A design method of a pure rolling variable-ratio rack-and-pinion pair

Abstract

The present application relates to a kind of pure rolling variable transmission ratio gear rack pair design method, in involute helical gear equidistant each radial section plane, establish involute tooth profile equation and its conjugate variable transmission ratio rack tooth profile equation;Calculate involute tooth profile and conjugate variable transmission ratio rack tooth profile pure rolling point;Based on the conjugate variable transmission ratio rack tooth profile and in its tooth profile pure rolling point, establish conjugate variable transmission ratio rack tooth profile equal gradient offset distance formula, establish pure rolling variable transmission ratio rack tooth profile equation;Equidistant set gear angle, according to variable transmission ratio curve calculation each angle under rack displacement, calculate out pure rolling variable transmission ratio rack tooth profile point cloud, fit tooth profile point cloud, establish pure rolling variable transmission ratio rack entity model;The present application avoids the drawbacks of traditional line contact variable transmission ratio gear pair design theory and method, realizes the pure rolling contact transmission between tooth surface, solves the problem of variable transmission ratio tooth surface wear.

Description

Technical Field

[0001] This invention relates to the field of gear transmission technology, specifically to a design method for a pure rolling variable transmission ratio gear rack pair. Background Technology

[0002] As a key component of the automotive steering system, the steering gear is a reduction transmission device composed of gear-based design parts. Its main function is to appropriately transform the steering angle and steering torque from the steering input end, causing the wheels to turn and achieving vehicle steering. The transmission ratio of the steering gear determines the transmission ratio of the steering system, which is an important factor affecting the ease of steering, handling stability, and sensitivity of the vehicle. The emergence of variable ratio gears (racks) meets the need for variable transmission ratios in mechanical steering gears. Unlike traditional mechanical fixed ratio steering gears, mechanical variable ratio steering gears use variable ratio gear and rack pairs. The gear (rack) tooth profile is non-involute, without changing its original structure and control method, thus achieving variable transmission ratio steering.

[0003] Currently, all gear-rack pairs in steering systems that achieve variable transmission ratios are of line contact type, lacking gear-rack pairs designed for pure rolling contact. Especially in automotive mechanical variable-ratio steering systems, under low-speed, heavy-load conditions, relative sliding between the tooth surfaces leads to tooth wear. This tooth wear prevents the steering gear-rack pair from steering according to the theoretical variable transmission ratio curve, severely impacting its service performance. Summary of the Invention

[0004] This invention addresses the shortcomings of existing technologies by providing a design method for pure rolling variable transmission ratio gear rack pairs. It overcomes the drawbacks of existing fixed transmission ratio pure rolling tooth profile design methods, which suffer from difficulties in surface forming, complex solid modeling, and poor accuracy. The invention proposes a method for constructing pure rolling variable transmission ratio tooth profiles based on involute conjugate tooth profiles. It also proposes the concept of using involute herringbone gears in pure rolling variable transmission ratio gear pairs, thus solving the problems of large axial forces and poor load-bearing capacity commonly found in pure rolling gear pairs.

[0005] This invention is achieved through the following technical solution, providing a design method for a pure rolling variable transmission ratio gear and rack pair, comprising the following steps:

[0006] S1: In a certain radial cross-section of the involute helical gear, establish the involute helical gear tooth profile equation; based on the kinematic relationship between the involute helical gear and the variable transmission ratio rack, establish the conjugate variable transmission ratio rack tooth profile equation in the cross-section, and establish the calculation equation for the pure rolling point on the involute tooth profile and the conjugate variable transmission ratio rack tooth profile.

[0007] Based on the tooth profile of the conjugate variable transmission ratio rack and the pure rolling point on its tooth profile, the formula for the equal gradient offset distance in the cross-sectional plane and the equation for the tooth profile of the variable transmission ratio rack after horizontal offset are established.

[0008] S2: Set the gear rotation angle step size at equal intervals within the gear rotation angle range, calculate the rack displacement according to the variable transmission ratio curve, calculate the conjugate variable transmission ratio rack tooth profile and the pure rolling points on the involute tooth profile and the conjugate variable transmission ratio rack tooth profile, and calculate the pure rolling variable transmission ratio rack tooth profile; generate several equally spaced radial cross-sections in the involute helical gear to generate the pure rolling variable transmission ratio rack tooth profile point cloud; fit the tooth profile point cloud to establish the pure rolling variable transmission ratio rack solid model.

[0009] As an optimization, in step S1, the involute helical gear tooth profile equation established in the cross-sectional plane is shown in formulas (1) and (2):

[0010]

[0011]

[0012] Wherein, formulas (1) and (2) are the left and right involute tooth profile equations of the involute helical gear in the said cross-section plane, respectively. These are the coordinates of the left and right involute tooth profile points within the stated cutting plane; μ = θ k +α k Here are the involute parameters, representing the involute pressure angle α. k With the angle of expansion θ k The sum of; σ L σ R θ represents the initial angles of the left and right involute curves, respectively; θ is the rotation angle of any radial section of the tooth profile of the involute helical gear relative to the middle radial section of the tooth profile along the gear axis; β is the helix angle of the involute helical gear, B is the tooth width, z is the number of teeth in the full circle, and r... a r and r b These are the tip circle radius, pitch circle radius, and base circle radius, respectively; α is the involute pressure angle; n represents the tooth number, i.e., n = 1, 2, 3 represent the left tooth, middle tooth, and right tooth of the involute helical gear.

[0013] As an optimization, in step S1, the established conjugate variable transmission ratio rack tooth profile equation in the cross-sectional plane is shown in formulas (3) and (4):

[0014]

[0015]

[0016] Among them, formulas (3) and (4) are respectively the variable transmission ratio rack tooth profile equations conjugate with the left and right involute tooth profiles in the said cross-section plane. These are the coordinates of the variable transmission ratio rack tooth profile points within the cross-sectional plane; is the gear rotation angle; s is the rack displacement of the variable transmission ratio, calculated by integrating the variable transmission ratio curve expression with the gear rotation angle as the integral variable. These are the unit normal vectors of the variable transmission ratio rack tooth profile points that are conjugate with the left and right involute tooth profiles, respectively; These are the relative velocities of the variable transmission ratio rack tooth profile points conjugate with the left and right involute tooth profiles, respectively. These represent the meshing equations for the variable transmission ratio rack tooth profiles that are conjugate with the left and right involute tooth profiles, respectively.

[0017] As an optimization, in step S1, the calculation equations for the pure rolling point on the involute tooth profile and the conjugate variable transmission ratio rack tooth profile are shown in formulas (5), (6), (7), and (8):

[0018]

[0019]

[0020]

[0021]

[0022] Among them, formulas (5) and (6) are the calculation equations for the pure rolling point on the left and right involute tooth profiles in the cutting plane, respectively; formulas (7) and (8) are the calculation equations for the pure rolling point on the variable transmission ratio rack tooth profile conjugate with the left and right involute tooth profiles in the cutting plane, respectively. These are the coordinates of the pure rolling points on the left and right involute tooth profiles within the cross-sectional plane, respectively. These are the coordinates of the pure rolling points on the variable transmission ratio rack tooth profiles that are conjugate to the left and right involute tooth profiles within the cross-sectional plane; The parameters for the pure rolling point tooth profiles on the left and right involute tooth profiles; s′ L ,s′ R For gear rotation angle The corresponding rack displacement; For a variable transmission ratio gear pair at a rotation angle of The transmission ratio at that time.

[0023] As an optimization, in step S1, the formulas for the equal gradient offset distance within the cut-off plane are established as shown in formulas (9) and (10):

[0024]

[0025]

[0026] Formulas (9) and (10) are formulas for the constant gradient offset distance of the tooth profile of the variable transmission ratio rack, which is conjugate with the left and right involute tooth profiles; Δt LΔt R λ represents the horizontal offset distance of the variable transmission ratio rack tooth profile, which is conjugate with the left and right involute tooth profiles; i This is the offset coefficient.

[0027] As an optimization, in step S1, the pure rolling variable transmission ratio rack tooth profile equation established after horizontal offset in the cross-sectional plane is shown in formulas (11) and (12):

[0028]

[0029]

[0030] Among them, formulas (11) and (12) respectively represent the equations of the pure rolling variable transmission ratio rack tooth profile after horizontal offset of the variable transmission ratio rack tooth profile conjugate with the left and right involute tooth profiles in the cross-section plane; The coordinates of the point on the pure rolling variable transmission ratio rack tooth profile after horizontal offset of the variable transmission ratio rack tooth profile conjugate with the left and right involute tooth profiles within the cross-sectional plane.

[0031] The beneficial effects of this invention are as follows: The pure rolling variable transmission ratio gear rack pair design method of this invention effectively avoids the drawbacks of the traditional mechanical variable transmission ratio steering gear variable transmission ratio tooth profile design theory and method, and overcomes the problem of severe tooth surface wear in existing variable transmission ratio gear pairs; based on the proposed pure rolling variable transmission ratio gear pair design method, a herringbone gear design method is proposed for pure rolling variable transmission ratio gear pairs, which reduces the problems of excessive contact stress and axial force that are common in pure rolling gear pairs. Attached Figure Description

[0032] Figure 1 This is a transmission ratio curve diagram of a pure rolling variable transmission ratio gear pair in a specific embodiment of the present invention;

[0033] Figure 2 This is a geometric diagram showing the relationship when the gear rotation angle is 0 degrees in the radial section of the middle section of the involute helical gear in a specific embodiment of the present invention.

[0034] Figure 3 This is a point cloud diagram of the pure rolling variable transmission ratio rack tooth profile calculated in a specific embodiment of the present invention;

[0035] Figure 4 This is a solid model diagram of a herringbone tooth pure rolling variable transmission ratio rack established in a specific embodiment of the present invention. Detailed Implementation

[0036] To clearly illustrate the technical features of this solution, the following detailed implementation method will be used to explain the solution.

[0037] like Figures 1-4As shown, the present invention provides a design method for a pure rolling variable transmission ratio gear and rack pair, comprising the following steps:

[0038] S1: In a certain radial cross-section of the involute helical gear, establish the involute helical gear tooth profile equation; based on the kinematic relationship between the involute helical gear and the variable transmission ratio rack, establish the conjugate variable transmission ratio rack tooth profile equation in the cross-section, and establish the calculation equation for the pure rolling point on the involute tooth profile and the conjugate variable transmission ratio rack tooth profile.

[0039] Based on the tooth profile of the conjugate variable transmission ratio rack and the pure rolling point on its tooth profile, the formula for the equal gradient offset distance in the cross-sectional plane and the equation for the tooth profile of the variable transmission ratio rack after horizontal offset are established.

[0040] The established equations for the involute helical gear tooth profile in the aforementioned cross-section plane are shown in formulas (1) and (2):

[0041]

[0042]

[0043] Wherein, formulas (1) and (2) are the left and right involute tooth profile equations of the involute helical gear in the said cross-section plane, respectively. These are the coordinates of the left and right involute tooth profile points within the stated cutting plane; μ = θ k +α k Here are the involute parameters, representing the involute pressure angle α. k With the angle of expansion θ k The sum of; σ L σ R θ represents the initial angles of the left and right involute curves, respectively; θ is the rotation angle of any radial section of the tooth profile of the involute helical gear relative to the middle radial section of the tooth profile along the gear axis; β is the helix angle of the involute helical gear, B is the tooth width, z is the number of teeth in the full circle, and r... a r and r b These are the tip circle radius, pitch circle radius, and base circle radius, respectively; α is the involute pressure angle; n represents the tooth number, i.e., n = 1, 2, 3 represent the left tooth, middle tooth, and right tooth of the involute helical gear.

[0044] The established conjugate variable transmission ratio rack tooth profile equations in the cross-sectional plane are shown in formulas (3) and (4):

[0045]

[0046]

[0047] Among them, formulas (3) and (4) are respectively the variable transmission ratio rack tooth profile equations conjugate with the left and right involute tooth profiles in the said cross-section plane. These are the coordinates of the variable transmission ratio rack tooth profile points within the cross-sectional plane; is the gear rotation angle; s is the rack displacement of the variable transmission ratio, calculated by integrating the variable transmission ratio curve expression with the gear rotation angle as the integral variable. These are the unit normal vectors of the variable transmission ratio rack tooth profile points that are conjugate with the left and right involute tooth profiles, respectively; These are the relative velocities of the variable transmission ratio rack tooth profile points conjugate with the left and right involute tooth profiles, respectively. These represent the meshing equations for the variable transmission ratio rack tooth profiles that are conjugate with the left and right involute tooth profiles, respectively.

[0048] The equations for calculating the pure rolling point on the involute tooth profile and the conjugate variable transmission ratio rack tooth profile are shown in formulas (5), (6), (7), and (8):

[0049]

[0050]

[0051]

[0052]

[0053] Among them, formulas (5) and (6) are the calculation equations for the pure rolling point on the left and right involute tooth profiles in the cutting plane, respectively; formulas (7) and (8) are the calculation equations for the pure rolling point on the variable transmission ratio rack tooth profile conjugate with the left and right involute tooth profiles in the cutting plane, respectively. These are the coordinates of the pure rolling points on the left and right involute tooth profiles within the cross-sectional plane, respectively. These are the coordinates of the pure rolling points on the variable transmission ratio rack tooth profiles that are conjugate to the left and right involute tooth profiles within the cross-sectional plane; The parameters for the pure rolling point tooth profiles on the left and right involute tooth profiles; s′ L ,s′ R For gear rotation angle The corresponding rack displacement; For a variable transmission ratio gear pair at a rotation angle of The transmission ratio at that time.

[0054] The established formulas for the isogradient offset distance within the cut-off plane are shown in formulas (9) and (10):

[0055]

[0056]

[0057] Formulas (9) and (10) are formulas for the constant gradient offset distance of the tooth profile of the variable transmission ratio rack, which is conjugate with the left and right involute tooth profiles; ΔtL Δt R λ represents the horizontal offset distance of the variable transmission ratio rack tooth profile, which is conjugate with the left and right involute tooth profiles; i This is the offset coefficient.

[0058] The established pure rolling variable transmission ratio rack tooth profile equations after horizontal offset within the cross-sectional plane are shown in formulas (11) and (12):

[0059]

[0060]

[0061] Among them, formulas (11) and (12) respectively represent the equations of the pure rolling variable transmission ratio rack tooth profile after horizontal offset of the variable transmission ratio rack tooth profile conjugate with the left and right involute tooth profiles in the cross-section plane; The coordinates of the point on the pure rolling variable transmission ratio rack tooth profile after horizontal offset of the variable transmission ratio rack tooth profile conjugate with the left and right involute tooth profiles within the cross-sectional plane.

[0062] S2: Set the gear rotation angle step size at equal intervals within the gear rotation angle range, calculate the rack displacement according to the variable transmission ratio curve, calculate the conjugate variable transmission ratio rack tooth profile and the pure rolling points on the involute tooth profile and the conjugate variable transmission ratio rack tooth profile, and calculate the pure rolling variable transmission ratio rack tooth profile; generate several equally spaced radial cross-sections in the involute helical gear to generate the pure rolling variable transmission ratio rack tooth profile point cloud; fit the tooth profile point cloud to establish the pure rolling variable transmission ratio rack solid model.

[0063] The pure rolling gear rack pair of this steering gear ratio changeover was experimentally tested. The variable transmission ratio curve of the recirculating ball steering gear ratio changeover is designed as follows:

[0064]

[0065] Among them, i z This is the gear ratio of the steering gear.

[0066] The recirculating ball steering gear consists of two stages of transmission pairs, where the transmission ratio of the screw and nut pair is: i s =360 / P. Where, i s P is the transmission ratio of the screw and nut pair, and P is the screw pitch. In this experiment, P = 15 mm.

[0067] The design of a recirculating ball ratio steering gear with a pure rolling variable transmission ratio gear and rack pair can be obtained, for example... Figure 1 The variable transmission ratio curve shown is:

[0068]

[0069] Where i is the transmission ratio of a pure rolling variable transmission ratio gear pair.

[0070] In this experiment, the initial radial section plane of the involute helical gear was taken as the middle radial section plane. The involute helical gear adopted a 3-tooth design. The specific design parameters are: the clearance between the tip of the involute helical gear tooth and the root of the variable transmission ratio rack tooth, and between the root of the involute helical gear tooth and the tip of the variable transmission ratio rack tooth, is C = 2.5 mm; the tooth width is B = 66 mm; the number of teeth in the full circle is z = 8; and the base circle radius is r. b = 39.5867mm, tooth tip circle radius is r a = 54.188mm, root circle radius is r f =35.688mm, helix angle β=30deg; draw a plane of symmetry for the middle tooth through the axis of the gear base cylinder, the distance from the starting point of the involute of the middle tooth to this plane is 10.0323mm, the distances from the starting point of the involute of the two side teeth to this plane are 19.9843mm and 34.1721mm respectively; the offset coefficient is λ1=0.005.

[0071] In this experiment, the initial involute angles of the left and right involutes of the middle teeth in the initial radial section of the involute helical gear are σ. L2 = -14.6804deg, σ R2 =14.6804deg, the initial angles of the left and right involutes of the left tooth are σ and σ, respectively. L1 = -59.6804deg, σ R1 = -30.3196deg, the initial angles of the left and right involutes of the right tooth are σ and σ, respectively. L3 =30.3196deg, σ R3 =59.6804deg.

[0072] In this experiment, within the rotation angle range of the involute helical gear constrained by the variable transmission ratio curve, the gear rotation angle step size was set at equal intervals.

[0073] In this experiment, after determining the solution range of the involute tooth profile parameters in the variable transmission ratio rack tooth profile solution model according to the aforementioned steps, the MATLAB software was used to complete the numerical solution of the pure rolling variable transmission ratio rack tooth profile equation.

[0074] In this experiment, several radial cross-sections were generated at equal intervals on the involute helical gear, with a spacing of Δl = 1.2 mm between adjacent cross-sections. The same operation was performed on the other cross-sections besides the initial cross-section.

[0075] In this experiment, Table 1 shows the coordinates of the pure rolling point of the left tooth profile of the middle tooth of the involute helical gear. The unit is "mm".

[0076] The final point cloud of the pure rolling tooth profile of the variable transmission ratio rack generated in this experiment is as follows: Figure 3 As shown.

[0077] In this experiment, the surface fitting module of CATIA software was used to fit the point cloud of the tooth profile of a pure rolling variable transmission ratio rack, and the tooth profile of a pure rolling variable transmission ratio rack was generated. According to the clearance requirements of pure rolling gear pairs, the clearance between the gear tooth tip and the root of the variable transmission ratio rack, and between the gear tooth root and the tooth tip of the variable transmission ratio rack were designed.

[0078] Table 1

[0079]

[0080]

[0081] In this experiment, Table 2 shows the coordinates of the pure rolling point of the rack tooth profile with variable transmission ratio, which is conjugate to the left tooth profile of the middle tooth of the involute helical gear. The unit is "mm".

[0082] Table 2

[0083] (-23.610,48.223,4.800) (-22.997,48.065,6.000) (-22.387,47.906,7.200) (-21.778,47.745,8.400) (-21.172,47.582,9.600) (-20.568,47.419,10.800) (-19.965,47.255,12.000) (-19.364,47.091,13.200) (-18.764,46.927,14.400) (-18.165,46.763,15.600) (-17.568,46.599,16.800) (-16.970,46.436,18.000) (-16.374,46.274,19.200) (-15.777,46.114,20.400) (-15.181,45.955,21.600) (-14.584,45.798,22.800) (-13.988,45.644,24.000) (-13.391,45.493,25.200) (-12.793,45.344,26.400) (-12.194,45.199,27.600) (-11.595,45.058,28.800) (-10.994,44.920,30.000) (-10.392,44.788,31.200) (-9.788,44.660,32.400) (-9.183,44.537,33.600) (-8.576,44.420,34.800) (-7.967,44.309,36.000) (-7.355,44.204,37.200) (-6.741,44.106,38.400) (-6.125,44.015,39.600) (-5.506,43.931,40.800) (-4.884,43.855,42.000) (-4.259,43.787,43.200) (-3.630,43.728,44.400) (-2.998,43.678,45.600) (-2.363,43.637,46.800) (-1.723,43.605,48.000) (-1.079,43.583,49.200) (-0.431,43.571,50.400) (0.222,43.569,51.600) (0.879,43.598,52.800) (1.542,43.598,54.000) (2.210,43.628,55.200) (2.884,43.670,56.400) (3.565,43.723,57.600) (4.251,43.787,58.800) (4.944,43.862,60.000) (5.644,43.949,61.200) (6.352,44.047,62.400) (7.067,44.157,63.600) (7.791,44.278,64.800) (8.523,44.410,66.000)

[0084] In this experiment, CATIA software was used to draw spline curves, smoothly connect the generated variable transmission ratio tooth profile and the variable transmission ratio rack tooth root line in the cross-sectional plane, design the transition curve at the tooth root of the variable transmission ratio rack, and fit it sequentially along the tooth width direction of the variable transmission ratio rack to form the transition surface at the tooth root of the variable transmission ratio rack. Finally, a solid model of the herringbone tooth pure rolling variable transmission ratio rack was established.

[0085] The solid model of the herringbone tooth pure rolling variable transmission ratio rack established in this experiment is as follows: Figure 4 As shown.

[0086] The present invention is not limited to the above embodiments. For example, the number of tooth profile points on the involute, the tooth width of the involute helical gear, and the offset coefficient in the offset distance calculation formula can be changed as needed.

[0087] Of course, the above description is not limited to the examples above. Technical features not described in this invention can be implemented by or using existing technology, and will not be repeated here. The above embodiments and drawings are only used to illustrate the technical solutions of this invention and are not intended to limit this invention. This invention has been described in detail with reference to preferred embodiments. Those skilled in the art should understand that any changes, modifications, additions or substitutions made by those skilled in the art within the scope of this invention do not depart from the spirit of this invention and should also fall within the scope of protection of the claims of this invention.

Claims

1. A design method for a pure rolling variable transmission ratio gear and rack pair, characterized in that, Includes the following steps: S1: Establish the tooth profile equation of the involute helical gear in a certain radial cross-section plane; Based on the kinematic relationship between the involute helical gear and the variable transmission ratio rack, the tooth profile equation of the conjugate variable transmission ratio rack in the cross-section is established, and the calculation equation of the pure rolling point on the involute tooth profile and the conjugate variable transmission ratio rack tooth profile is established. Based on the tooth profile of the conjugate variable transmission ratio rack and the pure rolling point on its tooth profile, the formula for the equal gradient offset distance in the cross-sectional plane and the equation for the tooth profile of the variable transmission ratio rack after horizontal offset are established. S2: Set the gear rotation angle step size at equal intervals within the gear rotation angle range, calculate the rack displacement according to the variable transmission ratio curve, calculate the conjugate variable transmission ratio rack tooth profile and the pure rolling points on the involute tooth profile and the conjugate variable transmission ratio rack tooth profile, and calculate the pure rolling variable transmission ratio rack tooth profile; generate several equally spaced radial cross-sections in the involute helical gear to generate the pure rolling variable transmission ratio rack tooth profile point cloud; fit the tooth profile point cloud to establish the pure rolling variable transmission ratio rack solid model; In step S1, the established equations for the involute helical gear tooth profile within the cross-sectional plane are shown in formulas (1) and (2): （1） （2） Wherein, formulas (1) and (2) are the left and right involute tooth profile equations of the involute helical gear in the said cross-section plane, respectively. , These are the coordinates of the left and right involute tooth profile points within the cutting plane, respectively. For involute parameters, representing the involute pressure angle. With the corner sum; , These are the initial angles of the left and right involutes, respectively; The rotation angle of any radial section tooth profile of an involute helical gear relative to the intermediate radial section tooth profile along the gear axis; The helix angle of an involute helical gear. For tooth width, The number of teeth is a whole circle. , and These are the tip circle radius, pitch circle radius, and base circle radius, respectively. The involute pressure angle; Indicates the gear tooth number, i.e. This indicates the left, middle, and right teeth of an involute helical gear; In step S1, the established conjugate variable transmission ratio rack tooth profile equations in the cross-sectional plane are shown in formulas (3) and (4): （3） （4） Among them, formulas (3) and (4) are respectively the variable transmission ratio rack tooth profile equations conjugate with the left and right involute tooth profiles in the said cross-section plane. , These are the coordinates of the variable transmission ratio rack tooth profile points within the cross-sectional plane; For gear rotation angle; The variable transmission ratio rack displacement is calculated by integrating the variable transmission ratio curve expression with the gear rotation angle as the integration variable. , These are the unit normal vectors of the variable transmission ratio rack tooth profile points that are conjugate with the left and right involute tooth profiles, respectively; , These are the relative velocities of the variable transmission ratio rack tooth profile points conjugate with the left and right involute tooth profiles, respectively. , These represent the meshing equations for the variable transmission ratio rack tooth profiles that are conjugate with the left and right involute tooth profiles, respectively. In step S1, the equations for calculating the pure rolling point on the involute tooth profile and the conjugate variable transmission ratio rack tooth profile are shown in formulas (5), (6), (7), and (8): （5） （6） （7） （8） Among them, formulas (5) and (6) are the calculation equations for the pure rolling point on the left and right involute tooth profiles in the cutting plane, respectively; formulas (7) and (8) are the calculation equations for the pure rolling point on the variable transmission ratio rack tooth profile conjugate with the left and right involute tooth profiles in the cutting plane, respectively. , These are the coordinates of the pure rolling points on the left and right involute tooth profiles within the cross-sectional plane, respectively. , These are the coordinates of the pure rolling points on the variable transmission ratio rack tooth profiles that are conjugate to the left and right involute tooth profiles within the cross-sectional plane; , These are the pure rolling point tooth profile parameters on the left and right involute tooth profiles; , For gear rotation angle , The corresponding rack displacement; , For a variable transmission ratio gear pair at a rotation angle of , The transmission ratio at that time.

2. The design method for a pure rolling variable transmission ratio gear and rack pair according to claim 1, characterized in that: In step S1, the formulas for the equal gradient offset distance within the cut-off plane are shown in formulas (9) and (10): （9） （10） Among them, formulas (9) and (10) are formulas for the equal gradient offset distance of the tooth profile of the variable transmission ratio rack that is conjugate with the left and right involute tooth profiles; , These are the horizontal offset distances of the variable transmission ratio rack tooth profiles that are conjugate with the left and right involute tooth profiles, respectively; This is the offset coefficient.

3. The design method for a pure rolling variable transmission ratio gear and rack pair according to claim 1, characterized in that: In step S1, the pure rolling variable transmission ratio rack tooth profile equation established after horizontal offset within the cross-sectional plane is shown in formulas (11) and (12): （11） （12） Among them, formulas (11) and (12) respectively represent the equations of the pure rolling variable transmission ratio rack tooth profile after horizontal offset of the variable transmission ratio rack tooth profile conjugate with the left and right involute tooth profiles in the cross-section plane; , The coordinates of the point on the pure rolling variable transmission ratio rack tooth profile after horizontal offset of the variable transmission ratio rack tooth profile conjugate with the left and right involute tooth profiles within the cross-sectional plane.

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