Topological modification method for cylindrical gear

Abstract

The invention relates to the technical field of gear machining processes, in particular to a topological modification method for a cylindrical gear. The topological modification method comprises the following steps: constructing a cylindrical gear time-varying meshing stiffness model; and establishing a nonlinear dynamic model of the cylindrical gear transmission system, and combining the nonlinear dynamic model with the cylindrical gear time-varying meshing stiffness model to obtain the vibration displacement of the cylindrical gear on each degree of freedom. And establishing a transformation matrix M21, and obtaining the comprehensive offset through the matrix M21 and the vibration displacement on each degree of freedom. And overlapping the tooth surface with a standard tooth surface to obtain a topological modified tooth surface. And optimizing the topological modified tooth surface to obtain a B spline curved surface, namely the optimized topological modified tooth surface of the cylindrical gear. According to the topological modification method, the tooth surface subjected to modification is a smooth curved surface, the situation that the modification amount of the designed topological modification tooth surface suddenly changes can be avoided, meanwhile, the designed topological modification tooth surface can be machined through the optimal design, and the purpose of design and manufacturing consistency is achieved.

Description

Technical Field

[0001] This invention relates to the field of gear processing technology, and specifically to a method for topological modification of cylindrical gears. Background Technology

[0002] With globalization, the upgrading of manufacturing, and the continuous development of new energy vehicles, people's demands for driving comfort are also increasing. Gears, as key components in automotive drive systems, directly affect vehicle noise and performance due to their design and manufacturing precision. To improve gear meshing characteristics and transmission efficiency, gear modification is generally used. Gear modification refers to the intentional, minute adjustment of the tooth surface to deviate from the theoretical tooth surface. While drum-shaped modification can improve uneven loading on the tooth surface, it also increases the maximum contact stress of the gear to some extent, leading to limitations in existing gear modification methods. Furthermore, existing gear modification methods often rely on empirical formulas to calculate modification parameters, resulting in poor design accuracy. Moreover, although the actual parameters of the modified gears designed using this method are within acceptance tolerances, the modified gears fail the meshing noise and vibration characteristic test. Additionally, the tooth surfaces designed using empirical formulas to calculate modification parameters are irregular and overly complex, making them unsuitable for machine tool processing. Summary of the Invention

[0003] To address the technical problem that existing gear modification methods result in overly complex tooth surface structures that cannot be machined on machine tools, this invention provides a topological modification method for cylindrical gears.

[0004] This invention employs the following technical solution: a topological modification method for cylindrical gears, comprising: constructing a time-varying meshing stiffness model of the cylindrical gear; establishing a nonlinear dynamic model of the cylindrical gear transmission system, and obtaining the vibration displacement of the cylindrical gear in each degree of freedom through the time-varying meshing stiffness model and the nonlinear dynamic model; and establishing a transformation matrix M between the coordinate system of the loaded gear and the coordinate system of the gear shaft. 21 Through matrix M 21 The vibration displacement of the cylindrical gear in each degree of freedom is used to obtain the comprehensive offset between the actual tooth surface position and the theoretical position of the cylindrical gear during meshing. This comprehensive offset is then superimposed on the standard tooth surface to obtain the topologically modified tooth surface. The entire tooth surface of the obtained topologically modified tooth surface is meshed along the tooth direction and tooth profile direction. n×m mesh points are selected on the tooth surface, and these n×m mesh points are used as the shape value points Q of the B-spline surface. i,jThe data points at the four corners of the defined value lattice become the four corner points of the entire surface, and the other data points become the common corner points of the corresponding adjacent surface patches. Then, the control vertices P of the B-spline surface are obtained through inverse calculation of the B-spline surface. k，d This yields the B-spline surface, which is the optimized topologically modified tooth surface of the cylindrical gear.

[0005] As a further improvement of the present invention, the construction method of the time-varying meshing stiffness model of cylindrical gear is as follows: After slicing the cylindrical gear, it is divided into a series of spur gear slices, and the stiffness model of the spur gear is established based on the beam deformation theory and the improved potential energy method; then, based on the principle of cumulative integration, the slices are integrated along the contact trace of the cylindrical gear to obtain the time-varying meshing stiffness model of cylindrical gear.

[0006] Typically, the stiffness model of the spur gear includes a bending stiffness model k. b Shear stiffness model k s Radial compressive stiffness model k a Hertzian contact stiffness model k h and gear body stiffness model k f The time-varying meshing stiffness model k of the cylindrical gear t The formula is as follows:

[0007]

[0008] In the formula, The bending stiffness of the loaded gear; The bending stiffness of the driven gear; The shear stiffness of the loaded gear; The shear stiffness of the driven gear; The radial compressive stiffness of the loaded gear; k is the radial compressive stiffness of the driven gear. h The Hertzian contact stiffness between the loaded gear and the driven gear; The stiffness of the gear body under load; This refers to the stiffness of the driven gear.

[0009] As a further improvement of the present invention, the vibration displacement of the cylindrical gear in each degree of freedom includes: the lateral vibration displacement of the gear shaft of the loaded gear and the driven gear in the x-axis direction, the vertical vibration displacement of the loaded gear and the driven gear in the y-axis direction, and the torsional vibration displacement of the loaded gear and the driven gear in the direction of the meshing line of the loaded tooth surface.

[0010] In a typical technical solution of the present invention, the nonlinear dynamic model of the cylindrical gear transmission system is constructed using the lumped mass method without considering other external excitations besides the input torque.

[0011] As a further improvement of the present invention, the comprehensive offset is obtained by the following method: the vibration displacement of the gear shaft of the loaded gear of the cylindrical gear in the x-axis direction and the vibration displacement in the y-axis direction are obtained by matrix M. 21 This is converted into the displacement on the tooth surface of the loaded gear at the corresponding meshing moment. Then, the normal offset of the loaded tooth surface is obtained through the normal vector at the meshing point of the loaded gear tooth surface. Finally, the torsional vibration displacement along the meshing line direction of the loaded tooth surface during the meshing process of the cylindrical gear is superimposed to obtain the comprehensive offset.

[0012] As a further improvement to the present invention, the transformation matrix M 21 The transformation relationship is as follows: M 21 =M 2R ·M RW ·M WW' ·M W'1 In the formula, M 2R The transformation relationship between the moving coordinate system and the fixed coordinate system of the gear shaft of the loaded gear; M RW This represents the transformation relationship between the fixed coordinate system of the gear shaft and the fixed coordinate system of the loaded gear; M WW' The transformation relationship between the fixed coordinate system of the loaded gear and the coordinate system after rotating the origin of the fixed coordinate system of the loaded gear by an angle u around the origin; M W'1 This represents the transformation relationship between the solid coordinate system of the loaded gear, rotated around the origin by an angle u, and the motion coordinate system of the loaded gear.

[0013] As a further improvement of the present invention, the inverse calculation of the B-spline surface yields the control vertex P of the B-spline surface. k,d The process is as follows: First, the model points are parameterized using the uniform parameter method; then, the control node vectors U and V are calculated based on the parameterized parameter sequence. Finally, a bilinear equation system is established, thus obtaining the control vertex P. k,d The inverse calculation formula for the B-spline surface is as follows:

[0014]

[0015] in, Let c be the B-spline basis functions. It is a B-spline basis function of degree q.

[0016] As a further improvement of the present invention, the transformation relationship M between the moving coordinate system of the gear shaft of the loaded gear and the fixed coordinate system of the gear shaft of the loaded gear is... 2R for:

[0017]

[0018] Where, xp The x-axis represents the vibration displacement of the gear shaft of the loaded gear; the y-axis represents the vibration displacement of the gear shaft in the x-axis direction. p The vibration displacement of the gear shaft of the loaded gear in the y-axis direction.

[0019] As a further improvement of the present invention, the transformation relationship M between the fixed coordinate system of the gear shaft of the loaded gear and the fixed coordinate system of the loaded gear is... RW for:

[0020]

[0021] Where l is the distance between the fixed coordinate system of the gear shaft of the loaded gear and the fixed coordinate system of the loaded gear.

[0022] As a further improvement of the present invention, the transformation relationship M between the fixed coordinate system of the loaded gear and the coordinate system after rotating the origin of the fixed coordinate system of the loaded gear by an angle u around the origin is provided. WW' for:

[0023]

[0024] Where u is the angle of rotation of the loaded gear around the origin of the coordinate system.

[0025] As a further improvement of the present invention, the transformation relationship M between the solid coordinate system of the loaded gear, rotated by an angle u around the origin, and the motion coordinate system of the loaded gear is... W'1 for:

[0026]

[0027] Where 'a' represents the distance the moving coordinate system of the loaded gear moves relative to the fixed coordinate system of the loaded gear along the z-axis.

[0028] As a further improvement of the present invention, the vibration displacement x of the gear shaft of the loaded gear in the x-axis direction is... p The vibration displacement y of the gear shaft of the loaded gear in the y-axis direction p Through matrix M 21 The formulas for calculating the displacements Δx1, Δy1, and Δz1 on the tooth surface of the loaded gear at the corresponding meshing moment are as follows:

[0029]

[0030] Where, r b β is the base circle radius of the cylindrical gear; b The base circle helix angle of the cylindrical gear.

[0031] The technical solution provided by this invention has the following beneficial effects:

[0032] (1) The topology modification method for cylindrical gears provided by this invention optimizes the designed topology-modified tooth surface into a B-spline surface by fitting it, making the designed topology-modified tooth surface a smooth surface. This allows the gears designed using the topology modification method of this embodiment to avoid abrupt changes in the modification amount of the designed topology-modified tooth surface (such as protrusions or concave points on the topology-modified tooth surface). At the same time, this optimization design enables the designed topology-modified tooth surface to be machined, achieving the goal of consistency between design and manufacturing.

[0033] (2) The topological modification method for cylindrical gears provided by this invention utilizes an improved potential energy method to establish a stiffness model during gear meshing. Then, the time-varying stiffness is used to solve the nonlinear dynamic equations of gear meshing, obtaining the vibration displacement during meshing. A transformation matrix is ​​then established to convert the displacement on the gear shaft into displacement on the tooth surface, thereby determining the magnitude of the tooth surface topological modification amount, thus obtaining the designed topologically modified tooth surface. This design method not only effectively replaces the method of solving for gear topological modification amounts using empirical formulas in engineering applications, but also allows for a more comprehensive analysis of the internal and external excitations affecting the dynamic response of cylindrical gears, thereby improving the design accuracy of the modified gear and contributing to improved meshing characteristics of the designed cylindrical gear. This reduces meshing impact on the designed gear and improves its smoothness during transmission. Attached Figure Description

[0034] Figure 1 The present invention provides a flowchart of the steps of a method for topological modification of a cylindrical gear.

[0035] Figure 2 This is a diagram of the stiffness model of the spur gear established in the test example of this invention.

[0036] Figure 3 This is a time-varying meshing stiffness diagram of a single tooth pair in the test example of this invention.

[0037] Figure 4 This is a schematic diagram of a simplified nonlinear dynamic model of a helical gear transmission system used in the test examples of this invention.

[0038] Figure 5 This is a diagram showing the lateral vibration displacement of the gear shaft of the loaded gear in the x-axis direction in the test example of this invention.

[0039] Figure 6 This is a diagram showing the lateral vibration displacement of the driven gear shaft in the x-axis direction in the test example of this invention.

[0040] Figure 7 This is a diagram showing the vertical vibration displacement of the gear shaft of the loaded gear in the y-axis direction in the test example of this invention.

[0041] Figure 8This is a diagram showing the vertical vibration displacement of the driven gear shaft in the y-axis direction in the test example of this invention.

[0042] Figure 9 This is a diagram showing the torsional vibration displacement transformation of the gear shaft of the loaded gear in the meshing direction of the tooth surface in the test example of the present invention.

[0043] Figure 10 This is a diagram showing the torsional vibration displacement of the driven gear shaft in the meshing direction of the tooth surface in the test example of this invention.

[0044] Figure 11 This is a schematic diagram of the coordinate systems in the helical gear in the test example of the present invention.

[0045] Figure 12 This is a mesh diagram of the helical gear constructed on the topologically modified tooth surface in the test example of this invention.

[0046] Figure 13 This is a schematic diagram of the modification amount of the helical gear in the optimized topological modified tooth surface in the tooth profile and tooth direction in the test example of the present invention. Detailed Implementation

[0047] The present invention will now be further described in conjunction with specific embodiments. It should be noted that, without conflict, the various embodiments or technical features described below can be arbitrarily combined to form new embodiments.

[0048] In the description of this invention, it should be noted that directional terms such as "center," "lateral," "longitudinal," "length," "width," "thickness," "upper," "lower," "front," "rear," "left," "right," "vertical," "horizontal," "top," "bottom," "inner," "outer," "clockwise," and "counterclockwise," etc., indicate the orientation and positional relationship based on the orientation or positional relationship shown in the accompanying drawings. These are used only for the convenience of describing the invention and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation. They should not be construed as limiting the specific scope of protection of this invention. The terms "first," "second," etc., in the specification and claims of this invention are used to distinguish similar objects and are not necessarily used to describe a specific order or sequence. The terms "comprising" and "having," and any variations thereof, in the specification and claims of this invention, are intended to cover non-exclusive inclusion. For example, a process, method, system, product, or device that includes a series of steps or units is not necessarily limited to those steps or units explicitly listed, but may include other steps or units not explicitly listed or inherent to these processes, methods, products, or devices.

[0049] Example 1

[0050] This embodiment provides a method for topological modification of cylindrical gears. Please refer to [reference needed]. Figure 1 It includes the following steps:

[0051] (I) Constructing a time-varying meshing stiffness model for cylindrical gears

[0052] First, the cylindrical gear is sliced ​​into a series of spur gear slices. Then, based on beam deformation theory and using the improved potential energy method, a stiffness model of the spur gear is established. The stiffness model of the spur gear includes the bending stiffness model k. b Shear stiffness model k s Radial compressive stiffness model k a Hertzian contact stiffness model k h and gear body stiffness model k f .

[0053] Specifically: Bending stiffness model k b The expression is:

[0054]

[0055] Shear stiffness model k s The expression is:

[0056]

[0057] Radial compressive stiffness model k a The expression is:

[0058]

[0059] Hertzian contact stiffness model k h The expression is:

[0060]

[0061] Gear body stiffness model k f The expression is:

[0062]

[0063] The parameters in the above five models are explained as follows: b is the tooth width; v represents Poisson's ratio; E represents the elastic modulus; α1 is the angle between the meshing force and the Y-axis; α2 is the half angle corresponding to the base circle; α5 is the angle between the root circle and the X-axis; r b L* is the base circle radius; L*, M*, P*, and Q* are all coefficients related to gear design; μ f S represents the distance between the intersection of the line of action of the meshing force and the line of symmetry of the gear teeth and the highest point of the root circle; f It represents the arc length corresponding to a single tooth on the root circle.

[0064] By establishing a spur gear stiffness model and integrating along the contact trace of the cylindrical gear using the cumulative integration principle, the time-varying meshing stiffness model k of the cylindrical gear can be obtained. t Its expression is as follows:

[0065]

[0066] In the formula, The bending stiffness of the loaded gear; The bending stiffness of the driven gear; The shear stiffness of the loaded gear; The shear stiffness of the driven gear; The radial compressive stiffness of the loaded gear; k is the radial compressive stiffness of the driven gear. h The Hertzian contact stiffness between the loaded gear and the driven gear; The stiffness of the gear body under load; This refers to the stiffness of the driven gear.

[0067] (II) Establish a nonlinear dynamic model of the cylindrical gear transmission system and solve for the vibration displacement of the cylindrical gear in each degree of freedom.

[0068] A nonlinear dynamic model of the cylindrical gear transmission system is established using the lumped mass method, without considering external excitations other than the input torque. The specific expression of the nonlinear dynamic model is as follows:

[0069]

[0070] Among them, c m k is the average damping of the meshing gear pair; t k is the time-varying meshing stiffness of the cylindrical gear at the moment of meshing. tp k represents the time-varying meshing stiffness of the loaded gear. tg The time-varying meshing stiffness of the driven gear; The lateral acceleration of the gear shaft of the loaded gear in the x-axis direction; The velocity of the gear shaft of the loaded gear in the x-axis direction; x p The vibration displacement of the gear shaft of the loaded gear in the x-axis direction; Let x be the lateral acceleration of the driven gear shaft in the x-axis direction; x represents the velocity of the driven gear shaft in the x-axis direction; g The vibration displacement of the driven gear shaft in the x-axis direction; The vertical vibration acceleration of the gear shaft of the loaded gear in the y-axis direction; The velocity of the gear shaft of the loaded gear in the y-axis direction; y p The vertical vibration displacement of the gear shaft of the loaded gear in the y-axis direction; The vertical vibration acceleration of the driven gear shaft in the y-axis direction; The velocity of the driven gear shaft in the y-axis direction is y; g The vertical vibration displacement of the driven gear shaft in the y-axis direction; This refers to the torsional acceleration of the loaded gear along the meshing line direction. θ is the velocity of the loaded gear along the meshing line of its teeth. p This refers to the torsional vibration displacement of the loaded gear along the meshing line direction on the tooth surface; The torsional acceleration of the driven gear in the direction of the meshing line on the tooth surface; θ is the velocity of the driven gear along the line of meshing on the tooth surface. g The torsional vibration displacement of the driven gear along the meshing line direction is m. p The mass of the loaded gear; I p m is the moment of inertia of the loaded gear. g I is the mass of the driven gear; g Let r be the moment of inertia of the driven gear. bp r is the base circle radius of the loaded gear. bg k is the base circle radius of the driven gear. px k represents the support stiffness of the loaded gear in the x-axis direction. gx k represents the support stiffness of the driven gear in the x-axis direction. py k represents the support stiffness of the loaded gear in the y-axis direction. gy c represents the support stiffness of the driven gear in the y-axis direction. px c is the support damping of the loaded gear in the x-axis direction; py c is the support damping of the loaded gear in the y-axis direction; gx c is the support damping of the driven gear in the x-axis direction; gy Support damping of the driven gear in the y-axis direction; T p T represents the input torque of the loaded gear. g q is the output torque of the driven gear; f(q) is the backlash function; α is the pressure angle; β is the helix angle.

[0071] Based on the time-varying meshing stiffness model k of cylindrical gears t The time-varying meshing stiffness value k of the cylindrical gear obtained by solving the equation. tSubstituting these values ​​into the aforementioned nonlinear dynamic model, we obtain the lateral vibration displacement of the gear shaft in the x-axis direction, the vertical vibration displacement of the gear shaft in the y-axis direction, the torsional vibration displacement of the gear shaft in the tooth meshing direction, the lateral vibration displacement of the driven gear shaft in the x-axis direction, the vertical vibration displacement of the driven gear shaft in the y-axis direction, and the torsional vibration displacement of the driven gear shaft in the tooth meshing direction.

[0072] (III) Design of topology-modified tooth surfaces

[0073] Establish the transformation matrix M between the coordinate system of the loaded gear and the coordinate system of the gear shaft. 21 Define the fixed coordinate system of the tooth surface of the loaded gear as S w (O w -x w y w , z w The fixed coordinate system of the gear shaft is S. R (O R -x R y R , z R The distance between the two fixed coordinate systems is l. The motion coordinate system of the loaded gear is defined as S1(O1-x1, y1, z1), and the motion coordinate system of the gear shaft (whether this is the shaft of the loaded gear or the driven gear) is defined as S2(O2-x2, y2, z2). The transformation matrix M is constructed... 21 The vibration displacement on the gear shaft (i.e., the vibration displacement of the loaded gear shaft in the x-axis direction and the vibration displacement of the loaded gear shaft in the y-axis direction) is converted into the displacement of the tooth surface point at the meshing position. Then, the normal offset of the loaded tooth surface is obtained through the normal vector at the meshing point of the loaded tooth surface. Finally, the torsional vibration displacement in the meshing line direction of the loaded tooth surface during the meshing process of the cylindrical gear is superimposed to obtain the comprehensive offset. Superimposing this comprehensive offset onto the standard tooth surface yields the designed topologically modified tooth surface.

[0074] Matrix M 21 The expression is as follows: M 21 =M 2R ·M RW ·M WW' ·M W'1 In the formula, M 2R The transformation relationship between the moving coordinate system and the fixed coordinate system of the gear shaft of the loaded gear; M RW The transformation relationship between the fixed coordinate system of the gear shaft and the fixed coordinate system of the loaded gear; M WW'The transformation relationship between the fixed coordinate system of the loaded gear and the coordinate system after rotating the origin of the fixed coordinate system of the loaded gear by an angle u around the origin; M W'1 This represents the transformation relationship between the solid coordinate system of the loaded gear, rotated around the origin by an angle u, and the motion coordinate system of the loaded gear.

[0075] In this embodiment, a spatial transformation matrix can be used to transform between different spatial coordinate systems according to the coordinate transformation method, thereby obtaining the transformation relationship between different coordinate systems. The transformation relationship between the coordinate system of the loaded gear and the coordinate system of the gear shaft in this embodiment is described below:

[0076] According to the principle of involute spiral surface formation, when the loaded gear rotates by an angle μ about the z-axis (i.e., the central axis of the loaded tooth surface), the distance that the moving coordinate system of the loaded gear moves relative to its fixed coordinate system in the z-axis direction is a, where a = r. b μ / tanβ b ;where r b Let β be the radius of the base circle. b The base circle helix angle.

[0077] The transformation relationship between the moving coordinate system and the fixed coordinate system of the gear shaft of the loaded gear can be represented by matrix M. 2R express,

[0078]

[0079] Where, x p The x-axis represents the vibration displacement of the gear shaft of the loaded gear; the y-axis represents the vibration displacement of the gear shaft in the x-axis direction. p The vibration displacement of the gear shaft of the loaded gear in the y-axis direction.

[0080] The transformation relationship between the fixed coordinate system of the gear shaft and the fixed coordinate system of the loaded gear can be represented by matrix M. RW express:

[0081]

[0082] Where l is the distance between the fixed coordinate system of the gear shaft of the loaded gear and the fixed coordinate system of the loaded gear.

[0083] The transformation relationship M between the fixed coordinate system of the loaded gear and the coordinate system after rotating the origin of the fixed coordinate system of the loaded gear by an angle u around the origin. WW' for:

[0084]

[0085] Where u is the angle of rotation of the loaded gear around the origin of the coordinate system (i.e., the central axis of the loaded tooth surface).

[0086] The transformation relationship M between the solid coordinate system of the loaded gear, rotated by an angle u around the origin, and the motion coordinate system of the loaded gear. W'1 for:

[0087]

[0088] Where 'a' represents the distance the moving coordinate system of the loaded gear moves relative to the fixed coordinate system of the loaded gear along the z-axis.

[0089] The matrix M is obtained through the above transformation. 21 The vibration displacement x of the gear shaft of the loaded gear in the x-axis direction p The vibration displacement y of the gear shaft of the loaded gear in the y-axis direction p Through matrix M 21 This is converted into displacements Δx1, Δy1, and Δz1 on the tooth surface of the loaded gear at the corresponding meshing moment. Then, the normal offset of the loaded tooth surface is obtained through the normal vector at the meshing point of the tooth surface. Finally, the torsional vibration displacement in the direction of the meshing line of the loaded tooth surface during the meshing process of the cylindrical gear is superimposed to obtain the comprehensive offset.

[0090] Specifically, the vibration displacement x of the gear shaft of the loaded gear in the x-axis direction. p The vibration displacement y of the gear shaft of the loaded gear in the y-axis direction p Through matrix M 21 The formulas for calculating the displacements Δx1, Δy1, and Δz1 on the tooth surface of the loaded gear at the corresponding meshing moment are as follows:

[0091]

[0092] Where, r b β is the radius of the base circle; b The helix angle of the base circle.

[0093] (iv) Optimize the topology-modified tooth surface

[0094] The designed topologically modified tooth surface is fitted into a B-spline surface. The core of this fitting is to construct a c×q degree B-spline surface using a given set of shape points to precisely approximate these points. This operation requires inverse calculation of node vectors and control vertices. The following describes how to fit the designed topologically modified tooth surface into a B-spline surface: The entire tooth surface is meshed along the tooth direction and tooth profile direction. n×m mesh points are selected and used as the shape points Q of the B-spline surface. i,j(i = 0, ..., n-1; j = 0, ..., m-1). The four data points at the four corners of the model value lattice become the four corner points of the entire surface, and the other data points become the common corner points of the corresponding adjacent surface patches. This ensures that the points in each row of the model value lattice lie on an isoparametric line of the surface. The model value points are parameterized using the uniform parameter method, and the formula is as follows:

[0095]

[0096] Then, the control node vector U = [μ0, μ1, ..., μ] is calculated inversely based on the parameterized parameter sequence. n+c+1 ] and V = [v0, v1, ..., v m+ q +1 ].

[0097] in:

[0098] The internal nodes are calculated based on an average distribution:

[0099] Finally, a system of bilinear equations was established to obtain the control vertex P. k,d This completes the inverse calculation process of B-spline surfaces. The formula for the inverse calculation of B-spline surfaces is as follows:

[0100]

[0101] in, Let c be the B-spline basis functions. It is a B-spline basis function of degree q.

[0102] The topology modification method for cylindrical gears provided in this embodiment utilizes an improved potential energy method to establish a stiffness model during gear meshing. Then, it solves the nonlinear dynamic equations of gear meshing using time-varying stiffness to obtain the vibration displacement during meshing. By establishing a transformation matrix, the displacement on the gear shaft is converted into displacement on the tooth surface, thereby determining the magnitude of the tooth surface topology modification amount, thus obtaining the designed topology-modified tooth surface. This design method not only effectively replaces the empirical formula method for solving gear topology modification amounts in engineering applications, but also allows for a more comprehensive analysis of the internal and external excitations affecting the dynamic response of cylindrical gears, thereby improving the design accuracy of the modified gear and contributing to improved meshing characteristics. This reduces meshing impact and improves the smoothness of the transmission process. Furthermore, due to the complexity of the topology-modified tooth surfaces designed in existing technologies, protrusions, concave points, or abrupt changes in tooth surface modification amounts are prone to appear on the designed topology-modified tooth surfaces. Based on the principle of continuous generating gear machining, it is impossible to machine gears with abrupt changes in tooth surface modification amounts. For example, if the amount of modification at the gear tooth surface machining position is normal at one moment, but the amount of modification changes abruptly at the next moment, and gear machining is continuous and cannot be abrupt, existing technologies cannot manufacture such complex gears. In this embodiment, the designed topologically modified tooth surface is fitted and optimized into a B-spline surface, making the designed topologically modified tooth surface a smooth surface. This allows the gear designed using the topological modification method of this embodiment to avoid abrupt changes in the modification amount of the designed topologically modified tooth surface (such as protrusions or concave points on the topologically modified tooth surface). Simultaneously, this optimized design allows the designed topologically modified tooth surface to be machined, achieving consistency between design and manufacturing.

[0103] Test case

[0104] Cylindrical gears are divided into spur gears and helical gears. The difference between them lies in the different changes in the contact trace during meshing. Ideally, the contact trace of a spur gear is a straight line parallel to the gear axis, and its theoretical length is equal to the tooth width. For helical gears, however, the contact trace changes continuously along the tooth width during meshing, generally undergoing a process of lengthening, stabilizing, and then shortening; the specific length change is also related to parameters such as the helix angle and the contact ratio. The length of a helical gear is generally less than the tooth width. Since the calculation method for spur gears is not as complex as that for helical gears, this test example uses a helical gear, which has a more complex calculation, to verify this topology modification method. Therefore, this test example provides a topology modification method for a helical gear with specific dimensions based on the topology modification method provided in Example 1. The parameters of a pair of helical gears are as follows: pinion teeth z1 = 35, gear teeth z2 = 72, normal module m. n=4.0, helix angle β b =30°, normal pressure angle α = 20°, tooth width b = 30mm. The topological modification method for helical gears includes the following steps:

[0105] First, the helical gear is sliced ​​into a series of spur gear slices. Then, based on beam deformation theory and using the improved potential energy method, a stiffness model of the spur gear is established. Please refer to [reference needed]. Figure 2 The stiffness model of spur gears includes the bending stiffness model k. b Shear stiffness model k s Radial compressive stiffness model k a Hertzian contact stiffness model k h and gear body stiffness model k f .

[0106] Specifically: Bending stiffness model k b The expression is:

[0107]

[0108] Shear stiffness model k s The expression is:

[0109]

[0110] Radial compressive stiffness model k a The expression is:

[0111]

[0112] Hertzian contact stiffness model k h The expression is:

[0113]

[0114] Gear body stiffness model k f The expression is:

[0115]

[0116] The parameters in the above five models are explained as follows: b is the tooth width; v represents Poisson's ratio; E represents the elastic modulus; α1 is the angle between the meshing force and the Y-axis; α2 is the half angle corresponding to the base circle; α5 is the angle between the root circle and the X-axis; r b L* is the base circle radius; L*, M*, P*, and Q* are all coefficients related to gear design; μ f S represents the distance between the intersection of the line of action of the meshing force and the line of symmetry of the gear teeth and the highest point of the root circle; f It represents the arc length corresponding to a single tooth on the root circle.

[0117] By establishing the spur gear stiffness model and integrating along the contact trace of the helical gear using the cumulative integration principle, the time-varying meshing stiffness model k of the helical gear can be obtained. t Its expression is as follows:

[0118]

[0119] In the formula, The bending stiffness of the loaded gear; The bending stiffness of the driven gear; The shear stiffness of the loaded gear; The shear stiffness of the driven gear; The radial compressive stiffness of the loaded gear; k is the radial compressive stiffness of the driven gear. h The Hertzian contact stiffness between the loaded gear and the driven gear; The stiffness of the gear body under load; This refers to the stiffness of the driven gear.

[0120] in, Figure 3 The diagram shows the time-varying meshing stiffness of a single tooth pair in the test example. Figure 3 The dashed line represents the meshing stiffness curve of a single tooth pair throughout the entire meshing cycle, while the solid line represents the superimposed composite meshing stiffness curve. It can be seen that the meshing stiffness curve of a single tooth pair first increases slowly, reaches a peak value, remains constant at that value, and then slowly decreases again. This trend is consistent with the changing pattern of the contact trace during the meshing of a helical gear pair, thus proving that the trend of the stiffness curve obtained in this embodiment is correct. Figure 3 The value of the solid line curve corresponds to k in the second step of the dynamic equation. t It changes over time.

[0121] To establish a nonlinear dynamic model of the helical gear transmission system using the lumped mass method and without considering external excitations other than the input torque, please refer to [reference needed]. Figure 4 , Figure 4 A simplified schematic diagram of the nonlinear dynamic model of a helical gear transmission system. The specific expression of the nonlinear dynamic model is as follows:

[0122]

[0123] Among them, c m k is the average damping of the meshing gear pair; t k is the time-varying meshing stiffness at the moment of engagement of the helical gears. tp k represents the time-varying meshing stiffness of the loaded gear. tg The time-varying meshing stiffness of the driven gear; The lateral acceleration of the gear shaft of the loaded gear in the x-axis direction; The velocity of the gear shaft of the loaded gear in the x-axis direction; x p The vibration displacement of the gear shaft of the loaded gear in the x-axis direction; Let x be the lateral acceleration of the driven gear shaft in the x-axis direction; x represents the velocity of the driven gear shaft in the x-axis direction; g The vibration displacement of the driven gear shaft in the x-axis direction; The vertical vibration acceleration of the gear shaft of the loaded gear in the y-axis direction; The velocity of the gear shaft of the loaded gear in the y-axis direction; y p The vertical vibration displacement of the gear shaft of the loaded gear in the y-axis direction; The vertical vibration acceleration of the driven gear shaft in the y-axis direction; The velocity of the driven gear shaft in the y-axis direction is y; g The vertical vibration displacement of the driven gear shaft in the y-axis direction; This refers to the torsional acceleration of the loaded gear along the meshing line direction. θ is the velocity of the loaded gear along the meshing line of its teeth. p This refers to the torsional vibration displacement of the loaded gear along the meshing line direction on the tooth surface; The torsional acceleration of the driven gear in the direction of the meshing line on the tooth surface; θ is the velocity of the driven gear along the line of meshing on the tooth surface. g The torsional vibration displacement of the driven gear along the meshing line direction is m. p The mass of the loaded gear; I p m is the moment of inertia of the loaded gear. g I is the mass of the driven gear; g Let r be the moment of inertia of the driven gear. bp r is the base circle radius of the loaded gear. bg k is the base circle radius of the driven gear. px k represents the support stiffness of the loaded gear in the x-axis direction. gx k represents the support stiffness of the driven gear in the x-axis direction. py k represents the support stiffness of the loaded gear in the y-axis direction. gy c represents the support stiffness of the driven gear in the y-axis direction. px c is the support damping of the loaded gear in the x-axis direction; py c is the support damping of the loaded gear in the y-axis direction; gx c is the support damping of the driven gear in the x-axis direction; gy Support damping of the driven gear in the y-axis direction; Tp T represents the input torque of the loaded gear. g q is the output torque of the driven gear; f(q) is the backlash function; α is the pressure angle; β is the helix angle.

[0124] Based on the helical gear time-varying meshing stiffness model k t The time-varying meshing stiffness value k of the helical gear obtained by solving the equation. t Substituting these values ​​into the aforementioned nonlinear dynamic model, we obtain the lateral vibration displacement of the loaded gear shaft in the x-axis direction, the vertical vibration displacement of the loaded gear shaft in the y-axis direction, the torsional vibration displacement of the loaded gear shaft in the tooth meshing direction, the lateral vibration displacement of the driven gear shaft in the x-axis direction, the vertical vibration displacement of the driven gear shaft in the y-axis direction, and the torsional vibration displacement of the driven gear shaft in the tooth meshing direction. Please refer to... Figures 5 to 10 ,in Figure 5 It represents the lateral vibration displacement of the gear shaft of the loaded gear in the x-axis direction; Figure 6 The lateral vibration displacement of the driven gear shaft in the x-axis direction; Figure 7 It represents the vertical vibration displacement of the gear shaft of the loaded gear in the y-axis direction; Figure 8 It represents the vertical vibration displacement of the driven gear shaft in the y-axis direction; Figure 9 It represents the torsional vibration displacement of the gear shaft of the loaded gear in the meshing direction of the tooth surface; Figure 10 This represents the torsional displacement of the driven gear shaft in the meshing direction of the gear teeth. Figures 5 to 10 The following analysis is conducted: Figures 5 to 10 The results are all obtained through solving a nonlinear dynamic model. Because the time-varying stiffness of the nonlinear dynamic model changes periodically, the obtained vibration displacement also changes periodically with time. Figures 5 to 10 The horizontal axis in the diagram corresponds to the time of one meshing cycle, during which the vibration displacement changes with time. This series of vibration displacements, obtained through a nonlinear dynamic model, is then superimposed to obtain different vibration values ​​at the same time point. These values ​​are then converted to the meshing point on the tooth surface using coordinate transformation, and the vibration displacement at the meshing point is used as the basis for calculating the modification amount at that point. The meshing point modification amount obtained through this calculation designs the topological tooth surface from the perspective of gear meshing transmission. Compared to the original method of determining the modification amount using empirical formulas, this improves design accuracy and better enhances transmission performance.

[0125] Please refer to Figure 11 Establish a fixed coordinate system S for the tooth surface of the loaded gear. w (O w -x w yw , z w And the fixed coordinate system S of the gear shaft R (O R -x R y R , z R The distance between the two fixed coordinate systems is l. Establish the motion coordinate system of the loaded gear as S1(O1-x1, y1, z1), and the motion coordinate system of the gear shaft (whether this is the shaft of the loaded gear or the driven gear) as S2(O2-x2, y2, z2). Through the constructed transformation matrix M... 21 The vibration displacement on the gear shaft (i.e., the vibration displacement of the loaded gear shaft in the x-axis direction and the vibration displacement of the loaded gear shaft in the y-axis direction) is converted into the displacement of the tooth surface point at the meshing position. Then, the normal offset of the loaded tooth surface is obtained through the normal vector at the meshing point of the loaded tooth surface. Finally, the torsional vibration displacement in the meshing line direction of the loaded tooth surface during helical gear meshing is superimposed to obtain the comprehensive offset. Superimposing this comprehensive offset onto the standard tooth surface yields the designed topologically modified tooth surface.

[0126] Matrix M 21 The expression is as follows: M 21 =M 2R ·M RW ·M WW' ·M W'1 In the formula, M 2R The transformation relationship between the moving coordinate system and the fixed coordinate system of the gear shaft of the loaded gear; M RW The transformation relationship between the fixed coordinate system of the gear shaft and the fixed coordinate system of the loaded gear; M WW' The transformation relationship between the fixed coordinate system of the loaded gear and the coordinate system after rotating the origin of the fixed coordinate system of the loaded gear by an angle u around the origin; M W'1 This represents the transformation relationship between the solid coordinate system of the loaded gear, rotated around the origin by an angle u, and the motion coordinate system of the loaded gear.

[0127] The transformation relationship between the moving coordinate system and the fixed coordinate system of the gear shaft of the loaded gear can be represented by matrix M. 2R express,

[0128]

[0129] Where, x p The x-axis represents the vibration displacement of the gear shaft of the loaded gear; the y-axis represents the vibration displacement of the gear shaft in the x-axis direction. p The vibration displacement of the gear shaft of the loaded gear in the y-axis direction.

[0130] The transformation relationship between the fixed coordinate system of the gear shaft and the fixed coordinate system of the loaded gear can be represented by matrix M. RW express:

[0131]

[0132] Where l is the distance between the solid coordinate system of the gear shaft of the loaded gear and the fixed coordinate system of the loaded gear.

[0133] The transformation relationship M between the fixed coordinate system of the loaded gear and the coordinate system after rotating the origin of the fixed coordinate system of the loaded gear by an angle u around the origin. WW' for:

[0134]

[0135] Where u is the angle of rotation of the loaded gear around the origin of the coordinate system.

[0136] The transformation relationship M between the solid coordinate system of the loaded gear, rotated by an angle u around the origin, and the motion coordinate system of the loaded gear. W'1 for:

[0137]

[0138] Where a is the distance the moving coordinate system of the loaded gear moves relative to the fixed coordinate system of the loaded gear in the z-axis direction.

[0139] The matrix M is obtained through the above transformation. 21 The vibration displacement x of the gear shaft of the loaded gear in the x-axis direction p The vibration displacement y of the gear shaft of the loaded gear in the y-axis direction p Through matrix M 21 This is converted into displacements Δx1, Δy1, and Δz1 on the tooth surface of the loaded gear at the corresponding meshing moment. Then, the normal offset of the loaded tooth surface is obtained through the normal vector at the meshing point of the loaded gear tooth surface. Finally, the torsional vibration displacement in the direction of the meshing line of the loaded tooth surface during the meshing process of the helical gear is superimposed to obtain the comprehensive offset.

[0140] Specifically, the vibration displacement x of the gear shaft of the loaded gear in the x-axis direction. p The vibration displacement y of the gear shaft of the loaded gear in the y-axis direction p Through matrix M 21 The formulas for calculating the displacements Δx1, Δy1, and Δz1 on the tooth surface of the loaded gear at the corresponding meshing moment are as follows:

[0141]

[0142] Where, r bβ is the radius of the base circle; b The helix angle of the base circle.

[0143] Considering that the tooth profile of a helical gear is a spiral surface, in this test case, only the parameter c = q = 3 is set to construct a 3×3 B-spline surface to accurately interpolate these points. Please refer to... Figure 12 The entire tooth surface of the designed topologically modified tooth surface is meshed along the tooth direction and tooth profile direction. n×m mesh points are selected on the tooth surface, and these n×m mesh points are used as the shape value points Q of the B-spline surface. i,j (i = 0, ..., n-1; j = 0, ..., m-1). The four data points at the four corners of the model value lattice become the four corner points of the entire surface, and the other data points become the common corner points of the corresponding adjacent surface patches. This ensures that the points in each row of the model value lattice lie on an isoparametric line of the surface. The model value points are parameterized using the uniform parameter method, and the formula is as follows:

[0144]

[0145] Then, the control node vector U = [μ0, μ1, ..., μ] is calculated inversely based on the parameterized parameter sequence. n+c+1 ] and V = [v0, v1, ..., v m+ q +1 ].

[0146] in:

[0147] The internal nodes are calculated based on an average distribution:

[0148] Finally, a system of bilinear equations was established to obtain the control vertex P. k,d This completes the inverse calculation process of B-spline surfaces. The formula for the inverse calculation of B-spline surfaces is as follows:

[0149]

[0150] in, Let c be the B-spline basis functions. It is a B-spline basis function of degree q.

[0151] The B-spline surface can be obtained by performing the above series of operations. Please refer to [reference needed]. Figure 13 As shown. Figure 13 The B-spline surface obtained by fitting can be called a uniform cubic B-spline surface. Through... Figure 13 Analysis shows that there are no abrupt changes in the profile modification amount on the fitted tooth surface. The overall pattern is that the profile modification amount is lower in the middle and higher on both sides. That is, the profile modification amount is larger during the engagement and disengagement times and smaller during the stable transmission time in the middle. However, the transition is relatively smooth and can be achieved.

[0152] The basic principles, main features, and advantages of this invention have been described above. Those skilled in the art should understand that this invention is not limited to the above embodiments. The embodiments and descriptions in the specification are merely principles of the invention. Various changes and modifications can be made without departing from the spirit and scope of the invention, and all such changes and modifications fall within the scope of the invention as claimed. The scope of protection claimed by this invention is defined by the appended claims and their equivalents.

Claims

1. A method for topological modification of cylindrical gears, characterized in that, It includes: Construct a time-varying meshing stiffness model for cylindrical gears; A nonlinear dynamic model of a cylindrical gear transmission system is established, and the vibration displacement of the cylindrical gear in each degree of freedom is obtained by using the time-varying meshing stiffness model and the nonlinear dynamic model of the cylindrical gear. Establish the transformation matrix M between the coordinate system of the loaded gear and the coordinate system of the gear shaft. 21 Through matrix M 21 The vibration displacement of the cylindrical gear in each degree of freedom is used to obtain the comprehensive offset between the actual tooth surface position and the theoretical position of the cylindrical gear during meshing; the comprehensive offset is superimposed on the standard tooth surface to obtain the topologically modified tooth surface; The entire tooth surface of the obtained topologically modified tooth surface is meshed along the tooth direction and tooth profile direction. n×m mesh points are selected on the tooth surface, and these n×m mesh points are used as the shape value points Q of the B-spline surface. i,j The data points at the four corners of the defined value lattice become the four corner points of the entire surface, and the other data points become the common corner points of the corresponding adjacent surface patches. Then, the control vertices P of the B-spline surface are obtained through inverse calculation of the B-spline surface. k，d This yields the B-spline surface, which is the optimized topologically modified tooth surface of the cylindrical gear.

2. The method for topological modification of cylindrical gears as described in claim 1, characterized in that, The construction method of the time-varying meshing stiffness model of cylindrical gear is as follows: After slicing the cylindrical gear into a series of spur gear slices, the stiffness model of the spur gear is established based on the beam deformation theory and the improved potential energy method; then, based on the principle of cumulative integration, the slices are integrated along the contact trace of the cylindrical gear to obtain the time-varying meshing stiffness model of the cylindrical gear.

3. The method for topological modification of cylindrical gears as described in claim 2, characterized in that, The stiffness model of the spur gear includes a bending stiffness model k. b Shear stiffness model k s Radial compressive stiffness model k a Hertzian contact stiffness model k h and gear body stiffness model k f The time-varying meshing stiffness model k of the cylindrical gear t The formula is as follows: In the formula, The bending stiffness of the loaded gear; The bending stiffness of the driven gear; The shear stiffness of the loaded gear; The shear stiffness of the driven gear; The radial compressive stiffness of the loaded gear; k is the radial compressive stiffness of the driven gear. h The Hertzian contact stiffness between the loaded gear and the driven gear; The stiffness of the gear body under load; This refers to the stiffness of the driven gear.

4. The method for topological modification of cylindrical gears as described in claim 1, characterized in that, The vibration displacements of cylindrical gears in each degree of freedom include: the lateral vibration displacement of the loaded gear and driven gear in the x-axis direction, the vertical vibration displacement of the loaded gear and driven gear in the y-axis direction, and the torsional vibration displacement of the loaded gear and driven gear in the direction of the meshing line of the loaded tooth surface.

5. The method for topological modification of cylindrical gears as described in claim 1, characterized in that: The nonlinear dynamic model of the cylindrical gear transmission system is constructed using the lumped mass method without considering external excitations other than the input torque.

6. The method for topological modification of cylindrical gears as described in claim 1, characterized in that, The overall offset is obtained by the following method: the vibration displacement of the cylindrical gear shaft in the x-axis direction and the vibration displacement in the y-axis direction are obtained through matrix M. 21 The displacement is converted into the displacement on the tooth surface of the loaded gear at the corresponding meshing moment; then the normal offset of the loaded tooth surface is obtained through the normal vector at the meshing point of the loaded gear tooth surface; finally, the torsional vibration displacement in the direction of the meshing line of the loaded tooth surface during the meshing process of the cylindrical gear is superimposed to obtain the comprehensive offset.

7. The method for topological modification of cylindrical gears as described in claim 1, characterized in that, Transformation matrix M 21 The transformation relationship is as follows: M 21 =M 2R ·M RW ·M WW' ·M W'1 , In the formula, M 2R The transformation relationship between the moving coordinate system and the fixed coordinate system of the gear shaft of the loaded gear; M RW This represents the transformation relationship between the fixed coordinate system of the gear shaft and the fixed coordinate system of the loaded gear; M WW' The transformation relationship between the fixed coordinate system of the loaded gear and the coordinate system after rotating the origin of the fixed coordinate system of the loaded gear by an angle u around the origin; M W'1 This represents the transformation relationship between the solid coordinate system of the loaded gear, rotated around the origin by an angle u, and the motion coordinate system of the loaded gear.

8. The method for topological modification of cylindrical gears as described in claim 1, characterized in that, The inverse calculation of the B-spline surface yields the control vertex P of the B-spline surface. k,d The process is as follows: First, the model points are parameterized using the uniform parameter method; then, the control node vectors U and V are calculated based on the parameterized parameter sequence; finally, a bilinear equation system is established, thus obtaining the control vertex P. k,d The inverse calculation formula for the B-spline surface is as follows: in, Let c be the B-spline basis functions. It is a B-spline basis function of degree q.

9. The method for topological modification of cylindrical gears as described in claim 1, characterized in that, The transformation relationship M between the moving coordinate system and the fixed coordinate system of the gear shaft of the loaded gear. 2R for: Where, x p The x-axis represents the vibration displacement of the gear shaft of the loaded gear; the y-axis represents the vibration displacement of the gear shaft in the x-axis direction. p The vibration displacement of the gear shaft of the loaded gear in the y-axis direction; And / or, the transformation relationship M between the fixed coordinate system of the gear shaft of the loaded gear and the fixed coordinate system of the loaded gear. RW for: Where l is the distance between the fixed coordinate system of the gear shaft of the loaded gear and the fixed coordinate system of the loaded gear; And / or, the transformation relationship M between the fixed coordinate system of the loaded gear and the coordinate system after rotating the origin of the fixed coordinate system of the loaded gear by an angle u around the origin. WW' for: Where u is the angle of rotation of the loaded gear around the origin of the coordinate system; And / or, the transformation relationship M between the solid coordinate system of the loaded gear, rotated by an angle u around the origin, and the motion coordinate system of the loaded gear. W'1 for: Where a is the distance the moving coordinate system of the loaded gear moves relative to the fixed coordinate system of the loaded gear in the z-axis direction.

10. The method for topological modification of cylindrical gears as described in claim 9, characterized in that, The vibration displacement x of the gear shaft of the loaded gear in the x-axis direction. p The vibration displacement y of the gear shaft of the loaded gear in the y-axis direction p Through matrix M 21 The formulas for calculating the displacements Δx1, Δy1, and Δz1 on the loaded tooth surface at the corresponding meshing moment are as follows: Where, r b β is the base circle radius of the cylindrical gear; b The base circle helix angle of the cylindrical gear.

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