JPH0252301A - Formation of secondary curved surface - Google Patents

Abstract

PURPOSE:To form an optional parabolic surface and a cylindrical surface mirror by the same mechanism by applying specific bending moment to both end edges of a plane mirror or support body which supports the plane mirror in contact. CONSTITUTION:The plane mirror 1 contacts the support body 2 across its mirror surface and the bending moment is applied to both end edges of the support body 2 as fulcrum 3 and 4 in said state by values Ma and Mb in the opposite directions to curve the support body 2. Here, the values Ma and Mb are expressed by equations I and II. In the equations I and II, P and Q are represented by equations III and IV, where E is the longitudinal elastic modulus of the plane mirror or support body including the plane mirror, I is its sectional secondary moment, l is the distance between both fulcrum, and F is the focal length of the secondary curved surface mirror. Further, D is the distance from the intersection of symmetrical surfaces of a perpendicular from the point where the inclination of the surface of the secondary curved surface mirror to the plane containing both end edges is zero to the symmetrical surface of the secondary curved surface mirror to the vertex of the secondary curved surface mirror. Consequently, the parabolic surface mirror with a large radius of curvature and the cylindrical surface mirror are formed by using the same mechanism.

Description

DETAILED DESCRIPTION OF THE INVENTION [Field of Industrial Application] The present invention relates to a method of forming a quadratic curved mirror such as a parabolic mirror or a cylindrical mirror.

[Conventional technology]

Conventionally, methods for forming this type of quadratic curved mirror include a polishing method in which a mirror base material is polished into a predetermined curved surface, and a bending method in which a plane mirror is bent into a predetermined curvature. For mirrors with a small radius of curvature, it is possible to form a mirror surface using the former polishing method, but for mirrors with a radius of curvature of several kilometers, the polishing method requires that the surface grain size of the mirror surface be reduced. The latter bending method was used instead.

[Problem B that the invention attempts to solve]

Most of the conventional methods for forming quadratic mirrors described above involve forming cylindrical mirrors using a double-end support beam method that applies equal bending moments to the mirror support or both fulcrums of the mirror. The disadvantage is that it only forms a shape, and it is impossible to form a parabolic mirror using the same mechanism.

An object of the present invention is to form a parabolic mirror and a cylindrical mirror with a large radius of curvature using the same mechanism, which could not be achieved using conventional bending methods.

[Means to solve the problem]

The method for forming a quadratic curved mirror of the present invention is based on two opposing parallel linear end edges of a plane mirror or a support that closely supports the plane mirror as fulcrums, and E is a plane mirror or The longitudinal elastic modulus of the support including the plane mirror, ■ is its second moment of area, 1 is the distance between both supports, F is the focal length of the quadratic curved mirror, and D is the 21st order with respect to the plane containing both edges. Values Ma and Mb expressed by the distance between the intersection of the perpendicular line drawn to the symmetry plane of the curved mirror with the symmetry plane and the vertex of the quadratic curved mirror from the point where the inclination of the surface of the curved mirror becomes zero. The plane mirror or support is curved by applying bending moments in opposite directions.

[For use]

The basic principle of the present invention will be explained with reference to the drawings.

Figure 1 shows the plane mirror 1 in close contact with the support 2 and bent into a curved surface due to the bending moment applied to both edges of the support, and Figure 2 shows the relationship between the two coordinate systems for a parabola. It is a graph showing a relationship.

The plane mirror 1 is in close contact with the support body 2 via the mirror surface, and in this state, bending moments of magnitudes Ma and Mb are applied to both the support points 3 and 4 with both end edges of the support body 2 as the fulcrums 3 and 4, respectively. By curving the support body 2 by applying in opposite directions, a cylindrical mirror or a parabolic mirror can be formed. Note that when applying a bending moment directly to both ends of the plane mirror 1 without using the support 2, the principle is the same as the above case. A method for forming a mirror will be explained.

The support 2 in FIG.
Since it can be considered as a dimensional problem, both fulcrums 3
．． The origin is the point where the inclination of the center line 5 of the support body 2 is zero with respect to the straight line connecting the two, the tangential direction of the center line 5 is the X axis, and the direction perpendicular to this is the Y axis (the curved side is the positive direction). Therefore, if the distance between both fulcrums 3.4 is 2, and the longitudinal elastic modulus and moment of elasticity of the support 2 including the plane mirror 1 are E and I, respectively, then a bending moment of magnitude Ma is applied to the fulcrum 3 in a clockwise direction. The shape of the center line 5 when a bending moment of magnitude Mb is applied counterclockwise to the support 2 at the fulcrum 4 can be expressed by the following equation.

Equation (1) can be easily obtained by parallel transformation of the coordinate system to the origin where the inclination of the center line 5 is zero, from the formula for calculating the deflection of a beam supported at both ends, which is known in the field of mechanics of materials, so the derivation will be omitted. On the other hand, as shown in Figure 2, a parabola is generally expressed using a focal length F, f(x,y)-y2-+
It is expressed as px=o. +, x = D, y = -2F With the position of the bottom as the origin, the origin (D.

-2F■)), the tangent direction of the parabola is the X axis, and the direction perpendicular to this is the Y axis.
=t(x) can be approximated by the following polynomial.

However, P and Q in formulas (3) and (4) are respectively expressed by the following formulas.

Equation (2) can also be derived by changing the coordinate system to the origin (
0, 0) Empty point (o, -z, rl) Move f in two parallel s and rotate until the X axis coincides with the tangential direction of the parabola at the new origin, convert it to a new coordinate system, and then convert Equation 7. Since it can be obtained by expanding into a polynomial regarding X and omitting the quartic term wind -L of X, details of the induction will be omitted. That is, equation (2) is a curve that approximates a parabola.

Therefore, by setting the coefficients of the quadratic and cubic terms regarding X in equations (1) and (2) equally, the bending moment conditional expression for forming a paraboloid can be obtained, and the magnitude of the bending moment at this time is Ma and Mb are each expressed by the following formula (
3) and (4).

3P+ (12Q-3P2)''/2 ,,, (3゜Ma= Mb=Ma-P...(4
) If Ma=Mb in equations (3) and (4), it can be easily inferred that a cylindrical mirror can be formed in the same manner as in the conventional method described above.

〔Example〕

Next, embodiments of the present invention will be described with reference to the drawings.

FIG. 3 is a graph showing the results of measuring the deviation of a parabolic mirror manufactured by an embodiment of the method for forming a quadratic mirror of the present invention from an ideal parabolic mirror.

First, a parabolic mirror was installed at a point 17 m from the light source point at an oblique incidence angle of 3.6 mrad, and the case of converting diverging light from the light source into parallel light was measured and confirmed. The parabolic mirror at this time is F = 0.2203]9mm, D = 16.
99978 m, Il, = 1.5 m, E I
=28186.6Nm2, and the magnitude of the bending moment required to bend the support 2 into this paraboloid is M
a=318. ONm, Mb=278.5Nm. In FIG. 3, the Y coordinate (μm) of the ideal parabolic mirror 6 is shown on the left Y axis, and the deviation (%) of the parabolic mirror 7 according to this embodiment from the Y coordinate of the ideal parabolic mirror 6 is shown on the right Y-axis, and it can be seen from this that a parabolic mirror can be formed with a deviation of at most 0.06% for a mirror with a length of 1.5 m. Also, if Ma=Mb=596.9Nm,
A cylindrical mirror of I can be formed with a radius of curvature of 4.722, which allows light to be focused further 17 m downstream from the mirror.

(Effects of the Invention) As explained above, the present invention enables arbitrary parabolic and Cylindrical mirrors can be formed using the same mechanism, making it possible to achieve parallelization and condensing of light with the same mechanism, especially for experimental purposes, and a large quadratic curved mirror with a radius of curvature of several kilometers. In the sense that it can form
It has the effect of being suitable for reflecting highly directional synchrotron radiation light.

[Brief explanation of the drawing]

Figure 1 shows a plane mirror bent into a curved surface by a humming moment applied to both edges while in close contact with a support, and Figure 2 shows the mutual relationship between the two coordinate systems of a parabola. The graph shown in FIG. 3 is a graph showing the results of measuring the deviation of a parabolic mirror manufactured by an embodiment of the method for forming a quadratic curved mirror of the present invention from an ideal parabolic mirror. 1...Flat mirror 2...Support, 3.4
...Fully point, 5 ...Center line, 6 ...
Ideal parabolic mirror 7 Parabolic mirror according to this embodiment Patent applicant
Nippon Telegraph and Telephone Corporation

Claims (1)

[Claims] 1. A method for forming a quadratic curved mirror, which forms a quadratic curved mirror such as a parabolic mirror or a cylindrical mirror by bending a plane mirror to a predetermined curvature, comprising: Ma={3P+(12Q-3P^2)^1^/^2}/, using the opposing parallel straight edge edges of the supports that are closely supported as fulcrums, respectively.
6, Mb=Ma-P where P=3EIl(FD)^1
^/^2/4(F+D)^3, Q=3F(EI)^2/
4(F+D)^3, E is the longitudinal elastic modulus of the plane mirror or the support including the plane mirror, I is its moment of inertia, l is the distance between both supporting points, F is the focal length of the quadratic mirror, D is the intersection of the perpendicular line drawn from the point where the inclination of the surface of the quadratic mirror with respect to the plane including both edges is zero to the symmetry plane of the quadratic curved mirror and the vertex of the quadratic curved mirror. A method for forming a quadratic curved mirror, characterized in that a plane mirror or a support is curved by applying bending moments having values Ma and Mb expressed by the distance between them in opposite directions.