A method of designing a variable width spiral spring and a spiral spring

Abstract

The application relates to a variable-width spiral spring design method and a spiral spring, and belongs to the technical field of spiral springs, which comprises the following steps: S1, determining the polar angle corresponding to a planar spiral line at different arc length positions; S2, determining the curvature radius corresponding to the planar spiral line at different polar angle positions; S3, designing the preliminary section width of the spiral spring according to different curvature radius positions; and S4, adjusting the preliminary width of the spiral spring according to the size of the radius-to-half-thickness ratio of different positions, so that the uniform distribution of bending stress is realized. The core of the application lies in that the corresponding width change law is designed according to the bending stress change law with the curvature radius disclosed by the curved beam theory. Through the method, the maximum bending stress of all sections can be equal, so that the load capacity and stability of the spiral spring are improved.

Description

Technical Field

[0001] This application relates to the technical field of spiral springs, and in particular to a design method for a variable width spiral spring and a spiral spring. Background Technology

[0002] A spiral spring, also known as a clockwork spring, is an elastic element that coils in a planar spiral shape, typically made of high-carbon steel or alloy spring steel. Its structural feature is that one end is fixed, while the other end rotates around a central axis under load, storing elastic potential energy. After unloading, it returns to its original shape due to elasticity. It is widely used in watches, energy storage devices, and actuators.

[0003] The structural feature of a spiral spring is that one end is fixed, while the other end can rotate around its central axis and store elastic potential energy when subjected to force. After unloading, it returns to its original shape due to elasticity. A planar spiral spring is made of slender spring material wound in a spiral, and it can store bending strain energy after applying a torsional torque. It has advantages such as low energy consumption and stable driving torque, and when combined with pneumatic drive, it can achieve rapid opening and closing operations.

[0004] Existing spiral springs with uniform width and thickness often experience breakage near the inner coil due to uneven strength along their length during use. Summary of the Invention

[0005] In order to achieve uniform bending stress along the helical direction of the spiral spring and reduce the occurrence of fracture failure near the inner ring of the spiral spring, this application provides a design method for a variable width spiral spring and a spiral spring.

[0006] This application provides a design method for a constant-strength variable-width spiral spring, which adopts the following technical solution:

[0007] S1. Determine the polar angle corresponding to the planar spiral at different arc length positions;

[0008] S2. Determine the radius of curvature of the planar spiral at different polar angle positions;

[0009] S3. Design the initial cross-sectional width of the spiral spring according to different curvature radii;

[0010] S4. Adjust the initial width of the spiral spring according to the ratio of radius to half thickness at different parts to achieve uniform distribution of bending stress.

[0011] Optionally, the planar spiral can be an Archimedean spiral or a logarithmic spiral.

[0012] Optionally, the profile of the spiral spring to be designed is based on the Archimedean spiral, and step S1 includes:

[0013] S11. Measure the arc length and apply the formula... First, determine the polar angle by reverse calculation. In the formula, parameter a = t / 2π, where t is the pitch of the spiral spring, i.e., the distance between adjacent profiles. That is, the polar angle;

[0014] In step S2, according to the formula The radius of curvature is obtained, with parameter a = t / 2π.

[0015] Optionally, the profile of the spiral spring to be designed is based on a logarithmic spiral, and step S1 includes:

[0016] S11. Measure the arc length and apply the formula... First, determine the polar radius using the reverse approach, where the parameter k = cotα represents the cotangent of the angle between the polar radius at any point on the helix and the tangent at that point. Then, determine the polar angle based on the polar radius. The polar radius of a logarithmic helix increases exponentially with the increase of the polar angle, and the distance between adjacent helices increases with the extension of the arc length. The expression is: In the formula, For any polar angle The corresponding polar radius of the helix. 'a' is the initial polar radius, i.e., when the polar angle... When, the distance from the helix to the pole, parameter k=cotα, represents the cotangent of the angle between the polar radius of any point on the helix and the tangent at that point, ρ2 and ρ1 are the polar radii of the helix at two different polar angles, and e is a constant;

[0017] In step S2, according to the formula The radius of curvature is obtained.

[0018] Optionally, step S3 includes:

[0019] S31. Determining the effect of circumferential curvature variation on the stress of a spiral spring: Define the bending radius coefficient ρ = R / b, where R and b are the mid-surface curvature radius and cross-sectional width of the spiral spring, respectively; analyze the stress and deformation of cross-sections at different curvature positions from the mandrel to the fixed end of the spiral spring, and substitute the definitions of cross-sectional area A and moment of inertia Jz into the equation. Define the bending radius coefficient When y = h / 2, the stress on the inner side is:

[0020]

[0021] Right now:

[0022] ;

[0023] When y = -h / 2, the stress on the outer side is:

[0024] ;

[0025] Right now:

[0026] ;

[0027] S32. According to the formula, based on the different radii of curvature of the parts... and formula Based on the magnitude of the revealed bending stress, the initial cross-sectional width of the spiral spring is designed, which is the cross-sectional width relative to a straight beam with a uniform cross-section.

[0028] In summary, this application includes at least one of the following beneficial technical effects:

[0029] 1. It can achieve uniform bending stress along the helix direction of the spiral spring. This reduces the occurrence of fracture near the inner coil of the spiral spring. It also improves material utilization and reduces structural weight. For applications in weight-sensitive structures such as those used in aerospace, it can reduce structural weight and enable more rational structural design.

[0030] 2. The constant-strength, variable-width spiral spring design method is particularly suitable for aerospace applications, such as satellites and spacecraft, where structural weight is highly critical. This design method effectively reduces structural weight while maintaining the performance of the spiral spring. Attached Figure Description

[0031] Figure 1 This is a schematic diagram of the initial state profile of a spiral spring;

[0032] Figure 2 This is a schematic diagram of the curved beam segment;

[0033] Figure 3 This is a schematic diagram of the planar bending deformation of a curved beam;

[0034] Figure 4 This is a schematic diagram of radial stress analysis of a micro-segment of a curved beam under pure bending conditions;

[0035] Figure 5 This is a schematic diagram of the beam cross-section;

[0036] Figure 6 This is a schematic diagram of the variable cross-sectional width of a constant bending stress spiral spring with different radii of curvature;

[0037] Figure 7 This is a schematic diagram of the Archimedean spiral;

[0038] Figure 8 These are schematic diagrams of Archimedean spirals with different values ​​of a = 0.1 to 0.5;

[0039] Figure 9 This is a schematic diagram showing the relationship between the arc length and polar angle of the Archimedean spiral for different values ​​of a = 0.1 to 0.6;

[0040] Figure 10 This is a schematic diagram showing the relationship between the radius of curvature and the polar angle of Archimedean spirals with different values ​​of a = 0.1 to 0.6;

[0041] Figure 11 This is a schematic diagram of a logarithmic spiral;

[0042] Figure 12 This is a schematic diagram showing the relationship between the arc length and polar angle of a logarithmic spiral with k=1;

[0043] Figure 13 This is a schematic diagram showing the relationship between the arc length and polar angle of a logarithmic spiral with k=2;

[0044] Figure 14 This is a schematic diagram showing the relationship between the arc length and polar angle of a logarithmic spiral with k=3;

[0045] Figure 15 This is a schematic diagram showing the relationship between the radius of curvature and the polar angle of a logarithmic spiral with k=1.

[0046] Figure 16 This is a schematic diagram showing the relationship between the radius of curvature and the polar angle of a logarithmic spiral with k=2.

[0047] Figure 17 This is a schematic diagram showing the relationship between the radius of curvature and the polar angle of a logarithmic spiral with k=3. Detailed Implementation

[0048] The following is in conjunction with the appendix Figures 1-17 This application will be described in further detail.

[0049] The core of this invention lies in designing a corresponding width variation law based on the law of bending stress variation with radius of curvature revealed by the curved beam theory. This method ensures that the maximum bending stress is equal across all cross-sections, thereby improving the load-bearing capacity and stability of the spiral spring.

[0050] To facilitate subsequent understanding, we will first explain the relationship between bending stress and radius of curvature based on the analysis of curved beam theory, and then explain bending stress. Finally, based on the stress results, we will design a method for designing a variable width spiral spring with equal strength to achieve uniform bending stress of the spiral spring along the helical direction, so that the maximum bending stress of each section is equal.

[0051] First: Reference Figure 1 The stress of a planar spiral spring is analyzed based on the curved beam theory of advanced materials mechanics. By establishing a series of curved beam models with different width-to-thickness ratios and bending radii, the theoretical results of bending stress in different parts are studied, demonstrating the applicability of the straight beam theory and its accuracy when applied to planar spiral springs for driving applications.

[0052] refer to Figure 2Assume the curved beam has a longitudinal plane of symmetry, and all loads applied to the beam are concentrated within this plane of symmetry. Therefore, after deformation, the axis of the curved beam will remain within this longitudinal plane of symmetry, i.e., the curved beam undergoes planar bending. Taking a partial cross-section of the curved beam, the axis of the curved beam is the x-axis, the plane of symmetry of the cross-section is the y-axis, and the z-axis is perpendicular to both the x and y axes and passes through the centroid of the cross-section. The load intensity q is distributed along the y and x axes. r and q φ The stress component at any point on the cross-section is σ. φ and τ σr The corresponding internal force: In the formula, N is the axial force, and M is the axial force. z For bending moment, Q r For shear force. To simplify the writing, M is... z and Q r The internal forces are written as M and Q, and the symbols for the internal forces still follow the convention for straight beams.

[0053] refer to Figure 3 When a curved beam undergoes planar bending, there is no torsional deformation, and the planar assumption is still satisfied. The planar bending deformation of the curved beam at this time can be represented by the circumferential displacement u and radial displacement v of the centroid of the cross-section along the circumferential x-axis and radial y-axis, as well as the rotation angle of the cross-section about the z-axis. These displacements and rotation angles will cause deformation of the longitudinal fibers of the curved beam.

[0054] Stress analysis of a curved beam under pure bending: Reference Figure 4 Curved beams, due to their initial curvature, also experience radial stress σ under pure bending conditions. r A small segment is taken from a purely curved beam with adjacent cross sections. Then, a portion of a curved surface parallel to the coordinate plane xz is cut from this small segment. Its stress condition is as follows: Figure 4 As shown in the image.

[0055] Expression of bending stress: Bending stress under pure bending conditions when subjected to bending moment M. The z-axis is the centroidal axis of the cross-section.

[0056] Central axis position: Reference Figure 5 Let e ​​be the distance from the neutral axis to the centroid of the cross section. Let the equation... where y=e,σ φ =0, thus determining the position of the neutral axis. If r represents the radius of curvature of the neutral layer, and ρ represents the radius of curvature of any fiber ab, then have , will Japanese style Substitution The position of the neutral axis on the cross section is obtained. .

[0057] Bending stress formula: For a rectangular beam with a wall thickness h much smaller than its cross-sectional width b, when bending about the z-axis, the tension zone shrinks and the compression zone area increases. Expansion, Moment of Inertia Since a planar spiral spring is a structure with continuously changing circumferential curvature, in order to study the influence of the change in circumferential curvature on the stress of the spiral spring, the bending radius coefficient ρ=R / b is defined, where R and b are the mid-surface curvature radius and cross-sectional width of the spiral spring, respectively. The stress and deformation of the planar spiral spring are analyzed by taking cross-sections at different curvature positions from the mandrel to the fixed end.

[0058] Substituting the definitions of cross-sectional area A and moment of inertia Jz into the equation... Define the bending radius coefficient Therefore, when y = h / 2, the stress on the inner side is:

[0059]

[0060] Right now:

[0061] ;

[0062] When y = -h / 2, the stress on the outer side is:

[0063] ;

[0064] Right now:

[0065] .

[0066] Bending stress under different bending radius coefficients: Based on the straight beam theory and advanced materials mechanics theory, the stress of a curved beam with uniform curvature under different bending radius conditions was calculated for a thickness b = 9 mm and bending radius coefficients ρ = 1, 3.805, 7.61, 15.22, 30.44, 60.88, 121.76, and infinite (straight beam). The theoretical values ​​of bending stress and radial stress from elasticity mechanics and advanced materials mechanics were dimensionless under different bending radius coefficients, using the straight beam theoretical stress value as the unit. The results are shown in Table 1.

[0067] Table 1. Stress Comparison of Curved Beams with Different Bending Radius Coefficients

[0068] unit:

[0069]

[0070] Table 1 shows that the bending radius coefficient affects the stress. As the bending radius coefficient ρ increases, the theoretical value of the bending stress of the curved beam gets closer and closer to that of the straight beam; therefore, calculating the bending stress of the curved beam using the straight beam theory is inaccurate. The calculation method using the curved beam theory to calculate the bending stress is more in line with reality.

[0071] This application discloses a design method for a constant strength variable width spiral spring.

[0072] Example 1: Archimedes' spiral:

[0073] refer to Figure 7 When analyzing spiral springs, the Archimedean spiral is required. To minimize contact, the profile of planar spiral springs used in actuators often adopts the Archimedean spiral. The cross-section of the spiral spring is a rectangle with a width of b and a thickness of h, and the z-axis is the axis of symmetry along the thickness direction of the cross-section. The cross-sectional dimensions of the spiral spring are very small relative to its length, conforming to the definition of a slender beam. Therefore, the influence of shear force on the bending deformation of the spiral spring can be ignored, and its deformation can be transformed into the bending problem of a slender curved beam.

[0074] Basic characteristics of the Archimedean spiral: The polar diameter of the Archimedean spiral increases linearly with the increase of the polar angle, and the spacing between adjacent profiles is equal. The expression is...

[0075]

[0076] In the formula, For any polar angle The corresponding helix radius. Parameter a = t / 2π, where t is the pitch of the spiral spring (i.e., the distance between adjacent profiles).

[0077] refer to Figure 7 , is the Archimedean spiral, a curve representing the trajectory traced by a moving point starting from the horizontal axis and moving at a constant speed v along a ray, while this ray rotates around the pole O at a constant angular velocity w. Where a = v / w.

[0078] arc length

[0079]

[0080] radius of curvature

[0081]

[0082] refer to Figure 8 For Archimedean spirals with different values ​​of a = 0.1 to 0.5.

[0083] Arc length and radius of curvature of the Archimedean spiral: Reference Figure 9 This provides the relationship between the arc length and polar angle of the Archimedean spiral under different parameters 'a', and allows you to look up the corresponding arc length when a given polar angle is given.

[0084] refer to Figure 10This paper presents the relationship between the curved beam radius and polar angle of the Archimedean spiral under different parameters 'a', allowing users to look up the corresponding radius of curvature for a given polar angle. This facilitates determining the radius of curvature from the graph and then further refining the calculation. Figure 6 Query the relative width.

[0085] refer to Figure 6 According to different curvature radii, Based on the revealed bending stress, the cross-sectional width of the spring is designed relative to that of a straight beam with a uniform cross-section. According to the stress variation law on the inner side in Table 1, and based on the ratio of radius to half thickness at different locations, the width of the spiral spring can be appropriately increased to achieve a uniform distribution of bending stress.

[0086] A design method for a constant strength variable width spiral spring includes: in this embodiment, the planar spiral is an Archimedean spiral.

[0087] S1: Determine the polar angle corresponding to the planar helix at different arc length positions. S11: First, measure the arc length; the measurement and calculation methods are existing techniques. Using the formula... The polar angle can be obtained by reverse calculation. The polar angle is the fundamental parameter for the spatial position of various parts of the positioning spring. S12. Based on the polar angle, the formula is used... The polar radius can be obtained.

[0088] S2: Determine the radius of curvature of the planar helix at different polar angle positions. According to the formula... The radius of curvature at the corresponding polar angle position can be obtained, which describes the degree of curvature of that part.

[0089] S3: Design the initial cross-sectional width of the spiral spring according to different radii of curvature: according to different radii of curvature... The magnitude of the revealed bending stress is used to design the cross-sectional width relative to a straight beam with a uniform cross-section.

[0090] S4: Adjust the initial width of the spiral spring according to the ratio of radius to half thickness at different locations to achieve a uniform distribution of bending stress: refer to Table 1 for the stress variation law on the inner side. Figure 6 By appropriately increasing the width of the spiral spring according to the ratio of radius to half thickness at different locations, a uniform distribution of bending stress can be achieved.

[0091] Example 2: The difference from Example 1 is that the planar spiral in this example is a logarithmic spiral.

[0092] S1: Determine the polar angle corresponding to the planar helix at different arc length positions. S11: First, measure the arc length; the measurement and calculation methods are existing techniques; then, according to the formula... The polar radii ρ1 and ρ2 are obtained by reverse calculation; S12, based on the polar radii, according to the formula The polar angle is obtained by reverse calculation, where, For any polar angle The corresponding polar radius of the helix, where 'a' is the initial polar radius, i.e., when the polar angle... When, the distance from the helix to the pole, parameter k=cotα, represents the cotangent of the angle between the polar radius of any point on the helix and the tangent at that point, ρ2 and ρ1 are the polar radii of the helix at two different polar angles, and e is a constant;

[0093] refer to Figure 11 In a logarithmic spiral, the angle between the curve and all rays passing through the poles is equal (α) (k=ctgα). As the curve approaches negative infinity, it rotates clockwise around the pole and approaches the pole.

[0094] In step S2, according to the formula The radius of curvature is obtained.

[0095] The arc length of a logarithmic spiral increases exponentially with increasing polar angle. (Reference) Figure 12 , Figure 13 and Figure 14 The arc lengths of logarithmic spirals with different parameters a = 0.1 to 0.6 for k = 1, 2, and 3 are given respectively. This facilitates the determination of the arc length at different polar angles.

[0096] The radius of curvature of a logarithmic spiral increases exponentially with increasing polar angle, and the distance between adjacent spirals increases with increasing arc length. (Refer to...) Figure 15 , Figure 16 and Figure 17 The figures show the radii of the curved beams for logarithmic spirals with different parameters a = 0.1 to 0.6 when k = 1, 2, and 3. This is to facilitate determining the radius of curvature from the figures and then... Figure 6 Query the relative width.

[0097] Example 3: A spiral spring is manufactured using a constant strength variable width spiral spring design method from Example 1 or Example 2.

[0098] The embodiments described in this specific implementation are preferred embodiments of this application and are not intended to limit the scope of protection of this application. Therefore, all equivalent changes made in accordance with the structure, shape and principle of this application should be covered within the scope of protection of this application.

Claims

1. A design method for a variable width spiral spring, characterized in that: include S1. Determine the polar angle corresponding to the planar spiral at different arc length positions; S2. Determine the radius of curvature of the planar spiral at different polar angle positions; S3. Design the initial cross-sectional width of the spiral spring according to different curvature radii; S4. Adjust the initial width of the spiral spring according to the ratio of radius to half thickness at different parts to achieve uniform distribution of bending stress. The planar spiral is either an Archimedean spiral or a logarithmic spiral; The profile of the spiral spring to be designed is based on the Archimedean spiral. Step S1 includes: S11. Measure the arc length and apply the formula... First, determine the polar angle by reverse calculation. In the formula, L is the arc length, and the parameter a = t / 2π, where t is the pitch of the spiral spring, i.e., the distance between adjacent profiles. That is, the polar angle; The profile of the spiral spring to be designed is based on a logarithmic spiral. Step S1 includes: S11. Measure the arc length and apply the formula... First, determine the polar radius using the formula, where L is the arc length and the parameter k = cotα represents the cotangent of the angle between the polar radius at any point on the helix and the tangent at that point. Then, determine the polar angle based on the polar radius. The polar radius of a logarithmic helix increases exponentially with the increase of the polar angle, and the distance between adjacent helices increases with the extension of the arc length. The expression is: In the formula, For any polar angle The corresponding polar radius of the helix, where 'a' is the initial polar radius, i.e., when the polar angle... When, the distance from the helix to the pole, parameter k=cotα, represents the cotangent of the angle between the polar radius of any point on the helix and the tangent at that point, ρ2 and ρ1 are the polar radii of the helix at two different polar angles, and e is a constant.

2. The design method for a variable width spiral spring according to claim 1, characterized in that: The profile of the spiral spring to be designed is based on the Archimedean spiral. In step S2, according to the formula... The radius of curvature is obtained, with parameter a = t / 2π.

3. The design method for a variable width spiral spring according to claim 1, characterized in that: The profile of the spiral spring to be designed is based on a logarithmic spiral. In step S2, according to the formula... The radius of curvature is obtained.

4. A method for designing a variable width spiral spring according to claim 2 or 3, characterized in that: Step S3 includes: S31. Determine the effect of circumferential curvature change of a spiral spring on its stress: Define the bending radius coefficient ρ = R / b, where R and b are the mid-surface curvature radius and cross-sectional width of the spiral spring, respectively; analyze the stress and deformation of cross-sections of the spiral spring at different curvature positions from the mandrel to the fixed end. Under the action of bending moment M, the bending stress during pure bending is: In the formula, M is the bending moment, A is the area of ​​the compression zone, R is the radius of curvature of the mid-surface of the spiral spring, and J... z It is the moment of inertia, and , In the formula, b is the cross-sectional width of the spiral spring, and h is the wall thickness. The areas A and J of the pressure zone are then considered. z Definition Substitution Define the bending radius coefficient When y = h / 2, the stress on the inner side is: ； Right now: ； When y = -h / 2, the stress on the outer side is: ； Right now: ； S32. According to the formula, based on the different radii of curvature of the parts... and formula Based on the magnitude of the revealed bending stress, the initial cross-sectional width of the spiral spring is designed, which is the cross-sectional width relative to a straight beam with a uniform cross-section.

5. A spiral spring, characterized in that: It is manufactured using the variable width spiral spring design method as described in claim 4.

Images