A design method for machining spiral bevel gears using a sinusoidal profile tool

Abstract

The present invention provides a design method for processing spiral bevel gears with a sinusoidal profile tool, including: establishing a sinusoidal profile coordinate system, a sinusoidal profile generating surface mathematical model, obtaining the sinusoidal profile generating surface position vector and unit normal vector; using gear pair geometric parameters and processing parameters to establish a spiral bevel gear cutting mathematical model, and obtaining the gear pair tooth surface equation based on differential geometry principles and meshing equations; through tooth surface division, the tooth surface is discretized by a rotation projection method, and the coordinates of discrete points on the tooth surface and their normal vectors are solved; a tooth surface mismatch topological model is established, a tooth surface mismatch diagram is constructed, the influence of the design parameters of the sinusoidal profile on the tooth surface mismatch amount is analyzed, and the sinusoidal profile design parameters are determined; and a spiral bevel gear pair is obtained and tested. The spiral bevel gear designed by the present invention can avoid edge contact and reduce the contact stress of the tooth surface, reduce the impact of the tooth pair when meshing in and out, increase the stability during operation, and reduce the noise during operation.

Description

Technical Field

[0001] The present invention relates to the technical field of gear processing, and in particular to a design method for processing spiral bevel gears using a sinusoidal profile tool. Background Art

[0002] Spiral bevel gears, with their low noise, smooth transmission, strong load-bearing capacity, and high reliability, have become essential transmission components in automobiles, construction machinery, and other equipment. Currently, the most widely used tool profile for machining spiral bevel gears is a straight line. Linear tools, with their simple structure and ease of manufacture, continue to dominate the gear manufacturing industry. However, gear pairs designed and machined using linear tools employing the "local conjugate principle" are prone to stress concentration and edge contact. This is particularly true during engagement and disengagement, which can cause significant impact and lead to high vibration and noise levels.

[0003] To address these issues, the tooth surface can be modified to improve meshing performance. Using specialized tools to modify the tooth profile is an effective method. Currently, the use of various tool profiles for tooth surface modification is gradually entering the gear manufacturing industry. These profiles primarily include parabolic profiles, circular arc profiles, or profiles composed of multiple curves. However, the mathematical models for these tool profiles have a limited number of modification parameters, which limits the flexibility and freedom of tooth surface optimization. Summary of the Invention

[0004] The purpose of the present invention is to provide a design method for processing spiral bevel gears using a sinusoidal profile tool in response to the deficiencies in the above-mentioned background technology, so that the processed spiral bevel gear pair can avoid poor contact characteristics while having greater processing flexibility and freedom.

[0005] In order to achieve the above object, the present invention provides a design method for machining spiral bevel gears using a sinusoidal profile tool, comprising the following steps:

[0006] S1, establish the sinusoidal tool profile coordinate system and the mathematical model of the sinusoidal tool profile generating surface, and obtain the position vector and unit normal vector of the sinusoidal tool profile generating surface;

[0007] S2, using the gear pair geometric parameters and processing parameters, establish the spiral bevel gear cutting mathematical model, based on the differential geometry principle and meshing equation, obtain the gear pair tooth surface equation;

[0008] S3, by dividing the tooth surface, the tooth surface is discretized by the rotation projection method, and the coordinates of the discrete points on the tooth surface and their normal vectors are solved;

[0009] S4, establish a tooth surface mismatch topology model, use the tooth surface data calculated by the straight tool profile as the reference tooth surface, and the tooth surface data processed by the sinusoidal tool profile as the tooth surface to be analyzed, calculate the projection of the vectors of the corresponding grid points on the reference tooth surface normal vector, and graphically construct a tooth surface mismatch map based on the mismatch amount of each grid point. Analyze the influence of the design parameters of the sinusoidal tool profile on the tooth surface mismatch amount, and determine the design parameters of the sinusoidal tool profile;

[0010] S5, obtaining the corresponding spiral bevel gear pair by solving the discrete points on the tooth surface according to the sinusoidal tool profile parameters determined in S4;

[0011] S6, test the tooth surface loading contact condition of the spiral bevel gear pair and finally confirm the design scheme.

[0012] Furthermore, in step S1, the coordinate system S0 {X0, Y0, Z0} is fixedly connected to the cutter head and coincides with the coordinate axis Z0. The sinusoidal tool profile includes part a and part b. Part a is the sine function y = Asin (ωs2 + T) + h as the main working part of the tool profile. Part b is an arc used to generate the tooth root fillet. The tooth root fillet radius is ρ2, α 02 is the tooth profile angle of the large wheel cutter, r 02 is the nominal radius of the large wheel cutter head, p w2 is the tool tip distance, and the tool tip radius is r c2 , where r c2 =r 02 ±0.5p w2 , s2 and λ2 are the parameters of the tool profile of parts a and b respectively, and θ2 is the tool profile angle of the large wheel;

[0013] The position vector and unit normal vector of the sinusoidal tool profile of part a are:

[0014]

[0015] The position vector and unit normal vector of the sinusoidal tool profile in part b are:

[0016]

[0017]

[0018] in,

[0019] Furthermore, in step S2, the coordinate system Sm{Xm, Ym, Zm} is rigidly connected to the machine tool, and the cradle axis revolves around Z m The axis rotates, the unit vector p1 in the direction of the wheel axis is [-cosγ m1 0 sinγ m1 ] T , the direction of the wheel axis g1=

[001] T Indicates that Xm O m Y m The plane is the machine tool plane, O m is the center of the machine tool, O0 is the center of the cutter head, O1 is the intersection point of the small wheel design, and the equation of the small wheel surface and the unit normal vector are expressed in the coordinate system S m middle:

[0020]

[0021] Where q=q1+Δq1, q1 is the initial angle of the cradle, and the cradle rotates around Z m Rotate Δq1 to reach position q;

[0022] Due to the existence of spiral motion, the expression of the wheel bed position is:

[0023] X B =X B1 -H l1 Δq1

[0024] Among them H l1 is the spiral motion coefficient;

[0025] From the small wheel intersection O1 to the machine center O m The position vector is expressed as:

[0026] m1=X B g1+X p p1-E m1

[001] T

[0027] Assume that the angular velocity scalar of the small wheel is 1, that is, ω p =g1, the small wheel speed is ω1=R a1 p1, where R a1 is the rolling ratio when the small wheel is formed, then the relative angular velocity and relative motion speed between the small wheel forming wheel and the workpiece are:

[0028]

[0029] v p1 =ω p1 ×r m01 -R a1 p1×m1-H l1 g 01

[0030] Substitute the above formula into the meshing equation (v p1 n m01=0), we can get s1 = f(θ1,q), and substituting this expression into the above formula, we can get the gear tooth surface equation and its unit vector. Finally, the gear tooth surface equation and unit vector are expressed in the coordinate system with the gear design intersection point O1 as the origin, which is:

[0031]

[0032] Furthermore, the projection of the vector of the corresponding grid point of the tooth surface being analyzed in step S4 on the normal vector of the reference tooth surface is h ij =(r i -r j )·n i , (i=1,2,3,4...;j=1).

[0033] Furthermore, the sinusoidal blade profile design parameters include amplitude, angular frequency, initial phase, and translation parameters.

[0034] Furthermore, in step S5, the tooth surface discrete points obtained by the determined sinusoidal tool profile parameters are imported into the software for three-dimensional modeling to obtain a spiral bevel gear pair model for finite element dynamics simulation analysis.

[0035] Furthermore, the tooth surface loading contact condition of the spiral bevel gear pair includes loading transmission error, tooth surface contact area, and maximum contact stress.

[0036] The above solution of the present invention has the following beneficial effects:

[0037] The present invention provides a design method for machining spiral bevel gears using a sinusoidal profile tool. The established sinusoidal tool profile mathematical model has multiple shaping parameters, and reasonable shaping parameters can be selected to actively design the optimal tooth surface. The sinusoidal tool profile can modify the tooth top and tooth root of the tooth surface to avoid edge contact and reduce the contact stress of the tooth surface. The gear pair machined by the sinusoidal tool profile, combined with the optimization of the machine tool adjustment parameters, can theoretically make its transmission error curve a sine function, with the slopes on both sides of the tooth pair transition being equal, that is, the relative angular velocity is 0, and the relative angular acceleration is very small, which can reduce the impact when the tooth pair meshes in and out, increase the stability during operation, and reduce the noise during operation.

[0038] Other beneficial effects of the present invention will be described in detail in the subsequent specific implementation section. BRIEF DESCRIPTION OF THE DRAWINGS

[0039] Figure 1 is a flow chart of the steps of the present invention;

[0040] Figure 2 Schematic diagram of the sinusoidal tool profile coordinate system in the present invention;

[0041] Figure 3Schematic diagram of the coordinate system for generating the tooth surface of the small gear in the present invention;

[0042] Figure 4 This is a topological diagram of the concave tooth surface of the small wheel in the present invention;

[0043] Figure 5 This is a topological diagram of the convex tooth surface of the small wheel in the present invention;

[0044] Figure 6 is the sinusoidal transmission error, angular velocity, and angular acceleration curve in the present invention;

[0045] Figure 7 Figure 1 is a diagram of the gear analysis loading results in the present invention (a and b are the contact conditions of the convex and concave surfaces of the first set of large wheels, and c and d are the contact conditions of the convex and concave surfaces of the second set of large wheels);

[0046] Figure 8(a) is the contact stress diagram of the tooth surface of the forward rotating large wheel in the present invention, Figure 8(b) is the contact stress diagram of the tooth surface of the reverse rotating large wheel in the present invention, Figure 8(c) is the contact stress diagram of the tooth surface of the forward rotating small wheel in the present invention, and Figure 8(d) is the contact stress diagram of the tooth surface of the reverse rotating small wheel in the present invention. DETAILED DESCRIPTION

[0047] In order to make the technical problems, technical solutions and advantages to be solved by the present invention clearer, the following will be described in detail with reference to the accompanying drawings and specific embodiments. Obviously, the embodiments described are part of the embodiments of the present invention, rather than all the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative work are within the scope of protection of the present invention. In addition, the technical features involved in the different embodiments of the present invention described below can be combined with each other as long as they do not conflict with each other.

[0048] In the description of the present invention, it should be noted that the terms "center," "upper," "lower," "left," "right," "vertical," "horizontal," "inner," and "outer," etc., indicating orientations or positional relationships, are based on the orientations or positional relationships shown in the accompanying drawings and are intended solely to facilitate and simplify the description of the present invention. They are not intended to indicate or imply that the devices or components referred to must have, be constructed, or operate in a specific orientation, and therefore should not be construed as limitations on the present invention. Furthermore, the terms "first," "second," and "third" are used for descriptive purposes only and should not be construed as indicating or implying relative importance.

[0049] In the description of the present invention, it should be noted that, unless otherwise expressly specified or limited, the terms "mounted," "connected," and "connected" should be understood broadly. For example, they may refer to a locking connection, a detachable connection, or an integral connection; they may refer to a mechanical connection or an electrical connection; they may refer to a direct connection or an indirect connection through an intermediate medium; and they may refer to internal communication between two components. Those skilled in the art will understand the specific meanings of the above terms in the present invention based on the specific circumstances.

[0050] The present invention relates to the field of gear processing technology. Currently, the most widely used tool profiles for processing spiral bevel gears are mainly straight lines. Linear tools have the advantages of simple structure and easy manufacturing, and still occupy a major position in the gear manufacturing industry. However, the gear pairs designed and processed by linear tools using the "local conjugate principle" are prone to stress concentration, edge contact and other problems, especially when the teeth are meshing in and out, which can easily cause large impacts, resulting in high vibration and noise. Therefore, the design of tooth surface modification using different tool profiles has gradually entered the field of gear manufacturing, mainly including parabolic profiles, circular arc profiles, or profiles composed of multiple curves. However, the mathematical models of these tool profiles have the problem of few modification parameters, which will affect the flexibility and freedom of tooth surface optimization. Based on this, an embodiment of the present invention provides a design method for processing spiral bevel gears using a sinusoidal profile tool, aiming to solve the above problems.

[0051] The tools involved are suitable for all spiral bevel gears processed by end milling and end hobbing. The following will take the spiral bevel gear as an example to explain its specific implementation steps. The steps are as follows: Figure 1 As shown:

[0052] S1, establish the sinusoidal tool profile coordinate system and the mathematical model of the sinusoidal tool profile production surface, such as Figure 2 As shown, the position vector and unit normal vector of the sinusoidal tool profile are derived.

[0053] The coordinate system S0{X0, Y0, Z0} is fixed to the cutter head and coincides with the coordinate axis Z0. The sinusoidal tool profile is mainly composed of two parts. Part a is the sine function y=Asin(ωs2+T)+h as the main working part of the tool profile, and part b is the arc used to generate the tooth root fillet. The tooth root fillet radius is ρ2, α 02 is the tooth profile angle of the large wheel cutter, r 02 is the nominal radius of the large wheel cutter head, p w2 is the tool tip distance, and the tool tip radius is r c2 , where r c2 =r 02 ±0.5p w2 , s2 and λ2 are the parameters of the tool profile of parts a and b respectively, θ2 is the rotation angle of the large wheel tool profile, A is the amplitude, ω is the angular frequency, T is the phase, and h is the translation parameter.

[0054] Therefore, the position vector and unit normal vector of the sinusoidal tool profile of part a are:

[0055]

[0056] The position vector and unit normal vector of the sinusoidal tool profile in part b are:

[0057]

[0058] in, In the above formula, the double symbol "±" represents the convex surface of the large wheel, and the lower symbol represents the concave surface of the large wheel.

[0059] S2, using the gear pair geometric parameters and processing parameters, establish the mathematical model of spiral bevel gear cutting, and obtain the gear pair tooth surface equation based on the differential geometry principle and meshing equation.

[0060] Taking the solution of the gear tooth surface equation as an example, the gear tooth surface is developed into a coordinate system as follows Figure 3 As shown, the coordinate system Sm{Xm, Ym, Zm} is rigidly connected to the machine tool, and the cradle axis revolves around Z m The axis rotates, the unit vector p1 in the direction of the wheel axis is [-cosγ m1 0 sinγ m1 ] T , the direction of the wheel axis g1=

[001] T Indicates that X m O m Y m The plane is the machine tool plane, O m is the center of the machine tool, O0 is the center of the cutter head, and O1 is the intersection point of the small wheel design. The equation of the small wheel generating surface and the unit normal vector are expressed in the coordinate system S m middle:

[0061]

[0062] Where q=q1+Δq1, q1 is the initial angle of the cradle, and the cradle rotates around Z m Rotate Δq1 to reach position q, r 01 Indicates the position vector of the small wheel sinusoidal blade profile, n 01 Represents the unit normal vector of the sinusoidal blade profile of the small wheel.

[0063] Due to the existence of spiral motion, the expression of the wheel bed position is:

[0064] X B =X B1 -H l1 Δq1

[0065] Among them H l1is the spiral motion coefficient;

[0066] From the small wheel intersection O1 to the machine center O m The position vector is expressed as:

[0067] m1=X B g1+X p p1-E m1

[001] T

[0068] Assume that the angular velocity scalar of the small wheel is 1, that is, ω p =g1, the small wheel speed is ω1=R a1 p1, where R a1 is the rolling ratio when the small wheel is formed, then the relative angular velocity ω between the small wheel forming wheel and the workpiece p1 With relative motion speed v p1 They are:

[0069]

[0070] v p1 =ω p1 ×r m01 -R a1 p1×m1-H l1 g1

[0071] Substitute the above formula into the meshing equation (v p1 n m01 =0), we can get s1 = f(θ1,q), and substituting this expression into the above formula, we can get the gear tooth surface equation and its unit vector. Finally, the gear tooth surface equation and unit vector are expressed in the coordinate system with the gear design intersection point O1 as the origin, which is:

[0072]

[0073] S3, through the tooth surface division, the tooth surface is discretized by the rotation projection method, and the coordinates of the discrete points on the tooth surface and their normal vectors are solved.

[0074] S4, establish the tooth surface mismatch topology model, take the tooth surface data calculated by the straight tool profile as the reference tooth surface, take the tooth surface data processed by the sinusoidal tool profile as the tooth surface to be analyzed, and calculate the projection of the vectors of the corresponding grid points on the reference tooth surface normal vector; that is, h ij =(r i -r j )·n i , (i=1,2,3,4...; j=1), the tooth surface mismatch diagram can be constructed by graphically displaying the mismatch amount of each grid point; the influence of the design parameters of the sinusoidal tool profile (amplitude A, angular frequency ω, initial phase φ, translation parameter h) on the tooth surface mismatch amount is analyzed, and then the sinusoidal tool profile design parameters are determined.

[0075] The following table shows the topological mismatch of the gear tooth surface reverse surface established with different amplitudes A, where Example 1 is the reference tooth surface, and Examples 2 to 4 are the topological tooth surfaces obtained with different amplitudes. Figure 4 、 Figure 5 Analysis shows that the mismatch between tooth surfaces can be changed by changing the amplitude A. Similarly, the target tooth surface can be obtained by changing the other three parameters.

[0076]

[0077] S5: The determined sinusoidal tool profile parameters are solved to obtain discrete points on the tooth surface, which are then imported into 3D software. Three-dimensional modeling is then performed on the gear pairs designed using the linear and sinusoidal tools. Finite element dynamics simulation is performed on the theoretically installed gear pairs. Two sets of comparative analysis are performed (the first set uses linear tools for both large and small wheels, while the second set uses linear tools for the large wheels and sinusoidal tools for the small wheels). The operating conditions of the two sets of examples under the same operating conditions are analyzed (including loading transmission error, tooth surface contact area, maximum contact stress, etc.), and the design solution is finally confirmed.

[0078] In addition, a pair of gear pairs can be actually processed by a fully CNC bevel gear machine, and red lead powder can be applied to the positive and reverse surfaces of the large and small wheels, and noise inspection and rolling inspection experiments can be carried out on them on the test bench.

[0079] In the method provided in this embodiment, the established sinusoidal tool profile mathematical model has multiple shaping parameters, and reasonable shaping parameters can be selected to actively design the optimal tooth surface; the sinusoidal tool profile can modify the tooth top and tooth root to avoid edge contact and reduce the contact stress of the tooth surface; the gear pair machined by the sinusoidal tool profile, combined with the optimization of the machine tool adjustment parameters, can theoretically make its transmission error curve a sine function, such as Figure 6 As shown, the slopes on both sides of the gear pair transition are equal, that is, the relative angular velocity is 0, and the relative angular acceleration is very small, which can reduce the impact when the gear pair engages and disengages, increase the smoothness during operation, and reduce the noise during operation.

[0080] This method has been proven feasible through experiments, simulations and use: the spiral bevel gears in which the large wheel is processed by the developing method and the small wheel is processed by the double spiral method are taken as the research objects. After consulting relevant materials, it is learned that in the gear pairs designed with modified tool profiles and linear tool profiles, the large wheel is not modified and the gear pair with modified tool profiles has better meshing performance. Therefore, based on finite element dynamics simulation, two groups of case analyses are set up for comparison (the large and small wheels of the first group are both processed by linear tools, and the large wheel of the second group is processed by linear tools and the small wheel is processed by sinusoidal tools). The working conditions are set to 100 rad / s and the load is 600 N.m. The changes in contact stress before and after modification are analyzed.

[0081] Figure 7 Figures 2 and 3 show the results of tooth surface loading analysis, where (a) and (b) show the contact conditions of the first set of large wheel convex and concave surfaces, and (c) and (d) show the contact conditions of the second set of large wheel convex and concave surfaces. The analysis shows that under heavy load, the first set of convex and concave surfaces both have tooth tip edge contact, and the contact point is located at the tooth tip edge line. The contact points of the second set of convex and concave tooth surfaces both move toward the tooth root, which is beneficial to improving the meshing performance of the gear pair and fully demonstrates the effectiveness of the sinusoidal tool profile in tooth surface modification.

[0082] Figure 8(a) shows the contact stress diagram for the forward-rotating large wheel tooth surface, Figure 8(b) shows the contact stress diagram for the reverse-rotating large wheel tooth surface, Figure 8(c) shows the contact stress diagram for the forward-rotating small wheel tooth surface, and Figure 8(d) shows the contact stress diagram for the reverse-rotating small wheel tooth surface. Analysis shows that the reshaped large and small wheel reverse contact surfaces are significantly reduced, and the change is more stable, which is beneficial to transmission stability and reduces operating noise. The maximum contact force during startup of the reshaped forward surface is also reduced, and its overall contact stress is reduced.

[0083] The above is a preferred embodiment of the present invention. It should be pointed out that for ordinary technicians in this technical field, several improvements and modifications can be made without departing from the principles of the present invention. These improvements and modifications should also be regarded as within the scope of protection of the present invention.

Claims

1. A design method for machining spiral bevel gears using a sinusoidal profile tool, characterized in that: The steps include: S1, establish the sinusoidal tool profile coordinate system and the sinusoidal tool profile generating surface mathematical model, and obtain the sinusoidal tool profile generating surface position vector and unit normal vector; S2, using the gear pair geometric parameters and processing parameters, establish the spiral bevel gear cutting mathematical model, based on the differential geometry principle and meshing equation, obtain the gear pair tooth surface equation; S3, through the tooth surface division, the tooth surface is discretized by the rotation projection method, and the coordinates of the discrete points on the tooth surface and their normal vectors are solved; S4, establish a tooth surface mismatch topology model, take the tooth surface data calculated by the straight tool profile as the reference tooth surface, take the tooth surface data processed by the sinusoidal tool profile as the analyzed tooth surface, calculate the projection of the vectors of the corresponding grid points of the two tooth surfaces on the normal vector of the reference tooth surface, and graphically construct a tooth surface mismatch diagram by graphing the mismatch amount of each grid point, analyze the influence of the design parameters of the sinusoidal tool profile on the tooth surface mismatch amount, and determine the design parameters of the sinusoidal tool profile; S5, obtaining the corresponding spiral bevel gear pair by solving the discrete points on the tooth surface according to the sinusoidal tool profile parameters determined in S4; S6, test the tooth surface loading contact condition of the spiral bevel gear pair and finally confirm the design scheme.

2. According to claim 1, a design method for machining spiral bevel gears with a sinusoidal profile tool is characterized in that: The design parameters of the sinusoidal tool profile include amplitude, angular frequency, initial phase, and translation parameters.

3. The design method for machining spiral bevel gears with a sinusoidal profile tool according to claim 1, characterized in that: In step S5, the discrete points on the tooth surface obtained by the determined sinusoidal tool profile parameters are imported into the software for three-dimensional modeling to obtain a spiral bevel gear pair model for finite element dynamics simulation analysis.

4. The design method for machining spiral bevel gears with a sinusoidal profile tool according to claim 3, characterized in that: The tooth surface loading contact conditions of spiral bevel gear pairs include loading transmission error, tooth surface contact area, and maximum contact stress.

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