Double-pressure-angle asymmetric tooth profile design and worm grinding wheel tooth grinding method

Abstract

The invention provides a double-pressure-angle asymmetric tooth profile design and worm grinding wheel tooth grinding method. A tooth profile equation of the asymmetric gear is deduced, the meshing relation between a rack tool and the gear is analyzed in detail, and a precise profile geometric model of the dual-pressure-angle asymmetric gear is further established. In the aspect of worm grinding wheel gear grinding machining, the contour design and finishing method of a worm grinding wheel and application of the worm grinding wheel in asymmetric gear machining are studied, and a worm grinding wheel geometric model and a diamond roller geometric model needed by double-pressure-angle asymmetric gear worm grinding wheel gear grinding are established. The method provides a theoretical basis and an accurate model for design and processing of the asymmetric gear, and is of great significance for improving the efficiency and reliability of a gear transmission system.

Description

Technical Field

[0001] The present invention relates to the field of mechanical manufacturing, and in particular to a double-pressure-angle asymmetric tooth profile design and a worm grinding wheel gear grinding method. Background Art

[0002] As an important transmission element, gears are increasingly used in a variety of fields such as aerospace, automobiles, and wind power generation. The pressure angle of a gear is a key geometric parameter in the gear meshing process. The angle between the normal pressure direction at the tooth profile meshing point and the linear velocity direction at that point is defined as the pressure angle. The pressure angle reflects the angle between the direction of the driving force acting on the gear and the velocity direction of the action point, which directly affects the force transmission efficiency. Although traditional symmetrical involute gears have good transmission efficiency due to their tooth profile design with the same pressure angle, their transmission performance and reliability cannot meet the requirements in some special applications, such as high speed, heavy load, and complex dynamic load conditions.

[0003] Asymmetric gears, as a new type of gear, utilize different pressure angles on the left and right tooth sides. This effectively improves the gear's load capacity, reduces meshing errors, and enhances overall transmission efficiency. They are particularly suitable for applications requiring higher strength and dynamic performance. Dual-pressure-angle asymmetric gears, an improved form of asymmetric gears, utilize different pressure angles on the drive and non-drive sides, significantly changing the gear's tooth profile. By properly designing the pressure angle, the tooth root thickness can be increased, improving the gear's load capacity and strength, and, to a certain extent, optimizing its dynamic characteristics.

[0004] However, the design of asymmetric gears is more complex than that of traditional gears, requiring more considerations, including tooth profile shape, pressure angle selection, and meshing characteristics. Currently, the industry primarily uses processes such as wire cutting, milling, and 3D printing to process asymmetric gears. These methods cannot meet the industry's demand for high-precision and high-efficiency machining of asymmetric gears.

[0005] Therefore, it is of great significance to develop a dual-pressure angle asymmetric tooth profile design and a worm grinding wheel gear grinding method. Summary of the Invention

[0006] The purpose of the present invention is to provide a double-pressure-angle asymmetric tooth profile design and a worm grinding wheel gear grinding method to solve the problems existing in the prior art.

[0007] The technical solution adopted to achieve the purpose of the present invention is as follows: 1. A dual-pressure angle asymmetric tooth profile design method, characterized in that: the gear tooth profile equation of the dual-pressure angle asymmetric gear is derived based on the meshing motion relationship between the rack tool and the forming gear.

[0008] Further, the following steps are included:

[0009] 1) Determine the relative motion relationship between the wheel blank and the rack cutter during processing;

[0010] 2) Derive the transformation relationship between the gear coordinate system and the rack tool coordinate system; establish fixed coordinate systems S f (x f o f y f ), rack tool motion coordinate system S r (x r o r y r ) and the formed gear workpiece motion coordinate system S w (x w o w y w );r b is the pitch radius of the formed gear; in the motion coordinate system S of the formed gear workpiece w (x w o w y w ) is the position vector of any point P in the w , in the rack tool motion coordinate system S r (x r o r y r ) The position vector of point P is r r , the formed gear is in the workpiece motion coordinate system S w (x w o w y w ) is the turning angle Then the workpiece motion coordinate system S w (x w o w y w ) and the fixed coordinate system S f (x f o f y f ) is:

[0011]

[0012] Rack tool motion coordinate system S r (x r o r y r ) and the fixed coordinate system S f (x f o f y f ) is:

[0013]

[0014] Workpiece motion coordinate system S w (x w o w y w ) and the rack tool motion coordinate system S r (x r o r y r ) is:

[0015]

[0016] 3) Establish the tooth profile equation of the rack tool; take the motion coordinate system S of the rack tool r (x r o r y r ) and tool reference axis x r Based on the normal tooth profile of the rack tool, it consists of five parts:

[0017] AB: Rack straight line segment used to machine the tooth profile on the driving side;

[0018] CE: Rack straight line segment used to machine the tooth root arc;

[0019] FG: Rack straight line segment used for machining the non-driving side tooth profile;

[0020] Used for machining the transition arc segment of the drive side tooth root transition curve;

[0021] Used for machining the transition arc segment of the tooth root transition curve on the non-driving side;

[0022] Among them, α m , α n are the tool tooth profile pressure angles for machining the driving tooth side and the non-driving tooth side respectively; θ m ,θ n are the center angles of the tooth root transition curves of the driving tooth side and the non-driving tooth side respectively; ρ m , ρ n are the curvature radii of the tooth root transition curves of the driving tooth side and the non-driving tooth side respectively; S is the tooth thickness; h a 、h f They are tooth top height and tooth root height respectively; is the tooth top height coefficient, m is the module, Indicates the head clearance coefficient; the calculation formulas of each parameter are as follows:

[0023]

[0024] S=πm / 2

[0025]

[0026] h m =h f -(1-sinα m )ρ m

[0027] h n =h f -(1-sinα n )ρ n (38)

[0028] From the geometric relationship, we can see that the calculation formulas of its various parts are as follows:

[0029] Rack straight line CE segment for machining tooth root arc:

[0030]

[0031] Where, l DE 、l CD is the length of straight line segments DE and CD.

[0032] Machining the rack straight line FG segment of the non-driving side tooth profile:

[0033]

[0034] Where, l FG 、l DE is the length of straight line segments FG and DE.

[0035] Machining the rack straight line AB segment of the drive side tooth profile:

[0036]

[0037] Where, l AB 、l CD is the length of the straight line segments AB and CD.

[0038] Machining the transition arc of the non-driving side tooth root transition curve part:

[0039]

[0040] Where, l DE is the length of straight line segment DE.

[0041] Machining the transition arc of the drive side tooth root transition curve part:

[0042]

[0043] Where, l CD is the length of the straight line segment CD;

[0044] Write equation (9) as the normal tooth profile curve equation of the rack tool r (α m ,α n ):

[0045]

[0046] In addition, the derivative of the above formula can be obtained as the rack tool motion coordinate system S r The tangent vector n of the tool tooth profile r for:

[0047]

[0048] The unit normal vector is:

[0049]

[0050] According to the geometric relationship, the unit normal vector of each segment of the tool tooth profile is:

[0051] n r_FG =[-cosα n -sinα n 0] T

[0052] n r_EF =[-sinθ n -cosθ n 0] T

[0053] n r_CE =

[010] T

[0054] n r_BC =[sinθ m -cosθ m 0] T

[0055] n r_AB =[-cosα m sinα m 0] T (47)

[0056] Where, the subscripts represent the corresponding line segments;

[0057] Write formula (13) as:

[0058]

[0059] 4) Derive the meshing equation between the rack tool and the gear; when the rack tool moves at a speed v rWhen translating to the left, the formed gear rotates counterclockwise at an angular velocity ω, and the instantaneous rotation center O is located at the same speed v r Vertical line O w O, its position satisfies the following vector equation:

[0060]

[0061] Since the motion between the tool and the gear is pure rolling along point O, the displacement of the tool and the rotation angle of the gear satisfy the following conditions:

[0062]

[0063] Rack tool motion coordinate system S r The vector radius of the moving point P is:

[0064]

[0065] Workpiece motion coordinate system S w The vector radius of the moving point P is:

[0066]

[0067] The relative sliding velocity at point P is:

[0068] v=v w -v r (53)

[0069] Where, v w Point P is in the workpiece motion coordinate system S w The speed in v r Point P is in the rack tool motion coordinate system S r The velocity in satisfies the following equation:

[0070]

[0071] From the above formula we can deduce:

[0072]

[0073] In order to ensure normal gear meshing, the relative velocity vector of any point on the gear tooth profile is required to be tangent to the normal vector, that is:

[0074] n r (α m ,α n )·v=0 (56)

[0075] The above formula (22) is expressed as:

[0076] f(α m ,α n)=0 (57)

[0077] Combining the normal equation and relative velocity equation of each curve equation of the rack tool, the meshing equation can be derived as follows:

[0078]

[0079] Where, the subscripts represent the corresponding line segments;

[0080] 5) According to the meshing relationship between the rack tool and the gear, the gear tooth profile curve equation is derived; by combining the gear meshing equation with the gear tooth profile equation after coordinate transformation, the following conjugate tooth profile equation between the gear and the tool can be obtained:

[0081]

[0082] Combining the conjugate tooth profile equation with the asymmetric tool equation for each segment of the gear tooth profile, the tooth profile equation for the formed double-pressure-angle asymmetric gear under asymmetric tool machining can be derived as follows:

[0083] Drive tooth side involute tooth root transition arc tooth profile segment:

[0084]

[0085] Involute tooth profile section on the driving tooth side:

[0086]

[0087] Involute tooth profile section on the non-driving tooth side:

[0088]

[0089] Involute tooth root transition arc tooth profile section on the non-driving tooth side:

[0090]

[0091] Root arc tooth profile segment:

[0092]

[0093] Furthermore, after step 5), there is also a step of using MATLAB software to draw the gear normal tooth profile under different gear normal pressure angles.

[0094] The present invention also discloses a double-pressure-angle asymmetric gear having a tooth profile curve designed according to the tooth profile design method.

[0095] The present invention also discloses a method for grinding the asymmetric gear with a worm wheel using a dual-pressure-angle gear, comprising the following steps:

[0096] 1) Use diamond roller to trim the tooth surface of the worm grinding wheel;

[0097] 2) Gear grinding: The worm grinding wheel and the workpiece gear are engaged with each other in a spatial staggered axis to realize the grinding of asymmetric gears with double pressure angles.

[0098] Furthermore, the axial section of the worm of the worm grinding wheel is a straight line, and the contour surface of the grinding wheel presents an Archimedean spiral surface; combined with the normal contour curve r of the rack tool r (α m ,α n ), the axial section curve r of the worm grinding wheel wg (α m ,α n ,u m ,u n ) is defined as:

[0099]

[0100] Among them, α m , α n Respectively represent the tooth profile angles of the driving side tooth profile and the non-driving side tooth profile; u m 、u n Respectively represent the straight generatrix of the driving side tooth profile and the non-driving side tooth profile, which are used to determine the position of any point M and N on the straight generatrix. The value range is r a 、r f They represent the root circle radius and the top circle radius of the worm grinding wheel respectively; H p =kπmcosλ, k, m, and λ are the number of worm grinding wheel heads, module, and lead angle respectively;

[0101] The axial section curve r of the worm grinding wheel wg (α m ,α n ,u m ,u n )Substitute into the vector rotation formula The equation of the right-hand helicoidal surface of the worm grinding wheel can be obtained as follows:

[0102] S wg (α m ,α n ,u m ,u n ,θ)=R z (θ)·r wg (α m ,α n ,u m ,u n ) (66)

[0103] Where θ represents the angle of rotation of the straight generatrix around the z-axis, and p is the helical parameter, which represents the axial displacement of the straight generatrix when it rotates one unit angle around the z-axis.

[0104] Combined with the axial tooth profile equation of the worm grinding wheel, the axial tooth profile equation of the diamond roller can be obtained: d (α m ,α n ,t m ,t n )for:

[0105]

[0106] Among them, α m , α n Respectively represent the tooth profile angles of the driving side and non-driving side of the diamond roller; t m , t n Respectively represent the tooth profile parameters of the driving side tooth profile and the non-driving side tooth profile, and their value range is Top thickness

[0107] The diamond roller axial tooth profile equation r d (α m ,α n ,t m ,t n )Substitute into the vector rotation formula The tooth surface equation of the diamond roller can be obtained as

[0108]

[0109] The technical effect of the present invention is unquestionable: it provides a theoretical basis and an accurate model for the design and processing of asymmetric gears, and is of great significance for improving the efficiency and reliability of gear transmission systems. BRIEF DESCRIPTION OF THE DRAWINGS

[0110] Figure 1 is the gear and rack tool coordinate system;

[0111] Figure 2 Schematic diagram of the normal tooth profile of the asymmetric rack tool;

[0112] Figure 3 Verify the rack tool tooth profile and gear tooth profile;

[0113] Figure 4 Schematic diagram of the spatial meshing relationship between the worm grinding wheel and the workpiece gear;

[0114] Figure 5 It is an Archimedean spiral;

[0115] Figure 6 This is the point cloud diagram of the worm sand contour surface;

[0116] Figure 7 It is the three-dimensional model of the worm grinding wheel;

[0117] Figure 8 This is a schematic diagram of diamond roller dressing;

[0118] Figure 9 This is a schematic diagram of the diamond roller;

[0119] Figure 10 This is the point cloud diagram of the diamond roller tooth profile;

[0120] Figure 11 This is a three-dimensional model of the diamond roller;

[0121] Figure 12 It is a CNC machining process based on the VERICUT system;

[0122] Figure 13 This is the model of the worm wheel gear grinding machine;

[0123] Figure 14 The linkage relationship between the axes of the worm wheel gear grinding machine;

[0124] Figure 15 Slotted worm grinding wheel for simulation;

[0125] Figure 16 This is the simulation result. DETAILED DESCRIPTION

[0126] The present invention will be further described below with reference to the following examples, but it should not be understood that the scope of the present invention is limited to the following examples. Without departing from the above technical ideas of the present invention, various substitutions and modifications can be made according to common technical knowledge and customary means in the art, and all should be included in the scope of protection of the present invention.

[0127] Example 1:

[0128] This embodiment provides a dual-pressure-angle asymmetric tooth profile design method, which derives the gear tooth profile equation of the dual-pressure-angle asymmetric gear based on the meshing motion relationship between the rack cutter and the forming gear. The method includes the following steps:

[0129] 1) Determine the relative motion relationship between the wheel blank and the rack cutter during processing;

[0130] 2) Derive the transformation relationship between the gear coordinate system and the rack tool coordinate system; establish fixed coordinate systems S f (x f o f y f ), rack tool motion coordinate system S r (x r o r yr ) and the formed gear workpiece motion coordinate system S w (x w o w y w );r b is the pitch radius of the formed gear; in the motion coordinate system S of the formed gear workpiece w (x w o w y w ) is the position vector of any point P in the w , in the rack tool motion coordinate system S r (x r o r y r ) The position vector of point P is r r , the formed gear is in the workpiece motion coordinate system S w (x w o w y w ) is the turning angle Then the workpiece motion coordinate system S w (x w o w y w ) and the fixed coordinate system S f (x f o f y f ) is:

[0131]

[0132] Rack tool motion coordinate system S r (x r o r y r ) and the fixed coordinate system S f (x f o f y f ) is:

[0133]

[0134] Workpiece motion coordinate system S w (x w o w y w ) and the rack tool motion coordinate system S r (x r o r y r ) is:

[0135]

[0136] 3) Establish the tooth profile equation of the rack tool; the double pressure angle asymmetric gear tooth profile equation obtained by the gear tool is the tooth profile equation that best conforms to the theoretical processing results. Figure 2 A schematic diagram of the tooth profile of a rack cutter on the normal plane is shown. The profile of an asymmetric gear cutter consists of multiple components, with straight and circular segments used to generate the gear's involute and tooth root arc, respectively. This design ensures that the cutter accurately replicates the complex geometry of asymmetric gears during machining. The meshing between the rack cutter and the formed gear follows specific meshing laws, making it feasible to derive the corresponding gear tooth profile equation based on the meshing kinematic relationship between the two. This process not only improves machining accuracy but also effectively reduces potential failures in gear transmissions.

[0137] The motion coordinate system S of the rack tool r (x r o r y r ) and tool reference axis x r Based on the normal tooth profile of the rack tool, it consists of five parts:

[0138] AB: Rack straight line segment used to machine the tooth profile on the driving side;

[0139] CE: Rack straight line segment used to machine the tooth root arc;

[0140] FG: Rack straight line segment used for machining the non-driving side tooth profile;

[0141] Used for machining the transition arc segment of the drive side tooth root transition curve;

[0142] Used for machining the transition arc segment of the tooth root transition curve on the non-driving side;

[0143] Among them, α m , α n are the tool tooth profile pressure angles for machining the driving tooth side and the non-driving tooth side respectively; θ m ,θ n are the center angles of the tooth root transition curves of the driving tooth side and the non-driving tooth side respectively; ρ m , ρ n are the curvature radii of the tooth root transition curves of the driving tooth side and the non-driving tooth side respectively; S is the tooth thickness; h a 、h f They are tooth top height and tooth root height respectively; is the tooth top height coefficient, m is the module, Indicates the head clearance coefficient; the calculation formulas of each parameter are as follows:

[0144]

[0145] S=πm / 2

[0146]

[0147] h m =h f -(1-sinα m )ρ m

[0148] h n =h f -(1-sinα n )ρ n (72)

[0149] From the geometric relationship, we can see that the calculation formulas of its various parts are as follows:

[0150] Rack straight line CE segment for machining tooth root arc:

[0151]

[0152] Where, l DE 、l CD is the length of straight line segments DE and CD.

[0153] Machining the rack straight line FG segment of the non-driving side tooth profile:

[0154]

[0155] Where, l FG 、l DE is the length of straight line segments FG and DE.

[0156] Machining the rack straight line AB segment of the drive side tooth profile:

[0157]

[0158] Where, l AB 、l CD is the length of the straight line segments AB and CD.

[0159] Machining the transition arc of the non-driving side tooth root transition curve part:

[0160]

[0161] Where, l DE is the length of straight line segment DE.

[0162] Machining the transition arc of the drive side tooth root transition curve part:

[0163]

[0164] Where, l CD is the length of the straight line segment CD;

[0165] Write equation (9) as the normal tooth profile curve equation of the rack tool r (α m ,α n ):

[0166]

[0167] In addition, the derivative of the above formula can be obtained as the rack tool motion coordinate system S r The tangent vector n of the tool tooth profile r for:

[0168]

[0169] The unit normal vector is:

[0170]

[0171] According to the geometric relationship, the unit normal vector of each segment of the tool tooth profile is:

[0172] n r_FG =[-cosα n -sinα n 0] T

[0173] n r_EF =[-sinθ n -cosθ n 0] T

[0174] n r_CE =

[010] T

[0175] n r_BC =[sinθ m -cosθ m 0] T

[0176] n r_AB =[-cosα m sinα m 0] T (81)

[0177] Where, the subscripts represent the corresponding line segments;

[0178] Write formula (13) as:

[0179]

[0180] 4) Derive the meshing equation between the rack tool and the gear; when the rack tool moves at a speed v r When translating to the left, the formed gear rotates counterclockwise at an angular velocity ω, and the instantaneous rotation center O is located at the same speed v r Vertical line O w O, its position satisfies the following vector equation:

[0181]

[0182] Since the motion between the tool and the gear is pure rolling along point O, the displacement of the tool and the rotation angle of the gear satisfy the following conditions:

[0183]

[0184] Rack tool motion coordinate system S r The vector radius of the moving point P is:

[0185]

[0186] Workpiece motion coordinate system S w The vector radius of the moving point P is:

[0187]

[0188] The relative sliding velocity at point P is:

[0189] v=v w -v r (87)

[0190] Where, v w Point P is in the workpiece motion coordinate system S w The speed in v r Point P is in the rack tool motion coordinate system S r The velocity in satisfies the following equation:

[0191]

[0192] From the above formula we can deduce:

[0193]

[0194] In order to ensure normal gear meshing, the relative velocity vector of any point on the gear tooth profile is required to be tangent to the normal vector, that is:

[0195] n r (α m ,α n )·v=0 (90)

[0196] The above formula (22) is expressed as:

[0197] f(α m ,α n )=0 (91)

[0198] Combining the normal equation and relative velocity equation of each curve equation of the rack tool, the meshing equation can be derived as follows:

[0199]

[0200] In the formula, the subscripts represent the corresponding line segments;

[0201] 5) According to the meshing relationship between the rack tool and the gear, the gear tooth profile curve equation is derived; by combining the gear meshing equation with the gear tooth profile equation after coordinate transformation, the following conjugate tooth profile equation between the gear and the tool can be obtained:

[0202]

[0203] Combining the conjugate tooth profile equation with the asymmetric tool equation for each segment of the gear tooth profile, the tooth profile equation for the formed double-pressure-angle asymmetric gear under asymmetric tool machining can be derived as follows:

[0204] Drive tooth side involute tooth root transition arc tooth profile segment:

[0205]

[0206] Involute tooth profile section on the driving tooth side:

[0207]

[0208] Involute tooth profile section on the non-driving tooth side:

[0209]

[0210] Involute tooth root transition arc tooth profile section on the non-driving tooth side:

[0211]

[0212] Root arc tooth profile segment:

[0213]

[0214] 6) Steps for plotting the gear normal tooth profile at different gear normal pressure angles using MATLAB software. Based on the derived equations, a corresponding program was developed using MATLAB software to plot the gear normal tooth profile at different gear normal pressure angles. Figure 3 The normal tooth profile of a single tooth is shown, representing an asymmetric gear with different pressure angles on both sides. Figure 3 a is the rack tool tooth profile. Figure 3b represents the gear tooth profile. The program's design allows for flexible tooth profile curves for varying pressure angles and root radiuses, making it ideal for the design and analysis of asymmetric gears. By adjusting input parameters, users can observe changes in the tooth profile, gaining a deeper understanding of the impact of different designs on gear performance. This improves design efficiency and facilitates the selection of the optimal gear structure for practical applications.

[0215] Table 1 Basic parameters for verification of asymmetric gear tooth profile

[0216]

[0217] Example 2:

[0218] This embodiment provides a dual-pressure angle asymmetric gear whose tooth profile curve is designed according to the tooth profile design method described in Example 1.

[0219] Example 3:

[0220] This embodiment provides a worm grinding wheel method for grinding asymmetric gears with dual pressure angles as described in Example 2, comprising the following steps:

[0221] 1) Use diamond roller to trim the tooth surface of the worm grinding wheel;

[0222] To ensure that the machined workpiece gear meets the predetermined accuracy and usage requirements, the axial section of the worm grinding wheel should be the same as the section at the rack tool end. The profile of the worm grinding wheel is a cylindrical helical surface with a constant pitch, which is different from the structure of an involute helical gear. Its axial section does not lie in the xoy plane, but in the yoz plane. The axial section moves helically around the z axis, so the geometric characteristics of the worm grinding wheel are significantly different from those of an involute helical gear. Figure 5 In this embodiment, the axial section of the worm of the worm grinding wheel is a straight line, and the contour surface of the grinding wheel presents an Archimedean spiral surface; combined with the normal contour curve r of the rack tool r (α m ,α n ), the axial section curve r of the worm grinding wheel wg (α m ,α n ,u m ,u n ) is defined as:

[0223]

[0224] Among them, α m , α n Respectively represent the tooth profile angles of the driving side tooth profile and the non-driving side tooth profile; u m 、u nRespectively represent the straight generatrix of the driving side tooth profile and the non-driving side tooth profile, which are used to determine the position of any point M and N on the straight generatrix. The value range is r a 、r f They represent the root circle radius and the top circle radius of the worm grinding wheel respectively; H p =kπmcosλ, k, m, and λ are the number of worm grinding wheel heads, module, and lead angle respectively;

[0225] The axial section curve r of the worm grinding wheel wg (α m ,α n ,u m ,u n )Substitute into the vector rotation formula The equation of the right-hand helicoidal surface of the worm grinding wheel can be obtained as follows:

[0226] S wg (α m ,α n ,u m ,u n ,θ)=R z (θ)·r wg (α m ,α n ,u m ,u n ) (100)

[0227] Where θ represents the angle of rotation of the straight generatrix around the z-axis, and p is the helical parameter, which represents the axial displacement of the straight generatrix when it rotates one unit angle around the z-axis.

[0228] Diamond roller plays a vital role in the grinding process of worm grinding wheel and is an indispensable key component. Before grinding asymmetric gears, the worm grinding wheel needs to be processed with diamond roller. The pressure angles on both sides of the tooth surface of the asymmetric gear are different, so the angles on both sides of the diamond roller are designed to be different to achieve the dressing of the worm grinding wheel of the asymmetric gear. Figure 8 In this way, by adjusting the tooth profile angle of the diamond roller, the processing requirements of asymmetric gears can be effectively met, thereby improving the processing accuracy of the worm grinding wheel.

[0229] It is known that the pressure angle of the driving tooth of the diamond roller is, the pressure angle of the non-driving tooth is, the top thickness is s, and the maximum outer radius is. By introducing variable tooth surface parameters, the tooth profile equation of the diamond roller can be established.

[0230] like Figure 9 As shown in the figure, during the worm wheel dressing process, the diamond roller rotates at high speed, so its rotation parameters do not need to be considered. The tooth surface and normal vector of the diamond roller can be effectively described by simply adjusting the parameters.

[0231] Combined with the axial tooth profile equation of the worm grinding wheel, the axial tooth profile equation of the diamond roller can be obtained: d (α m ,α n ,t m ,t n )for:

[0232]

[0233] Among them, α m , α n Respectively represent the tooth profile angles of the driving side and non-driving side of the diamond roller; t m , t n Respectively represent the tooth profile parameters of the driving side tooth profile and the non-driving side tooth profile, and their value range is Top thickness

[0234] The diamond roller axial tooth profile equation r d (α m ,α n ,t m ,t n )Substitute into the vector rotation formula The tooth surface equation of the diamond roller can be obtained as

[0235]

[0236] 2) Gear grinding; see Figure 4 The worm grinding wheel and the workpiece gear are engaged with each other in a spatial staggered axis to realize the grinding of asymmetric gears with double pressure angles.

[0237] Example 4:

[0238] Worm grinding wheels, a common tool used in gear machining, present numerous challenges when grinding asymmetric gears, such as complex gear geometry and difficult tool trimming. Therefore, accurate worm grinding wheel modeling and trimming are crucial for improving the machining accuracy of asymmetric gears. This example demonstrates the accuracy of the design in Example 3 and intuitively demonstrates the use of the design of the present invention, demonstrating its practical application.

[0239] According to the mathematical expression of the worm grinding wheel profile S wg (α m ,α n ,u m ,u n ,θ), constructing a geometric model of the worm grinding wheel, laying the foundation for the subsequent machine tool model. A discrete point cloud of the tooth surface was generated using MATLAB and imported into SOLIDWORKS to create a profile surface model of the worm grinding wheel. The basic parameters of the worm grinding wheel are shown in Table 2.

[0240] Table 2 Worm grinding wheel related parameters

[0241]

[0242] According to the above basic parameters, the value range of the double parameters in the worm grinding wheel profile equation can be determined as follows:

[0243]

[0244] Based on the double parameter equation of the right profile of the worm grinding wheel, the model of the right profile of the worm grinding wheel was established using the MATLAB program, as shown in Figure 6 In order to create a 3D model of the worm grinding wheel, the point cloud data of the grinding wheel profile generated in MATLAB was imported into SOLIDWORKS and converted into a solid. The established 3D model is shown in Figure 7 shown.

[0245] Based on the above relationship, select α m =25°、α n =20°, m=1.5, r d =100mm, r a =150mm parameter, the diamond roller tooth profile point cloud can be drawn in MATLAB as follows Figure 10 Similarly, by importing the above point cloud data into SOLIDWORKS, the three-dimensional model of the diamond roller can be established, as shown in Figure 11 shown.

[0246] VERICUT is CNC machining simulation software developed by CGTECH in the United States. It supports Windows and UNIX platforms. Using 3D display and virtual reality technology, VERICUT accurately simulates the CNC machining process, displays tool paths, and provides highly realistic machine tool simulation, helping to identify potential problems and avoid losses.

[0247] The system includes CNC program verification, motion simulation, path optimization, tool library, and physical comparison modules, supporting simulation of various machine tools, including CNC turning, milling, machining centers, and wire EDM. VERICUT optimizes CNC programs through simulation, improving machining efficiency, extending tool life, enhancing workpiece surface quality, and reducing errors. It also supports dimensional measurement and model storage.

[0248] Worm Grinding Wheel Gear Grinding Simulation Process

[0249] The NC machining process of worm wheel gear grinding based on VERICUT system is as follows Figure 12Before conducting CNC machining simulation, it is necessary to first establish the machine tool and kinematic models, then construct the tool, workpiece, and fixture models, define the tool motion trajectory, and write the CNC program. Next, configure the CNC system and related parameters, and finally conduct simulation tests.

[0250] The specific simulation steps are as follows:

[0251] (1) Virtual CNC machine tool modeling:

[0252] like Figure 13 As shown in the figure, a worm wheel gear grinding machine is created in SOLIDWORKS software. For a worm wheel gear grinding machine, the following axes need to be created based on the kinematic relationship: feed axis XYZ axis, workpiece rotation axis C axis, worm wheel rotation axis A axis, and worm wheel angular axis B axis. After completing the kinematic axis relationship, create a kinematic tree for the machine tool and import the model created in SOLIDWORKS into the kinematic tree, as shown in the figure. Figure 14 Set the initial position of the machine tool and generate the machine tool file and control file.

[0253] (2) Blank and fixture modeling: According to the machine tool modeling process, the fixture and blank models are established.

[0254] (3) Tool modeling and tool cutting edge definition:

[0255] When a worm grinding wheel is used to grind a gear, it is necessary to load the worm grinding wheel as a tool. Since the simulation software cannot recognize the tool shape of the worm grinding wheel, the working surface of the worm grinding wheel needs to be defined as the tool profile for grinding. When defining the worm grinding wheel as a tool, the worm grinding wheel is usually slotted to facilitate identification of the cutting edge. The obtained closed curves of the front and rear blades can be used to simulate the grinding effect of the worm grinding wheel and realize the normal grinding simulation of the gear. Figure 15 Shown is a slotted worm grinding wheel for simulation.

[0256] (4) CNC system parameter setting: Set the workpiece programming origin and tool compensation data, etc. to ensure the correct operation of the machining program.

[0257] (5) NC program: Before loading the NC program, define the tool list in the tool library, ensure that the G code corresponds to the tool number, and then perform simulation.

[0258] (6) Simulation result analysis and optimization: Observe the workpiece model, dimension measurement, and waste calculation, and check for collisions and interferences. Use the AUTO-DIFF module to compare the processed model with the design model, and adjust the program and parameters to solve overcutting and undercutting problems.

[0259] The tooth profile after simulation processing is as follows Figure 16As shown in the figure, compared with the theoretical design profile of asymmetric gears with dual pressure angles, the gear grinding simulation using the designed asymmetric worm grinding wheel demonstrates good overall performance, essentially consistent with the theoretical design profile. This validates the feasibility of the asymmetric worm grinding wheel designed in this paper for gear grinding. The simulated drive tooth surface error meets actual production requirements, laying a solid theoretical foundation for subsequent asymmetric gear grinding experiments. This not only provides guidance for actual processing but also lays the foundation for further research and development, ensuring the efficiency and reliability of asymmetric gears in various applications.

Claims

1. A method for designing a dual-pressure-angle asymmetric tooth profile, characterized by: The gear tooth profile equation of the double pressure angle asymmetric gear is derived based on the meshing motion relationship between the rack cutter and the forming gear.

2. The method for designing a dual-pressure-angle asymmetric tooth profile according to claim 1, characterized in that: The following steps are involved: 1) Determine the relative motion relationship between the wheel blank and the rack cutter during processing; 2) Derive the transformation relationship between the gear coordinate system and the rack tool coordinate system; establish fixed coordinate systems S f (x f o f y f ), rack tool motion coordinate system S r (x r o r y r ) and the formed gear workpiece motion coordinate system S w (x w o w y w );r b is the pitch radius of the formed gear; in the motion coordinate system S of the formed gear workpiece w (x w o w y w ) is the position vector of any point P in the w , in the rack tool motion coordinate system S r (x r o r y r ) The position vector of point P is r r , the formed gear is in the workpiece motion coordinate system S w (x w o w y w ) is the turning angle Then the workpiece motion coordinate system S w (x w o w y w ) and the fixed coordinate system S f (x f o f y f ) is: Rack tool motion coordinate system S r (x r o r y r ) and the fixed coordinate system S f (x f o f y f ) is: Workpiece motion coordinate system S w (x w o w y w ) and the rack tool motion coordinate system S r (x r o r y r ) is: 3) Establish the tooth profile equation of the rack tool; take the motion coordinate system S of the rack tool r (x r o r y r ) and tool reference axis x r Based on the normal tooth profile of the rack tool, it consists of five parts: AB: Rack straight line segment used to machine the tooth profile on the driving side; CE: Rack straight line segment used to machine the tooth root arc; FG: Rack straight line segment used for machining the non-driving side tooth profile; Used for machining the transition arc segment of the drive side tooth root transition curve; Used for machining the transition arc segment of the tooth root transition curve on the non-driving side; Among them, α m , α n are the tool tooth profile pressure angles for machining the driving tooth side and the non-driving tooth side respectively; θ m ,θ n are the center angles of the tooth root transition curves of the driving tooth side and the non-driving tooth side respectively; ρ m , ρ n are the curvature radii of the tooth root transition curves of the driving tooth side and the non-driving tooth side respectively; S is the tooth thickness; h a 、h f They are tooth top height and tooth root height respectively; is the tooth top height coefficient, m is the module, Indicates the head clearance coefficient; the calculation formulas of each parameter are as follows: S=πm / 2 h m =h f -(1-sina m )r m h n =h f -(1-sina n )r n (4) From the geometric relationship, we can see that the calculation formulas of its various parts are as follows: Rack straight line CE segment for machining tooth root arc: Where, l DE 、l CD is the length of straight line segments DE and CD. Machining the rack straight line FG segment of the non-driving side tooth profile: Where, l FG 、l DE is the length of straight line segments FG and DE. Machining the rack straight line AB segment of the drive side tooth profile: Where, l AB 、l CD is the length of the straight line segments AB and CD. Machining the transition arc of the non-driving side tooth root transition curve part: Where, l DE is the length of straight line segment DE. Machining the transition arc of the drive side tooth root transition curve part: Where, l CD is the length of the straight line segment CD; Write equation (9) as the normal tooth profile curve equation of the rack tool r (α m ,α n ): In addition, the derivative of the above formula can be obtained as the rack tool motion coordinate system S r The tangent vector n of the tool tooth profile r for: The unit normal vector is: According to the geometric relationship, the unit normal vector of each segment of the tool tooth profile is: n r_FG =[-cosα n -sinα n 0] T n r_EF =[-sinθ n -cosθ n 0] T n r_CE ＝[010] T n r_BC =[sinθ m -cosθ m 0] T n r_AB =[-cosα m sinα m 0] T (13) Where, the subscripts represent the corresponding line segments; Write formula (13) as: 4) Derive the meshing equation between the rack tool and the gear; when the rack tool moves at a speed v r When translating to the left, the formed gear rotates counterclockwise at an angular velocity ω, and the instantaneous rotation center O is located at the same speed v r Vertical line O w O, its position satisfies the following vector equation: Since the motion between the tool and the gear is pure rolling along point O, the displacement of the tool and the rotation angle of the gear satisfy the following conditions: Rack tool motion coordinate system S r The vector radius of the moving point P is: Workpiece motion coordinate system S w The vector radius of the moving point P is: The relative sliding velocity at point P is: v=v w -v r (19) Where, v w Point P is in the workpiece motion coordinate system S w The speed in v r Point P is in the rack tool motion coordinate system S r The velocity in satisfies the following equation: From the above formula we can deduce: In order to ensure normal gear meshing, the relative velocity vector of any point on the gear tooth profile is required to be tangent to the normal vector, that is: n r (a m ,a n )·v=0 (22) The above formula (22) is expressed as: f(α m ,α n )＝0 (23) Combining the normal equation and relative velocity equation of each curve equation of the rack tool, the meshing equation can be derived as follows: Where, the subscripts represent the corresponding line segments; 5) According to the meshing relationship between the rack tool and the gear, the gear tooth profile curve equation is derived; by combining the gear meshing equation with the gear tooth profile equation after coordinate transformation, the following conjugate tooth profile equation between the gear and the tool can be obtained: Combining the conjugate tooth profile equation with the asymmetric tool equation for each segment of the gear tooth profile, the tooth profile equation for the formed double-pressure-angle asymmetric gear under asymmetric tool machining can be derived as follows: Drive tooth side involute tooth root transition arc tooth profile segment: Involute tooth profile section on the driving tooth side: Involute tooth profile section on the non-driving tooth side: Involute tooth root transition arc tooth profile section on the non-driving tooth side: Root arc tooth profile segment:

3. The method for designing a dual-pressure-angle asymmetric tooth profile according to claim 2, characterized in that: After step 5), there is also a step of drawing the gear normal tooth profile under different gear normal pressure angles using MATLAB software.

4. A dual pressure angle asymmetric gear, characterized by: The double-pressure-angle asymmetric gear has a tooth profile curve designed according to the tooth profile design method according to any one of claims 1 to 3.

5. The worm grinding wheel method for double pressure angle asymmetric gear according to claim 4, characterized in that: The following steps are involved: 1) Use diamond roller to trim the tooth surface of the worm grinding wheel; 2) Gear grinding: The worm grinding wheel and the workpiece gear are engaged with each other in a spatial staggered axis to realize the grinding of asymmetric gears with double pressure angles.

6. The worm grinding wheel method for dual-pressure-angle asymmetric gears according to claim 1, characterized in that: The axial section of the worm of the worm grinding wheel is a straight line, and the contour surface of the grinding wheel presents an Archimedean spiral surface; combined with the normal contour curve r of the rack tool r (α m ,α n ), the axial section curve r of the worm grinding wheel wg (α m ,α n ,u m ,u n ) is defined as: Among them, α m , α n Respectively represent the tooth profile angles of the driving side tooth profile and the non-driving side tooth profile; u m 、u n Respectively represent the straight generatrix of the driving side tooth profile and the non-driving side tooth profile, which are used to determine the position of any point M and N on the straight generatrix. The value range is r a 、r f They represent the root circle radius and the top circle radius of the worm grinding wheel respectively; H p =kπmcosλ, k, m, and λ are the number of worm grinding wheel heads, module, and lead angle respectively; The axial section curve r of the worm grinding wheel wg (α m ,α n ,u m ,u n )Substitute into the vector rotation formula The equation of the right-hand helicoidal surface of the worm grinding wheel can be obtained as follows: S wg (a m ,a n ,u m ,u n ,θ)=R z (θ)·r wg (a m ,a n ,u m ,u n ) (32) Where θ represents the angle of rotation of the straight generatrix around the z-axis, and p is the helical parameter, which represents the axial displacement of the straight generatrix when it rotates one unit angle around the z-axis. Combined with the axial tooth profile equation of the worm grinding wheel, the axial tooth profile equation of the diamond roller can be obtained: d (α m ,α n ,t m ,t n )for: The diamond roller axial tooth profile equation r d (α m ,α n ,t m ,t n )Substitute into the vector rotation formula The tooth surface equation of the diamond roller can be obtained as follows: Among them, α m , α n Respectively represent the tooth profile angles of the driving side and non-driving side of the diamond roller; t m , t n Respectively represent the tooth profile parameters of the driving side tooth profile and the non-driving side tooth profile, and their value range is Top thickness

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