Interconversion calculation method for cross-ball distance and arc tooth thickness of asymmetric deflection helical gear
In the design of asymmetric displaced helical gears, the method of calculating the span spherical distance and arc teeth thickness using involute lines and the plane coordinates of the sphere centers is solved, and efficient design and processing and manufacturing are achieved.
Patent Information
- Application Number
- CN202411990463.7
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2024-12-31
- Publication Date
- 2025-06-03
AI Technical Summary
In the design and processing and manufacturing of asymmetric displacement helical gears, it is difficult to efficiently calculate the spherical distance and arc teeth thickness, resulting in low design efficiency and remodeling after modifying parameters, which has certain limitations.
A method for calculating the mutual conversion of asymmetric displaced helical gears and arc teeth thickness. Using the involute linear properties and the contact properties of the spatial sphere and the spiral surface, the span spherical distance and arc teeth thickness are calculated through the plane coordinates of the sphere center and the involute expansion angle between the spiral surface tangent points of the sphere and the spiral surface.
This method does not require three-dimensional model to assist measurement, improves the design efficiency of asymmetric displacement helical gears, facilitates the evaluation of tooth thickness machining accuracy, and simplifies the design process.
Smart Images
Figure CN120086988A_ABST
Abstract
Description
Technical Field
[0001] The present invention relates to a calculation method for mutual conversion between the over-ball distance and the arc tooth thickness of an asymmetric modified helical gear. Background Art
[0002] Due to different pressure angles on both sides, the asymmetric modified helical gear has higher load-carrying capacity, lower tooth root bending stress, and longer service life compared with the traditional symmetric helical gear. It is applicable to transmission systems with large loads and high speeds in new energy vehicles, aerospace, etc.
[0003] During the processing and manufacturing of an asymmetric modified helical gear, since the base circles corresponding to the tooth profiles on both sides are different, it is impossible to calculate the actual tooth thickness by measuring the common normal, and only the over-ball distance of the gear can be measured to calculate the actual tooth thickness and evaluate the machining accuracy of the gear. During the gear design process, it is necessary to calculate the range of the over-ball distance according to the given upper and lower deviations of the tooth thickness. However, the parameters of the helical surfaces on both sides of the asymmetric modified helical gear are different, and the solution of the spatial contact points between the measuring balls and the helical surfaces on both sides is relatively complex. The traditional accurate calculation method of the over-ball distance needs to establish accurate models of the asymmetric modified helical gear and the measuring balls for measurement. Although this method can obtain accurate over-ball distance values, the design efficiency is low, and it is necessary to re-model after modifying the tooth thickness parameters, which has certain limitations. Summary of the Invention
[0004] Aiming at the deficiencies existing in the prior art, the purpose of the present invention is to provide a calculation method for mutual conversion between the over-ball distance and the arc tooth thickness of an asymmetric modified helical gear, aiming to improve the design efficiency of the asymmetric modified helical gear, facilitate the designer to evaluate the machining accuracy of the tooth thickness of the asymmetric modified helical gear, and simplify the design process of the asymmetric modified helical gear.
[0005] In order to solve the above technical problems, the present invention is implemented by adopting the following technical solutions:
[0006] A calculation method for mutual conversion between the over-ball distance and the arc tooth thickness of an asymmetric modified helical gear, calculating the over-ball distance M for the arc tooth thickness of the asymmetric modified helical gear:
[0007]
[0008] In the formula, r m is the radius of the circle where the center O dp of the measuring ball corresponding to the over-ball distance M is located; π is the pi; z is the number of teeth; i is the tooth number coefficient, when the number of teeth is odd, i = 1, when the number of teeth is even, i = 0; d p is the diameter of the measuring ball.
[0009] The present invention further includes the following technical features:
[0010] Specifically, for the calculation of the actual face arc tooth thickness S of an asymmetric modified helical gear by measuring the distance across balls tc and the actual normal arc tooth thickness S nc :
[0011] S tc = πm t - r(σ l + σ r + invα tl + invα tr ) (16)
[0012] S nc = πm n - r(σ l + σ r + invα tl + invα tr )cosβ (17)
[0013] In the formula, m t is the face module, m n is the normal module, r is the pitch diameter, σ l and σ r are the initial involute spread angles of the left and right involutes corresponding to the actual arc tooth thickness; invα tl and invα tr are the involute functions of the small and large pressure angle sides of the asymmetric gear respectively; α tl and α tr are the face pressure angles of the involutes on the small and large pressure angle sides of the asymmetric gear respectively; β is the helix angle.
[0014] Specifically, r m in formula (1) is:
[0015]
[0016] In the formula, x dp and y dp are the plane coordinates O dp of the center of the measuring ball O dp (x dp , y dp ) in the cross-sectional plane of the gear shaft where the center of the ball is located; among them, O dp (x dp , y dp ) is solved by the following formula:
[0017]
[0018] In the above formula, x l and y l are the coordinates of the center of the ball (x l , yl ); x r , y r are the coordinates of the center of the measuring ball when it is tangent to the helical surface on the large pressure angle side of the asymmetric gear (x r , y r ); when the coordinates of the two centers of the balls coincide, it is considered that the measuring ball is tangent to both helical surfaces at the same time.
[0019] Specifically, each variable in Equation (3) is expressed as:
[0020]
[0021] In the formula, x inv_l , y inv_l are the equations of the involute on the small pressure angle side of the asymmetric gear, and x inv_r , y inv_r are the equations of the involute on the large pressure angle side of the asymmetric gear; x n_inv_l , y n_inv_l are the elements in the normal vector N inv_l of the involute on the small pressure angle side of the asymmetric gear; x n_inv_r , y n_inv_r are the elements in the normal vector N inv_r of the involute on the large pressure angle side of the asymmetric gear.
[0022] Specifically, the equation of the involute on the small pressure angle side of the asymmetric gear:
[0023]
[0024] The equation of the involute on the large pressure angle side of the asymmetric gear:
[0025]
[0026] In the formula, r bl , r br are the base circle radii corresponding to the involutes on the small and large pressure angle sides respectively; μ l , μ r are the involute expansion angles corresponding to the tangent points of the measuring ball and the helical surfaces on the small and large pressure angle sides on the end face tooth profile; σ l , σ r are the initial expansion angles of the involutes on the small and large pressure angle sides respectively.
[0027] The normal vector N inv_l of the involute on the small pressure angle side of the asymmetric gear:
[0028]
[0029] The normal vector N inv_r of the involute on the large pressure angle side of the asymmetric gear:
[0030]
[0031] In the formula, β b is the base circle helix angle;
[0032] By combining equations (5) to (11) and solving for μ l and μ r , x dp , y dp and r m can be obtained, and then the distance across balls M can be obtained.
[0033] Specifically, when the initial involute development angles σ l and σ r of the involutes on the small and large pressure angle sides are as in formula (12), the distance across balls obtained is the upper limit S ts of the face arc tooth thickness, corresponding to the maximum distance across balls M max ;
[0034]
[0035] In the above formula, m n is the normal module; r is the pitch circle radius; β is the helix angle; invα tl and invα tr are the involute functions of the involutes on the small and large pressure angle sides of the asymmetric gear respectively, α tl and α tr are the face pressure angles of the involutes on the small and large pressure angle sides of the asymmetric gear respectively; α nl and α nr are the normal pressure angles of the involutes on the small and large pressure angle sides of the asymmetric gear respectively.
[0036] Specifically, when the initial involute development angles σ l and σ r of the involutes on the small and large pressure angle sides are as in formula (13), the distance across balls obtained is the lower limit S ti of the face arc tooth thickness, corresponding to the minimum distance across balls M min ;
[0037]
[0038] In the above formula, m n is the normal module; r is the pitch circle radius; β is the helix angle; invα tl and invα tr are the involute functions of the involutes on the small and large pressure angle sides of the asymmetric gear respectively, α tl and α tr are the face pressure angles of the involutes on the small and large pressure angle sides of the asymmetric gear respectively; α nl and α nrThey are the normal pressure angles of the involutes on the small and large pressure angle sides of the asymmetric gear, respectively.
[0039] Specifically, in equations (16) and (17), the initial involute development angles σ l , σ r on the left and right sides corresponding to the actual arc tooth thickness are obtained through the following process: Substitute equations (5) to (11) and the following equation (15) into equation (3) to solve for the initial involute development angles σ l , σ r on the left and right sides corresponding to the actual arc tooth thickness;
[0040] The involute development angles μ l , μ r of the tangent points of the measuring balls and the two helical surfaces corresponding to the end face tooth profile are:
[0041]
[0042] Compared with the prior art, the present invention has the following technical effects:
[0043] The calculation method for the mutual conversion of the cross-ball distance and the arc tooth thickness of the asymmetric modified helical gear proposed by the present invention is a numerical calculation method proposed by utilizing the properties of involutes and the contact properties between a spatial sphere and a spatial helical surface. It does not require the auxiliary measurement of the cross-ball distance through a three-dimensional model, and can effectively improve the design efficiency of the asymmetric modified helical gear.
[0044] The calculation method proposed by the present invention can quickly calculate the actual tooth thickness through the measured cross-ball distance during the machining and manufacturing stage of the gear, facilitating the designer to evaluate the machining accuracy of the asymmetric modified helical gear. BRIEF DESCRIPTION OF THE DRAWINGS
[0045] Figure 1 is a schematic diagram of the cross-ball distance measurement of the asymmetric modified helical gear.
[0046] Figure 2 is a schematic diagram of the gear shaft cross-section where the center of the measuring ball is located.
[0047] Figure 3 is a schematic diagram of the involute of the tooth surface of the asymmetric modified helical gear.
[0048] Figure 4 is a schematic diagram of the tooth thickness of the asymmetric modified helical gear.
[0049] Figure 5 is a schematic diagram of the asymmetric gear model corresponding to the upper limit of the end face arc tooth thickness.
[0050] Figure 6 is a schematic diagram of the maximum cross-ball distance measurement of the asymmetric gear model corresponding to the upper limit of the end face arc tooth thickness.
[0051] Figure 7Schematic diagram of an asymmetric gear model corresponding to the lower limit of the arc tooth thickness at the end face.
[0052] Figure 8 Schematic diagram of the minimum ball span measurement of an asymmetric gear model corresponding to the lower limit of the arc tooth thickness at the end face.
[0053] Figure 9 Schematic diagram of an asymmetric gear model corresponding to the actual arc tooth thickness at the end face.
[0054] Figure 10 Schematic diagram of the ball span measurement of an asymmetric gear model corresponding to the actual arc tooth thickness at the end face.
[0055] The meanings of the various labels in the figure are as follows:
[0056] 1. Measuring ball; 2. Asymmetric modified helical gear. Specific implementation mode
[0057] The following are specific embodiments of the present invention. It should be noted that the present invention is not limited to the following specific embodiments, and all equivalent transformations made on the basis of the technical solution of this application fall within the protection scope of the present invention.
[0058] Embodiment:
[0059] As Figures 1 to 10 shown, this embodiment provides a calculation method for the mutual conversion of the ball span and the arc tooth thickness of an asymmetric modified helical gear. The figure shows an asymmetric modified helical gear 2 and a measuring ball 1. This method includes:
[0060] 1. Method for calculating the ball span of an asymmetric modified helical gear according to the arc tooth thickness:
[0061] Step 1: Given the normal theoretical arc tooth thickness S n and the end face theoretical arc tooth thickness S t on the premise of, determine the upper deviation ΔS ns and the lower deviation ΔS ni of the normal arc tooth thickness according to the operating conditions and the selected design standards, and calculate the upper limit S ns and the lower limit S ni of the normal arc tooth thickness, as well as the corresponding upper limit S ts and the lower limit S ti of the end face arc tooth thickness;
[0062] Step 2: Select the measuring ball diameter d p , generally preferably take 1.728m n and round up. The measuring ball diameter d p can be adjusted according to the actual situation. m n is the normal module;
[0063] Step 3: Calculate the theoretical cross-ball distance M of the asymmetric helical gear:
[0064]
[0065] In the above formula, r m is the center O of the measuring ball corresponding to the cross-ball distance M dp The radius of the circle; π is the circumference of the circle; z is the number of teeth; i is the tooth coefficient, when the number of teeth is odd, i=1, when the number of teeth is even, i=0.
[0066] In formula (1), r m The calculation formula is:
[0067]
[0068] In the above formula, x dp ,y dp The center of the ball O dp Plane coordinate O in the gear shaft section dp (x dp ,y dp ),like Figure 2 As shown;
[0069] In formula (2), O dp (x dp ,y dp ) can be solved by the following formula:
[0070]
[0071] In the above formula, x l ,y l is the coordinate of the center of the sphere when the measuring sphere is tangent to the helical surface on the side with small pressure angle of the asymmetric gear (x l ,y l ); x r ,y r is the coordinate of the center of the sphere when the measuring sphere is tangent to the helical surface on the side with large pressure angle of the asymmetric gear (x r ,y r ); When the coordinates of the two sphere centers coincide, it can be considered that the measuring sphere is tangent to the spiral surfaces on both sides at the same time.
[0072] The variables in formula (3) are expressed as:
[0073]
[0074] In formula (5), x inv_l ,y inv_l The equation of the involute on the small pressure angle side of the asymmetric gear is:
[0075]
[0076] In Equation (6), x inv_r , y inv_r are the equations of the involutes on the large pressure angle side of the asymmetric gear:
[0077]
[0078] In Equations (7) and (8), r bl , r br are the base circle radii corresponding to the involutes on the small and large pressure angle sides respectively; μ l , μ r are the involute expansion angles corresponding to the tangent points of the measuring balls with the helical surfaces on the small and large pressure angle sides on the end face tooth profile respectively; σ l , σ r are the initial expansion angles of the involutes on the small and large pressure angle sides respectively:
[0079]
[0080] In the above formula, m n is the normal module; r is the pitch circle radius; β is the helix angle; invα tl , invα tr are the involute functions on the small and large pressure angle sides of the asymmetric gear respectively, α tl , α tr are the end face pressure angles of the involutes on the small and large pressure angle sides of the asymmetric gear respectively; α nl , α nr are the normal pressure angles of the involutes on the small and large pressure angle sides of the asymmetric gear respectively.
[0081] In Equation (5), x n_inv_l , y n_inv_l are the elements in the normal vector N inv_l of the involute on the small pressure angle side of the asymmetric gear:
[0082]
[0083] In Equation (6), x n_inv_r , y n_inv_r are the elements in the normal vector N inv_r of the involute on the large pressure angle side of the asymmetric gear:
[0084]
[0085] In Equations (10) and (11), β b is the base circle helix angle.
[0086] It can be seen from Equations (5) to (11) that in Equation (3), there are only two unknowns μ l , μ r , that is, within the range of the involute expansion angle of the asymmetric gear tooth profile, μl , μ r There is a unique solution. Solving equation (3) can obtain the exact value of the theoretical cross-ball distance of the asymmetric modified helical gear.
[0087] Step 4: Calculate the upper limit S of the face arc tooth thickness ts The corresponding maximum cross-ball distance M max :
[0088] Express σ in equation (9) l , σ r as:
[0089]
[0090] The obtained cross-ball distance M = M max ;
[0091] Step 5: Calculate the lower limit S of the face arc tooth thickness ti The corresponding minimum cross-ball distance M min :
[0092] Express σ in equation (9) l , σ r as:
[0093]
[0094] The obtained cross-ball distance M = M min .
[0095] 2. Method for calculating the actual arc tooth thickness of an asymmetric modified helical gear based on the measured cross-ball distance:
[0096] Step 1: Select a measuring ball with an appropriate diameter d p to measure the cross-ball distance M of the asymmetric modified helical gear c ;
[0097] Step 2: Calculate the radius r of the circle where the center of the measuring ball is located mc :
[0098]
[0099] In the above formula, i is the tooth number coefficient. When the number of teeth is odd, i = 1; when the number of teeth is even, i = 0.
[0100] Step 3: Calculate the involute expansion angles μ l , μ r corresponding to the tangent points of the measuring ball and the two helical surfaces on the face tooth profile:
[0101]
[0102] Step 4: Substitute Equations (5) to (11) and Equation (15) into Equation (3) to solve for the initial involute development angles σ l and σ r ;
[0103] Step 5: Calculate the measured ball span M c corresponding to the actual face arc tooth thickness S tc and the actual normal arc tooth thickness S nc :
[0104] S tc = πm t - r(σ l + σ r + invα tl + invα tr ) (16)
[0105] S nc = πm n - r(σ l + σ r + invα tl + invα tr )cosβ (17)
[0106] In the above equations, m t is the face module, and m n is the normal module.
[0107] 3. Calculation example for the mutual conversion of the ball span and arc tooth thickness of an asymmetrically modified helical gear:
[0108] The gear parameters after design are as follows in the table:
[0109] Table 1 Gear parameters
[0110]
[0111]
[0112] Based on the upper limit S ts of the face arc tooth thickness, the relevant parameters for calculating the maximum ball span M max are as follows in the table:
[0113] Table 2 Relevant parameters
[0114]
[0115] Based on the lower limit S ti of the face arc tooth thickness, the relevant parameters for calculating the minimum ball span M min are as follows in the table:
[0116] Table 3 Relevant parameters
[0117]
[0118] According to the measured ball span M c Calculate the actual face arc tooth thickness S tc and the relevant parameters of the actual normal arc tooth thickness Sn nc are as follows in the table:
[0119] Table 4 Relevant parameters
[0120]
[0121] The method for mutual conversion calculation of the ball span and arc tooth thickness of an asymmetric modified helical gear provided by the present invention is also applicable to an asymmetric helical gear without modification design.
[0122] The preferred embodiments of the present invention have been described in detail above in conjunction with the accompanying drawings. However, the present invention is not limited to the specific details in the above embodiments. Within the scope of the technical concept of the present invention, various simple modifications can be made to the technical solutions of the present invention, and these simple modifications all fall within the protection scope of the present invention.
[0123] In addition, it should be noted that, in the case of no contradiction, the various specific technical features described in the above specific embodiments can be combined in any appropriate manner. To avoid unnecessary repetition, the present invention will not separately describe various possible combination manners.
[0124] Furthermore, any combination can be made between various different embodiments of the present invention, as long as it does not violate the idea of the present invention, and it should also be regarded as the content disclosed by the present invention.
Claims
1. A method for calculating the mutual conversion between the cross-ball distance and arc tooth thickness of an asymmetric helical gear, characterized in that: Calculation of arc tooth thickness of asymmetric helical gear across the ball distance M: In the formula, r m is the center O of the measuring ball corresponding to the cross-ball distance M dp The radius of the circle; π is the circumference ratio; z is the number of teeth; i is the tooth number coefficient, when there are odd teeth, i=1, when there are even teeth, i=0; d p Measuring ball diameter.
2. The method for calculating the mutual conversion between the cross-ball distance and the arc tooth thickness of an asymmetric helical gear according to claim 1, characterized in that: Calculate the actual end face arc tooth thickness S by measuring the cross-ball distance of asymmetric helical gears tc , actual normal arc tooth thickness S nc : S tc =πm t -r(s l +s r +invα tl +invα tr ) (16) S nc =πm n -r(s l +s r +invα tl +invα tr )cosβ (17) In the formula, m t is the end modulus, m n is the normal modulus, r is the pitch circle diameter, σ l , σ r is the initial expansion angle of the involute on the left and right sides corresponding to the actual arc tooth thickness; invα tl 、invα tr are the involute functions of the small and large pressure angle sides of the asymmetric gear respectively; α tl , α tr are the end face pressure angles of the involute on the small and large pressure angle sides of the asymmetric gear; β is the helix angle.
3. The method for calculating the mutual conversion between the cross-ball distance and the arc tooth thickness of an asymmetric helical gear according to claim 2, characterized in that: In formula (1), r m for: In the formula, x dp ,y dp The center of the ball O dp Plane coordinate O in the gear shaft section dp (x dp ,y dp ); Among them, O dp (x dp ,y dp ) can be solved by the following formula: In the above formula, x l ,y l is the coordinate of the center of the sphere when the measuring sphere is tangent to the helical surface on the side with small pressure angle of the asymmetric gear (x l ,y l ); x r ,y r is the coordinate of the center of the sphere when the measuring sphere is tangent to the helical surface on the side with large pressure angle of the asymmetric gear (x r ,y r ); When the coordinates of the two sphere centers coincide, it is considered that the measuring sphere is tangent to the helical surfaces on both sides at the same time.
4. The method for calculating the mutual conversion between the cross-ball distance and the arc tooth thickness of an asymmetric helical gear according to claim 3, characterized in that: The variables in formula (3) are expressed as: In the formula, x inv_l ,y inv_l is the equation of the involute on the small pressure angle side of the asymmetric gear, x inv_r ,y inv_r is the equation of the involute on the large pressure angle side of the asymmetric gear; x n_inv_l ,y n_inv_l is the normal vector N of the involute on the small pressure angle side of the asymmetric gear inv_l The elements in x n_inv_r ,y n_inv_r is the normal vector N of the involute on the large pressure angle side of the asymmetric gear inv_r The elements in .
5. The method for calculating the mutual conversion between the cross-ball distance and the arc tooth thickness of an asymmetric helical gear according to claim 4, characterized in that: The equation of the involute on the small pressure angle side of the asymmetric gear is: The equation of the involute on the large pressure angle side of the asymmetric gear is: In the formula, r bl 、r br are the base circle radii corresponding to the involutes on the small and large pressure angle sides, respectively; μ l , μ r are the involute angles of the tangent points between the measuring ball and the helical surfaces with small and large pressure angles on the end face tooth profile; σ l , σ r are the initial development angles of the involutes on the small and large pressure angle sides, respectively; The involute normal vector N of the asymmetric gear with small pressure angle invl : The involute normal vector N of the asymmetric gear with large pressure angle inv_r : In the formula, β b is the base circle helix angle; Combining equations (5) to (11), we can solve μ l , μ r , then we can get x dp ,y dp and r m , thus obtaining the cross-ball distance M.
6. The method for calculating the mutual conversion between the cross-ball distance and the arc tooth thickness of an asymmetric helical gear according to claim 5, characterized in that: The initial expansion angles σ of the involutes on the small and large pressure angle sides l , σ r When the formula (12) is used, the span distance is the upper limit of the end face arc tooth thickness S ts The corresponding maximum span distance M max ; In the above formula, m n is the normal module; r is the pitch circle radius; β is the helix angle; invα tl 、invα tr are the involute functions of the small and large pressure angle sides of the asymmetric gear, α tl , α tr are the end face pressure angles of the involutes on the small and large pressure angle sides of the asymmetric gear; α nl , α nr are the normal pressure angles of the involutes on the small and large pressure angle sides of the asymmetric gear, respectively.
7. The method for calculating the mutual conversion between the cross-ball distance and the arc tooth thickness of an asymmetric helical gear according to claim 5, characterized in that: The initial expansion angles σ of the involutes on the small and large pressure angle sides l , σ r When the formula (13) is used, the span distance is the lower limit of the end face arc tooth thickness S ti The corresponding minimum span M min ; In the above formula, m n is the normal module; r is the pitch circle radius; β is the helix angle; invα tl 、invα tr are the involute functions of the small and large pressure angle sides of the asymmetric gear, α tl , α tr are the end face pressure angles of the involutes on the small and large pressure angle sides of the asymmetric gear; α nl , α nr are the normal pressure angles of the involutes on the small and large pressure angle sides of the asymmetric gear, respectively.
8. The method for calculating the mutual conversion between the cross-ball distance and the arc tooth thickness of an asymmetric helical gear according to claim 5, characterized in that: In equations (16) and (17), the initial angles σ of the involutes on the left and right sides corresponding to the actual arc tooth thickness are l , σ r The solution process is: Substitute equations (5) to (11) and the following equation (15) into equation (3) to obtain the initial angle σ of the involute on the left and right sides corresponding to the actual arc tooth thickness: l , σ r ; The involute angle μ corresponding to the tangent point between the measuring ball and the helical surfaces on both sides on the end face tooth profile l , μ r :