Pure rolling pinion-and-rack mechanism having parabolic function-based tooth profiles and hyperbolic tooth traces

Abstract

The present invention relates to the technical field of pinion transmission, and provides a pure rolling pinion-and-rack mechanism having parabolic function-based tooth profiles and hyperbolic tooth traces, comprising a pinion and a rack. A transverse tooth profile of each of the pinion and the rack consist of a transverse working tooth profile curve and a root transition curve, and the transverse tooth profiles of the pinion and the rack are both bilaterally symmetrical; the transverse working tooth profile curves of the pinion and the rack are parabolas; the tooth surfaces of the pinion and the rack have a hyperbolic tooth trace structure; and at least one pair of tooth meshing points of the pinion and the rack are located at nodes to achieve pure rolling meshing contact, and a meshing line formed by a movement trajectory of the meshing points of the pinion and the rack form two contact lines on the tooth surfaces of the pinion and the rack, respectively. The present invention has the advantage effects that: the contact lines are axisymmetric hyperbolas after being unfolded on a pitch cylinder surface to eliminate axial force, the theoretical value of the relative sliding speed of all meshing points on the contact lines is zero, the tooth profile tooth surface is a meshing tooth surface without relative sliding, the friction loss and noise are significantly reduced, and the transmission efficiency is greatly improved.

Description

A parabola function tooth profile hyperbolic tooth trace pure rolling gear rack mechanism TECHNICAL FIELD

[0001] The present application relates to the technical field of gear transmission, and particularly relates to a parabola function tooth profile hyperbolic tooth trace pure rolling gear rack mechanism. BACKGROUND

[0002] Traditional involute gears are widely used in various mechanical equipment, especially in high-speed and high-load transmission occasions, including application in automobile transmission systems, automobile gearboxes, ship engines and machine tool headstocks and other industrial equipment, and the main function of gears is to transmit motion and power.

[0003] Chinese patent with application number 201710016238.7 discloses a concave-convex meshing arc gear rack mechanism without relative sliding, and Chinese patent with application number 201710016207.1 discloses a convex-concave meshing arc gear rack mechanism without relative sliding, the tooth profiles of these gears and racks are all arc tooth profiles, and the relative sliding of the tooth surfaces is large when the gears and racks are in meshing transmission, which causes problems such as friction and wear, gluing and plastic deformation, and it is difficult to meet the precise transmission requirements of high-end equipment. In addition, the heat and noise generated by sliding friction also put higher requirements on the working environment and equipment maintenance. SUMMARY

[0004] In view of this, in order to solve the problem of large relative sliding of the tooth surfaces and serious friction and wear in the gear rack mechanism of the existing gear transmission technology, embodiments of the present application provide a parabola function tooth profile hyperbolic tooth trace pure rolling gear rack mechanism.

[0005] Embodiments of the present application provide a parabola function tooth profile hyperbolic tooth trace pure rolling gear rack mechanism, which comprises a gear and a rack, the end face tooth profile of the gear and the rack is composed of an end face working tooth profile curve and a dedendum transition curve, and the end face tooth profile of the gear and the rack is symmetrical on both sides; the end face working tooth profile curve of the gear and the rack is a parabola; the tooth surfaces of the gear and the rack have a hyperbolic tooth trace structure; at least one pair of gear tooth meshing points of the gear and the rack are located at nodes to realize pure rolling meshing contact, and the meshing lines formed by the trajectories of the meshing points of the gear and the rack form two contact lines on the tooth surfaces of the gear and the rack, respectively.

[0006] Further, the contact line of the gear is a curve having a hyperbolic line type after being developed along a pitch cylinder, the contact line of the rack is a curve having a hyperbolic line type after being developed along a pitch surface, and the tooth surfaces of the gear and the rack are formed by regularly scanning the end face tooth profile along the contact lines.

[0007] Furthermore, the right-side working tooth profile curves of both the gear and the rack are formed by parabolas, and the tooth root transition curve is composed of Hermite curves; the range of values ​​for the end-face working tooth profile curve is determined by controlling the start and end points of the working tooth profile curve according to specific control points; wherein the gear tooth tip control point is determined by the intersection of the tooth tip circle and the parabolic curve, and the starting control point of the tooth root transition curve is the intersection point P formed by the intersection of the parabolic curve and the tooth root transition start circle. G4 The contact control point of the tooth root transition curve is formed by the tooth root circle and the 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.

[0008] Furthermore, the contact line between the gear and the rack is determined according to the following method:

[0009] 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 meshing lines of the gear and the rack coincide, and the z2 axis is on the rack, at a distance z. k Shaft has distance, 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.

[0010] In coordinate system O k -x k ,y k ,z k In this context, let the parametric equation of the line of action of the meshing point of the gear and the rack be:

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

[0012] When the engagement point moves along the engagement line, a contact line is formed on the tooth surface of the gear and the rack respectively; according to the coordinate transformation principle, the coordinate transformation matrix of the three space coordinate systems O0-x0, y0, z0, O1-x1, y1, z1 and O2-x2, y2, z2 is:

[0013] M 1k = M 10 × M 0k (3)

[0014] Wherein,

[0015] In formula (4) and (6), R1 is the pitch circle radius of the gear, is the rotation angle of the gear;

[0016] The parametric equation of the contact line of the tooth surface of the gear is obtained from formula (1) and (5):

[0017] The parametric equation of the contact line of the tooth surface of the rack is obtained from formula (1) and (4):

[0018] Further, the end surface tooth profile of the gear and the rack is determined by the following method:

[0019] In the coordinate system O k -x k ,y k ,z k , the parametric equation of the right side working tooth profile of the gear is:

[0020] In the coordinate system O k -x k ,y k ,z k , the parametric equation of the right side working tooth profile of the rack is:

[0021] In the coordinate system O k -x k ,y k ,z k , the parametric equation of the left side working tooth profile of the rack is:

[0022] In the coordinate system O1-x1, y1, z1, the parametric equation of the right side working tooth profile of the gear is:

[0023] In the coordinate system O1-x1, y1, z1, the parametric equation of the left side working tooth profile of the gear is:

[0024] Further, the tooth surfaces of the gear and the rack are determined as follows:

[0025] The tooth surface of the gear is formed by sweeping along the motion law of the meshing point M, and the parametric equation of the left working tooth surface of the gear is:

[0026] The parametric equation of the right working tooth surface of the gear is:

[0027] The tooth surface of the rack is formed by realizing the motion trajectory along the rack contact curve, and the parametric equation of the left working tooth surface of the rack is:

[0028] The parametric equation of the right working tooth surface of the rack is:

[0029] Further, the dedendum transition curves of the gear and the rack are determined as follows:

[0030] The right dedendum of the gear end face uses a Hermite curve as a transition curve, wherein the Hermite curve is determined by points P F3 and P F4 , and tangent vectors T F3 and T F4 of points P F3 and P F4 respectively, P F3 is determined by the right working tooth profile curve of the gear and the start radius R h1 of the dedendum transition fillet, P F4 is determined by the dedendum circle radius R f1 and the inclined line passing through point P F3 with a slope of 1, and the parametric equation of the Hermite curve is:

[0031] The right dedendum of the rack end face uses a Hermite curve as a transition curve, wherein the Hermite curve is determined by points P G3 and P G4 , and tangent vectors T G3 and T G4 of points P G3 and P G4 respectively, P G3 is determined by the right working tooth profile curve of the gear and the start radius R h2 of the dedendum transition fillet, P G4 is determined by the dedendum circle radius R f2 and the inclined line passing through point P G3And the slope of the slope of 1 is determined, the Hermite curve parameter equation is:

[0032] wherein,

[0033] The technical scheme provided by the embodiment of the application has the beneficial effects that:

[0034] 1. The gear rack mechanism based on the parabola function tooth profile hyperbolic tooth line pure rolling, based on the active design method of the meshing line parameter equation, the meshing point is set at the node, the meshing line is constructed through the movement law of the meshing point, and the gear contact line is a hyperbola after being developed on the pitch cylinder surface; similarly, the rack contact line is a hyperbola at the rack pitch surface, and the contact line is an axisymmetric hyperbola after being developed on the pitch cylinder surface to eliminate the axial force, the relative sliding speed theoretical value of all meshing points on the contact line is zero, the tooth profile tooth surface is a meshing tooth surface without relative sliding, the friction loss and noise are significantly reduced, the transmission efficiency is greatly improved, the failure forms such as tooth surface wear and plastic deformation are reduced, and the service life of the transmission system is prolonged.

[0035] 2. The gear rack mechanism based on the parabola function tooth profile hyperbolic tooth line pure rolling, the gear tooth profile and the rack tooth profile are both tooth profiles established based on the parabola function, and the gear rack tooth root uses the Hermite curve, so that the bending strength of the tooth root is enhanced, the gear is not easy to break, and the service life of the gear rack is enhanced.

[0036] 3. The gear rack mechanism based on the parabola function tooth profile hyperbolic tooth line pure rolling, the gear coincidence degree is designed freely, the structure shape of the tooth profile can be determined by setting the coincidence degree value, the uniform distribution of load is realized, and the dynamic characteristics are improved. BRIEF DESCRIPTION OF DRAWINGS

[0037] Fig. 1 is a schematic view of the gear rack mechanism based on the parabola function tooth profile hyperbolic tooth line pure rolling in the embodiment 1 of the application;

[0038] Fig. 2 is a schematic view of the space meshing coordinate system of the gear rack mechanism based on the parabola function tooth profile hyperbolic tooth line pure rolling in the embodiment 1;

[0039] Fig. 3 is a schematic view of the end surface of the gear rack mechanism based on the parabola function tooth profile hyperbolic tooth line pure rolling in the embodiment 1;

[0040] Fig. 4 is a schematic view of the tooth profile of the gear and the rack in the embodiment 1;

[0041] Fig. 5 is a schematic view of the gear in the embodiment 1;

[0042] Fig. 6 is a schematic view of the rack in the embodiment 1;

[0043] Fig. 7 is a schematic diagram of a parabolic function profile hypoid gear rack mechanism in Example 2.

[0044] In the figure: 1, driver; 2, input shaft; 3, coupling; 4, output shaft; 5, gear; 6, rack; 7, meshing line; 8, pitch cylinder; 9, rack pitch surface; 10, gear contact line; 11, rack contact line; 12, root transition curve; 13, end face working profile curve. DETAILED DESCRIPTION

[0045] In order to make the objects, technical solutions and advantages of the present application clearer, the following further describes the embodiments of the present application with reference to the accompanying drawings. The following introduces a relatively preferred one of the multiple possible embodiments of the present application, which is intended to provide a basic understanding of the present application, but is not intended to identify key or decisive elements or limit the scope of protection.

[0046] In all the examples shown and discussed herein, any specific values should be interpreted as merely exemplary and not as a limitation. Thus, other examples of the exemplary embodiments can have different values.

[0047] Techniques, methods, and equipment known to those of ordinary skill in the relevant art can not be discussed in detail, but should be considered part of the specification where appropriate.

[0048] It should be noted that like reference numerals and letters refer to like items in the following drawings, and thus once an item is defined in one drawing, it need not be discussed further in subsequent drawings. It is to be understood, however, that the sizes of the components shown in the drawings can not be drawn to scale for ease of description.

[0049] In the description of the present application, it should be noted that the circuits and electronic components and modules involved in the present application are all prior art, which can be implemented by those skilled in the art without further description.

[0050] It should be further noted that unless otherwise explicitly specified and limited, the terms "mounting", "connecting" should be interpreted broadly, for example, it can be fixed connection, or detachable connection, or integral connection; it can be mechanical connection, or electrical connection; it can be direct connection, or indirect connection through intermediate medium; it can be internal communication of two elements. For those of ordinary skill in the art, the specific meaning of the above terms in the present application can be understood according to the specific circumstances.

[0051] Example 1:

[0052] The embodiment of the present application provides a kind of based on parabola function profile hyperbolic curve tooth trace pure rolling gear rack mechanism, including gear 5 and rack 6, the end face profile of the gear 5 and the rack 6 is composed of end face working tooth profile curve 13 and dedendum transition curve 12, the end face profile of the gear 5 and the rack 6 are both left and right two sides symmetry;The end face working tooth profile curve 13 of the gear 5 and the rack 6 is parabola;Gear 5 and the rack 6 have hyperbolic curve tooth line structure;At least one pair of gear teeth engagement point of the gear 5 and the rack 6 is located at node to realize pure rolling meshing contact, the trajectory formed by the engagement point of the gear 5 and the rack 6 is formed two contact lines in the gear surface of the gear 5 and the rack 6 respectively, gear contact line 10, rack contact line 11.The gear contact line 10 is the curve that has hyperbolic curve line type after being developed along pitch cylinder 8, the rack contact line 11 is the curve that has hyperbolic curve line type after being developed along rack pitch surface 9, and the gear surface of the gear 5 and the rack 6 is that end face profile is formed along contact line regular scanning.

[0053] Please refer to Figure 1, the embodiment of the present application provides a kind of end face parabola profile hyperbolic curve tooth line gear rack mechanism, the coincidence degree of the gear 5 and the rack 6 is ε=1.6, gear 5 and rack 6 form a pair of gear rack pair, gear 5 is connected output shaft 4, input shaft 2 is fixedly connected with output shaft 4 by shaft coupling 3, input shaft 2 is fixedly connected with driver 1, rack 6 is connected with driven load.

[0054] Please refer to Figure 1~6, the pitch cylinder 8 radius of gear is R1, the rack 6 tip circle radius is R a1 , dedendum circle radius is R f1 , gear dedendum cylinder outer surface is uniformly distributed with hyperbolic curve tooth structure, the gear contact line 10 is axisymmetric hyperbolic curve after being developed along gear pitch cylinder surface.The rack contact line 11 is axisymmetric hyperbolic curve after being developed along rack pitch surface 9.The gear end face profile is sequentially composed of parabola 11, left side dedendum transition curve 12, i.e. Hermite curve from tooth tip to dedendum.

[0055] Please refer to the space meshing coordinate system schematic diagram of the embodiment of the present application based on parabola function profile hyperbolic curve tooth line pure rolling gear rack mechanism in Figure 2, the end face working profile of the gear 5 and the rack 6 and dedendum transition fillet are left and right two sides symmetry, end face left profile is obtained by symmetry of end face right profile, and left dedendum transition fillet can be obtained by symmetry of right dedendum transition fillet.The tooth tip control point P U3 of the right side end face working profile of the gear 5 is determined by the intersection of tip circle radius R a1 and end face working profile;The dedendum transition start control point P F3 of the right side end face working profile of the gear 5 is determined by the intersection of dedendum transition start circle radius R h1The point P at the tooth root control of the tooth root transition curve 12 on the right end face of gear 5 is determined by the intersection with the working tooth profile of 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 6. U2 The length R of the tooth tip position of rack 6 a2 The starting control point P for the tooth root transition of the working tooth profile on the right end face of rack 6 is determined by the intersection with the end face working tooth profile. G3 The length R of the transition from the root of the 6th tooth of the rack 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 rack 6 is also defined. 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.

[0056] 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.

[0057] The contact lines of the gear 5 and the rack 6, specifically the gear contact line 10 and the rack contact line 11, are determined according to the following method:

[0058] 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 5. k The axis z2 coincides with the meshing line of the gear 5 and the rack 6, and the axis z2 is on the rack 6, at a distance z k Shaft has distance, 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 5, and the coordinate system O2-x2,y2,z2 is fixed to the rack 6. The gear 5 rotates around the z0 axis with a uniform angular velocity ω1, and the rack 6 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 5, and the coordinate system O2-x2,y2,z2 moves along the y2 axis with the rack 6.

[0059] In coordinate system O k -x k ,y k ,z k In this context, let the parametric equation of the line of action 7 of the meshing point motion of the gear 5 and the rack 6 be:

[0060] The relationship between the rotation angle of the gear 5 and the movement of the rack 6 is:

[0061] When the meshing point M moves along the meshing line 7, gear contact lines 10 and rack contact lines 11 are formed on the tooth surfaces of the gear 5 and the rack 6, respectively; according to the principle of coordinate transformation, the coordinate transformation matrix of the three space coordinate systems O0-x0,y0,z0, O1-x1,y1,z1 and O2-x2,y2,z2 is:

[0062] M 1k = M 10 × M 0k (3)

[0063] wherein,

[0064] In formula (4) and (6), R1 is the pitch circle radius of the gear 5, is the rotation angle of the gear 5;

[0065] The parametric equation of the gear contact line 10 of the tooth surface of the gear 5 is obtained from formula (1) and (5):

[0066] The parametric equation of the rack contact line 11 of the tooth surface of the rack 6 is obtained from formula (1) and (4):

[0067] Further, the end surface tooth profile of the gear 5 and the rack 6 is determined by the following method:

[0068] In the coordinate system O k -x k ,y k ,z k , the parametric equation of the right side working tooth profile of the gear 5 is:

[0069] The parametric equation of the right side working tooth profile of the rack 6 is:

[0070] In the coordinate system O k -x k ,y k ,z k , the parametric equation of the left side working tooth profile of the rack 6 is:

[0071] In the coordinate system O1-x1,y1,z1, the parametric equation of the right side working tooth profile of the gear 5 is:

[0072] The parameter equation of the left side working tooth profile of the gear 5 in the coordinate system O1-x1, y1, z1 is:

[0073] Further, the tooth surfaces of the gear 5 and the rack 6 are determined by the following method:

[0074] The tooth surface of the gear 5 is formed by sweeping along the motion law of the meshing point M, and the parameter equation of the left side working tooth surface of the gear 5 is:

[0075] The parameter equation of the right side working tooth surface of the gear 5 is:

[0076] The tooth surface of the rack 6 is formed by realizing the motion trajectory along the rack contact curve, and the parameter equation of the left side working tooth surface of the rack 6 is:

[0077] The parameter equation of the right side working tooth surface of the rack 6 is:

[0078] Further, the dedendum transition curve 12 of the gear 5 and the rack 6 is determined by the following method:

[0079] The right side dedendum of the end surface of the gear 5 uses a Hermite curve as the transition curve, wherein the Hermite curve is determined by the tangent vectors T F3 and T F4 of points P F3 and P F4 , the point P F3 is determined by the right side working tooth profile curve of the gear 5 and the start radius R F4 of the dedendum fillet, the point P F3 is determined by the dedendum radius R h1 and the inclined line passing through the point P F4 with a slope of 1, and the parameter equation of the Hermite curve is:

[0080] The right side dedendum of the end surface of the rack 6 uses a Hermite curve as the transition curve, wherein the Hermite curve is determined by the tangent vectors T G3 and T G4 of points P G3 and P G4 , the point P G3 is determined by the right side working tooth profile curve of the gear 5 and the start radius R G4 of the dedendum fillet, the point P G3 is determined by the dedendum radius R h2 and the inclined line passing through the point P G4 with a slope of 1, and the parameter equation of the Hermite curve is:h2 P G4 is determined by the dedendum radius R f2 and the line passing through point P G3 with a slope of 1, and the Hermite curve parameter equation is:

[0081] wherein,

[0082] In all the above formulas:

[0083] p - parabola parameter;

[0084] x - motion parameter variable of the meshing point M, and x ∈ [0, Δx];

[0085] Δx - maximum value of the motion parameter variable of the meshing point;

[0086] - linear proportionality coefficient of the meshing point motion;

[0087] i - coincidence degree

[0088] m t - face module;

[0089] Z1 - number of gear teeth;

[0090] Z2 - number of rack teeth passed after one rotation of the gear;

[0091] T H - Hermite type line parameter, 0.2 ≤ T H ≤ 1.5;

[0092] t H - value range of the Hermite type line, 0 ≤ t H ≤ 1;

[0093] P F3 - intersection point of the gear transition fillet start radius and the gear working tooth profile parameter equation;

[0094] P F4 - intersection point of the dedendum radius and the line passing through point P F3 with a slope of 1;

[0095] P G3 - intersection point of the rack transition fillet start length and the rack working tooth profile parameter equation;

[0096] P G4 - intersection point of the rack dedendum position length and the line passing through point P G3 with a slope of 1;

[0097] T P1- unit tangent vector of point P1;

[0098] T P2 - unit tangent vector of point P2;

[0099] T G1 - unit tangent vector of point G1;

[0100] T G2 - unit tangent vector of point G2;

[0101] x P (P F3 ) - x coordinate of point P F3 ;

[0102] y p (P F3 ) - y coordinate of point P F3 ;

[0103] z p (P F3 ) - z coordinate of point P F3 ;

[0104] x p (P F4 ) - x coordinate of point P F4 ;

[0105] y p (P F4 ) - y coordinate of point P F4 ;

[0106] z p (P F4 ) - z coordinate of point P F4 ;

[0107] x G (P G3 ) - x coordinate of point P G3 ;

[0108] y G (P G3 ) - y coordinate of point P G3 ;

[0109] z G )P G3 ) - z coordinate of point P G3 ;

[0110] x G (P G4 ) - x coordinate of point P G4 ;

[0111] y G (P G4 ) - y coordinate of point PG4 the y coordinate of point P

[0112] z G the z coordinate of point P G4 - the z coordinate of point P G4

[0113] Δd - face width coefficient

[0114] b - the tooth width of the gear, b = Δd x 2R1; (21)

[0115] α t - the face pressure angle, α t = 20°

[0116] - the addendum coefficient,

[0117] - the dedendum coefficient,

[0118] R1 - the pitch radius of the gear, R1 = m t Z1 / 2; (22)

[0119] R2 - the pitch line of the rack

[0120] a - the center distance of the rack and gear, a = R1 + R2; (23)

[0121] h a - the addendum,

[0122] h f - the dedendum,

[0123] R a1 - the addendum circle radius of the gear, R a1 = R1 + h a ; (26)

[0124] R f1 - the dedendum circle radius of the gear, R f1 = R1 - h f ; (27)

[0125] R h1 - the start radius of the transition fillet of the gear, R h1 = R1 - h a ; (28)

[0126] R a2 ​- rack tooth top position length, R a2 = R2 + h a ; (29)

[0127] R f2 - rack tooth root position length, R f2 = R2 - h f ; (30)

[0128] R h2 - rack transition fillet start length, R h2 = R2 - h a ; (31)

[0129] ε - degree of overlap,

[0130] p t - face pitch, p t = πm t ; (33)

[0131] The relevant parameters are respectively Z1 = 16, i = 1, m t = 2, b = 32 mm, α t = 20°, and Δx = 0.2, R1 = 16 mm, R2 = 38 mm are obtained;

[0132] Then, the contact line parameter equation and the face profile parameter equation of the gear and the rack in the example are obtained by substituting the above values into equations (1) - (33), the gear and the rack tooth surface structure are obtained, and the assembly can be performed according to the correct center distance.

[0133] Example 2

[0134] As shown in FIG. 7, the example 2 of the present application also provides another parabola function profile hyperbolic curve tooth line pure rolling gear and rack mechanism, the gear 5 is connected to the output shaft 4, the output shaft 4 is connected to the input shaft 2 through the coupling 3, the input shaft 2 and the driver 1 are fixedly connected, the rack 6 is connected to the driven load. In this embodiment, the gear 5 has 20 teeth. The rack 6 has 30 teeth, and the designed degree of overlap ε = 2. When the output shaft 4 drives the gear 5 to rotate, since the two pairs of adjacent gear and rack are in meshing state, the pre-set degree of overlap ε = 2 of the parabola function profile hyperbolic curve tooth line pure rolling gear and rack mechanism is ensured, so that at least two pairs of teeth are simultaneously engaged in transmission at each moment, thereby realizing the continuous and stable meshing transmission of the parabola function profile hyperbolic curve tooth line pure rolling gear and rack mechanism in the rotary motion.

[0135] The relevant parameters take values as follows: Z1=20, i=1, m t =2, ε=2, b=40mm, α t =20°, and Δx=0.2, R1=20mm, R2=38mm are obtained.

[0136] The above numerical values are substituted into the equations (1)-(33) to obtain the contact line parameter equations and the end surface tooth profile parameter equations of the gear and the rack in the example, and then the double-curve rack and pinion mechanism based on the parabolic function tooth profile and the hyperbolic tooth line pure rolling is obtained according to the motion law of the meshing points, and the assembly can be performed according to the correct center distance.

[0137] In this document, the front, rear, upper, lower and other orientation words are defined with the parts in the drawings and the positions of the parts relative to each other, just to express the technical solution clearly and conveniently. It should be understood that they are relative concepts, which can be changed accordingly according to different ways of use and placement, and the use of the orientation words should not limit the scope of the application.

[0138] In the case of no conflict, the above embodiments and features in the embodiments can be combined with each other. The above description is only the preferred embodiments of the present application, and is not used to limit the present application. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present application should be included in the protection scope of the present application.

Claims

1. A parabolic function profile hyperbolic curve tooth trace pure rolling gear rack mechanism, characterized in that: The gear and the rack include end face tooth profiles which are composed of end face working tooth profile curves and dedendum transition curves, and 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 parabolas; the tooth surfaces of the gear and the rack have hyperbolic tooth line structures; at least one pair of gear tooth engagement points of the gear and the rack are located at nodes to realize pure rolling engagement contact, and the engagement lines formed by the trajectories of the engagement points of the gear and the rack form two contact lines on the tooth surfaces of the gear and the rack respectively.

2. A parabolic function profile hypoid gear rack mechanism with pure rolling based on double curve tooth profile, according to claim 1, characterized in that: The contact line of the gear is a curve having a hyperbolic line type after being developed along a pitch cylinder, the contact line of the rack is a curve having a hyperbolic line type after being developed along a pitch surface, and the tooth surfaces of the gear and the rack are formed by regularly scanning the end face tooth profiles along the contact lines.

3. A parabolic function profile hypoid gear rack mechanism with pure rolling based on double curve tooth profile according to claim 1, characterized in that: The right side working tooth profile curve of the gear and the rack is formed by a parabola, and the dedendum transition curve is composed of a Hermite curve; the end face working tooth profile curve is valued in a range according to specific control points to control the start point and the end point of the working tooth profile curve; wherein the gear tooth top control point is determined by the intersection point of the addendum circle and the parabola curve, and the dedendum transition curve start control point is formed by the intersection point P of the parabola curve and the dedendum transition start circle G4 , and the dedendum transition curve contact control point is formed by the intersection point P of the dedendum circle and the inclined line with the point P G4 and the slope of 1 G3 ; the dedendum curve is formed by connecting the point P G4 and the point P G3 according to the Hermite curve equation.

4. A parabolic function profile hypoid gear rack mechanism with pure rolling based on double curve tooth trace of gear teeth profile according to claim 1, characterized in that: The contact lines of the gear and the rack are determined by the following method: O0-x0,y0,z0, O1-x1,y1,z1 and O2-x2,y2,z2 are established k -x k ,y k ,z k , O1-x1,y1,z1 and O2-x2,y2,z2, z0 axis and z1 axis coincide with the rotation axis of the gear, z2 axis coincides with the meshing line of the gear and the rack, z3 axis is on the rack, and the distance between z3 axis and z2 axis is k k ​​ distance, 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 the coordinate system O k -x k ,y k ,z k The engagement line parameter equation of the engagement point motion of the gear and the rack is: The relationship between the gear rotation angle and the rack motion is: When the engagement points move along the engagement lines, the contact lines are formed on the tooth surfaces of the gear and the rack respectively; according to the principle of coordinate transformation, the coordinate transformation matrixes of three space coordinate systems O0-x0, y0, z0, O1-x1, y1, z1 and O2-x2, y2, z2 are as follows: M 1k = M 10 x M 0k (3) wherein, In formulae (4) and (6), R1is the pitch circle radius of the gear, is the rotation angle of the gear; The parametric equation of the contact line of the gear tooth surface is obtained from the equations of (1) and (5) as follows: The parametric equation of the contact line of the rack tooth surface is obtained from the equations of (1) and (4) as follows:

5. A parabolic function profile hypoid gear rack mechanism with pure rolling based on double curve tooth trace of gear teeth profile according to claim 4, characterized in that: The end face tooth profiles of the gear and the rack are determined by the following method: In the coordinate system O k - x k , y k , z k The right side working profile parameter equation of the gear: In the coordinate system O k - x k , y k , z k The right side working profile parameter equation of the rack: In the coordinate system O k - x k , y k , z k The left side working profile parameter equation of the rack: The right side working profile parameter equation of the gear in the coordinate system O1-x1,y1,z1: The parameter equation of the left side working profile of the gear in the coordinate system O1-x1, y1, z1 is:

6. A parabolic function profile hypoid gear rack mechanism with pure rolling based on double curve tooth trace of gear teeth profile according to claim 5, characterized in that: The tooth surfaces of the gear and the rack are determined by the following method: The tooth surface of the gear is formed by sweeping along the movement law of the meshing point M, and the parameter equation of the working left tooth surface of the gear is: The parametric equation of the working right tooth surface of the gear: The rack tooth surface is formed along the motion trajectory of the rack contact curve, and a parameter equation of the working left tooth surface of the rack is: The parametric equation of the working right tooth surface of the rack:

7. A parabolic function profile hypoid gear rack mechanism with pure rolling based on double curve tooth trace of gear teeth, according to claim 6, characterized in that: The dedendum transition curves of the gear and the rack are determined by 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: The right side tooth root of the rack end face uses a Hermite curve as a transition curve, wherein the Hermite curve is determined by points P G3 and P G4 , and tangent vectors T G3 and T G4 of the points P G3 and P G4 respectively, the point P G3 is determined by a right side working tooth profile curve of the gear and a tooth root transition fillet start radius R h2 , the point P G4 is determined by a tooth root circle radius R f2 and a slant line passing through the point P G3 with a slope of 1, and a parameter equation of the Hermite curve is as follows: wherein, In all the above formulas: p is a parabola parameter; x is a motion parameter variable of the engagement point M, and x∈[0, Δx]; Δx is the maximum value of the motion parameter variable of the engagement point; is a linear proportionality coefficient of the motion of the engagement point; i is the coincidence degree m t - end module Z1 is the number of gear teeth; Z2 is the number of rack teeth after one rotation of the gear; T H are the Hermite type line parameters; t H The range of values for the Hermite type line; Δd is a face width coefficient; b is the tooth width of the gear, b=Δd×2R1; (21) a t - end face pressure angle; is a addendum coefficient; is a top clearance coefficient; R1- is the pitch radius of the gear, R1= m t Z1 / 2; (22) R2 is the pitch line of the rack; a is the center distance of the rack and the gear, a=R1+R2; (23) h a - addendum, h f - dedendum height, R a1 - gear tooth 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 - rack tooth tip position length, R a2 = R2+ h a ; (29) R f2 - rack tooth root position length, R f2 = R2 - h f ; (30) R h2 - the start length of the rack transition fillet, R h2 = R2 - h a ; (31) ε-polymerization degree, p t - face pitch, p t = πm t ; (33).

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