Method for processing large-face grinding wheel with diagonal modification of helical gear

Abstract

The application discloses a helical gear diagonal modification large-plane grinding wheel processing method, relates to the gear transmission technical field, and aims at the problem that the helical gear diagonal modification cannot be realized by the large-plane grinding wheel in the prior art. The application adopts the generating method to process the diagonal modification helical gear, and realizes the diagonal modification by the additional movement of the gear workpiece in the tangential direction. The large-plane grinding wheel has the advantages of good stability, linear contact, high linear velocity, simple grinding wheel dressing and the like, and therefore, the diagonal modification grinding precision is stable, and the grinding efficiency is high.

Description

Technical Field

[0001] This invention relates to the field of gear transmission technology, specifically to a method for machining a large flat grinding wheel with diagonal profile for helical gears. Background Technology

[0002] Helical gears, due to their smooth transmission and high load-bearing capacity, are widely used in new energy vehicles, wind power equipment, ships, helicopters, and other applications. With the rapid development of national defense and industry, higher demands are being placed on vibration and noise reduction in gear systems. Gear modification is a recognized effective way to achieve this. Gear modification techniques include three types: tooth profile modification, tooth direction modification, and diagonal modification. Compared to the other two methods, diagonal modification only modifies the tooth surface by selecting appropriate modification amounts, modification areas, and modification curves at the tooth tip engagement and tooth root engagement parts, leaving the middle parts unmodified or minimally modified. This effectively reduces the impact force during engagement and disengagement, and provides a larger effective load-bearing contact area with less reduction in gear overlap. Therefore, achieving vibration and noise reduction in helical gear transmission pairs through diagonal modification is essential.

[0003] Due to the characteristics of diagonal gear modification, there is relatively little research on the design and machining of diagonal gear modification. The design of diagonal gear modification mainly includes two methods: The first is the diagonal gear modification design formula proposed by Wang Xianfa in "Calculation of the Optimal Modification Amount and Selection of Modification Area for Diagonal Gear Modification of Narrow Helical Gears," and his work "Research on Diagonal Gear Modification Method and Noise Reduction Effect of Narrow Helical Gears" experimentally proved that diagonally modified gears have a better noise reduction effect than unmodified gears. Based on Wang Xianfa's diagonal gear modification formula, Jiang Jinke proposed two methods in "Research on Tooth Surface Design and CNC Machining Technology of High-Speed ​​Involute Cylindrical Gears": plane grinding and conical grinding, and provided detailed theoretical derivations. He also conducted diagonal gear modification grinding experiments using a QMK50 five-axis linkage conical grinding machine, verifying that the modification amount basically met the design requirements, but the accuracy was not high. The second method is the calculation method for the diagonal gear modification amount of helical gear pairs disclosed in Chinese Patent No. [CN103577713A], which has not yet been experimentally verified.

[0004] Because precision equipment such as ships and helicopters have high requirements for gear precision, generally requiring grade 4, with the highest precision reaching grade 3, milling, gear shaping, or shaving cannot meet the requirements for high-precision gear machining and profile modification. Gear grinding is the main method for achieving high-precision gear machining with hardened tooth surfaces. Among these methods, large-plane grinding wheel generating grinding has the advantages of high machining accuracy, profile modification accuracy, and high machining efficiency compared to forming grinding wheel grinding and worm grinding wheel generating grinding. It can be used to achieve high-precision diagonal profile modification grinding of helical gears. Chinese Patent No. [CN113798602A] discloses a method for achieving high-precision involute tooth profile drum-shaped modification using a large-plane grinding wheel. However, the high-precision diagonal profile modification of helical gears using a large-plane grinding wheel has not yet been studied, and its research has important theoretical and practical significance. Summary of the Invention

[0005] The purpose of this invention is to address the problem that diagonal modification of helical gears cannot be achieved using large-plane grinding wheels in the existing technology, and to propose a method for machining helical gears with large-plane grinding wheels that allows for diagonal modification.

[0006] The technical solution adopted by the present invention to solve the above-mentioned technical problems is as follows:

[0007] A method for machining a large flat grinding wheel with diagonal profile for helical gears includes the following steps:

[0008] Step 1: Obtain the gear to be modified, and determine the modification area according to the rotation direction of the gear to be modified. Specifically, determining the modification area according to the rotation direction of the gear to be modified involves:

[0009] If the gear to be modified has a left-hand helix direction, the modification area of ​​the left tooth surface is located at the root of the front end of the left tooth surface and the top of the rear end of the left tooth surface, and the modification area of ​​the right tooth surface is located at the root of the rear end of the right tooth surface and the top of the front end of the right tooth surface.

[0010] If the gear to be modified has a right-hand helix, the modification area of ​​the left tooth surface is located at the top of the front end of the left tooth surface and the root of the rear end of the left tooth surface, and the modification area of ​​the right tooth surface is located at the root of the front end of the right tooth surface and the top of the rear end of the right tooth surface.

[0011] Among them, the left tooth surface and the right tooth surface are respectively, when facing the tooth, with the tooth groove as the reference, the tooth surface located to the left of the tooth groove is the left tooth surface, and the tooth surface located to the right of the tooth groove is the right tooth surface;

[0012] Step 2: Determine the termination positions of the tooth tip modification and tooth root modification. The termination position of the tooth tip modification is the tooth tip circle. The steps for determining the termination position of the tooth root modification are as follows:

[0013] First, based on the gear meshing principle, determine the intersection point of the driven gear's addendum circle and the line of meshing, and use the distance r from this intersection point to the center of the driving gear's circle. k1 Using a radius of 1, we obtain a circle. Extending this circle, we obtain a cylinder with the same thickness as the gear. The intersection of the cylindrical surface and the tooth surface is the end point of the tooth root modification.

[0014] Step 3: Given the modification height on the tooth tip and tooth root end faces, and combined with the meshing equation, obtain the modification length at the modification termination position of the tooth tip and tooth root. The modification height on the tooth tip and tooth root end faces and the modification length at the modification termination position of the tooth tip and tooth root are the modification areas.

[0015] Step 4: Discretize the tooth tip / root modified tooth surface and determine the distance between each discrete point in the tooth tip / root modified area and the tooth tip boundary line. Then, based on the distance between each discrete point in the tooth tip / root modified area and the tooth tip / root boundary line, and combined with the order of the modification curve, obtain the modification amount E1 of a certain discrete point of the tooth tip / root modified tooth surface. The tooth tip boundary line is the boundary line between the standard tooth surface and the tooth tip modified tooth surface, and the tooth root boundary line is the boundary line between the standard tooth surface and the tooth root modified tooth surface.

[0016] Step 5: Based on the rack cutter model, and using homogeneous coordinate transformation and gear meshing equation, establish a large-plane grinding wheel model. In the large-plane grinding wheel model, the working end face of the grinding wheel coincides with the side face of the helical rack cutter. The working end face of the grinding wheel is used to machine the working tooth profile of the gear, and the transition arc at the circumference of the grinding wheel is used to machine the transition surface of the gear.

[0017] Step Six: Based on the rack cutter model, tooth tip modification amount, and tooth root modification amount, and according to the generating method for machining standard involute gears and the principle of diagonal modification, by making the movement distance of the gear workpiece in the tangential direction of the gear and the law change of the instantaneous machining angle of the gear workpiece shaft, it is equivalent to the additional motion of the large flat grinding wheel that is equal in size and opposite in direction, thereby realizing diagonal modification machining and thus realizing helical gear machining.

[0018] Furthermore, the distance r k1 Represented as:

[0019]

[0020] Where, r b1 The base circle radius r of the driving wheel is... b1 =r p1 cosα t r b2 The base circle radius r of the driven gear is... b2 =r p2 cosα t r p1 The pitch circle radius of the driving wheel is represented by r. p1 =m n z1 / 2cosβ,r p2 The radius of the driven gear's pitch circle, r p2 =m n z² / 2cosβ, α t α represents the end-face pressure angle. t =arctan(tanα) n / cosβ), α n The normal pressure angle of the driving and driven wheels is represented by m. nβ represents the normal module of the driving and driven gears, z1 represents the number of teeth of the driving gear, z2 represents the number of teeth of the driven gear, K1 represents the intersection of the tooth tip circle and the line of action of the driven gear, and N1, N2, and N1 represent the normal vectors of the theoretical tooth surface point coordinates.

[0021] Furthermore, in step three, given the modification heights on the tooth tip and root end faces, and combining them with the meshing equation, the specific steps for obtaining the modification lengths at the termination positions of the tooth tip and root modification are as follows:

[0022] △ABC and △DEF represent the tooth tip modification area and the tooth root modification area, respectively. AB and DE represent the modification height on the gear end face, BC and EF represent the rotational projection of the tooth surface contact line, i.e. the modification start line, AC and DF represent the modification length on the modification end line, W represents the tooth width, G represents the intersection of the perpendicular line drawn from A and BC, and H represents the intersection of the perpendicular line drawn from D and EF.

[0023] Obtain the rotational projection of the tooth surface, and then take x. a The shaft is located on the center line of the tooth width, z a The shaft is located on the gear axis, z a1 The axis is located on the pitch circle, x a2 The axis is located in the normal direction of the tooth tip contact line, x a3 The shaft is located in the normal direction of the tooth root contact line;

[0024] In the tooth tip modification region △ABC, given the modification height AB on the tooth tip face, then in z a O a x a The coordinates of point B in the coordinate system are:

[0025]

[0026] Where, x ai z ai (i = A, B, C...) represent coordinate system z. a O a x a The x and z coordinate components corresponding to different points in the middle;

[0027] Let z a O a x a Point B in the coordinate system corresponds to the gear moving coordinate system S. a -x a y a z a position vector R in B 、N B R B and N B Represented as:

[0028]

[0029] Where R0(u,l) and N0(u,l) represent the coordinates of point B in the rack cutter reference coordinate system S. b -x b y b z b In the figure, position vector and normal vector, u and l represent the rack cutter tooth surface parameters, M db (β) indicates that point B is located in the rack cutter reference coordinate system S. b -x b y b z b To the rack cutter moving coordinate system S d -x d y d z d coordinate transformation, This indicates that point B originates from the moving coordinate system S of the rack cutter. d -x d y d z d To the gear moving coordinate system S a -x a y a z a coordinate transformation;

[0030] Obtain point B in the rack cutter moving coordinate system S d -x d y d z d position vector R in d (u,l) and normal vector N d (u,l), position vector R d (u,l)=M db (β)R0(u,l), normal vector N d (u,l)=M db (β)N0(u,l);

[0031] After that, R d (u,l), N d (u,l) and Substituting into the meshing equation, the meshing equation is expressed as:

[0032]

[0033] Where i = x, y, z, R Bi These represent the coordinates of point B in the gear-driven moving coordinate system S. a -x a y a z a Position vector coordinate components along the x, y, and z axes, Rdi N di These represent the coordinates of point B in the rack cutter's moving coordinate system S. d -x d y d z d Position vector and normal vector coordinate components along the x, y, and z axes. This represents the instantaneous machining angle of the gear corresponding to point B, where, It is an expression about u and l, obtained through the meshing equation. and z aB =R Bz Solving for the two unknown parameters u and l:

[0034] In z a O a x a In the coordinate system, based on the approximate similarity of the gear machining angle corresponding to any point on the tooth surface contact line projection BC and the x-coordinate of point C on the tooth tip modification termination line, that is... x aC =r a1 Substitute these values ​​into the meshing equation, solve for the unknowns u and l, and then calculate z. ac Then the shaping length AC is expressed as:

[0035]

[0036] In the tooth root modification region △DEF, given the modification height DE on the tooth root end face, then in z a O a x a The coordinates of point E in the coordinate system are represented as follows:

[0037]

[0038] Where, x ai z ai (i = A, B, C...) represent coordinate system z. a O a x a The x and z coordinate components corresponding to different points in the middle;

[0039] Let point E be in the gear moving coordinate system S a -x a y a z a The position vector and normal vector in the figure are R and R, respectively. E N E From the formula:

[0040] Point E is obtained in the rack cutter moving coordinate system S d -x d y d zd position vector R in d (u,l) and normal vector N d (u,l), then, R d (u,l), N d (u,l) and formula Substitute into the meshing equation and solve for the machining angle corresponding to the tooth surface contact line EF. Based on the fact that the machining angle is the same at any point on EF and the tooth root modification termination point F is at z a O a x a The x-coordinate in the coordinate system has x aE =r k1 Substitute it into the meshing equation and solve for z. aF Then the shaping length DF is expressed as:

[0041]

[0042] Furthermore, the steps for obtaining the modification amount E1 at a discrete point on the tooth tip / root modified tooth surface are as follows:

[0043] Let R ai (z ai ,x ai ) represents z a O a x a A discrete point, i = A, B, C, D, E, F, G, H…, is located in the tooth tip and root shaping region of the coordinate system.

[0044] The modification amount E1 at a discrete point on the tooth tip / root modified tooth surface is expressed as:

[0045]

[0046] Among them, y a y f k represents the maximum modification amount at the tooth tip and tooth root, respectively. a k f These represent the number of shaping steps in the tooth tip shaping region and the tooth root shaping region, respectively. The perpendicular distance from point A to BC is AG = ACsin(β). a ), O a2 The perpendicular distance from point to BC is x a2G The perpendicular distance from point D to EF is DH = DFsin(β). f ), O a3 The perpendicular distance x from the point to EF a3H ,β a =arctan(AB / AC), β f=arctan(DE / DF), R ai (z ai ,x ai ) in z a2 O a2 x a2 The coordinates in the coordinate system are R a2i (z a2i ,x a2i ), R ai (z ai ,x ai ) in z a3 O a3 x a3 The coordinates in the coordinate system are R a3i (z a3i ,x a3i ), This indicates the instantaneous processing angle.

[0047] Furthermore, the relationship between the radii of circles at different positions of the grinding wheel in the large-plane grinding wheel model is expressed as follows:

[0048]

[0049] Where, r s0 O represents the center of the working end face of the grinding wheel. s The distance r to the boundary point L between the working end face and the circumferential transition arc. s1 Represents the center O s The distance r to any point within the working face of the grinding wheel s2 LM represents the radius of the transition arc at the circumference of the grinding wheel, where LM = 0.1m. n α n The normal pressure angle of the gear being machined, m n This represents the normal modulus of the master and slave wheels.

[0050] Furthermore, in the large planar grinding wheel model, the origin of the moving coordinate system is taken at the center O of the working end face of the grinding wheel. s At this point, the end face of the grinding wheel is represented as:

[0051]

[0052] Among them, R s1 (r s1 ,θ), N s1 (r s1 ,θ) represent the position vector and normal vector of the large flat grinding wheel end face, respectively, and θ represents r s1 With O s The included angle L ranges from -90° to 90°, O s L represents the coordinate system S that does not move with the grinding wheel. s -x s ys z s An imaginary fixed line that rotates. Represents the partial differential symbol.

[0053] Furthermore, the surface at the circumferential transition arc of the grinding wheel in the large planar grinding wheel model is represented as follows:

[0054]

[0055] Among them, R s2 (θ,δ), N s2 (θ, δ) represent the position vector and normal vector at the circumferential transition arc of the large-plane grinding wheel, respectively, and δ represents x. s O s y s The angle between NL and NP1 corresponding to a moving point P1 on the circular arc LP in the plane, from L to P, ranges from 0° to (90°-α). n ).

[0056] Furthermore, the helical gear machining includes: a large-plane grinding wheel process for grinding the standard tooth surface and a diagonal shaping process.

[0057] Furthermore, the process of grinding the standard tooth surface with a large flat grinding wheel specifically involves: positioning the working end face of the grinding wheel and the circumferential transition arc surface in the grinding wheel's moving coordinate system S. s -x s y s z s The position vector R represented in the middle s And Phaya N s Position vector R s And Phaya N s The coordinates are transformed into the moving coordinate system S1-x1y1z1 of the workpiece gear through matrix transformation and translation and rotation transformation respectively. The nonlinear meshing equation is solved by the fsolve function in MATLAB to solve the unknown parameters, and then the position vector R1 and normal vector N1 of the theoretical tooth surface point coordinates are obtained.

[0058] Matrix transformation is expressed as:

[0059]

[0060] in, r0 represents the installation radius of the grinding wheel, r0 = r s –((1.25+a)m n ) / cos(α n ), where 'a' represents the root cutting amount.

[0061] Indicates instantaneous machining angle The distance the corresponding gear moves in the tangential direction of the gear.

[0062] The nonlinear meshing equation is expressed as:

[0063]

[0064] in, vl represents the vertical distance from the point with position vector R1 in the gear moving coordinate system S1-x1y1z1 to the gear axis, and hl represents the vertical distance from the point with position vector R1 on the tooth surface in the gear moving coordinate system S1-x1y1z1 to the gear end section.

[0065] The relationship between the large flat grinding wheel and the additional distance traveled in the tangential direction of the gear at a certain instantaneous machining angle is shown in the following formula:

[0066]

[0067] in, β represents the additional travel distance of the large-plane grinding wheel in the tangential direction of the gear. b The base cylinder helix angle β of the gear b =arctan(tan(β)cos(α) t )).

[0068] Furthermore, the diagonal shaping process specifically includes:

[0069] When the form is not modified, the simultaneous equation is:

[0070]

[0071]

[0072]

[0073]

[0074] By solving the equations simultaneously, the position vector R1 and normal vector N1 of the gear standard involute tooth surface machined by a large flat grinding wheel can be obtained;

[0075] When refining the form, use the following simultaneous forms:

[0076]

[0077]

[0078]

[0079]

[0080]

[0081] By solving the equations simultaneously, the working tooth profile position vector R2 and normal vector N2 of the gear after the large-plane grinding wheel is modified can be obtained;

[0082] The modification amount E2 in the direction of the normal vector N1 at the standard tooth surface coordinate point is obtained by solving R1, N1, and R2. E2 is expressed as:

[0083] E2=(R 2x -R 1x )N 1x +(R 2y -R 1y )N 1y +(R 2z -R 1z )N 1z

[0084] in

[0085]

[0086]

[0087] When performing tooth crest and root shaping, k x The subscripts x and y are a and f, respectively. x The subscript x is taken as a or f. These represent the machining angles corresponding to the end point and the beginning point of the shaping process, respectively.

[0088] The beneficial effects of this invention are:

[0089] This application proposes a large-flat grinding wheel to replace the hypothetical rack cutter, employing a generating method to machine diagonally modified helical gears. The diagonal modification is achieved through additional motion of the gear workpiece in the tangential direction. The large-flat grinding wheel offers advantages such as good stability, line contact, high linear velocity, and simple wheel dressing, resulting in stable diagonal modification grinding accuracy and high grinding efficiency. Furthermore, since the large-flat grinding wheel eliminates the need for axial feed motion, grinding efficiency is significantly improved, and the machine tool mechanism is simplified. By pre-setting additional motion, different types of diagonal modification curves, such as linear, quadratic parabolic, and quadratic parabolic curves, can be easily achieved during grinding. Finally, this method can be implemented on existing CNC gear shaving cutter grinding machines using CNC programming to achieve different forms of helical gear diagonal modification, reducing processing costs. Attached Figure Description

[0090] Figure 1 This is a schematic diagram of the left and right tooth surface modification areas of the left helical tooth in this application;

[0091] Figure 2 This is a schematic diagram of the left and right tooth surface modification areas of the right helical tooth in this application;

[0092] Figure 3This is a schematic diagram illustrating the solution for the termination position of the drive gear tooth root modification in this application.

[0093] Figure 4 This is a three-dimensional diagonal modification diagram of the left tooth surface of the right-hand helical tooth in this application;

[0094] Figure 5 This is a schematic diagram of the diagonal modification of the rotating projection surface of the left tooth surface of the right-hand helical tooth in this application.

[0095] Figure 6 This is a schematic diagram of the coordinate transformation of the tooth surface generated by the rack cutter method in this application;

[0096] Figure 7 A schematic diagram of a large flat grinding wheel replacing a rack cutter in this application;

[0097] Figure 8 This is a schematic diagram of the cross-sectional shape of the large flat grinding wheel in this application;

[0098] Figure 9 A schematic diagram of coordinate system transformation for generating the left tooth surface of a right-hand helical tooth from a large planar grinding wheel in this application;

[0099] Figure 10 A three-dimensional diagonal shaping diagram of a straight line for the large plane grinding wheel on the left tooth surface of the right-hand helical tooth in this application;

[0100] Figure 11 A two-dimensional diagonal shaping diagram of a straight line for the large plane grinding wheel on the left tooth surface of the right-hand helical tooth in this application;

[0101] Figure 12 A three-dimensional diagonal shaping diagram of the second parabola for the large planar surface grinding wheel on the left tooth surface of the right-hand helical tooth in this application;

[0102] Figure 13 A three-dimensional diagonal shaping diagram of the second parabola for the large planar surface grinding wheel on the left tooth surface of the right-hand helical tooth in this application;

[0103] Figure 14 This is a flowchart of this application. Detailed Implementation

[0104] It should be noted that, where there is no conflict, the various embodiments disclosed in this application can be combined with each other.

[0105] Specific implementation method one: Refer to Figure 1 This embodiment describes a method for machining a large flat surface grinding wheel with diagonal profile for helical gears, comprising the following steps:

[0106] Step 1: Obtain the gear to be modified, and determine the modification area according to the rotation direction of the gear to be modified. Specifically, determining the modification area according to the rotation direction of the gear to be modified involves:

[0107] If the gear to be modified has a left-hand helix direction, the modification area of ​​the left tooth surface is located at the root of the front end of the left tooth surface and the top of the rear end of the left tooth surface, and the modification area of ​​the right tooth surface is located at the root of the rear end of the right tooth surface and the top of the front end of the right tooth surface.

[0108] If the gear to be modified has a right-hand helix, the modification area of ​​the left tooth surface is located at the top of the front end of the left tooth surface and the root of the rear end of the left tooth surface, and the modification area of ​​the right tooth surface is located at the root of the front end of the right tooth surface and the top of the rear end of the right tooth surface.

[0109] Among them, the left tooth surface and the right tooth surface are respectively, when facing the tooth, with the tooth groove as the reference, the tooth surface located to the left of the tooth groove is the left tooth surface, and the tooth surface located to the right of the tooth groove is the right tooth surface;

[0110] Step 2: Determine the termination positions of the tooth tip modification and tooth root modification. The termination position of the tooth tip modification is the tooth tip circle. The steps for determining the termination position of the tooth root modification are as follows:

[0111] First, based on the gear meshing principle, determine the intersection point of the driven gear's addendum circle and the line of meshing, and use the distance r from this intersection point to the center of the driving gear's circle. k1 Using a radius of 1, we obtain a circle. Extending this circle, we obtain a cylinder with the same thickness as the gear. The intersection of the cylindrical surface and the tooth surface is the end point of the tooth root modification.

[0112] Step 3: Given the modification height on the tooth tip and tooth root end faces, and combined with the meshing equation, obtain the modification length at the modification termination position of the tooth tip and tooth root. The modification height on the tooth tip and tooth root end faces and the modification length at the modification termination position of the tooth tip and tooth root are the modification areas.

[0113] Step 4: Discretize the tooth tip / root modified tooth surface and determine the distance between each discrete point in the tooth tip / root modified area and the tooth tip boundary line. Then, based on the distance between each discrete point in the tooth tip / root modified area and the tooth tip / root boundary line, and combined with the order of the modification curve, obtain the modification amount E1 of a certain discrete point of the tooth tip / root modified tooth surface. The tooth tip boundary line is the boundary line between the standard tooth surface and the tooth tip modified tooth surface, and the tooth root boundary line is the boundary line between the standard tooth surface and the tooth root modified tooth surface.

[0114] Step 5: Based on the rack cutter model, and using homogeneous coordinate transformation and gear meshing equation, establish a large-plane grinding wheel model. In the large-plane grinding wheel model, the working end face of the grinding wheel coincides with the side face of the helical rack cutter. The working end face of the grinding wheel is used to machine the working tooth profile of the gear, and the transition arc at the circumference of the grinding wheel is used to machine the transition surface of the gear.

[0115] Step Six: Based on the rack cutter model, tooth tip modification amount, and tooth root modification amount, and according to the generating method for machining standard involute gears and the principle of diagonal modification, by making the movement distance of the gear workpiece in the tangential direction of the gear and the law change of the instantaneous machining angle of the gear workpiece shaft, it is equivalent to the additional motion of the large flat grinding wheel that is equal in size and opposite in direction, thereby realizing diagonal modification machining and thus realizing helical gear machining.

[0116] (1) Diagonal modification is divided into positive diagonal modification and negative diagonal modification. Positive diagonal modification refers to modification in the meshing and disengagement areas of the tooth surface, while negative diagonal modification refers to modification at both ends of the tooth in the contact line direction. This invention only discusses positive diagonal modification. The diagonal modification area is determined by the gear's helical direction and the left and right tooth surfaces. Here, the left and right tooth surfaces are defined as follows: when facing the tooth, with the tooth groove as a reference, the tooth surface located to the left of the tooth groove is the left tooth surface, and the tooth surface located to the right of the tooth groove is the right tooth surface. Specifically, taking a helical gear as an example, the tooth tip and tooth root modification areas of the left and right tooth surfaces of a left-hand helical gear are as follows: Figure 1 As shown in the shaded area, the modification area for the left tooth surface of the left helical tooth is located at the root of the front end and the top of the rear end, while the modification area for the right tooth surface of the left helical tooth is located at the root of the rear end and the top of the front end; the modification areas for the tooth tip and root of the left and right tooth surfaces of the right helical active helical tooth are as follows... Figure 2 As shown in the shaded area, the left tooth surface modification area of ​​the right helical tooth is located at the top of the front end and the root of the rear end, while the right tooth surface modification area of ​​the right helical tooth is located at the root of the front end and the top of the rear end.

[0117] The termination point of the tooth tip modification is the tip circle. If a tooth tip chamfer is considered, the corresponding chamfer length should be subtracted. The termination point of the tooth root modification is determined under the condition of ensuring the minimum meshing length of the involute. According to the gear meshing principle, in Figure 3 The termination point of the tooth root modification of the driving gear is point K1, which is the intersection of the addendum circle of the driven gear and the line of action, with a radius r. k1 The expression is:

[0118]

[0119] In the formula, the base circle radii of the driving and driven wheels are r and r, respectively. b1 =r p1 cosα t r b2 =r p2 cosα t The pitch circle radii are r p1 =m n z1 / 2cosβ, r p2 =m n z² / 2cosβ, end face pressure angle α t =arctan(tanα) n / cosβ), r k1α is the distance from the actual meshing start point K1 of the driving wheel 1 to the center O1. n Pressure angle of the master and driven wheel normal surfaces, m n β is the normal module of the master and driven gears, β is the pitch circle helix angle of the master and driven gears, and z1 and z2 are the number of teeth of the master and driven gears.

[0120] (2) Given the modification heights on the tooth tip and root end faces, determine the boundary lines of the standard tooth surface, the modified tooth surface at the tooth tip, and the modified tooth surface at the tooth root, and then determine the modification lengths at the tooth tip and root, thereby determining the modification area. Using a diagonal modification method with the same modification amount along the contact line, taking the left tooth surface of the active right-hand helical tooth as an example, the three-dimensional modification diagram and the rotating projection plane modification diagram of the diagonal modification are respectively as follows: Figure 4 , Figure 5 As shown. In Figure 5 In the diagram, △ABC and △DEF represent the tooth tip and tooth root modification areas, respectively; AB and DE represent the modification heights on the tooth end faces; BC and EF represent the rotational projections of the tooth surface contact lines, i.e., the modification starting lines; AC and DF represent the modification lengths on the modification ending lines; W represents the tooth width; point G is the intersection of the perpendicular lines drawn from point A and BC; point H is the intersection of the perpendicular lines drawn from point D and EF; and x is taken as... a The shaft is located on the center line of the tooth width, z a The shaft is located on the gear axis, z a1 The axis is located on the pitch circle, x a2 The axis is located in the normal direction of the tooth tip contact line, x a3 The shaft is located in the normal direction of the tooth root contact line.

[0121] In the tooth tip modification region △ABC, given the modification height AB on the tooth tip face, then in z a O a x a The coordinates of point B in the coordinate system are:

[0122]

[0123] In the formula x ai z ai (i = A, B, C...) represent respectively... Figure 5 In the z-coordinate system a O a x a The x and z coordinate components corresponding to different points in the equation.

[0124] set up Figure 4 z a O a x a Point B in the coordinate system corresponds to Figure 5 Medium gear moving coordinate system S a -x a y a za The position vector and normal vector in the figure are R and R, respectively. B N B It can be calculated from equation (3):

[0125]

[0126] In the formula, R0(u,l) and N0(u,l) represent the positions of point B. Figure 6 Reference coordinate system S for the middle rack cutter b -x b y b z b In the figure, position vector and normal vector, u and l are the rack cutter tooth surface parameters; R d (u,l), N d (u,l) represents the coordinate system S of the rack cutter moving coordinate system. d -x d y d z d In the context of position vectors and normal vectors, position vector R d =M db R0, Dharma Arrow N d =M db N0, M db (β) indicates that point B is located in the rack cutter reference coordinate system S. b -x b y b z b To the rack cutter moving coordinate system S d -x d y d z d coordinate transformation, This indicates that point B originates from the moving coordinate system S of the rack cutter. d -x d y d z d To the gear moving coordinate system S a -x a y a z a Coordinate transformation.

[0127] Will R d (u,l), N d Substituting (u,l) and equation (2) into the meshing equation (4), we can solve for the instantaneous gear machining angle corresponding to point B. in It is an expression for u and l. The two unknown parameters u and l can be solved by the first two equations in equation (4). The equation has a solution:

[0128]

[0129] In the formula R Bi(i = x, y, z) represent the coordinates of point B in the gear-driven coordinate system S. a -x a y a z a Position vector coordinate components along the x, y, and z axes, R di N di (i = x, y, z) represent the coordinates of point B in the moving coordinate system S of the rack cutter. d -x d y d z d Position vector and normal vector coordinate components along the x, y, and z axes.

[0130] exist Figure 5 z a O a x a In the coordinate system, based on the approximate similarity of the gear machining angle corresponding to any point on the tooth surface contact line projection BC and the x-coordinate of point C on the tooth tip modification termination line, that is... x aC =r a1 Substituting these values ​​into the meshing equation (4) allows us to solve for the unknowns u and l, and then determine z. ac Then the shaping length AC:

[0131]

[0132] To facilitate the calculation of the profile length along the contact line normal direction in the tooth tip region, the projected helix angle of the tooth tip profile starting line is approximately determined:

[0133] β a =arctan(AB / AC) (6)

[0134] In the tooth root modification region △DEF, given the modification height DE on the tooth root end face, in Figure 5 z a O a x a The coordinates of point E in the coordinate system are represented as follows:

[0135]

[0136] Let point E be at Figure 4 Gear moving coordinate system S a -x a y a z a The position vector and normal vector in the figure are R and R, respectively. E N E R can be calculated and expressed by equation (3). d (u,l), N d (u,l) indicates that point E is located at... Figure 6 rack and pinion moving coordinate system Sd -x d y d z d The position vector and normal vector in the middle will R d (u,l), N d Substituting (u,l) and equation (7) into the meshing equation (4) to solve for the machining angle corresponding to the tooth surface contact line EF. Based on the fact that the machining angle is the same at any point on EF and the tooth root modification termination point F is at... Figure 5 z a O a x a The x-coordinate in the coordinate system has x aE =r k1 Substitute it into the meshing equation (4) to solve for z. aF Then the shaping length DF:

[0137]

[0138] To facilitate the calculation of the modification length of the tooth root region along the normal direction of the contact line, the projected helix angle corresponding to the tooth surface contact line EF is approximately determined:

[0139] β f =arctan(DE / DF) (9)

[0140] (3) Discretize the tooth tip and root profiles as follows: Figure 5 As shown, the tooth tip modification amount is calculated based on the distance between discrete points in the tooth tip modification region and the tooth tip boundary line, as well as the order of the modification curve. Similarly, the tooth root modification amount is calculated based on the distance between discrete points in the tooth root modification region and the tooth root boundary line, as well as the order of the modification curve. According to the gear meshing principle, the order of machining the left tooth surface of a right-hand helical gear with a rack cutter is first the tooth root, then the tooth tip. Therefore, the instantaneous machining angle gradually increases from the tooth root to the tooth tip. Figure 5 The dividing line BC corresponding to the tooth tip modification line The boundary criterion between the tooth tip modification area and the intermediate standard tooth surface is the tooth root modification boundary line EF. This serves as the boundary criterion between the root shaping region and the standard tooth surface. Let R be... ai (z ai ,x ai )(i=A,B,C,D,E,F,G,H...) means Figure 5 Chinese z a O a x a If a discrete point is located in the tooth tip and tooth root modification region of the coordinate system, then the modification amount E1 at a certain point on the entire tooth surface is:

[0141]

[0142] In the formula R a2i (z a2i ,x a2i ) is R ai (z ai ,x ai ) in z a2 O a2 x a2 Coordinate representation in a coordinate system, R a3i (z a3i ,x a3i ) is R ai (z ai ,x ai ) in z a3 O a3 x a3 Coordinate representation in a coordinate system, Figure 4 middle y a y f k represents the maximum modification amount at the tooth tip and tooth root, respectively. a k f These represent the number of shaping operations in the tooth tip shaping region and the tooth root shaping region, respectively. They can be either 1 or 2 times. The perpendicular distance from point A to BC is AG = ACsin(β). a ), O a2 The perpendicular distance from point to BC is x a2G The perpendicular distance from point D to EF is DH = DFsin(β). f ), O a3 The perpendicular distance x from the point to EF a3H .

[0143] (4) This invention proposes a method for machining helical gears using a generating method, which replaces the hypothetical rack cutter with a large flat grinding wheel. The analysis is based on machining the left tooth surface of a right-hand helical gear. Figure 7 This diagram illustrates a large-face grinding wheel replacing a rack cutter. The working face of the grinding wheel coincides with the side face of the helical rack cutter to machine the working tooth profile of the gear. The transition arc at the circumference of the grinding wheel machines the transition surface of the gear. Because the large-face grinding wheel has a large diameter, to ensure the chord length of the grinding wheel's working face can fully machine the tooth surface, the tool setting point is low, resulting in a certain amount of undercut at the gear root. The amount of undercut is calculated based on the width of the gear being machined. To ensure the gear root is an arc, a pre-machining hob is needed to pre-machine the undercut portion at the gear root.

[0144] like Figure 7 Grinding wheel model and Figure 8 As shown in the cross-section of the grinding wheel, the relationship between the radii of the circles at different positions on the grinding wheel can be expressed as:

[0145]

[0146] In the formula r s0Center O of the working end face of the grinding wheel s The distance r to the boundary point L between the working end face and the circumferential transition arc. s1 Center O s The distance r to any point within the working face of the grinding wheel s2 Let LM be the radius of the transition arc at the circumference of the grinding wheel. For ease of calculation, we take LM = 0.1m. n Calculate r s2 α n The normal pressure angle of the gear being machined.

[0147] The model of the grinding wheel and the moving coordinate system are as follows: Figure 7 As shown in the schematic diagram of the grinding wheel cross-section, Figure 8 As shown, when modeling the grinding wheel, the origin of the moving coordinate system is taken at the center O of the working end face of the grinding wheel. s At this point, the end face of the grinding wheel is represented as:

[0148]

[0149] In the formula R s1 (r s1 ,θ), N s1 (r s1 ,θ) represent the position vector and normal vector of the large flat grinding wheel end face, respectively, where θ is r s1 With O s The included angle L ranges from -90° to 90°, O s L is a coordinate system that does not move with the grinding wheel. s -x s y s z s An imaginary fixed line that rotates.

[0150] The surface at the circumferential transition arc of the grinding wheel is represented as follows:

[0151]

[0152] In the formula R s2 (θ,δ), N s2 (θ, δ) represent the position vector and normal vector at the circumferential transition arc of the large-plane grinding wheel, respectively. δ is the position vector at the circumferential transition arc of the large-plane grinding wheel. Figure 8 Chinese x s O s y s The angle between NL and NP1 corresponding to a moving point P1 on the circular arc LP in the plane, from L to P, ranges from 0° to (90°-α). n ).

[0153] The process of grinding standard tooth surfaces with a large flat grinding wheel is as follows: Figure 9 The coordinate system transformation of the large flat grinding wheel into the left tooth surface of the right-hand helical tooth is shown in the diagram. The working end face of the grinding wheel and the circumferential transition arc surface are transformed in the moving coordinate system S of the grinding wheel.s -x s y s z s The position vector R represented in the middle s And Phaya N s The coordinates are transformed into the moving coordinate system S1-x1y1z1 of the workpiece gear through the matrix transformation formula (14) and the nonlinear meshing equation (15) is solved by the "fsolve" function in MATLAB to solve the unknown parameters, and then the position vector R1 and normal vector N1 of the theoretical tooth surface point coordinates are obtained.

[0154]

[0155] In the formula r0 = r s –((1.25+a)m n ) / cos(α n r0 represents the installation radius of the grinding wheel, and a represents the root cut amount.

[0156] Indicates instantaneous machining angle The distance the corresponding gear travels in the tangential direction of the gear.

[0157]

[0158] In the formula vl represents the vertical distance from the point with position vector R1 in the gear moving coordinate system S1-x1y1z1 to the gear axis, and hl represents the vertical distance from the point with position vector R1 on the tooth surface in the gear moving coordinate system S1-x1y1z1 to the gear end section.

[0159] (5) When grinding gears with a large flat grinding wheel, the large flat grinding wheel remains stationary, and the gear needs to rotate around its own axis and translate in the tangential direction to complete the generating motion. Based on the generating method for gear machining and the principle of diagonal shaping, the large flat grinding wheel achieves diagonal shaping by changing the additional rotation angle of the gear workpiece or by the additional motion of the gear in the tangential direction, causing the amount of machining of the grinding wheel at a certain instant to change according to a certain rule to achieve diagonal shaping. This paper studies the additional motion of the gear in the tangential direction. For ease of study, theoretically, it can be equivalently transformed into the motion of the large flat grinding wheel, whose direction of motion is opposite to and equal in magnitude to the additional motion of the gear in the tangential direction. Figure 5 Taking the left tooth surface of a right-hand helical gear as an example, the relationship between the large flat grinding wheel and the additional moving distance in the tangential direction of the gear at a certain instantaneous machining angle is shown in equation (16):

[0160]

[0161] In the formula β represents the additional travel distance of the large-plane grinding wheel in the tangential direction of the gear. b The base cylinder helix angle β of the gear b =arctan(tan(β)cos(α) t )).

[0162] The diagonal modification process is as follows: when no modification is performed, equations (11), (12), (14), and (15) are combined to solve for the position vector R1 and normal vector N1 of the standard involute tooth surface of the gear machined by the large flat grinding wheel; when modification is performed, equations (11), (12), (16), (17), and (18) are combined to solve for the position vector R2 and normal vector N2 of the working tooth profile of the gear after modification by the large flat grinding wheel; and R1, N1, and R2 are substituted into equation (19) to solve for the modification amount E2 in the direction of the normal vector N1 at the coordinate point of the standard tooth surface.

[0163]

[0164] In the formula M cs (r0), M bc (α n M ab (β) Same as in equation (14).

[0165]

[0166] In the formula

[0167] When shaping the tooth crest and root.

[0168] k x The subscripts x and y are a and f, respectively. x The subscript x is taken as a or f. These represent the machining angles corresponding to the end point and the beginning point of the shaping process, respectively.

[0169] Calculate the diagonal modification amount of the gear in the normal direction N1 of the unmodified tooth surface when machining with a large flat grinding wheel:

[0170] E2=(R 2x -R 1x )N 1x +(R 2y -R 1y )N 1y +(R 2z -R 1z )N 1z (19)

[0171] To facilitate the representation of three-dimensional profiles of different coordinate points on the working tooth surface, Figure 4The left tooth surface of the right-hand helical tooth is in Figure 5 The rotating projection surface is digitally discretized, and M1 represents... Figure 5 In the middle tooth height direction r a1 to r k1 The number of equally spaced discrete points, M2 represents the number of equally spaced discrete points from -W / 2 to W / 2 in the tooth width direction, and E2 represents the value of the diagonal modification amount in the normal direction of the standard tooth surface.

[0172] To facilitate the representation of the two-way profile diagram, the midpoint of the tooth width passing through the pitch circle is taken as the origin. Figure 4 The left tooth surface of the right-hand helical tooth is in Figure 5 The contact line normal in the tooth tip direction of the rotating projection surface is the negative half-axis of the abscissa, and the contact line normal in the tooth root direction is the positive half-axis of the abscissa. The normal lengths of the tooth tip and tooth root modification areas are discretized into M3 points, and the ordinate is the modification amount of the discretized points.

[0173] Example:

[0174] The basic parameters of the helical gear pair are as follows: the pinion is right-handed and the gear is left-handed; the normal module is 5mm; the displacement coefficient is 0; the pinion has 30 teeth; the gear has 72 teeth; the normal pressure angle is 20°; the pitch circle helix angle is 33.273°; and the tooth width is 40mm. Taking the left tooth surface of the right-handed pinion as an example, the maximum modification amount at the tooth tip and tooth root modification termination position is taken as 15μm; the modification height of the tooth tip and tooth root end face is 5mm respectively; the modification number is taken as 1 time and 2 times respectively; and the grinding wheel radius r is... s Take 400mm.

[0175] This invention does not consider the tooth tip chamfer; by substituting the basic gear parameters into equation (1), the tooth tip modification termination height r can be calculated. a1 = 94.7058mm, tooth root modification termination height r k1 = 85.6664mm, with M1=10, M2=20, M3=10, the profile modification amount at different points on the working tooth surface of the large flat grinding wheel is simulated and analyzed using MATLAB programming. The three profile diagrams of the left tooth surface of the right-hand helical tooth are as follows: Figure 10 , Figure 12 The diagrams shown are as follows: Figure 11 , Figure 13 As shown.

[0176] For shaping Figures 11-13The results are analyzed as follows: the simulation shaping effect of one-time and two-time diagonal shaping on the large flat surface grinding wheel is relatively ideal. When the shaping is done once, the difference in shaping amount between the tooth apex and the tooth root shaping termination point is 0.18μm and 0.17μm, respectively, with errors of 1.2% and 1.13%. When the shaping is done twice, the difference in shaping amount between the tooth apex and the tooth root shaping termination point is 0.63μm and 0.66μm, respectively, with errors of 4.2% and 4.4%. Therefore, the machining effect of one-time and two-time diagonal shaping can be considered ideal.

[0177] It should be noted that the specific embodiments are merely explanations and illustrations of the technical solution of the present invention and should not be used to limit the scope of protection. Any modifications made in accordance with the claims and specification of the present invention that are only partial should still fall within the protection scope of the present invention.

Claims

1. A method for machining a large flat grinding wheel with diagonal profile for helical gears, characterized in that... Includes the following steps: Step 1: Obtain the gear to be modified, and determine the modification area according to the rotation direction of the gear to be modified. Specifically, determining the modification area according to the rotation direction of the gear to be modified involves: If the gear to be modified has a left-hand helix direction, the modification area of ​​the left tooth surface is located at the root of the front end of the left tooth surface and the top of the rear end of the left tooth surface, and the modification area of ​​the right tooth surface is located at the root of the rear end of the right tooth surface and the top of the front end of the right tooth surface. If the gear to be modified has a right-hand helix, the modification area of ​​the left tooth surface is located at the top of the front end of the left tooth surface and the root of the rear end of the left tooth surface, and the modification area of ​​the right tooth surface is located at the root of the front end of the right tooth surface and the top of the rear end of the right tooth surface. Among them, the left tooth surface and the right tooth surface are respectively, when facing the tooth, with the tooth groove as the reference, the tooth surface located to the left of the tooth groove is the left tooth surface, and the tooth surface located to the right of the tooth groove is the right tooth surface; Step 2: Determine the termination positions of the tooth tip modification and tooth root modification. The termination position of the tooth tip modification is the tooth tip circle. The steps for determining the termination position of the tooth root modification are as follows: First, based on the gear meshing principle, determine the intersection point of the driven gear's addendum circle and the line of meshing, and use the distance r from this intersection point to the center of the driving gear's circle. k1 Using a radius of 1, we obtain a circle. Extending this circle, we obtain a cylinder with the same thickness as the gear. The intersection of the cylindrical surface and the tooth surface is the end point of the tooth root modification. Step 3: Given the modification height on the tooth tip and tooth root end faces, and combined with the meshing equation, obtain the modification length at the modification termination position of the tooth tip and tooth root. The modification height on the tooth tip and tooth root end faces and the modification length at the modification termination position of the tooth tip and tooth root are the modification areas. Step 4: Discretize the tooth tip / root modified tooth surface and determine the distance between each discrete point in the tooth tip / root modified area and the tooth tip boundary line. Then, based on the distance between each discrete point in the tooth tip / root modified area and the tooth tip / root boundary line, and combined with the order of the modification curve, obtain the modification amount E1 of a certain discrete point of the tooth tip / root modified tooth surface. The tooth tip boundary line is the boundary line between the standard tooth surface and the tooth tip modified tooth surface, and the tooth root boundary line is the boundary line between the standard tooth surface and the tooth root modified tooth surface. Step 5: Based on the rack cutter model, and using homogeneous coordinate transformation and gear meshing equation, establish a large-plane grinding wheel model. In the large-plane grinding wheel model, the working end face of the grinding wheel coincides with the side face of the helical rack cutter. The working end face of the grinding wheel is used to machine the working tooth profile of the gear, and the transition arc at the circumference of the grinding wheel is used to machine the transition surface of the gear. Step Six: Based on the rack cutter model, tooth tip modification amount, and tooth root modification amount, and according to the generating method for machining standard involute gears and the principle of diagonal modification, by making the movement distance of the gear workpiece in the tangential direction of the gear and the law change of the instantaneous machining angle of the gear workpiece shaft, it is equivalent to the additional motion of the large flat grinding wheel that is equal in size and opposite in direction, thereby realizing diagonal modification machining and thus realizing helical gear machining; In step three, given the modification heights on the tooth tip and root end faces, and combining them with the meshing equation, the specific steps for obtaining the modification lengths at the termination positions of the tooth tip and root modification are as follows: △ABC and △DEF represent the tooth tip modification area and the tooth root modification area, respectively. AB and DE represent the modification height on the gear end face, BC and EF represent the rotational projection of the tooth surface contact line, i.e. the modification start line, AC and DF represent the modification length on the modification end line, W represents the tooth width, G represents the intersection of the perpendicular line drawn from A and BC, and H represents the intersection of the perpendicular line drawn from D and EF. Obtain the rotational projection of the tooth surface, and then take x. a The shaft is located on the center line of the tooth width, z a The shaft is located on the gear axis, z a1 The axis is located on the pitch circle, x a2 The axis is located in the normal direction of the tooth tip contact line, x a3 The shaft is located in the normal direction of the tooth root contact line; In the tooth tip modification region △ABC, given the modification height AB on the tooth tip face, then in z a O a x a The coordinates of point B in the coordinate system are: Where, x ai、 z ai (i=A,B,C…) represent coordinate system z. a O a x a The x and z coordinate components corresponding to different points in the middle; Let z a O a x a Point B in the coordinate system corresponds to the gear moving coordinate system S. a -x a y a z a position vector R in B 、N B R B and N B Represented as: Where R0(u,l) and N0(u,l) represent the coordinates of point B in the rack cutter reference coordinate system S. b -x b y b z b In the figure, position vector and normal vector, u and l represent the rack cutter tooth surface parameters, M db (β) indicates that point B is located in the rack cutter reference coordinate system S. b -x b y b z b To the rack cutter moving coordinate system S d -x d y d z d coordinate transformation, M ad (φ) indicates that point B moves from the rack cutter's coordinate system S. d -x d y d z d To the gear moving coordinate system S a -x a y a z a coordinate transformation; Obtain point B in the rack cutter moving coordinate system S d -x d y d z d position vector R in d (u,l) and normal vector N d (u,l), position vector R d (u,l)=M db (β)R0(u,l), normal vector N d (u,l)= ; After that, R B (u,l,φ),R d (u,l), N d (u,l) and Substituting into the meshing equation, the meshing equation is expressed as: Where i = x, y, z, R Bi These represent the coordinates of point B in the gear-driven moving coordinate system S. a -x a y a z a Position vector coordinate components along the x, y, and z axes, R di N di These represent the coordinates of point B in the rack cutter's moving coordinate system S. d -x d y d z d Position vector and normal vector coordinate components along the x, y, and z axes, φ B This represents the instantaneous machining angle of the gear corresponding to point B, where φ B It is an expression about u and l, obtained through the meshing equation. and Solving for the two unknown parameters u and l: In z a O a x a In the coordinate system, based on the approximate similarity of the gear machining angle corresponding to any point on the tooth surface contact line projection BC and the x-coordinate of point C on the tooth tip modification termination line, i.e., φ C =φ B x aC =r a1 Substitute these values ​​into the meshing equation, solve for the unknowns u and l, and then calculate z. ac Then the shaping length AC is expressed as: In the tooth root modification region △DEF, given the modification height DE on the tooth root end face, then in z a O a x a The coordinates of point E in the coordinate system are represented as follows: Where, x ai、 z ai (i=A,B,C…) represent coordinate system z. a O a x a The x and z coordinate components corresponding to different points in the middle; Let point E be in the gear moving coordinate system S a -x a y a z a The position vector and normal vector in the figure are R and R, respectively. E N E From the formula: Point E is obtained in the rack cutter moving coordinate system S d -x d y d z d position vector R in d (u,l) and normal vector N d (u,l), then, R E (u,l,φ),R d (u,l), N d (u,l) and formula Substitute into the meshing equation and solve for the machining angle φ corresponding to the tooth surface contact line EF. E Based on the fact that the machining angle is the same at any point on EF and the tooth root modification termination point F is at z a O a x a The x-coordinate in the coordinate system has φ F =φ E x aE =r k1 Substitute it into the meshing equation and solve for z. aF Then the shaping length DF is expressed as: ； The steps for obtaining the modification amount E1 at a discrete point on the tooth tip / root modified tooth surface are as follows: Let R ai (z) ai ,x ai ) represents z a O a x a A discrete point, i = A, B, C, D, E, F, G, H…, is located in the tooth tip and root shaping region of the coordinate system. The modification amount E1 at a discrete point on the tooth tip / root modified tooth surface is expressed as: Among them, y a y f k represents the maximum modification amount at the tooth tip and tooth root, respectively. a k f These represent the number of shaping operations in the tooth tip shaping region and the tooth root shaping region, respectively. The perpendicular distance from point A to BC is AG = ACsin(β). a ), O a2 The perpendicular distance from point to BC is x a2G The perpendicular distance from point D to EF is DH = DFsin(β). f ), O a3 The perpendicular distance x from the point to EF a3H , , R ai (z) ai ,x ai ) in z a2 O a2 x a2 The coordinates in the coordinate system are R a2i (z) a2i ,x a2i ), R ai (z) ai ,x ai ) in z a3 O a3 x a3 The coordinates in the coordinate system are R a3i (z) a3i ,x a3i ), φ represents the instantaneous machining angle.

2. The method for machining a large flat surface grinding wheel with diagonal modification of helical gears according to claim 1, characterized in that... The distance r k1 Represented as: Where, r b1 The base circle radius r of the driving wheel is... b1 =r p1 cosα t r b2 The base circle radius r of the driven gear is... b2 =r p2 cosα t r p1 The pitch circle radius of the driving wheel is represented by r. p1 =m n z1 / 2cosβ,r p2 The radius of the driven gear's pitch circle, r p2 =m n z² / 2cosβ, α t α represents the end-face pressure angle. t =arctan(tanα n / cosβ), α n The normal pressure angle of the driving and driven wheels is represented by m. n Let z1 represent the normal module of the driving and driven gears, β represent the pitch circle helix angle of the driving and driven gears, z1 represent the number of teeth on the driving gear, and z2 represent the number of teeth on the driven gear. This indicates the intersection of the addendum circle of the driven gear and the line of action. , The normal vector representing the coordinates of the theoretical tooth surface point.

3. The method for machining a large flat surface grinding wheel with diagonal modification of helical gears according to claim 2, characterized in that... The relationship between the radii of circles at different positions of the grinding wheel in the large planar grinding wheel model is expressed as follows: Where, r s0 O represents the center of the working end face of the grinding wheel. s The distance r to the boundary point L between the working end face and the circumferential transition arc. s1 Represents the center O s The distance r to any point within the working face of the grinding wheel s2 LM represents the radius of the transition arc at the circumference of the grinding wheel, where LM = 0.1m. n α n The normal pressure angle of the gear being machined, m n This represents the normal modulus of the master and slave wheels.

4. The method for machining a large flat surface grinding wheel with diagonal modification of helical gears according to claim 3, characterized in that... In the large-plane grinding wheel model, the origin of the moving coordinate system is taken at the center O of the working end face of the grinding wheel. s At this point, the end face of the grinding wheel is represented as: Among them, R s1 (r s1 ,θ), N s1 (r s1 ,θ) represent the position vector and normal vector of the large flat grinding wheel end face, respectively, and θ represents r s1 With O s The included angle L ranges from -90° to 90°, O s L represents the coordinate system S that does not move with the grinding wheel. s -x s y s z s An imaginary fixed line that rotates. Represents the partial differential symbol.

5. A method for machining a large flat surface grinding wheel with diagonal modification of helical gears according to claim 3, characterized in that... The surface at the circumferential transition arc of the grinding wheel in the large flat grinding wheel model is represented as follows: Among them, R s2 (θ,δ), N s2 (θ, δ) represent the position vector and normal vector at the circumferential transition arc of the large-plane grinding wheel, respectively, and δ represents x. s O s y s The angle between NL and NP1 corresponding to a moving point P1 on the circular arc LP in the plane, from L to P, ranges from 0° to (90°-α). n ).

6. A method for machining a large flat surface grinding wheel with diagonal modification of helical gears according to claim 5, characterized in that... The helical gear machining process includes: grinding the standard tooth surface with a large flat grinding wheel and a diagonal shaping process.

7. A method for machining a large flat surface grinding wheel with diagonal modification of helical gears according to claim 6, characterized in that... The specific process of grinding standard tooth surfaces with a large-plane grinding wheel is as follows: The working end face of the grinding wheel and the circumferential transition arc surface are aligned in the grinding wheel's moving coordinate system S. s -x s y s z s The position vector R represented in the middle s And Phaya N s Position vector R s And Phaya N s The coordinates are transformed into the moving coordinate system S1-x1y1z1 of the workpiece gear through matrix transformation and translation and rotation transformation respectively. The nonlinear meshing equation is solved by the fsolve function in MATLAB to solve the unknown parameters, and then the position vector R1 and normal vector N1 of the theoretical tooth surface point coordinates are obtained. Matrix transformation is expressed as: in, r0 represents the installation radius of the grinding wheel, r0 = r s –(1.25+a)m n ) / cos(α n ), where 'a' represents the root cutting amount. , , , d φ d represents the distance the gear moves in the tangential direction corresponding to the instantaneous machining angle φ. φ =r p1 φ; The nonlinear meshing equation is expressed as: in, vl represents the vertical distance from the point with position vector R1 in the gear moving coordinate system S1-x1y1z1 to the gear axis, and hl represents the vertical distance from the point with position vector R1 on the tooth surface in the gear moving coordinate system S1-x1y1z1 to the gear end section. The relationship between the large flat grinding wheel and the additional distance traveled in the tangential direction of the gear at a certain instantaneous machining angle is shown in the following formula: Where, d sφ β represents the additional travel distance of the large-plane grinding wheel in the tangential direction of the gear. b The base cylinder helix angle β of the gear b =arctan(tan(β)cos(α t )).

8. A method for machining a large flat surface grinding wheel with diagonal modification of helical gears according to claim 7, characterized in that... The diagonal shaping process is specifically as follows: When the form is not modified, the simultaneous equation is: By solving the equations simultaneously, the position vector R1 and normal vector N1 of the gear standard involute tooth surface machined by a large flat grinding wheel can be obtained; When refining the form, use the following simultaneous forms: By solving the equations simultaneously, the working tooth profile position vector R2 and normal vector N2 of the gear after the large-plane grinding wheel is modified can be obtained; The modification amount E2 in the direction of the normal vector N1 at the standard tooth surface coordinate point is obtained by solving R1, N1, and R2. E2 is expressed as: in , ， ， When performing tooth crest and root shaping, k x The subscripts x and y are a and f, respectively. x The subscript x is taken as a, f, φ p φ q These represent the machining angles corresponding to the end point and the beginning point of the shaping process, respectively.

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