible groove 33 also serves as the external interface of the entire device, which is convenient for the integrated assembly of the entire device with other components.

[0048] During implementation, the angle between the two flexible legs is optimized and designed based on the size of the frame 1 and the number of flexible legs 3 selected, and the finite element analysis results are used, but the angle is within the range of 0°<α<180°; the parameters of the first flexible structure 32 and the parameters of the second flexible structure 34. For example, the groove depth, groove width, thinnest thickness and other data need to be iteratively optimized through the optimization algorithm to find the best value. The best value of the flexible structure is related to the size, size, material and mechanical conditions of the frame 1, and is further optimized according to the actual situation during use.

[0049] Furthermore, the airborne free-form surface reflector flexible support device provided in this embodiment further includes trimming pads 4 respectively arranged between the mirror frame 1 and the connecting member. The trimming pads 4 are used to ensure that the coplanarity of the flexible legs and the external mounting interface 35 is less than 0.01 mm by trimming the thickness thereof, thereby ensuring the installation position accuracy and attitude accuracy of the entire device when it is integrated with other components.

[0050] Example 3

[0051] Based on Example 2, see Figure 5 The flexible baffle 5 provided in this embodiment includes a sheet body and a horizontal flexible groove and a vertical flexible groove respectively arranged on the sheet body; the horizontal flexible groove is axially perpendicular to the vertical flexible groove; the sheet body is connected to the back of the frame 1; the horizontal flexible groove and the vertical flexible groove are both one or more.

[0052] This embodiment provides a preferred implementation of a flexible baffle 5 .

[0053] During implementation, the sheet body is a rectangular structure, and the horizontal flexible grooves include a first horizontal flexible groove 52, a second horizontal flexible groove 53, a third horizontal flexible groove 54 and a fourth horizontal flexible groove 55; the vertical flexible grooves include a first vertical flexible groove 56; the first horizontal flexible groove 52 is axially parallel to the second horizontal flexible groove 53, and the first vertical flexible groove 56 extends axially from the edge of the flexible baffle 5 to the center of the flexible baffle 5, passes through the second horizontal flexible groove 53 and then connects with the first horizontal flexible groove 52; the third horizontal flexible groove 54 and the fourth horizontal flexible groove 55 are symmetrically distributed along the axial direction of the first vertical flexible groove 56 and extend to the edge of the flexible baffle 5, and the third horizontal flexible groove 54 is axially parallel to the first horizontal flexible groove 52.

[0054] Specifically, the flexible baffle 5 is formed by wire cutting and has a rectangular shape. The flexible baffle 5 is provided with two mounting interfaces 51, and the frame 1 is provided with three baffle connecting holes 15, through which the frame 1 and the flexible baffle 5 are connected and fixed.

[0055] Specifically, the first horizontal flexible groove 52 is parallel to the second horizontal flexible groove 53, the first vertical flexible groove 56 is axially perpendicular to the second horizontal flexible groove 53, the first vertical flexible groove 56 starts from the center point of one of the long sides of the flexible baffle 5, extends along the short side direction to the center of the flexible baffle 5, first passes through the center of the second horizontal flexible groove 53, and merges and connects with the middle of the first horizontal flexible groove 52. The third horizontal flexible groove 54 is axially coaxial with the fourth horizontal flexible groove 55, the third horizontal flexible groove 54 and the fourth horizontal flexible groove 55 are symmetrically arranged on both sides of the first vertical flexible groove 56 and extend outward from the first vertical flexible groove 56 along the long side of the flexible baffle 5 to the edge of the short side of the flexible baffle 5, the third horizontal flexible groove 54 is axially parallel to the first horizontal flexible groove 52, and the two mounting interfaces 51 are respectively distributed on the outside of the two ends of the first horizontal flexible groove 52.

[0056] In this embodiment, a first horizontal flexible groove 52, a second horizontal flexible groove 53, a third horizontal flexible groove 54, a fourth horizontal flexible groove 55 and a first vertical flexible groove 56 are arranged on the flexible baffle 5. The main function of these five flexible grooves is to make the flexible baffle 5 flexible in multiple directions, which is convenient for releasing complex temperature stress and can also overcome the disadvantage that the common single-flexibility baffle is insufficient in releasing stress; it can better release temperature stress and prevent the flexible baffle 5 from squeezing the free-form surface reflector 2 due to the inconsistent thermal expansion coefficients of different materials when the temperature changes drastically, thereby increasing the shear load of the bonding epoxy glue and affecting the reliability of the entire device.

[0057] The present invention proposes a flexible support device for an airborne free-form surface reflector, which uses a small-point porous epoxy adhesive bonding method to fix the free-form surface reflector, thereby reducing the influence of the epoxy adhesive curing stress on the mirror group surface shape, and is provided with a reasonable glue overflow groove to better ensure the thickness uniformity of the glue layer and improve the mechanical stability of the device; multiple flexible legs are provided to facilitate the stress release of the entire device in a complex mechanical and thermal environment, thereby improving the mechanical and thermal stability of the device; multiple back flexible baffles are provided to assist in supporting the fixation of the free-form surface reflector and to release residual stress, the device has a compact structure, and the reliability of the device is improved. In addition, through a reasonable lightweight design, the weight of the device is reduced, thereby reducing the launch cost.

Claims

1. A flexible support device for an airborne free-form surface reflector, characterized in that: The spectacles include a spectacles frame (1) and a plurality of flexible legs (3) disposed on the back of the spectacles frame (1); the spectacles frame (1) is a polygonal axisymmetric structure with a central square hole (17) opened at the center; The flexible leg (3) comprises a connecting piece and two flexible legs respectively connected to the connecting piece; the flexible legs comprise a first flexible structure (32), a leg, a second flexible structure (34) and a flexible leg external mounting interface (35) which are sequentially connected; the first flexible structure (32) is connected to the back of the frame (1) via the connecting piece; a flexible groove (33) is also provided on the leg; and a trimming pad (4) is also provided between the frame (1) and the connecting piece; There are three flexible legs (3), the three flexible legs (3) are distributed on the back of the frame (1), and the connecting line of the three flexible legs (3) forms an isosceles triangle structure; The airborne free-form surface reflector flexible support device further comprises a plurality of flexible baffles (5) disposed on the back of the mirror frame (1); the plurality of flexible baffles (5) and the plurality of flexible legs (3) are arranged in an alternating manner; The flexible baffle (5) comprises a sheet body and a horizontal flexible groove and a vertical flexible groove respectively arranged on the sheet body; the axial direction of the horizontal flexible groove is perpendicular to the axial direction of the vertical flexible groove; the sheet body is connected to the back of the frame (1); the horizontal flexible groove and the vertical flexible groove are both one or more; The horizontal flexible grooves include a first horizontal flexible groove (52), a second horizontal flexible groove (53), a third horizontal flexible groove (54) and a fourth horizontal flexible groove (55); the vertical flexible grooves include a first vertical flexible groove (56); the first horizontal flexible groove (52) is axially parallel to the second horizontal flexible groove (53), and the first vertical flexible groove (56) is axially extended from the edge of the flexible baffle (5) to the center of the flexible baffle (5) and passes through the second horizontal flexible groove (53) and then communicates with the first horizontal flexible groove (52); the third horizontal flexible groove (54) and the fourth horizontal flexible groove (55) are axially symmetrically distributed along the first vertical flexible groove (56) and extend to the edge of the flexible baffle (5), and the third horizontal flexible groove (54) is axially parallel to the first horizontal flexible groove (52); There are three flexible baffles (5), which are distributed on the back of the frame (1), and the center points of the three flexible baffles (5) are connected to form an equilateral triangle structure; the vertex of the isosceles triangle structure is located below the base of the equilateral triangle structure, and the line connecting the vertex of the isosceles triangle structure and the vertex of the equilateral triangle structure is parallel to the vertical axis of the frame (1), and the base of the isosceles triangle structure is located below the base of the equilateral triangle structure; The airborne free-form surface reflector flexible support device further comprises a plurality of small glue injection holes (11) arranged on the outer side wall of the mirror frame (1) and connected to the central square hole (17); A glue overflow groove is provided on the hole wall of the central square hole (17), and the glue overflow groove is in a cross shape; There are two overflow glue grooves, which are symmetrically distributed on the wall of the central square hole (17) along the central axis of the mirror frame (1) in the vertical direction.

Citation Information

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