A pure rolling gear rack mechanism based on cosine tooth profile and parabolic tooth trace

Abstract

This invention provides a pure rolling gear and rack mechanism based on a cosine-profile parabolic tooth line, relating to the field of gear rotation technology. It includes a gear and a rack, the end face tooth profiles of which are composed of end face working tooth profile curves and tooth root transition curves. Both the end face tooth profiles of the gear and rack are symmetrical on both sides. The end face working tooth profile curves of the gear and rack are cosine curves. The tooth surfaces of the gear and rack have a parabolic tooth line structure. At least one pair of teeth of the gear and rack have a meshing point located at a node to achieve pure rolling meshing contact. The meshing lines formed by the movement trajectories of the meshing points of the gear and rack create two contact lines on the tooth surfaces of the gear and rack, respectively. The beneficial effects of this invention are: the use of symmetrical tooth lines in both the gear and rack eliminates the axial force present during gear and rack meshing, which helps to improve bearing service life.

Description

Technical Field

[0001] This invention relates to the field of gear rotation technology, and in particular to a pure rolling gear rack mechanism based on a cosine tooth profile parabolic tooth line. Background Technology

[0002] Traditional involute gears are widely used in various mechanical equipment, especially in high-speed and high-load transmission applications, including in industrial equipment such as automotive transmission systems, automotive gearboxes, marine engines, and machine tool headstocks. The main function of gears is to transmit motion and power. For example, Chinese patent application number 201710016238.7 discloses "a non-relative sliding concave-convex meshing circular arc gear rack mechanism", and Chinese patent application number 201710016207.1 discloses "a non-relative sliding convex-concave meshing circular arc gear rack mechanism". In these gear rack mechanisms, the meshing point is designed near the upper edge of the concave circular arc tooth profile, which easily leads to stress concentration and tooth breakage. At the same time, because their tooth lines are not symmetrical, there is axial force on the gear during transmission, which increases the requirements for gear bearings and affects the accuracy and life of the transmission system. Summary of the Invention

[0003] In view of this, in order to solve the problems of axial force in existing gear racks and the stress concentration caused by circular arc concave tooth profiles, embodiments of the present invention provide a pure rolling gear rack mechanism based on a cosine tooth profile parabolic tooth line.

[0004] An embodiment of the present invention provides a pure rolling gear and rack mechanism based on a cosine tooth profile parabolic tooth line, comprising a gear and a rack. The end face tooth profiles of the gear and the rack are composed of end face working tooth profile curves and tooth root transition curves. The end face tooth profiles of the gear and the rack are symmetrical on both sides. The end face working tooth profile curves of the gear and the rack are cosine curves. The tooth surfaces of the gear and the rack have parabolic tooth line structures. At least one pair of gear teeth meshing points of the gear and the rack are located at nodes to achieve pure rolling meshing contact. The meshing lines formed by the movement trajectories of the meshing points of the gear and the rack respectively form two contact lines on the tooth surfaces of the gear and the rack.

[0005] Furthermore, the tooth surface structure of the gear and the rack are respectively formed by their respective tooth profile curves moving along the contact line with the contact point, and the contact line is a parabola symmetrical about the z-axis after being unfolded along the pitch cylinder surface of the gear, and the rack is a parabola at the pitch surface.

[0006] Furthermore, the working tooth profile curves on the right end faces of both the gear and the rack are formed by cosine function curves, and the tooth root transition curves are composed of Hermite curves; the shape of the working tooth profile curves on the end faces is controlled by the intersection point of the addendum circle and the cosine function curve, and the shape of the tooth root transition curve is determined by the intersection point P of the tooth root transition start circle and the cosine function curve. G4 Control and the tooth root circle passing through point P G4 And the intersection point of the oblique lines with a slope of 1 forms P G3 Connect point P according to the Hermite curve equation. G4 and point P G3 Forming the tooth root curve.

[0007] Furthermore, the contact line between the gear and the rack is determined according to the following method:

[0008] Establish O0-x0,y0,z0, O k -x k ,y k ,z k In the four spatial coordinate systems O1-x1,y1,z1 and O2-x2,y2,z2, the z0 axis and z1 axis coincide with the rotation axis of the gear. k The axis z2 coincides with the meshing line of the gear and the rack, and the axis z2 is on the rack, at a distance z k Shaft has The distance of R1, z k The distance between the axis and the z0 axis is R1; the coordinate system O0-x0,y0,z0 is fixed to the gear, and the coordinate system O2-x2,y2,z2 is fixed to the rack. The gear rotates around the z0 axis with a uniform angular velocity ω1, and the rack moves along the y2 axis with a uniform linear velocity v1. After a period of time from the initial position, the coordinate system O0-x0,y0,z0 rotates around the z0 axis with the gear, and the coordinate system O2-x2,y2,z2 moves along the y2 axis with the rack.

[0009] In coordinate system O k -x k ,y k ,z k In the equation, the parametric equation of the line of action of the meshing point of the gear and rack is:

[0010]

[0011] The relationship between the gear rotation angle and the rack motion is as follows:

[0012]

[0013] When the meshing point moves along the meshing line, contact lines are formed on the tooth surfaces of the gear and the rack, respectively. According to the coordinate transformation principle, the coordinate transformation matrices for the three spatial coordinate systems O0-x0,y0,z0, O1-x1,y1,z1, and O2-x2,y2,z2 are as follows:

[0014] M 1k =M 10 ×M 0k (3)

[0015]

[0016] in,

[0017]

[0018] In equations (4) and (6), R1 is the pitch circle radius of the gear. It is the gear rotation angle;

[0019] The parametric equation of the contact line of the gear tooth surface obtained from equations (1) and (5) is as follows:

[0020]

[0021] The parametric equations of the contact line of the rack tooth surface obtained from equations (1) and (4) are as follows:

[0022]

[0023] Furthermore, the end face tooth profiles of the gear and the rack are determined by the following method:

[0024] In coordinate system O k -x k ,y k ,z k The parametric equation of the working tooth profile curve on the right end face of the gear described in the figure is as follows:

[0025]

[0026] In coordinate system O k -x k ,y k ,z k The parametric equation of the working tooth profile curve on the right end face of the rack described in the figure is as follows:

[0027]

[0028] In coordinate system O k -x k ,y k ,z k The parametric equation for the working tooth profile curve on the left end face of the rack described in the figure is as follows:

[0029]

[0030] The parametric equations for the working tooth profile curve on the right side of the gear in the coordinate system O1-x1,y1,z1 are as follows:

[0031]

[0032] The parametric equations for the working tooth profile curve on the left side of the gear in the coordinate system O1-x1,y1,z1 are as follows:

[0033]

[0034] Furthermore, the tooth surfaces of the gear and the rack are determined according to the following method:

[0035] The tooth surface of the gear is formed by sweeping along the motion law of the meshing point M. The parametric equation of the working left tooth surface of the gear is as follows:

[0036]

[0037] The parametric equations for the working right tooth surface of the gear are as follows:

[0038]

[0039] The tooth surface of the rack is formed by a motion trajectory along the rack contact curve. The parametric equation of the working left tooth surface of the rack is as follows:

[0040]

[0041] The parametric equations for the working right tooth surface of the rack are as follows:

[0042]

[0043] Furthermore, the tooth root transition curves of the gear and the rack are determined according to the following method:

[0044] The right tooth root of the gear end face uses the Hermite curve as a transition curve, where the Hermite curve starts from point P. F3 and P F4 and point P F3 and P F4 The tangent vectors are T F3 and T F4 Decision, P F3 The point is formed by the working tooth profile curve on the right side of the gear and the starting radius R of the tooth root transition fillet. h1 Decision, P F4 The point is formed by the radius R of the tooth root circle. f1 and passing through point P F3Furthermore, the slope of the Hermite curve is determined by a slope of 1, and the parametric equation is:

[0045]

[0046] The right tooth root of the rack end face uses the Hermite curve as a transition curve, where the Hermite curve starts from point P. G3 and P G4 and point P G3 and P G4 The tangent vectors are T G3 and T G4 Decision, P G3 The point is formed by the working tooth profile curve on the right side of the gear and the starting radius R of the tooth root transition fillet. h2 Decision, P G4 The point is formed by the radius R of the tooth root circle. f2 and passing through point R G3 Furthermore, the slope of the Hermite curve is determined by a slope of 1, and the parametric equation is:

[0047]

[0048] in,

[0049]

[0050] The beneficial effects of the technical solutions provided by the embodiments of the present invention are as follows:

[0051] 1. The present invention relates to a pure rolling gear and rack mechanism based on a cosine tooth profile parabolic tooth line. Based on an active design method using the parametric equation of the meshing line, a meshing point is set at the node, and the meshing line is constructed using the motion law of the meshing point. Furthermore, the gear contact line, when unfolded on the pitch cylinder surface, is a parabola; similarly, the rack contact line is a parabola at the rack pitch surface, and after unfolding on the pitch cylinder surface, it becomes an axisymmetric parabola to eliminate axial force, greatly improving bearing life and reducing maintenance costs.

[0052] 2. The present invention provides a pure rolling gear and rack mechanism based on a cosine tooth profile parabolic tooth line. Both the gear tooth profile and the rack tooth profile are established based on cosine functions. Furthermore, the gear and rack tooth roots use Hermite curves to enhance the bending strength of the tooth roots, making the gear less prone to tooth root breakage and increasing the service life of the gear and rack.

[0053] 3. The present invention provides a pure rolling gear and rack mechanism based on a cosine tooth profile parabolic tooth line. The gear contact ratio can be freely designed, and the structural shape of the tooth profile can be determined by setting the contact ratio value, thereby achieving uniform load distribution and improving dynamic characteristics. Attached Figure Description

[0054] Figure 1 This is a schematic diagram of a pure rolling gear and rack mechanism based on a cosine tooth profile parabolic tooth line in Embodiment 1 of the present invention;

[0055] Figure 2 This is a schematic diagram of the spatial meshing coordinate system of a pure rolling gear and rack mechanism based on a cosine tooth profile parabola tooth line in Embodiment 1;

[0056] Figure 3 This is a schematic diagram of the end face of a pure rolling gear rack mechanism based on a cosine tooth profile parabolic tooth line in Embodiment 1;

[0057] Figure 4 This is a schematic diagram of the tooth profiles of the gear and rack in Example 1;

[0058] Figure 5 This is a schematic diagram of the gears in Example 1;

[0059] Figure 6 This is a schematic diagram of the rack in Example 1;

[0060] Figure 7 This is a schematic diagram of a pure rolling gear rack mechanism based on a cosine tooth profile parabolic tooth line in Embodiment 2.

[0061] In the diagram: 1. Driver; 2. Input shaft; 3. Coupling; 4. Output shaft; 5. Gear; 6. Rack; 7. Meshing line kk; 8. Pitch cylinder; 9. Rack pitch surface; 10. Gear contact line; 11. Rack contact line; 12. Tooth root transition curve; 13. End face working tooth profile curve. Detailed Implementation

[0062] To make the objectives, technical solutions, and advantages of the present invention clearer, the embodiments of the present invention will be further described below with reference to the accompanying drawings. The following description presents a preferred embodiment of the various possible embodiments of the present invention, intended to provide a basic understanding of the invention, but not intended to identify key or decisive elements of the invention or to limit the scope of protection sought.

[0063] In all examples shown and discussed herein, any specific values ​​should be interpreted as merely exemplary and not as limitations. Therefore, other examples of exemplary embodiments may have different values.

[0064] Techniques, methods, and equipment known to those skilled in the art may not be discussed in detail, but where appropriate, such techniques, methods, and equipment should be considered part of the specification.

[0065] It should be noted that similar labels and letters in the following figures indicate similar items; therefore, once an item is defined in one figure, it does not need to be discussed further in subsequent figures. Also, it should be understood that, for ease of description, the dimensions of the various parts shown in the figures are not drawn to actual scale.

[0066] In the description of this invention, it should be noted that the circuits, electronic components and modules involved in this invention are all prior art, which can be fully implemented by those skilled in the art, and need not be elaborated upon.

[0067] It should be further noted that, unless otherwise explicitly specified and limited, the terms "installation" and "connection" should be interpreted broadly. For example, they can refer to a fixed connection, a detachable connection, or an integral connection; they can refer to a mechanical connection or an electrical connection; they can refer to a direct connection or an indirect connection through an intermediate medium; and they can refer to the internal connection of two components. Those skilled in the art can understand the specific meaning of the above terms in this invention based on the specific circumstances.

[0068] Example 1

[0069] This invention provides a pure rolling gear and rack mechanism based on a cosine tooth profile parabolic tooth line, comprising a gear 5 and a rack 6. The end face tooth profiles of the gear 5 and the rack 6 are composed of an end face working tooth profile curve 13 and a tooth root transition curve 12. The end face tooth profiles of the gear 5 and the rack 6 are symmetrical on both sides. The end face working tooth profile curve 13 of the gear 5 and the rack 6 is a cosine curve. The tooth surfaces of the gear 5 and the rack 6 have a parabolic tooth line structure. At least one pair of gear teeth of the gear 5 and the rack 6 are located at a node to achieve pure rolling meshing contact. The meshing lines formed by the movement trajectories of the meshing points of the gear 5 and the rack 6 respectively form two contact lines on the tooth surfaces of the gear 5 and the rack 6: a gear contact line 10 and a rack contact line 11. The tooth surface structure of the gear 5 and the rack 6 is formed by their respective tooth profile curves moving along the contact line as the contact point moves. The contact line, when unfolded along the pitch cylinder surface of the gear 5, is a parabola symmetrical about the z-axis. The rack 6 is a parabola at the pitch surface.

[0070] like Figure 1 As shown, in the pure rolling gear and rack mechanism based on cosine tooth profile parabola tooth line in this embodiment 1, the overlap ratio of gear 5 and rack 6 is ε=1.6. Gear 5 and rack 6 form a gear and rack pair. Gear 5 is connected to output shaft 4. Input shaft 2 is fixedly connected to output shaft 4 through coupling 3. Input shaft 2 is fixedly connected to driver 1. Rack 6 is connected to the driven load.

[0071] Please refer to Figures 1-6 The pitch cylinder of the gear has a radius of R1, and the addendum circle of the rack has a radius of R.a1 The radius of the tooth root circle is R f1 The gear tooth root cylinder has evenly distributed parabolic tooth lines on its outer surface. The contact line 10 of the gear, when unfolded along the gear pitch cylinder, is an axisymmetric parabola. The rack contact line 11, when unfolded along the gear pitch surface 9, is also an axisymmetric parabola. The gear end face tooth profile, from the tooth tip to the tooth root, is composed of a cosine curve and the left end face tooth root transition curve 12, i.e., the Hermite curve.

[0072] Please refer to Figure 2 This invention provides a schematic diagram of the spatial meshing coordinate system of a parabolic gear and rack mechanism with a cosine tooth profile on the end face. The working tooth profiles and root transition fillets of the gear and rack are symmetrical on both sides. The left tooth profile is obtained symmetrically from the right tooth profile, and similarly, the left root transition fillet is obtained symmetrically from the right root transition fillet. The tooth tip control point P of the working tooth profile on the right end face of the gear is shown. U3 From the tooth tip circle radius R a1 The starting control point P for the tooth root transition of the working tooth profile on the right end face of the gear is determined by the intersection with the end face working tooth profile. F3 Radius R of the transition circle from the tooth root h1 The point P at the tooth root control of the tooth root transition curve 12 on the right end face of the gear is determined by the intersection with the working tooth profile on the end face. F4 The radius of the tooth root circle and the point P F3 And the intersection of the oblique lines with a slope of 1. Similarly, the control point P of the working tooth profile tooth tip on the right end face of the rack. U2 The length R of the rack tooth tip position a2 The starting control point P for the tooth root transition of the working tooth profile on the right end face of the rack is determined by the intersection with the end face working tooth profile. G3 Length R of the transition from the root of the rack tooth h2 The tooth root control point P is determined by the intersection with the working tooth profile on the end face; the tooth root transition curve 12 on the right end face of the rack is defined by the intersection with the tooth profile on the end face. G4 Length R from the starting position of the tooth root transition h2 and passing through point P G3 And the intersection of oblique lines with a slope of 1.

[0073] Gear 5 rotates under the drive of driver 1, causing rack 6 to translate, thus realizing the transmission of motion and power between gear 5 and rack 6. In this embodiment, driver 1 is an electric motor.

[0074] The gear contact line 10 and rack contact line 11 are determined by the following method: Establish O0-x0,y0,z0, O k -x k ,y k ,z k In the four spatial coordinate systems O1-x1,y1,z1 and O2-x2,y2,z2, the z0 axis and z1 axis coincide with the rotation axis of the gear.k The axis z2 coincides with the meshing line of the gear and the rack, and the axis z2 is on the rack, at a distance z k Shaft has The distance of R1, z k The distance between the axis and the z0 axis is R1; the coordinate system O0-x0,y0,z0 is fixed to the gear, and the coordinate system O2-x2,y2,z2 is fixed to the rack. The gear rotates around the z0 axis with a uniform angular velocity ω1, and the rack moves along the y2 axis with a uniform linear velocity v1. After a period of time from the initial position, the coordinate system O0-x0,y0,z0 rotates around the z0 axis with the gear, and the coordinate system O2-x2,y2,z2 moves along the y2 axis with the rack.

[0075] In coordinate system O k -x k ,y k ,z k In the equation, the parametric equation of the line of action of the meshing point of the gear and rack is:

[0076]

[0077] The relationship between the gear rotation angle and the rack motion is as follows:

[0078]

[0079] When the meshing point M moves along the meshing line 7, gear contact line 10 and rack contact line 11 are formed on the tooth surfaces of the gear and rack, respectively. According to the coordinate transformation principle, the coordinate transformation matrices for forming three spatial coordinate systems, O0-x0,y0,z0, O1-x1,y1,z1 and O2-x2,y2,z2, are as follows:

[0080] M 1k =M 10 ×M 0k (3)

[0081]

[0082] in,

[0083]

[0084] In equations (4) and (6), R1 is the pitch circle radius of the gear. It is the gear rotation angle;

[0085] The parametric equation of the contact line of the gear tooth surface obtained from equations (1) and (5) is as follows:

[0086]

[0087] The parametric equations of the contact line of the rack tooth surface obtained from equations (1) and (4) are as follows:

[0088]

[0089] Furthermore, the end face tooth profiles of the gear and the rack are determined by the following method:

[0090] In coordinate system O k -x k ,y k ,z k The parametric equation of the working tooth profile curve 13 on the right end face of the gear described in the figure is as follows:

[0091]

[0092] In coordinate system O k -x k ,y k ,z k The working tooth profile curve 13 parametric equation of the right end face of the rack described in the figure is as follows:

[0093]

[0094] In coordinate system O k -x k ,y k ,z k The working tooth profile curve 13 parametric equation of the left end face of the rack described in the figure is as follows:

[0095]

[0096] In coordinate system O1-x1,y1,z1, the parametric equation of the working tooth profile curve 13 on the right side of the gear is as follows:

[0097]

[0098] In coordinate system O1-x1,y1,z1, the parametric equation of the working tooth profile curve 13 on the left side of the gear is as follows:

[0099]

[0100] Furthermore, the tooth surfaces of the gear and the rack are determined according to the following method:

[0101] The tooth surface of the gear is formed by sweeping along the motion law of the meshing point M. The parametric equation of the working left tooth surface of the gear is as follows:

[0102]

[0103] The parametric equations for the working right tooth surface of the gear are as follows:

[0104]

[0105] The tooth surface of the rack is formed by a motion trajectory along the rack contact curve. The parametric equation of the working left tooth surface of the rack is as follows:

[0106]

[0107] The parametric equations for the working right tooth surface of the rack are as follows:

[0108]

[0109] Furthermore, the tooth root transition curves of the gear 5 and the rack 6 are determined according to the following method:

[0110] The right tooth root of the gear end face uses the Hermite curve as a transition curve, where the Hermite curve starts from point P. F3 and P F4 and point P F3 and P F4 The tangent vectors are T F3 and T F4 Decision, P F3 The point is formed by the working tooth profile curve on the right side of the gear and the starting radius R of the tooth root transition fillet. h1 Decision, P F4 The point is formed by the radius R of the tooth root circle. f1 and passing through point P F3 Furthermore, the slope of the Hermite curve is determined by a slope of 1, and the parametric equation is:

[0111]

[0112] The right tooth root of the rack end face uses the Hermite curve as a transition curve, where the Hermite curve starts from point P. G3 and P G4 and point P G3 and P G4 The tangent vectors are T G3 and T G4 Decision, P G3 The point is formed by the working tooth profile curve on the right side of the gear and the starting radius R of the tooth root transition fillet. h2 Decision, P G4 The point is formed by the radius R of the tooth root circle. f2 and passing through point P G3 Furthermore, the slope of the Hermite curve is determined by a slope of 1, and the parametric equation is:

[0113]

[0114] in,

[0115]

[0116] In all the above formulas:

[0117] t-cosine function parametric equation parameters;

[0118] p - Sets the parameter value for the cosine function;

[0119] σ is the motion parameter variable of the engagement point M, and σ∈[0,Δσ];

[0120] Δσ - the maximum value of the motion parameter variable at the engagement point;

[0121] - is the linear proportionality coefficient for the motion at the meshing point;

[0122] i-transmission ratio

[0123] m t - End face module;

[0124] Z1 - Number of gear teeth;

[0125] Z2 - The number of rack teeth the gear passes through after one revolution;

[0126] T H For Hermite profile parameters, 0.2 ≤ T H ≤1.5;

[0127] t H The range of values ​​for the Hermite type line is 0 ≤ t. H ≤1;

[0128] P F3 - The intersection point of the gear transition fillet starting radius and the gear working tooth profile parametric equation;

[0129] P F4 - Radius of the tooth root circle and the point P it passes through F3 And the intersection of oblique lines with a slope of 1;

[0130] P G3 - The intersection point of the starting length of the rack transition fillet and the working tooth profile parametric equation of the rack;

[0131] P G4 -Length of rack tooth root position and point P passing through it G3 And the intersection of the slopes with a gradient of 1;

[0132] T P1 - The unit tangent vector at point P1;

[0133] T P2 - The unit tangent vector at point P2;

[0134] TG1 - The unit tangent vector at point G1;

[0135] T G2 - The unit tangent vector at point G2;

[0136] x P (P F3 )-Point P F3 The x-coordinate;

[0137] y p (P F3 )-Point P F3 y-coordinate;

[0138] z p (P F3 )-Point P F3 The z-coordinate;

[0139] x p (P F4 )-Point P F4 The x-coordinate;

[0140] y p (P F4 )-Point P F4 The y-coordinate;

[0141] z p (P F4 )-Point P F4 The z-coordinate;

[0142] x G (P G3 )-Point P G3 The x-coordinate;

[0143] y G (P G3 )-Point P G3 y-coordinate;

[0144] z G (P G3 )-Point P G3 The z-coordinate;

[0145] x G (P G4 )-Point P G4 The x-coordinate;

[0146] y G (P G4 )-Point P G4 y-coordinate;

[0147] Z G (P G4 )-Point P G4 The z-coordinate;

[0148] Δd - Width coefficient;

[0149] b - the width of the gear teeth, b = Δd × 2R1; (21)

[0150] α t - End face pressure angle, α t =20°;

[0151] -Tooth tip height coefficient,

[0152] -Pithole coefficient,

[0153] R1 is the pitch circle radius of the gear, R1 = m t Z1 / 2; (22)

[0154] R2 is the pitch line of the rack.

[0155] a - Center distance between rack and gear, a = R1 + R2; (23)

[0156] h a -Tooth tip height,

[0157] h f -Tooth root height,

[0158] R a1 - Gear tip circle radius, R a1 =R1+h a (26)

[0159] R f1 - Gear root circle radius, R f1 =R1-h f (27)

[0160] R h1 - Gear transition fillet start radius, R h1 =R1-h a (28)

[0161] R a2 - Length of rack tooth tip position, R a2 =R2+h a (29)

[0162] R f2 - Length of rack tooth root position, R f2 =R2-h f (30)

[0163] Rh2 -Starting length of the rack transition fillet, R h2 =R2-h a (31)

[0164] ε-coincidence,

[0165] p t -End face tooth pitch, p t =πm t (33)

[0166] The relevant parameters are taken as Z1 = 16, i = 1, m t =2, b = 32 mm, α t =20°, we get Δσ=0.2, R1=16mm, R2=38mm;

[0167] Then, by substituting the above values ​​into equations (1) to (33), we can obtain the contact line parameter equations and end face tooth profile parameter equations of the gear and rack in this example, obtain the gear and rack tooth surface structure, and assemble them according to the correct center distance.

[0168] Example 2:

[0169] like Figure 7 As shown, Embodiment 2 of the present invention also provides another pure rolling gear and rack mechanism based on a cosine tooth profile parabolic tooth line, wherein gear 5 is connected to output shaft 4, output shaft 4 is connected to input shaft 2 through coupling 3, input shaft 2 is fixedly connected to driver 1, and rack 6 is connected to the driven load. In this embodiment, gear 5 has 20 teeth. Rack 6 has 30 teeth, and the designed overlap ratio ε = 2. When output shaft 4 drives gear 5 to rotate, since both pairs of adjacent gears and racks that mount gear 5 and rack 6 are in a meshing state, and the pre-set overlap ratio ε = 2 of the pure rolling gear and rack mechanism based on a cosine tooth profile parabolic tooth line ensures that at every instant, at least two pairs of teeth participate in meshing transmission simultaneously, thereby realizing continuous and stable meshing transmission of the pure rolling gear and rack mechanism based on a cosine tooth profile parabolic tooth line during rotational motion.

[0170] The relevant parameters are set to the following values: Z1 = 20, i = 1, m t =2, ε=2, b = 40 mm, α t =20°, we get Δσ = 0.2, R1 = 20mm, R2 = 38mm;

[0171] Substituting the above values ​​into equations (1) to (33) yields the contact line parameter equations and end face tooth profile parameter equations for gear 5 and rack 6 in this example. Then, based on the motion law of the meshing point, the gear tooth structure of the gear and rack is obtained, and they can be assembled according to the correct center distance.

[0172] In this document, the directional terms such as front, back, top, and bottom are defined based on the position of the components in the accompanying drawings and their relative positions to each other, solely for the purpose of clarity and convenience in expressing the technical solution. It should be understood that these are relative concepts and can vary depending on different methods of use and placement; the use of these directional terms should not limit the scope of protection claimed in this application.

[0173] Where there is no conflict, the embodiments and features described above can be combined with each other. The above descriptions are merely preferred embodiments of the present invention and are not intended to limit the invention. Any modifications, equivalent substitutions, or improvements made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A pure rolling gear and rack mechanism based on a cosine tooth profile parabolic tooth line, characterized in that: The device includes a gear and a rack. The end face tooth profiles of the gear and the rack are composed of end face working tooth profile curves and tooth root transition curves. The end face tooth profiles of the gear and the rack are symmetrical on both sides. The end face working tooth profile curves of the gear and the rack are cosine curves. The tooth surfaces of the gear and the rack have parabolic tooth line structures. At least one pair of gear teeth meshing points of the gear and the rack are located at the node to achieve pure rolling meshing contact. The meshing lines formed by the movement trajectories of the meshing points of the gear and the rack respectively form two contact lines on the tooth surfaces of the gear and the rack. The tooth surface structures of the gear and the rack are formed by their respective tooth profile curves moving along the contact lines with the contact points. The contact lines, when unfolded along the pitch cylinder surface of the gear, are all parabolas symmetrical about the z-axis. The rack is a parabola at the pitch surface. The working tooth profile curves on the right end faces of both the gear and the rack are formed by cosine function curves, and the tooth root transition curves on the right end faces of both the gear and the rack are composed of Hermite curves. The shape of the working tooth profile curves on the end faces is controlled by the intersection point of the addendum circle and the cosine function curve, and the shape of the tooth root transition curve is determined by the intersection point P of the tooth root transition start circle and the cosine function curve. G4 Control and the tooth root circle passing through point P G4 And the intersection point of the oblique lines with a slope of 1 forms P G3 Connect point P according to the Hermite curve equation. G4 and point P G3 Forming a tooth root curve; The contact line between the gear and the rack is determined according to the following method: Establish O0-x0,y0,z0, O k -x k ,y k ,z k In the four spatial coordinate systems O1-x1,y1,z1 and O2-x2,y2,z2, the z0 axis and z1 axis coincide with the rotation axis of the gear. k The axis z2 coincides with the meshing line of the gear and the rack, and the axis z2 is on the rack, at a distance z k The distance along the axis, z k The distance between the axis and the z0 axis is R1; the coordinate system O0-x0,y0,z0 is fixed to the gear, and the coordinate system O2-x2,y2,z2 is fixed to the rack. The gear rotates around the z0 axis with a uniform angular velocity ω1, and the rack moves along the y2 axis with a uniform linear velocity v1. After a period of time from the initial position, the coordinate system O0-x0,y0,z0 rotates around the z0 axis with the gear, and the coordinate system O2-x2,y2,z2 moves along the y2 axis with the rack. In coordinate system O k -x k ,y k ,z k In the equation, the parametric equation of the line of action of the meshing point of the gear and rack is: 0≤ ≤ （1） The relationship between the gear rotation angle and the rack motion is as follows: （2） When the meshing point moves along the meshing line, contact lines are formed on the tooth surfaces of the gear and the rack, respectively. According to the coordinate transformation principle, the coordinate transformation matrices for the three spatial coordinate systems O0-x0,y0,z0, O1-x1,y1,z1, and O2-x2,y2,z2 are as follows: M 1k =M 10 ×M 0k (3) （4） in, （5） （6） In equations (4) and (6), R1 is the pitch circle radius of the gear, and φ1 is the gear rotation angle; The parametric equation of the contact line of the gear tooth surface obtained from equations (1) and (5) is as follows: （7） The parametric equations of the contact line of the rack tooth surface obtained from equations (1) and (4) are as follows: （8）； The end face tooth profiles of the gear and the rack are determined by the following method: In coordinate system O k -x k ,y k ,z k The parametric equation of the working tooth profile curve on the right end face of the gear described in the figure is as follows: （9） In coordinate system O k -x k ,y k ,z k The parametric equation of the working tooth profile curve on the right end face of the rack described in the figure is as follows: （10） In coordinate system O k -x k ,y k ,z k The parametric equation for the working tooth profile curve on the left end face of the rack described in the figure is as follows: （11） The parametric equations for the working tooth profile curve on the right side of the gear in the coordinate system O1-x1,y1,z1 are as follows: （12） The parametric equations for the working tooth profile curve on the left side of the gear in the coordinate system O1-x1,y1,z1 are as follows: （13） The tooth surfaces of the gear and the rack are determined according to the following method: The tooth surface of the gear is formed by sweeping along the motion law of the meshing point M. The parametric equation of the working left tooth surface of the gear is as follows: （14） The parametric equations for the working right tooth surface of the gear are as follows: （15） The tooth surface of the rack is formed by a motion trajectory along the rack contact curve. The parametric equation of the working left tooth surface of the rack is as follows: （16） The parametric equations for the working right tooth surface of the rack are as follows: （17）； The tooth root transition curves of the gear and the rack are determined according to the following method: The right tooth root of the gear end face uses the Hermite curve as a transition curve, where the Hermite curve starts from point P. F3 and P F4 and point P F3 and P F4 The tangent vectors are T F3 and T F4 Decision, P F3 The point is formed by the working tooth profile curve on the right side of the gear and the starting radius R of the tooth root transition fillet. h1 Decision, P F4 The point is formed by the radius R of the tooth root circle. f1 and passing through point P F3 Furthermore, the slope of the Hermite curve is determined by a slope of 1, and the parametric equation is: （18） The right tooth root of the rack end face uses the Hermite curve as a transition curve, where the Hermite curve starts from point P. G3 and P G4 and point P G3 and P G4 The tangent vectors are T G3 and T G4 Decision, P G3 The point is formed by the working tooth profile curve on the right side of the gear and the starting radius R of the tooth root transition fillet. h2 Decision, P G4 The point is formed by the radius R of the tooth root circle. f2 and passing through point P G3 Furthermore, the slope of the Hermite curve is determined by a slope of 1, and the parametric equation is: （19） in, （20） In all the above formulas: t-cosine function parametric equation parameters p - Sets the parameter value for the cosine function; σ is the motion parameter variable of the engagement point M, and σ∈[0,Δσ]; Δσ - the maximum value of the motion parameter variable at the engagement point; k φ - is the linear proportionality coefficient for the motion at the meshing point; m t - End face module; Z1 - Number of gear teeth; Z2 - The number of rack teeth the gear passes through after one revolution; T H For Hermite profile parameters; t H The range of values ​​for the Hermite type line; Δd - Width coefficient; b - the width of the gear teeth, b = Δd × 2R1; (21) αt - End face pressure angle; h an * -Tooth tip height coefficient; c n * -Potential coefficient; R1 is the pitch circle radius of the gear, R1 = m t Z1 / 2; (22) R2 is the pitch line of the rack; a - Center distance between rack and gear, a = R1 + R2; (23) h a -Tooth tip height, h a =h an * m t ;(twenty four) h f - Tooth root height, h f =(h an * +c n * )m t ; (25) R a1 - Gear tip circle radius, R a1 =R1+h a (26) R f1 - Gear root circle radius, R f1 =R1-h f (27) R h1 - Gear transition fillet start radius, R h1 =R1-h a (28) R a2 - Length of rack tooth tip position, R a2 =R2+h a (29) R f2 - Length of rack tooth root position, R f2 =R2-h f (30) R h2 -Starting length of the rack transition fillet, R h2 =R2-h a (31) ε-degree of polymerization, (32) p t -End face tooth pitch, p t =πm t ; (33).

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