A method of profile modification of a tooth profile of an involute gear

Abstract

The application belongs to the technical field of gear tooth profile modification, and discloses a tooth profile modification method of involute gear, which uses involute to replace traditional tooth profile modification curve, and uses a transition arc to connect the unmodified involute and the modified involute; first, the tooth profile modification parameters and the basic parameters of the gear are given; the intersection between the unmodified involute profile and the modified profile is calculated according to the modification length; the intersection between the modified profile and the addendum circle is calculated according to the maximum modification amount; the center of the transition arc, the radius of the transition arc, the base circle radius of the modified involute, the starting angle and the developed angles of the two endpoints are calculated according to the tangent condition and the modified involute equation; finally, the transition arc and a new involute are constructed as the modified profile line. Through the method, the accurate dressing of the grinding wheel in the gear grinding process can be realized, the direction of the normal pressure between the tooth profiles in the tooth profile modification part is ensured to be unchanged, the smooth meshing of the gear pair is beneficial, and technical support for subsequent finishing can be well provided.

Description

Technical Field

[0001] This invention relates to the field of gear tooth profile modification technology, and particularly to a method for modifying the tooth profile of an involute gear. Background Technology

[0002] Gear transmission, as a typical mechanical transmission method, is widely used in aerospace, shipbuilding, and instrumentation. During gear transmission, due to manufacturing errors and elastic deformation during meshing, engagement and disengagement impacts are unavoidable, necessitating gear profile modification. Gear profile modification curves are mainly linear and curved. Traditional formulas for gear profile modification curves include Walker's formula, the Hidaka, Terauchi, Fujii formula, and the Akikawa formula. Patent "A Modification Design Method for Spur Gear Tooth Surface, CN108953549B" uses spline curves for tooth profile modification, achieving second-order continuity for all points within the tooth surface. Patent "A Harmonic Drive Involute Tooth Profile Modification Method, CN107882950B" uses the tangency between the tooth profile modification curve and the involute profile as a constraint. It scales and rotates the motion trajectory curve of the flexible gear tooth center point to obtain the modification curve for the harmonic involute tooth profile, resulting in a smooth transition of the flexible gear tooth surface and improved meshing performance. Considering that grinding is usually the final process in gear finishing, and the power transmission performance of gear pairs, current modification methods have the following problems:

[0003] (1) Using the existing tooth profile modification curve for tooth profile modification, the accuracy of the sand profile is not easy to control during the grinding process, which ultimately results in the inaccurate machining of the tooth profile modification part.

[0004] (2) In the tooth profile modification section, it is impossible to accurately control the direction of the positive pressure between the tooth profiles, which is not conducive to the smooth meshing of the gear pair. Summary of the Invention

[0005] To address the shortcomings of existing technologies, this invention proposes a method for modifying the tooth profile of involute gears. By using an involute as the tooth profile modification curve, both the modified and unmodified portions of the gear tooth profile are involutes, providing a reference for subsequent finishing processes.

[0006] The technical solution adopted in this invention is as follows: a method for modifying the tooth profile of an involute gear, wherein the tooth profile of the involute gear includes an unmodified involute, a transition arc, and a modified involute; the unmodified involute and the modified involute are smoothly transitioned through a transition arc; the modified portion of the involute gear includes a transition arc and a modified involute; the steps for solving the modified portion are as follows:

[0007] First, given the tooth profile modification parameters, including the maximum modification amount and modification length, as well as the basic gear parameters, including module, number of teeth, addendum coefficient, clearance coefficient, and pressure angle, etc.; then, calculate the intersection point A between the unmodified involute profile and the modified part profile based on the modification length; further, calculate the intersection point C between the modified part profile and the addendum circle based on the maximum modification amount; further still, based on the tangency condition and by simultaneously establishing the modified involute equations, calculate the center coordinates and radius of the transition arc, as well as the base circle radius and starting angle of the modified involute; calculate the development angles of the intersection points B and C of the transition arc and the modified involute; finally, construct the transition arc between A and B, and construct a new involute between B and C.

[0008] Step 1: Given the involute gear tooth profile modification parameters, including the maximum modification amount δ max and shaping length L;

[0009] Step 2: Given the basic parameters of the involute gear, including the normal module m n normal pressure angle α n Normal surface tooth tip height coefficient Normal porosity coefficient Helix angle β, number of teeth z;

[0010] The end face parameters are calculated based on the basic parameters of the involute gear; the end face parameters include: end face module m. t End face pressure angle α t Pitch circle radius r, end face tooth addendum coefficient End face clearance coefficient

[0011] The specific calculation formula is as follows:

[0012]

[0013] In the end face, the base circle radius of the involute gear is:

[0014] r b =m t zcosα t / 2

[0015] Tooth tip circle radius r a With the root circle radius r f They are respectively:

[0016]

[0017] The initial angle σ0 of the unshaped involute is expressed as:

[0018] σ0=πm t / 4r-(tanα t -α t )

[0019] Let M be a point on the unmodified involute of the involute gear. Draw a tangent line from point M to the base circle of the gear, with the point of tangency being A1. Let O be the center of the base circle of the involute gear, and E be the starting point of the unmodified involute. The angle between OE and OA1 is the development angle at point M. The coordinates of point M are then represented as:

[0020]

[0021] Step 3: Calculate the intersection point A between the unshaped involute profile and the shaped profile based on the shaping length L;

[0022]

[0023] Where, r a The radius of the tooth tip circle, Let A be the development angle of point A on the unshaped involute contour.

[0024] Step 4: Based on the maximum reshaping amount δ max Calculate the intersection point C between the modified profile and the addendum circle.

[0025]

[0026] in, The development angle at the intersection of the unmodified involute profile and the tooth tip circle;

[0027] Step 5: Calculate the radius and center of the transition arc, the base circle radius and starting angle of the modified involute; the development angles at points B and C where the transition arc intersects the modified involute.

[0028] Represent the transition arc using parametric equations, where the radius of the transition arc is r. s The center of the transition arc is O. s The arc angle parameters of points A and B are θ and θ, respectively. A and θ B Then the following equations hold:

[0029]

[0030]

[0031] On the transition arc, the tangent vector at point A is represented as:

[0032]

[0033] On the unshaped involute, the tangent vector at point A is represented as:

[0034]

[0035] For the unshaped involute and the transition arc to be tangent at point A, the following equation holds:

[0036]

[0037] Similarly, for the transition arc and the modified involute to be tangent at point B, the following equation holds:

[0038]

[0039] Construct a new involute between points B and C as a modified involute, the base circle radius of which is r. b1 The initial angle is σ 01 The unfolding angle of point B on the involute is... The unfolding angle of point C is... Then any point M1 on the involute curve is represented as:

[0040]

[0041] Points B and C on the newly constructed involute are represented by the following formula:

[0042]

[0043]

[0044] The system of equations is obtained as follows:

[0045]

[0046] x B y B In the expression, O s By x B y A It means that θ B From σ 01 , This means solving the above system of equations and filtering the calculation results; after the solution is completed, r b1 σ 01 , and r s Given the variable, any point on the new involute... Represented as:

[0047]

[0048] r b1 σ 01 , and r s The unknown variable is one of four equations, which constitutes a system of indeterminate equations. The system of equations is solved using numerical methods.

[0049] The beneficial effects of this invention are as follows: This invention is based on involute gear tooth profile modification, using the involute as the tooth profile modification curve. This ensures that both the modified and unmodified parts of the gear tooth profile are involutes, which is beneficial for dressing the grinding wheel during the finishing process. Moreover, the involute is separable. When both the driving gear and the driven gear use the involute as the tooth profile modification curve, the instantaneous transmission ratio can be kept constant during the meshing of the modified tooth profile parts in the transmission process. The direction of the normal force between the tooth profiles is along the common normal direction of the contact point. The common normal is the common tangent line on the same side of the two base circles and there is only one. Therefore, using the involute as the tooth profile modification curve can ensure that the direction of the normal force between the tooth profiles remains unchanged, that is, it is always tangent to the base circle. Attached Figure Description

[0050] Figure 1 A schematic diagram of the tooth profile modification method for involute gears;

[0051] The symbols shown in the diagram are explained as follows: O is the origin of the coordinate system. Point M represents any point on the unmodified involute profile, and point M1 represents any point on the modified involute profile. Point A represents the intersection of the modified profile and the unmodified profile; point A1 is the point of tangency between the tangent line to the base circle passing through point M and the base circle. Point C represents the intersection of the modified profile and the addendum circle. Point D represents the intersection of the unmodified involute profile and the addendum circle. b r is the base circle radius of the unshaped involute profile. b1 σ0 is the base circle radius of the modified involute; σ0 is the starting angle of the unmodified involute. 01 This is the starting angle of the involute curve in the shaping section; For the variable of the unshaped involute or the development angle of point M, This refers to the variable of the involute in the modified part or the development angle of point M1;

[0052] Figure 2(a) shows an embodiment of the present invention. The solid line is the unmodified involute outline, the dotted line is the transition arc portion, and the dash line is the modified involute portion.

[0053] Figure 2(b) is an enlarged view of the transition arc of the tooth profile on the right side of the tooth groove;

[0054] Figure 2(c) is an enlarged view of the transition arc of the tooth profile on the left side of the tooth groove;

[0055] Figure 3 The pressure angle variation curves on the tooth profile are obtained by using parabolic, circular, cubic, and involute curves respectively. Detailed Implementation

[0056] The specific implementation of the present invention will now be described in detail with reference to the accompanying drawings and technical solutions.

[0057] This invention proposes a method for modifying the tooth profile of involute gears. A new involute curve is used as the tooth profile modification curve of the involute gear, so that both the unmodified tooth profile and the modified tooth profile are involutes. This ensures that the direction of the positive pressure at a certain point on the gear tooth profile is always tangent to the base circle of the unmodified or modified involute curve during transmission, making the gear meshing transmission smoother.

[0058] To ensure the continuity of curvature between the unmodified and modified involute curves at their intersection point and to avoid vibration and impact, a transition arc is used to connect the two curves, thus ensuring a smooth transition of the tooth surface. Figure 1 As shown, the contour includes segment AB (transition arc) and segment BC (contouring involute).

[0059] First, given the tooth profile modification parameters, including the maximum modification amount and modification length, as well as the basic gear parameters, including module, number of teeth, addendum coefficient, clearance coefficient, and pressure angle, etc.; then, calculate the intersection point A between the unmodified involute profile and the modified profile based on the modification length (e.g., ...). Figure 1 (as shown); further, calculate the intersection point C between the profile of the modified part and the addendum circle based on the maximum modification amount (as shown). Figure 1 (As shown); further, based on the tangency condition, and by simultaneously establishing the equations of the modified involute, calculate the coordinates of the center of the transition arc, its radius, and the base circle radius, initial angle, and development angles of the modified involute at points B and C; finally, construct a transition arc between A and B, and construct a new involute between B and C (as shown). Figure 1 (As shown).

[0060] This method uses an involute curve instead of the traditional tooth profile shaping curve, which provides excellent technical support for subsequent finishing. Taking an involute helical gear as an example, the specific steps of this invention are described in detail below:

[0061] Step 1. Given the tooth profile modification parameters, including the maximum modification amount and modification length, use δ respectively. max And L represents. Point D (e.g.) Figure 1 (As shown) is the intersection of the unmodified involute and the addendum circle, δ max Let L be the length of the tooth tip arc between points C and D, and L be the difference in distance between points C and A from the origin.

[0062] Step 2. Given the basic parameters of the involute helical gear, mainly including the normal module m. n The normal pressure angle is Δ n The normal surface tooth tip height coefficient is Normal porosity coefficient is The helix angle is α, and the number of teeth is z; when the involute gear is an involute spur gear, the helix angle is 0.

[0063] The end face parameters can be calculated from the basic parameters of helical gears. These end face parameters include: the end face module is m. t The end face pressure angle is Δ t The pitch circle radius is r, and the end face tooth addendum coefficient is... The end face clearance coefficient is

[0064] The specific calculation formula is as follows:

[0065]

[0066] In the end face, the base circle radius of the helical gear is:

[0067] r b =m t zcosΔ t / 2

[0068] Tooth tip circle radius r a With the root circle radius r f They are respectively:

[0069]

[0070] The initial angle σ0 of the unshaped involute can be expressed as:

[0071] σ0=πm t / 4r-(tanα t -α t )

[0072] Taking the tooth profile of a helical gear end face as an example, let M be a point on the involute. Draw a tangent line to the base circle through point M, with the point of tangency being A1. O is the center of the base circle of the helical gear, and E is the starting point of the involute. The angle between OE and OA1 is called the development angle at point M. If M can be represented as:

[0073]

[0074] Step 3. Calculate the intersection point A between the unshaped involute profile and the shaped profile based on the shaping length.

[0075]

[0076] Where, r a The radius of the tooth tip circle, Let be the development angle of point A on the unshaped involute contour.

[0077] Step 4. Calculate the intersection point C between the profile of the modified section and the addendum circle based on the maximum modification amount.

[0078]

[0079] in, It is the development angle at the intersection of the unmodified involute tooth profile and the tooth tip circle.

[0080] Step 5. Further, calculate the radius and center of the transition arc, as well as the base circle radius, starting angle, and development angles of points B and C of the modified involute.

[0081] Express the transition arc using parametric equations, and let the radius of the transition arc be r. s The center of the transition arc is O. s The parameters for points A and B are θ. A and θ B Then the following equations hold:

[0082]

[0083]

[0084] On the transition arc, the tangent vector at point A can be expressed as:

[0085]

[0086] On the unshaped involute, the tangent vector at point A can be expressed as:

[0087]

[0088] For the unshaped involute and the transition arc to be tangent at point A, the following equation holds:

[0089]

[0090] Similarly, for the transition arc and the modified involute to be tangent at point B, the following equation holds:

[0091]

[0092] Next, a new involute is constructed between B and C as the tooth profile modification curve, and the base circle radius of the modification involute is set to r. b1 The initial angle is σ 01 The unfolding angle of point B on the involute is... The unfolding angle of point C is... Then any point M1 on the involute can be represented as:

[0093]

[0094] Points B and C on the modified involute can be represented by the following formula:

[0095]

[0096]

[0097] Therefore, we get:

[0098]

[0099] x B y B In the expression, O s Can be derived from x B y A It means that θ B Can be derived from σ 01 , If we express that, then in this system of equations, r b1 σ 01 , and r s Since the unknown variable is one of four equations, forming a system of indeterminate equations, numerical methods can be used to solve the problem, and the results can be filtered. After solving, r b1 σ 01 , and r s As a known variable, any point on the involute curve is then considered. It can be represented as:

[0100]

[0101] The following section uses an involute helical gear, which is widely used in the wind power industry, to calculate the tooth profile modification curve proposed in this invention. However, the solution of this invention is not limited to this; it can also be applied to cylindrical spur gears, bevel gears, etc.

[0102] Based on the above solution method, the specific steps are as follows:

[0103] The preferred main parameters of the involute helical gear are as follows: number of teeth z = 26, normal module m n =17mm, normal pressure angle α n =20°, helix angle β=6°, addendum coefficient is Normal porosity coefficient is The preferred shaping parameters are as follows: maximum shaping amount δ max =0.25mm, shaping height L=20mm.

[0104] Calculate the gear end face parameters, the radii of each circle on the gear, and the initial position angle. The calculation results are as follows: the end face module is m. t =17.09mm, end face pressure angle is α t =20.1°, pitch circle radius is r = 222.22mm, and end face tooth addendum coefficient is The end face clearance coefficient is The base circle radius of the helical gear is r b =208.68mm, tooth tip circle radius is r a＝ 239.22mm, root circle radius is r f＝ 200.97mm, initial position angle is σ0=2.59°.

[0105] Calculate the intersection point A between the unshaped involute profile and the shaped profile, according to the formula. Substituting the numerical values, we get The coordinates of point A are (12.21, 218.88).

[0106] Calculate the intersection point C between the modified profile and the addendum circle, according to the formula. Substituting the numerical values, we get The coordinates of point C are (22.91, 238.12).

[0107] Solve for the parameters of the transition arc and the modified involute, and solve the system of equations:

[0108]

[0109] Where, x B y B It can be represented as

[0110]

[0111] Preferably, the initial values ​​(angle parameters expressed in radians) are given as follows:

[0112]

[0113] We obtain r separately. b1 =207.76mm, σ 01 =2.51°, r s =10.2mm.

[0114] According to the formula The parameters of points A and B on the transition arc are obtained as follows: θ A =158.97°, θ B =158.07°

[0115] Therefore, any point on the transition arc can be represented as:

[0116]

[0117] Any point on the contour line It can be represented as:

[0118]

[0119] At this point, the contour solution is complete, and the modified tooth profile is shown in Figure 2.

[0120] To further illustrate the advantages of this invention, the pressure angles at various points on the tooth profile were calculated when the tooth profile modification curves were parabolic, circular, cubic, and involute. Figure 3 As shown.

[0121] According to mechanical principles, a smaller pressure angle is better for a mechanism to have good force transmission characteristics. It can be seen that, in the tooth profile modification section, using an involute as the tooth profile modification curve results in a smoother pressure angle change compared to the other three types of tooth profile modification curves. Furthermore, at the tooth tip, using an involute as the tooth profile modification curve can achieve a smaller pressure angle. Considering the motion principle of gear machining tools and the force transmission characteristics of gear pairs, using an involute as the tooth profile modification curve is beneficial for grinding wheel dressing during gear finishing and for better force transmission performance of the gear pair during meshing.

[0122] The above embodiments are only for better explaining the present invention. The method of the present invention is not limited to external gear modification, but is also applicable to internal gear modification.

[0123] Those skilled in the art will readily understand that the above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A method for modifying the tooth profile of an involute gear, characterized in that, The tooth profile of the involute gear includes an unmodified involute, a transition arc, and a modified involute; the unmodified involute and the modified involute are smoothly transitioned through a transition arc; the modified part of the involute gear includes a transition arc and a modified involute. The steps to solve the shape-correction part are as follows: Step 1: Given the involute gear tooth profile modification parameters, including the maximum modification amount δ max and shaping length L; Step 2: Given the basic parameters of the involute gear, including the normal module m n normal pressure angle α n Normal surface tooth tip height coefficient Normal porosity coefficient Helix angle β, number of teeth z; The end face parameters are calculated based on the basic parameters of the involute gear; the end face parameters include: end face module m. t End face pressure angle α t Pitch circle radius r, end face tooth addendum coefficient End face clearance coefficient The specific calculation formula is as follows: In the end face, the base circle radius of the involute gear is: r b =m t zcosα t / 2 Tooth tip circle radius r a With the root circle radius r f They are respectively: The initial angle σ0 of the unshaped involute is expressed as: σ0=πm t / 4r-(tanα t -a t ) Let M be a point on the unmodified involute of the involute gear. Draw a tangent line from point M to the base circle of the gear, with the point of tangency being A1. Let O be the center of the base circle of the involute gear, and E be the starting point of the unmodified involute. The angle between OE and OA1 is the development angle at point M. The coordinates of point M are then represented as: Step 3: Calculate the intersection point A between the unshaped involute profile and the shaped profile based on the shaping length L; Where, r a The radius of the tooth tip circle, Let A be the development angle of point A on the unshaped involute contour. Step 4: Based on the maximum reshaping amount δ max Calculate the intersection point C between the modified profile and the addendum circle. in, The development angle at the intersection of the unmodified involute profile and the tooth tip circle; Step 5: Calculate the radius and center of the transition arc, the base circle radius and starting angle of the modified involute; the development angles at points B and C where the transition arc intersects the modified involute. Represent the transition arc using parametric equations, where the radius of the transition arc is r. s The center of the transition arc is O. s The arc angle parameters of points A and B are θ and θ, respectively. A and θ B Then the following equations hold: On the transition arc, the tangent vector at point A is represented as: On the unshaped involute, the tangent vector at point A is represented as: For the unshaped involute and the transition arc to be tangent at point A, the following equation holds: Similarly, for the transition arc and the modified involute to be tangent at point B, the following equation holds: Construct a new involute between points B and C as a modified involute, the base circle radius of which is r. b1 The initial angle is σ 01 The unfolding angle of point B on the involute is... The unfolding angle of point C is... Then any point M1 on the involute curve is represented as: Points B and C on the newly constructed involute are represented by the following formula: The system of equations is obtained as follows: x B y B In the expression, O s By x B y A It means that θ B From σ 01 , This means solving the above system of equations and filtering the calculation results; after the solution is completed, r b1 σ 01 , and r s Given the variable, any point on the new involute... Represented as:

2. The method for modifying the tooth profile of an involute gear according to claim 1, characterized in that, The system of equations is solved using numerical methods.

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