A pure rolling gear rack mechanism based on parabolic function tooth profile and hyperbolic tooth curve

Abstract

The present invention provides a pure rolling gear rack mechanism based on a parabolic function tooth profile hyperbolic tooth line, which relates to the field of gear transmission technology, including a gear and a rack, wherein the end face tooth profile of the gear and the rack is composed of an end face working tooth profile curve and a tooth root transition curve, and the end face tooth profiles of the gear and the rack are symmetrical on both sides; the end face working tooth profile curve of the gear and the rack is a parabola; the tooth surface of the gear and the rack has a hyperbolic tooth line structure; at least one pair of gear tooth meshing points of the gear and the rack are located at a node to achieve pure rolling meshing contact, and the meshing lines formed by the moving trajectories of the gear and rack meshing points form two contact lines on the tooth surfaces of the gear and the rack, respectively. Beneficial effects of the present invention: the gear and the rack both use symmetrical tooth lines to eliminate the axial force existing when the gear and the rack are meshing, which is beneficial to improving the service life of the bearing.

Description

Technical Field

[0001] The invention relates to the technical field of gear transmission, and in particular to a pure rolling gear rack mechanism based on a parabolic function tooth profile and a hyperbolic tooth line. Background Art

[0002] Traditional involute gears are widely used in various mechanical equipment, especially in high-speed and high-load transmission occasions, including automotive transmission systems, automotive gearboxes, ship engines, machine tool headstocks and other industrial equipment. The main function of gears is to transmit motion and power. For example, the Chinese patent with application number 201710016238.7 discloses "a concave-convex meshing circular arc gear rack mechanism without relative sliding", and the Chinese patent with application number 201710016207.1 discloses "a convex-concave meshing circular arc gear rack mechanism without relative sliding". The meshing point design in these gear rack mechanisms is located near the upper edge of the circular arc concave tooth profile, which is prone to stress concentration and tooth fracture. At the same time, because the tooth line does not use a symmetrical tooth line, there is an axial force on the gear during transmission, which increases the use requirements of the gear bearing and affects the accuracy and life of the transmission system. Summary of the invention

[0003] In view of this, in order to solve the problem of axial force in the gear rack mechanism of the existing gear transmission technology and the problem that the arc concave tooth profile is prone to stress concentration, an embodiment of the present invention provides a pure rolling gear rack mechanism based on a parabolic function tooth profile and a hyperbolic tooth line.

[0004] An embodiment of the present invention provides a pure rolling gear and rack mechanism based on a parabolic function tooth profile and a hyperbolic tooth line, comprising a gear and a rack, wherein the end face tooth profiles of the gear and the rack are composed of an end face working tooth profile curve and a tooth root transition curve, 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 a hyperbolic tooth line structure; at least one pair of gear tooth meshing points of the gear and the rack are located at a node to achieve pure rolling meshing contact, and the meshing lines formed by the moving 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.

[0005] Furthermore, the contact line of the gear is a curve with a hyperbolic shape after being unfolded along the pitch cylinder, the contact line of the rack is a curve with a hyperbolic shape after being unfolded along the 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 line.

[0006] Further, the right working tooth profile curves of the gear and the rack are both formed by parabolas, and the tooth root transition curve is composed of a Hermite curve; the value range of the end face working tooth profile curve is based on the control of the starting point and end point of the working tooth profile curve according to a specific control point; wherein the gear tooth top control point is determined by the intersection point of the tooth top circle and the parabola curve, and the tooth root transition curve start control point is formed by the intersection point P formed by the parabola curve and the tooth root transition start circle. G4 The contact control point of the tooth root transition curve is the tooth root circle and the point P G4 The intersection 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 Form 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 the 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, with a distance z k Axis Distance of R1, z k The distance between the axis and the z0 axis is R1; the coordinate system O0-x0, y0, z0 is fixedly connected to the gear, the coordinate system O2-x2, y2, z2 is fixedly connected to the rack, the gear rotates around the z0 axis at a uniform angular velocity ω1, and the rack moves along the y2 axis at a uniform linear velocity v1. After a period of time from the starting 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 the coordinate system O k -x k ,y k ,z k In the above, the meshing line parameter equation of the meshing point motion of the gear and the rack is assumed to be:

[0010]

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

[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 of the three spatial coordinate systems O0-x0, y0, z0, O1-x1, y1, z1 and O2-x2, y2, z2 are formed as follows:

[0014] M 1k =M 10 ×M 0k (3)

[0015]

[0016] in,

[0017]

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

[0019] The parameter equation of the contact line of the gear tooth surface is obtained by equations (1) and (5):

[0020]

[0021] The parametric equation of the contact line of the rack tooth surface is obtained by equations (1) and (4):

[0022]

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

[0024] In the coordinate system O k -x k ,y k ,z k The right working tooth profile parameter equation of the gear described in:

[0025]

[0026] In the coordinate system O k -x k ,y k ,z k The parametric equation of the right working tooth profile of the rack described in:

[0027]

[0028] In the coordinate system O k -x k ,y k ,z k The parameter equation of the left working tooth profile of the rack described in:

[0029]

[0030] The parametric equation of the right working tooth profile of the gear in the coordinate system O1-x1, y1, z1 is:

[0031]

[0032] The parametric equation of the left working tooth profile of the gear in the coordinate system O1-x1, y1, z1 is:

[0033]

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

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

[0036]

[0037] The parametric equation of the working right tooth surface of the gear:

[0038]

[0039] The formation of the rack tooth surface is achieved along the motion trajectory of the rack contact curve. The parameter equation of the working left tooth surface of the rack is:

[0040]

[0041] The parametric equation of the working right tooth surface of the rack:

[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 the transition curve, where the Hermite curve is 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 the radius R of the working tooth profile curve on the right side of the gear and the root transition fillet h1 Decision, P F4 The point is the radius of the tooth root circle R f1 and passing through point P F3The Hermite curve parameter equation is determined by the slope of 1:

[0045]

[0046] The right tooth root of the rack end face uses the Hermite curve as the transition curve, where the Hermite curve is 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 the radius R of the working tooth profile curve on the right side of the gear and the root transition fillet h2 Decision, P G4 The point is the radius of the tooth root circle R f2 and passing through point P G3 The Hermite curve parameter equation is determined by the slope of 1:

[0047]

[0048] in,

[0049]

[0050] The beneficial effects brought about by the technical solution provided by the embodiment of the present invention are:

[0051] 1. The present invention is a pure rolling gear rack mechanism based on a parabolic function tooth profile and hyperbolic tooth line. Based on an active design method of a meshing line parameter equation, the meshing point is set at a node, and the meshing line is constructed according to the motion law of the meshing point. The gear contact line is a hyperbola after being unfolded on the pitch cylindrical surface. Similarly, the rack contact line is a hyperbola at the rack pitch surface, and the contact line is an axisymmetric hyperbola after being unfolded on the pitch cylindrical surface to eliminate the axial force, which can significantly reduce the friction loss of the bearing, thereby improving the transmission efficiency.

[0052] 2. The present invention is a pure rolling gear rack mechanism based on a parabolic function tooth profile and a hyperbolic tooth line. Both the gear tooth profile and the rack tooth profile are tooth profiles established based on a parabolic function, and the gear rack tooth root uses a Hermite curve to enhance the tooth root bending strength, making it less likely for the gear tooth root to break, thereby enhancing the service life of the gear rack.

[0053] 3. The present invention is a pure rolling gear rack mechanism based on a parabolic function tooth profile and a hyperbolic tooth line. The gear overlap can be designed freely. The structural shape of the tooth profile can be determined by setting the overlap value, thereby achieving uniform load distribution and improving dynamic characteristics. BRIEF DESCRIPTION OF THE DRAWINGS

[0054] Figure 1 Schematic diagram of a pure rolling rack and pinion mechanism based on a parabolic function tooth profile and a hyperbolic tooth line in Embodiment 1 of the present invention;

[0055] Figure 2 It is a schematic diagram of a spatial meshing coordinate system of a pure rolling gear rack mechanism based on a parabolic function tooth profile and a hyperbolic tooth line in Example 1;

[0056] Figure 3 It is a schematic end view of a pure rolling rack and pinion mechanism based on a parabolic function tooth profile and a hyperbolic tooth line in Example 1;

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

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

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

[0060] Figure 7 It is a schematic diagram of a pure rolling rack and pinion mechanism based on a parabolic function tooth profile and a hyperbolic tooth line in Example 2.

[0061] 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. tooth root transition curve; 13. end face working tooth profile curve. DETAILED DESCRIPTION

[0062] In order to make the purpose, technical solution and advantages of the present invention clearer, the following will further describe the embodiments of the present invention in conjunction with the accompanying drawings. The following is a preferred embodiment of the present invention, which is intended to provide a basic understanding of the present invention, but is not intended to confirm the key or decisive elements of the present invention or to limit the scope of protection.

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

[0064] Technologies, methods, and equipment known to ordinary technicians in the relevant art may not be discussed in detail, but where appropriate, the technologies, methods, and equipment should be considered as part of the specification.

[0065] It should be noted that similar reference numerals and letters represent similar items in the following drawings, so once an item is defined in one drawing, it does not need to be further discussed in the subsequent drawings. At the same time, it should be understood that for ease of description, the sizes of the various parts shown in the drawings are not drawn according to the actual proportional relationship.

[0066] In the description of the present invention, it should be noted that the circuits, electronic components and modules involved in the present invention are all prior art and can be fully implemented by those skilled in the art without further elaboration.

[0067] It should be further explained that, unless otherwise clearly specified and limited, the terms "installation" and "connection" should be understood in a broad sense, for example, it can be a fixed connection, a detachable connection, or an integral connection; it can be a mechanical connection or an electrical connection; it can be a direct connection or an indirect connection through an intermediate medium, or it can be the internal communication of two components. For ordinary technicians in this field, the specific meanings of the above terms in the present invention can be understood according to specific circumstances.

[0068] Embodiment 1:

[0069] An embodiment of the present invention provides a pure rolling gear rack mechanism based on a parabolic function tooth profile and a hyperbolic tooth line, comprising a gear 5 and a rack 6, wherein 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, and 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 parabola; the tooth surfaces of the gear 5 and the rack 6 have a hyperbolic tooth line structure; at least one pair of gear tooth meshing points of the gear 5 and the rack 6 are located at a node to achieve pure rolling meshing contact, and the meshing lines formed by the moving trajectories of the meshing points of the gear 5 and the rack 6 form two contact lines on the tooth surfaces of the gear 5 and the rack 6, namely, a gear contact line 10 and a rack contact line 11. The gear contact line 10 is a curve having a hyperbolic shape after being unfolded along the pitch cylinder 8, the rack contact line 11 is a curve having a hyperbolic shape after being unfolded along the rack pitch surface 9, and the tooth surfaces of the gear 5 and the rack 6 are formed by scanning the end face tooth profile along the contact line regularity.

[0070] Please refer to Figure 1 In a gear rack mechanism with an end parabolic tooth profile and a hyperbolic tooth line provided in Example 1 of the present invention, the overlap degree between the gear 5 and the rack 6 is ε=1.6, the gear 5 and the rack 6 form a pair of gear rack pairs, the gear 5 is connected to the output shaft 4, the input shaft 2 is fixedly connected to the output shaft 4 through the coupling 3, the input shaft 2 is fixedly connected to the driver 1, and the rack 6 is connected to the driven load.

[0071] Please refer to Figures 1 to 6, the radius of the pitch cylinder 8 of the gear is R1, and the radius of the top circle of the rack 6 is R a1 , the root radius is R f1 The outer surface of the gear tooth root cylinder is uniformly distributed with gear teeth of hyperbolic tooth line structure. The gear contact line 10 is an axisymmetric hyperbola after being unfolded along the gear pitch cylindrical surface. The rack contact line 11 is an axisymmetric hyperbola after being unfolded along the rack pitch surface 9. The gear end face tooth profile is composed of a parabola 11 and a left end face tooth root transition curve 12, i.e., a Hermite curve, from the tooth top to the tooth root.

[0072] Please refer to Figure 2 The present invention is a schematic diagram of a spatial meshing coordinate system of a pure rolling gear rack mechanism based on a parabolic function tooth profile hyperbolic tooth line. The end face working tooth profile and the tooth root transition fillet of the gear 5 and the rack 6 are symmetrical on both sides. The left side tooth profile of the end face is symmetrically obtained by the right side tooth profile of the end face. Similarly, the left side tooth root transition fillet can be symmetrically obtained by the right side tooth root transition fillet. The tooth top control point P of the right side end face working tooth profile of the gear 5 is U3 The radius of the tooth tip circle R a1 The tooth root transition start control point P of the right end face working tooth profile of gear 5 is determined by the intersection with the end face working tooth profile; F3 Radius R of the circle starting from tooth root transition h1 The tooth root control point P of the tooth root transition curve 12 on the right side of gear 5 is determined by the intersection with the end face working tooth profile. F4 The root circle radius and the point P F3 And the intersection of the oblique lines with a slope of 1. Similarly, the tooth top control point P of the working tooth profile on the right end face of the rack 6 U2 The length R of the rack 6 tooth top position a2 The tooth root transition start control point P of the right end face working tooth profile of rack 6 is determined by the intersection with the end face working tooth profile; G3 Length R of the transition starting position from the tooth root of rack 6 h2 The tooth root control point P of the right end tooth root transition curve 12 of the rack 6 is determined by the intersection with the end working tooth profile; G4 Length R from tooth root transition start position h2 and passing point P G3 The intersection of the oblique lines with slope 1.

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

[0074] The contact line between the gear 5 and the rack 6, the gear contact line 10, and the rack contact line 11 are determined according to the following method:

[0075] 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 the z1 axis coincide with the rotation axis of the gear 5. k The axis coincides with the meshing line between the gear 5 and the rack 6, and the z2 axis is on the rack 6, with a distance z k Axis Distance of R1, z k The distance between the axis and the z0 axis is R1; the coordinate system O0-x0, y0, z0 is fixedly connected to the gear 5, the coordinate system O2-x2, y2, z2 is fixedly connected to the rack 6, the gear 5 rotates around the z0 axis at a uniform angular velocity ω1, and the rack 6 moves along the y2 axis at a uniform linear velocity v1. After a period of time from the starting 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;

[0076] In the coordinate system O k -x k ,y k ,z k In the figure, the parameter equation of the meshing line 7 of the meshing point motion of the gear 5 and the rack 6 is:

[0077]

[0078] The relationship between the rotation angle of the gear 5 and the movement of the rack 6 is:

[0079]

[0080] When the meshing point M moves along the meshing line 7, a gear contact line 10 and a rack contact line 11 are formed on the tooth surfaces of the gear 5 and the rack 6, respectively. According to the coordinate transformation principle, the coordinate transformation matrices of the three spatial coordinate systems O0-x0, y0, z0, O1-x1, y1, z1 and O2-x2, y2, z2 are formed as follows:

[0081] M 1k =M 10 ×M 0k (3)

[0082]

[0083] in,

[0084]

[0085] In equations (4) and (6), R1 is the pitch radius of gear 5, is the rotation angle of gear 5;

[0086] The parameter equation of the contact line of the gear 5 tooth surface 10 is obtained by equations (1) and (5):

[0087]

[0088] The parameter equation of the contact line of the tooth surface of the rack 6 and the rack contact line 11 is obtained by equations (1) and (4):

[0089]

[0090] Furthermore, the end face tooth profiles of the gear 5 and the rack 6 are determined by the following method:

[0091] In the coordinate system O k -x k ,y k ,z k The right working tooth profile parameter equation of gear 5 is as follows:

[0092]

[0093] The parameter equation of the right working tooth profile of the rack 6 is:

[0094]

[0095] In the coordinate system O k -x k ,y k ,z k The parameter equation of the left working tooth profile of the rack 6 is:

[0096]

[0097] The parametric equation of the right working tooth profile of the gear 5 in the coordinate system O1-x1, y1, z1 is:

[0098]

[0099] The parameter equation of the left working tooth profile of the gear 5 in the coordinate system O1-x1, y1, z1 is:

[0100]

[0101] Furthermore, the tooth surfaces of the gear 5 and the rack 6 are determined as follows:

[0102] The tooth surface of the gear 5 is formed by sweeping along the motion law of the meshing point M. The parameter equation of the working left tooth surface of the gear 5 is:

[0103]

[0104] The parameter equation of the working right tooth surface of the gear 5 is:

[0105]

[0106] The formation of the tooth surface of the rack 6 is achieved along the motion trajectory of the rack contact curve. The parameter equation of the working left tooth surface of the rack 6 is:

[0107]

[0108] The parameter equation of the working right tooth surface of the rack 6 is:

[0109]

[0110] Furthermore, the tooth root transition curve 12 of the gear 5 and the rack 6 is determined as follows:

[0111] The right tooth root of the end face of the gear 5 uses the Hermite curve as the transition curve, wherein the Hermite curve is 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 the radius R of the working tooth profile curve on the right side of gear 5 and the root transition fillet h1 Decision, P F4 The point is the radius of the tooth root circle R f1 and passing through point P F3 The Hermite curve parameter equation is determined by the slope of 1:

[0112]

[0113] The right tooth root of the rack 6 uses a Hermite curve as a transition curve, wherein the Hermite curve is 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 the radius R of the working tooth profile curve on the right side of gear 5 and the root transition fillet h2 Decision, P G4 The point is the radius of the tooth root circle R f2 and passing through point P G3 The Hermite curve parameter equation is determined by the slope of 1:

[0114]

[0115] in,

[0116]

[0117] In all the above formulas:

[0118] t-parabolic function parameter equation parameter;

[0119] p - set the parabola parameter value;

[0120] σ-motion parameter variable of meshing point M, and σ∈[0,Δσ];

[0121] Δσ-the maximum value of the motion parameter variable of the meshing point;

[0122] - is the linear proportional coefficient of the meshing point movement;

[0123] i-coincidence

[0124] m t -end modulus;

[0125] Z1-number of gear teeth;

[0126] Z2-number of teeth on the rack after the gear rotates one circle;

[0127] T H is the Hermite type line parameter, 0.2≤T H ≤1.5;

[0128] t H is the value range of the Hermite type line, 0≤t H ≤1;

[0129] P F3 -The intersection point of the starting radius of the gear transition fillet and the gear working tooth profile parameter equation;

[0130] P F4 - Radius of tooth root circle and point P F3 The intersection of the oblique lines with a slope of 1;

[0131] P G3 -The intersection point of the rack transition fillet starting length and the rack working tooth profile parameter equation;

[0132] P G4 -The length of the rack tooth root and the point P G3 The intersection of the lines with a slope of 1;

[0133] T P1 -Unit tangent vector of point P1;

[0134] TP2 -Unit tangent vector of point P2;

[0135] T G1 -Unit tangent vector of point G1;

[0136] T G2 -Unit tangent vector of point G2;

[0137] x P (P F3 )-Point P F3 The x coordinate of

[0138] y p (P F3 )-Point P F3 The y coordinate of

[0139] z p (P F3 )-Point P F3 The z coordinate of

[0140] x p (P F4 )-Point P F4 The x coordinate of

[0141] y p (P F4 )-Point P F4 The y coordinate of

[0142] z p (P F4 )-Point P F4 The z coordinate of

[0143] x G (P G3 )-Point P G3 The x coordinate of

[0144] y G (P G3 )-Point P G3 The y coordinate of

[0145] z G (P G3 )-Point P G3 The z coordinate of

[0146] x G (P G4 )-Point P G4 The x coordinate of

[0147] y G (P G4 )-Point P G4 The y coordinate of

[0148] ZG (P G4 )-Point P G4 The z coordinate of

[0149] Δd-face width coefficient;

[0150] b-gear tooth width, b = Δd × 2R1; (21)

[0151] α t - End pressure angle, α t =20°;

[0152] - tooth top height coefficient,

[0153] -Headspace coefficient,

[0154] R1- is the pitch radius of the gear, R1 = m t Z1 / 2; (22)

[0155] R2- is the pitch line of the rack;

[0156] a-center distance between rack and gear, a=R1+R2; (23)

[0157] h a - Tooth top height,

[0158] h f -Tooth root height,

[0159] R a1 - Gear tooth tip radius, R a1 =R1+h a ; (26)

[0160] R f1 -Gear tooth root radius, R f1 =R1-h f ; (27)

[0161] R h1 - Gear transition fillet start radius, R h1 =R1-h a ; (28)

[0162] R a2 - Length of rack tooth tip, R a2 =R2+h a ; (29)

[0163] R f2 - Length of rack tooth root, R f2=R2-h f ; (30)

[0164] R h2 - Rack transition fillet start length, R h2 =R2-h a ; (31)

[0165] ε-coincidence,

[0166] p t - Face pitch, p t =πm t ; (33)

[0167] The relevant parameters are Z1=16, i=1, m t =2, b=32mm,α t =20°, Δσ=0.2, R1=16mm, R2=38mm;

[0168] Then, substituting the above values ​​into equations (1)-(33) can obtain the contact line parameter equation and the end face tooth profile parameter equation of the gear and rack in this example, and obtain the gear and rack tooth surface structure, which can be assembled according to the correct center distance.

[0169] Embodiment 2:

[0170] like Figure 7 As shown, embodiment 2 of the present invention also provides another pure rolling rack-and-pinion mechanism based on parabolic function tooth profile and hyperbolic tooth line, wherein 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 is fixedly connected to the driver 1, and the rack 6 is connected to the driven load. In this embodiment, the number of teeth of the gear 5 is 20. The number of teeth of the rack 6 is 30, and the designed overlap ε=2. When the output shaft 4 drives the gear 5 to rotate, since the two pairs of adjacent gear racks where the gear 5 and the rack 6 are installed are in a meshing state, the preset overlap ε=2 of the pure rolling rack-and-pinion mechanism based on parabolic function tooth profile and hyperbolic tooth line is ensured, thereby ensuring that at each instant, at least two pairs of gear teeth are simultaneously involved in the meshing transmission, thereby realizing the continuous and stable meshing transmission of the pure rolling rack-and-pinion mechanism based on parabolic function tooth profile and hyperbolic tooth line in the rotational motion.

[0171] The relevant parameters are: Z1 = 20, i = 1, m t =2,ε=2, b=40mm,α t =20°, Δσ=0.2, R1=20mm, R2=38mm;

[0172] Substituting the above values ​​into equations (1)-(33) can obtain the contact line parameter equation and the end tooth profile parameter equation of the gear and rack in this example. Then, according to the motion law of the meshing point, a pure rolling gear rack mechanism with a hyperbolic tooth line based on a parabolic function tooth profile can be obtained, and it can be assembled according to the correct center distance.

[0173] In this article, the directional words such as front, back, top, and bottom involved are defined by the positions of the parts in the drawings and the positions of the parts relative to each other, just for the sake of clarity and convenience in expressing the technical solution. It should be understood that they are relative concepts and can change accordingly according to different ways of use and placement, and the use of the directional words should not limit the scope of protection claimed in this application.

[0174] In the absence of conflict, the above embodiments and features in the embodiments may be combined with each other. The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention shall be included in the protection scope of the present invention.

Claims

1. A pure rolling gear rack mechanism based on a parabolic function tooth profile and a hyperbolic tooth line, characterized in that: It comprises a gear and a rack, wherein the end face tooth profiles of the gear and the rack are composed of an end face working tooth profile curve and a tooth root transition curve, 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 parabolic; the tooth surfaces of the gear and the rack have a hyperbolic tooth line structure; at least one pair of gear tooth meshing points of the gear and the rack are located at a node to achieve pure rolling meshing contact, and the meshing lines formed by the moving 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; The right working tooth profile curves of the gear and the rack are both formed by parabolas, and the tooth root transition curve is formed by Hermit e Curve composition; the range of the end working tooth profile curve is based on the specific control point to control the starting and ending points of the working tooth profile curve; the gear tooth top control point is determined by the intersection point of the tooth top circle and the parabola curve, and the tooth root transition curve start control point is formed by the intersection of the parabola curve and the tooth root transition start circle to form the intersection point P G4 The contact control point of the tooth root transition curve is the tooth root circle and the point P G4 The intersection 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 tooth root curve; The contact line between the gear and the rack is determined by the following method: Create 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 the z1 axis coincide with the rotation axis of the gear. k The axis coincides with the meshing line between the gear and the rack, and the z2 axis is on the rack, at a distance z k Axis The distance, z k The distance between the axis and the z0 axis is R1; the coordinate system O0-x0, y0, z0 is fixedly connected to the gear, the coordinate system O2-x2, y2, z2 is fixedly connected to the rack, the gear rotates around the z0 axis at a uniform angular velocity ω1, and the rack moves along the y2 axis at a uniform linear velocity v1. After a period of time from the starting 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 In the equation, the meshing line parameter equation of the meshing point motion of the gear and the rack is: The relationship between the gear rotation angle and the rack motion is: 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 of the three spatial coordinate systems O0-x0, y0, z0, O1-x1, y1, z1 and O2-x2, y2, z2 are formed as follows: M 1k =M 10 ×M 0k (3) in, In formulas (4) and (6), R1 is the pitch radius of the gear, is the gear rotation angle; The parameter equation of the contact line of the gear tooth surface is obtained by equations (1) and (5): The parametric equation of the contact line of the rack tooth surface is obtained by equations (1) and (4): 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 working tooth profile parameter equation of the gear described in: In the coordinate system O k -x k ,y k , z k The parametric equation of the right working tooth profile of the rack described in: In the coordinate system O k -x k ,y k , z k The parameter equation of the left working tooth profile of the rack described in: The parametric equation of the right working tooth profile of the gear in the coordinate system O1--x1, y1, z1 is: The parametric equation of the left working tooth profile of the gear in the coordinate system O1-x1, y1, z1 is: The tooth surfaces of the gear and the rack are determined as follows: The gear tooth surface is formed by sweeping along the motion law of the meshing point M. 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 is: The formation of the rack tooth surface is achieved along the motion trajectory of the rack contact curve. The parameter equation of the working left tooth surface of the rack is: The parametric equation of the working right tooth surface of the rack: The tooth root transition curves of the gear and the rack are determined as follows: The right tooth root of the gear end face is Hermit e The curve is used as a transition curve, where the Hermite curve is 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 the radius R of the working tooth profile curve on the right side of the gear and the root transition fillet h1 Decision, P F4 The point is the radius of the tooth root circle R f1 and passing through point P F3 And the slope is 1, Hermit e The curve parameter equation is: The right tooth root of the rack end face uses the Hermite curve as the transition curve, wherein the Hermite curve is 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 the radius R of the working tooth profile curve on the right side of the gear and the root transition fillet h2 Decision, P G4 The point is the radius of the tooth root circle R f2 and passing through point P G3 And the slope is 1, Hermit e The curve parameter equation is: in, In all the above formulas: t-cycloid function parameter equation parameters; p - set the parabola parameter value; σ-motion parameter variable of meshing point M, and σ∈[0,Δσ]; Δ σ - Maximum value of the motion parameter variable of the meshing point; - is the linear proportional coefficient of the meshing point movement; m t -end modulus; Z1-number of gear teeth; T H is the Hermite type line parameter; t H The value range of the Hermite type line; Δd-face width coefficient; b-gear tooth width, b=Δd×2R1; α t - end face pressure angle; R1- is the pitch circle radius of the gear, R1 = mtZ1 / 2.

2. A pure rolling gear rack mechanism based on a parabolic function tooth profile and a hyperbolic tooth line as claimed in claim 1, characterized in that: The contact line of the gear is a curve with a hyperbolic shape after being unfolded along the pitch cylinder, the contact line of the rack is a curve with a hyperbolic shape after being unfolded along the 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 line.

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