Pure rolling gear rack mechanism based on cosine tooth profile and parabola tooth trace

Abstract

The invention provides a pure rolling gear and rack mechanism based on a cosine tooth profile and a parabola tooth trace, and relates to the technical field of gear rotation, the pure rolling gear and rack mechanism comprises 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, and the end face tooth profiles of the gear and the rack are both symmetrical on the left side and the right side; 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 trace structures; at least one pair of gear tooth meshing points of the gear and the rack are located at nodes to achieve pure rolling meshing contact, and meshing lines formed by moving tracks 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 gear rack has the beneficial effects that the gear rack adopts symmetrical tooth lines, so that the axial force existing when the gear rack is meshed is eliminated, and the service life of a bearing is prolonged.

Description

Technical Field

[0001] The invention relates to the technical field of gear rotation, and in particular to a pure rolling gear rack mechanism based on a cosine tooth profile and a parabola 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 existing gear rack 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 cosine tooth profile and a parabolic tooth curve.

[0004] An embodiment of the present invention provides a pure rolling gear rack mechanism based on a cosine tooth profile and a parabolic 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 cosine curves; the tooth surfaces of the gear and the rack have a parabolic 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 tooth surface structures of the gear and the rack are respectively formed by the movement of their respective tooth profile curves 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 cylindrical surface of the gear, and the rack is a parabola at the pitch surface.

[0006] Further, the right side end face working tooth profile curves of the gear and the rack are both formed by cosine function curves, and the tooth root transition curve is composed of Hermite curves; the shape of the end face working tooth profile curve is controlled by the intersection point of the tooth addendum circle and the cosine function curve, and the shape of the tooth root transition curve is controlled by the intersection point P of the tooth root transition start circle and the cosine function curve. G4 Control and by the root circle and through 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 axis coincides with the meshing line between the gear and the rack, 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 equation, the meshing line parameter equation of the meshing point motion of the gear and rack is:

[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 parameter equation of the working tooth profile curve on the right side of the gear is:

[0025]

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

[0027]

[0028] In the coordinate system O k -x k ,y k ,z k The parameter equation of the end working tooth profile curve on the left side of the rack is:

[0029]

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

[0031]

[0032] The parametric equation of the end working tooth profile curve of the left side 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 tooth surface of the rack is achieved along the motion trajectory of the rack contact curve. The parameter equation of the left working 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 R 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 cosine tooth profile and a parabolic tooth line. Based on an active design method of a meshing line parameter equation, a meshing point is set at a node, and a meshing line is constructed according to the motion law of the meshing point. The gear contact line is a parabola after being unfolded on the pitch cylindrical surface. Similarly, the rack contact line is a parabola at the rack pitch surface, and the contact line is an axisymmetric parabola after being unfolded on the pitch cylindrical surface to eliminate axial force, thereby greatly improving the service life of the bearing and reducing maintenance costs.

[0052] 2. The present invention is a pure rolling gear rack mechanism based on a cosine tooth profile and a parabolic tooth line. Both the gear tooth profile and the rack tooth profile are tooth profiles established based on a cosine 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 cosine tooth profile and a parabolic 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 cosine tooth profile and parabolic tooth trace 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 cosine tooth profile parabola tooth line in Example 1;

[0056] Figure 3 It is a schematic end view of a pure rolling gear rack mechanism based on a cosine tooth profile and parabolic 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 gear rack mechanism based on a cosine tooth profile and parabolic 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 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 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 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] Example 1

[0069] The present invention provides a pure rolling gear rack mechanism based on a cosine tooth profile and a parabolic tooth curve, 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 cosine curve; the tooth surfaces of the gear 5 and the rack 6 have a parabolic tooth curve 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 respectively 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 tooth surface structures of the gear 5 and the rack 6 are respectively formed by the movement of their respective tooth profile curves 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 cylindrical surface of the gear 5, and the rack 6 is a parabola at the pitch surface.

[0070] like Figure 1 As shown, in the pure rolling gear rack mechanism based on cosine tooth profile parabolic tooth line in this embodiment 1, the overlap degree of gear 5 and rack 6 is ε=1.6, gear 5 and rack 6 form a pair of gear rack pairs, 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, and 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 rack tooth top circle is Ra1 , the root circle radius is R f1 The outer surface of the gear tooth root cylinder is evenly distributed with gear teeth of parabolic tooth line structure. The gear contact line 10 is an axisymmetric parabola when unfolded along the gear pitch cylindrical surface. The rack contact line 11 is an axisymmetric parabola when unfolded along the gear pitch surface 9. The gear end face tooth profile is composed of a cosine curve 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 Schematic diagram of the spatial meshing coordinate system of a parabolic toothed gear rack mechanism with an end face cosine tooth profile of the present invention, the end face working tooth profile and the tooth root transition fillet of the gear and rack 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, and 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 U3 The radius of the tooth tip circle R a1 The tooth root transition start control point P of the right side end face working tooth profile 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 the gear face 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 U2 The rack tooth tip length R a2 The tooth root transition start control point P of the right end face working tooth profile of the rack is determined by the intersection with the end face working tooth profile; G3 Length R of the transition starting position from the rack tooth root h2 The tooth root control point P of the tooth root transition curve 12 on the right side of the rack is determined by the intersection with the end face 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 gear contact line 10 and the 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 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, 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;

[0075] In the coordinate system O k -x k ,y k ,z k In the equation, the meshing line 7 parameter equation of the meshing point motion of the gear and rack is:

[0076]

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

[0078]

[0079] 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 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:

[0080] M 1k =M 10 ×M 0k (3)

[0081]

[0082] in,

[0083]

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

[0085] The parameter equation of the contact line of the gear tooth surface is obtained by equations (1) and (5):

[0086]

[0087] The parametric equation of the contact line of the rack tooth surface is obtained by equations (1) and (4):

[0088]

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

[0090] In the coordinate system O k -x k ,y k ,z k The parameter equation of the working tooth profile curve 13 of the right end face of the gear is:

[0091]

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

[0093]

[0094] In the coordinate system O k -x k ,y k ,z k The parameter equation of the end working tooth profile curve 13 on the left side of the rack is:

[0095]

[0096] The parameter equation of the end working tooth profile curve 13 of the right side of the gear in the coordinate system O1-x1, y1, z1 is:

[0097]

[0098] The parameter equation of the end working tooth profile curve 13 of the left side of the gear in the coordinate system O1-x1, y1, z1 is:

[0099]

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

[0101] 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:

[0102]

[0103] The parametric equation of the working right tooth surface of the gear:

[0104]

[0105] The formation of the tooth surface of the rack is achieved along the motion trajectory of the rack contact curve. The parameter equation of the left working tooth surface of the rack is:

[0106]

[0107] The parametric equation of the working right tooth surface of the rack:

[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 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 F3 The Hermite curve parameter equation is determined by the slope of 1:

[0111]

[0112] 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 The Hermite curve parameter equation is determined by the slope of 1:

[0113]

[0114] in,

[0115]

[0116] In all the above formulas:

[0117] t-cosine function parameter equation parameters;

[0118] p-sets the cosine function parameter value;

[0119] σ-motion parameter variable of meshing point M, and σ∈[0,Δσ];

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

[0121] - is the linear proportional coefficient of the meshing point movement;

[0122] i-gear ratio

[0123] m t -end modulus;

[0124] Z1-number of gear teeth;

[0125] Z2-number of teeth on the rack after the gear rotates one circle;

[0126] T H is the Hermite type line parameter, 0.2≤T H ≤1.5;

[0127] t H is the value range of the Hermite type line, 0≤t H ≤1;

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

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

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

[0131] P G4 -The length of the rack tooth root and the point P G3 The intersection of the lines with a slope of 1;

[0132] T P1 -Unit tangent vector of point P1;

[0133] T P2 -Unit tangent vector of point P2;

[0134] TG1 -Unit tangent vector of point G1;

[0135] T G2 -Unit tangent vector of point G2;

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

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

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

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

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

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

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

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

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

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

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

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

[0148] Δd-face width coefficient;

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

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

[0151] - tooth top height coefficient,

[0152] -Headspace coefficient,

[0153] R1- is the pitch 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 top height,

[0157] h f -Tooth root height,

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

[0159] R f1 -Gear tooth root 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, R a2 =R2+h a ; (29)

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

[0163] Rh2 - Rack transition fillet start length, R h2 =R2-h a ; (31)

[0164] ε-coincidence,

[0165] p t - Face pitch, p t =πm t ; (33)

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

[0167] Then, 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, and obtain the gear and rack tooth surface structure, which can be assembled according to the correct center distance.

[0168] Embodiment 2:

[0169] like Figure 7 As shown, embodiment 2 of the present invention also provides another pure rolling gear rack mechanism based on cosine tooth profile parabola 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 driven load. In this embodiment, the number of teeth of gear 5 is 20. The number of teeth of rack 6 is 30, and the designed overlap ε=2. When output shaft 4 drives gear 5 to rotate, since the two pairs of adjacent gear racks where gear 5 and rack 6 are installed are in meshing state, the preset overlap ε=2 of pure rolling gear rack mechanism based on cosine tooth profile parabola tooth line ensures that at every instant, at least two pairs of gear teeth participate in meshing transmission at the same time, thereby realizing continuous and stable meshing transmission of pure rolling gear rack mechanism based on cosine tooth profile parabola tooth line in rotational motion.

[0170] 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;

[0171] 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 5 and the rack in this example. Then, according to the motion law of the meshing point, the gear tooth structure of the gear and rack can be obtained, and they can be assembled according to the correct center distance.

[0172] 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.

[0173] 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 cosine tooth profile parabolic tooth line, characterized by: It comprises 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 profile 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 cosine curve; the tooth surface of the gear and the rack has a parabolic tooth curve structure; at least one pair of gear tooth meshing points of the gear and the rack are located at a node to realize 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.

2. A pure rolling gear rack mechanism based on cosine tooth profile parabolic tooth curve according to claim 1, characterized in that: The tooth surface structures of the gear and the rack are respectively formed by the movement of their respective tooth profile curves 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 cylindrical surface of the gear, and the rack is a parabola at the pitch surface.

3. A pure rolling gear rack mechanism based on cosine tooth profile parabolic tooth curve as claimed in claim 1, characterized in that: The right side end working tooth profile curves of the gear and the rack are both formed by cosine function curves, and the tooth root transition curve is composed of Hermite curves; the shape of the end working tooth profile curve is controlled by the intersection point of the tooth top circle and the cosine function curve, and the shape of the tooth root transition curve is controlled by the intersection point P of the tooth root transition start circle and the cosine function curve. G4 Control and by the root circle and through 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.

4. A pure rolling gear rack mechanism based on cosine tooth profile parabolic tooth curve as claimed in claim 3, characterized in that: The contact line between the gear and the rack is 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 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, with 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 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):

5. A pure rolling gear rack mechanism based on cosine tooth profile parabolic tooth curve as claimed in 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 parameter equation of the working tooth profile curve of the right end face of the gear described in is: In the coordinate system O k -x k ,y k ,z k The parameter equation of the working tooth profile curve on the right end face of the rack is: In the coordinate system O k -x k ,y k ,z k The parameter equation of the end working tooth profile curve on the left side of the rack is: The parametric equation of the end working tooth profile curve of the right side of the gear in the coordinate system O1-x1, y1, z1 is: The parametric equation of the end working tooth profile curve of the left side of the gear in the coordinate system O1-x1, y1, z1 is:

6. A pure rolling gear rack mechanism based on cosine tooth profile and parabola tooth curve as claimed in claim 5, characterized in that: 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 tooth surface of the rack is achieved along the motion trajectory of the rack contact curve. The parameter equation of the left working tooth surface of the rack is: The parametric equation of the working right tooth surface of the rack:

7. A pure rolling gear rack mechanism based on cosine tooth profile and parabola tooth trace as claimed in claim 6, characterized in that: The tooth root transition curves of the gear and the rack are determined as follows: 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 F3 The Hermite curve parameter equation is determined by the slope of 1: 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 The Hermite curve parameter equation is determined by the slope of 1: in, In all the above formulas: t-cosine function parameter equation parameters p-sets the cosine function parameter value; σ-motion parameter variable of meshing point M, and σ∈[0,Δσ]; Δσ-the 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; Z2-number of teeth on the rack after the gear rotates one circle; 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; (21) α t - end face pressure angle; -tooth addendum coefficient; -Head 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-center distance between rack and gear, a=R1+R2; (23) h a - Tooth top height, h f -Tooth root height, R a1 - Gear tooth tip radius, R a1 =R1+h a ; (26) R f1 -Gear tooth root 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, R a2 =R2+h a ; (29) R f2 - Length of rack tooth root, R f2 =R2-h f ; (30) R h2 - Rack transition fillet start length, R h2 =R2-h a ; (31) ε-degree of polymerization, p t - Face pitch, p t =πm t ; (33).

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