Pure rolling gear rack mechanism based on hyperbolic cosine tooth profile and arc tangent tooth profile

Abstract

The invention provides a pure rolling gear and rack mechanism based on a hyperbolic cosine tooth profile and an arc tangent tooth profile, and relates to the technical field of gear transmission, 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 both composed of end face working tooth profile curves and tooth root transition curves, and the end face working tooth profile curves 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 hyperbolic cosine curves; tooth surfaces of the gear and the rack are provided with arc tangent curve 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 moving tracks of the meshing points of the gear and the rack form meshing lines to 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, the bearing burden is favorably reduced, and the maintenance cost is reduced.

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 hyperbolic cosine tooth profile and inverse tangent 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 application number 201710016238.7 discloses "a concave-convex meshing circular arc gear rack mechanism without relative sliding", and the Chinese patent application number 201710016207.1 discloses "a convex-concave meshing circular arc gear rack mechanism without relative sliding". The tooth profiles of these gear racks are all circular arc tooth profiles, but the meshing point design in these gear rack mechanisms is located near the upper edge of the circular arc concave tooth profile, which is easy to cause stress concentration and cause gear 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 that the arc concave tooth profile of the existing gear rack is prone to stress concentration and axial force near the upper edge, an embodiment of the present invention provides a pure rolling gear rack mechanism based on the inverse tangent tooth line of the hyperbolic cosine tooth profile.

[0004] An embodiment of the present invention provides a pure rolling gear and rack mechanism based on a hyperbolic cosine tooth profile and an inverse tangent tooth line, comprising a gear and a rack, wherein the end face tooth profiles of the gear and the rack are both composed of an end face working tooth profile curve and a tooth root transition curve, and the end face working tooth profile curves 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 hyperbolic cosine curves; the tooth surfaces of the gear and the rack have an inverse tangent curve 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 moving trajectories of the meshing points of the gear and the rack form a meshing line to 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 end face working tooth profile curves along the contact line as the contact point moves, and the contact line is an inverse tangent function curve symmetrical about the z-axis after being unfolded along the pitch cylindrical surface of the gear, and the rack is an inverse tangent function curve unfolded along the rack pitch surface.

[0006] Furthermore, the right side end face working tooth profile curves of the gear and the rack are both formed by a hyperbolic cosine function curve, and the right side tooth root transition curves of the gear and the rack are composed of a Hermite curve; the shape of the end face working tooth profile curve is controlled by the intersection point of the tooth top circle and the hyperbolic 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 hyperbolic cosine function curve. G4 control, and 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] Construct 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 two meshing points move along the meshing line, contact lines are formed on the tooth surfaces of the gear 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 parametric equation of the contact line of the gear tooth surface is obtained from equations (1) and (5):

[0020]

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

[0022]

[0023] Furthermore, the end surface working tooth profile curves of the gear and the rack are determined by the following method:

[0024] A coordinate system O is established at the meshing point between the gear and the rack end face. k -x k ,y k ,z k , the parametric equation of the working tooth profile curve on the right side of the gear is obtained:

[0025]

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

[0027]

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

[0029]

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

[0031]

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

[0033]

[0034] Furthermore, the tooth surface of the gear is formed by sweeping along the motion law of the meshing point M, and the parameter equation of the working left tooth surface of the gear is:

[0035]

[0036] The parametric equation of the working right tooth surface of the gear:

[0037]

[0038] The formation of the tooth surface of the rack is achieved by movement along the rack contact curve. The parameter equation of the working left tooth surface of the rack is:

[0039]

[0040] The parametric equation of the working right tooth surface of the rack:

[0041]

[0042] Furthermore, the right tooth root of the end face of the gear uses a Hermite curve as a 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 right working tooth profile curve of the gear and the radius R of the tooth 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:

[0043]

[0044] The right tooth root of the end face of the rack uses a Hermite curve as a transition curve, wherein the Hermite curve is from point P G3 and P G4 , and P G3 and P G4 The tangent vectors are P G3 and T G4 Decision, P G3 The point is the right working tooth profile curve of the gear and the radius R of the tooth 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:

[0045]

[0046] in,

[0047]

[0048] The beneficial effects brought about by the technical solution provided by the embodiment of the present invention are:

[0049] 1. The present invention discloses a pure rolling gear rack mechanism based on a hyperbolic cosine tooth profile and an inverse tangent tooth line, and an active design method based on 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 an inverse tangent curve after being unfolded on the pitch cylindrical surface. Similarly, the rack contact line is an inverse tangent curve at the rack pitch surface, and the contact line is an axisymmetric inverse tangent function curve after being unfolded on the pitch cylindrical surface to eliminate axial force, reduce bearing burden, and reduce maintenance costs.

[0050] 2. The present invention is a pure rolling gear rack mechanism based on a hyperbolic cosine tooth profile and an inverse tangent tooth line. Both the gear tooth profile and the rack tooth profile are tooth profiles established based on a hyperbolic 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.

[0051] 3. The pure rolling gear rack mechanism based on the hyperbolic cosine tooth profile and inverse tangent tooth line of the present invention has a free design of gear overlap. 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

[0052] Figure 1 Schematic diagram of a pure rolling rack and pinion mechanism based on a hyperbolic cosine tooth profile and inverse tangent tooth line in Example 1 of the present invention;

[0053] Figure 2 Schematic diagram of a spatial meshing coordinate system of a pure rolling gear rack mechanism based on a hyperbolic cosine tooth profile inverse tangent tooth line in Example 1;

[0054] Figure 3 It is a schematic end view of a pure rolling rack and pinion mechanism based on a hyperbolic cosine tooth profile and an inverse tangent tooth line in Example 1;

[0055] Figure 4 is a schematic diagram of the tooth profile of the gear and rack in Example 1;

[0056] Figure 5 is a schematic diagram of the gears in Example 1;

[0057] Figure 6 is a schematic diagram of the rack in Example 1;

[0058] Figure 7 It is a schematic diagram of a pure rolling rack and pinion mechanism based on a hyperbolic cosine tooth profile and inverse tangent tooth line in Example 2.

[0059] 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

[0060] 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.

[0061] 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.

[0062] 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.

[0063] 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.

[0064] 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.

[0065] 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.

[0066] Embodiment 1:

[0067] The present invention provides a pure rolling gear rack mechanism based on a hyperbolic cosine tooth profile and an inverse tangent 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 both composed of an end face working tooth profile curve 13 and a tooth root transition curve 12, and the end face working tooth profile curves 13 of the gear 5 and the rack 6 are both symmetrical on both sides; the end face working tooth profile curves 13 of the gear 5 and the rack 6 are hyperbolic cosine curves; the tooth surfaces of the gear 5 and the rack 6 have an inverse tangent curve 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 moving trajectories of the meshing points of the gear 5 and the rack 6 form a meshing line, and two contact lines are formed on the tooth surfaces of the gear 5 and the rack 6, namely, a gear contact line 10 and a rack contact line 11.

[0068] The tooth surface structures of the gear 5 and the rack 6 are respectively formed by the respective end face working tooth profile curves 13 moving along the gear contact line 10 and the rack contact line 11 as the contact point moves, and the gear contact line 10 and the rack contact line 11 are both z-axis symmetrical inverse tangent function curves after being unfolded along the pitch cylindrical surface of the gear 5, and the rack 6 is an inverse tangent function curve unfolded along the rack pitch surface.

[0069] The right side end surface working tooth profile curves 13 of the gear 5 and the rack 6 are both formed by hyperbolic cosine function curves, and the right side tooth root transition curves 12 of the gear 5 and the rack 6 are composed of Hermite curves; the shape of the end surface working tooth profile curve 13 is controlled by the intersection point of the tooth top circle and the hyperbolic cosine function curve, and the shape of the tooth root transition curve 12 is controlled by the intersection point P of the tooth root transition start circle and the hyperbolic cosine function curve. G4 control, and 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 G3Form the tooth root curve.

[0070] like Figure 1 As shown, in this embodiment, the pure rolling gear rack mechanism based on the hyperbolic cosine tooth profile inverse tangent tooth line includes a gear 5 and a rack 6, and the overlap 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. The gear 5 rotates under the drive of the driver 1, so that the rack 6 performs translational motion, 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.

[0071] refer to Figures 1 to 6 , the radius of the pitch cylinder 8 of gear 5 is R1, and the radius of the top circle of rack 6 is R a1 , the root circle radius is R f1 The outer surface of the tooth root cylinder of gear 5 is uniformly distributed with gear teeth of inverse tangent tooth line structure. The gear contact line 10 of gear 5 is an axisymmetric inverse tangent curve after being unfolded along the pitch cylindrical surface of gear 5. The rack contact line 11 of rack 6 is an axisymmetric inverse tangent curve after being unfolded along the gear rack pitch surface 9. The end face tooth profile of gear 5 is composed of a hyperbolic cosine curve 11 and a left end face tooth root transition curve 12, i.e., a Hermite curve 12, from the tooth top to the tooth root.

[0072] The end face working tooth profile and tooth root transition fillet of the gear 5 and rack 6 are symmetrical on both sides. The left side tooth profile of the end face is obtained by symmetry of the right side tooth profile of the end face. Similarly, the left side tooth root transition fillet can be obtained by symmetry of 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 PG3 The intersection of the oblique lines with slope 1.

[0073] refer to Figure 2 In the schematic diagram of the spatial meshing coordinate system of a pure rolling gear rack mechanism based on a hyperbolic cosine tooth profile inverse tangent tooth line of the present invention, the gear contact line 10 and the rack contact line 11 of the gear 5 and the rack 6 are determined according to the following method:

[0074] Construct 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 7 of the gear 5 and the rack 6, and the z2 axis is on the rack 6, the 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;

[0075] In the coordinate system O k -x k ,y k ,z k In the figure, the meshing line parameter equation of the meshing point M between gear 5 and rack 6 is:

[0076]

[0077] The relationship between the rotation angle of the gear 5 and the movement of the rack 6 is:

[0078]

[0079] When the two meshing points M move along the meshing line 7, contact lines 10 and 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:

[0080] M 1k =M 10 ×M 0k(3)

[0081]

[0082] in,

[0083]

[0084] In equations (4) and (6), R1 is the pitch radius of gear 5, is the rotation angle of gear 5;

[0085] The parameter equation of the gear contact line 10 of the gear 5 tooth surface is obtained from equations (1) and (5):

[0086]

[0087] The parameter equation of the rack contact line 11 of the tooth surface of the rack 6 is obtained by equations (1) and (4):

[0088]

[0089] The end surface working tooth profile curve 13 of the gear 5 and the rack 6 is determined by the following method:

[0090] A coordinate system O is established at the meshing point M of the end faces of the gear 5 and the rack 6. k -x k ,y k ,z k , the parameter equation of the working tooth profile curve on the right side of the gear 5 is obtained:

[0091]

[0092] The rack 6 is in the coordinate system O k -x k ,y k ,z k The parametric equation of the working tooth profile curve on the right side:

[0093]

[0094] The rack 6 is in the coordinate system O k -x k ,y k ,z k The parametric equation of the working tooth profile curve on the left side:

[0095]

[0096] The parameter equation of the working tooth profile curve of the gear 5 on the right side in the coordinate system O1-x1, y1, z1 is:

[0097]

[0098] The parameter equation of the working tooth profile curve of the gear 5 on the left side in the coordinate system O1-x1, y1, z1 is:

[0099]

[0100] 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:

[0101]

[0102] The parameter equation of the working right tooth surface of the gear 5 is:

[0103]

[0104] The formation of the tooth surface of the rack 6 is achieved by the movement along the contact curve of the rack 6. The parameter equation of the working left tooth surface of the rack 6 is:

[0105]

[0106] The parameter equation of the working right tooth surface of the rack 6 is:

[0107]

[0108] The right tooth root of the end face of the gear 5 uses the Hermite curve as a 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 right working tooth profile curve of the gear 5 and the tooth 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:

[0109]

[0110] The right tooth root of the end face 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 P G3 and P G4 The tangent vectors are P G3 and T G4 Decision, PG3 The point is the radius R of the right working tooth profile curve of the gear 5 and the tooth 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:

[0111]

[0112] in,

[0113]

[0114] In all the above formulas:

[0115] t - parameter of the hyperbolic cosine function equation;

[0116] p - set the hyperbolic cosine function parameter value;

[0117] σ-motion parameter variable of meshing point M, and σ∈[0,Δσ];

[0118] Δσ-the maximum value of the motion parameter variable of the meshing point;

[0119] - is the linear proportional coefficient of the meshing point movement;

[0120] i-gear ratio

[0121] m t -end modulus;

[0122] Z1-number of gear teeth;

[0123] Z2-number of teeth on the rack after the gear rotates one circle;

[0124] T H is the Hermite type line parameter, 0.2≤T H ≤1.5;

[0125] t H is the value range of the Hermite type line, 0≤t H ≤1;

[0126] P F3 -The intersection point of the starting radius of the gear transition fillet and the gear working tooth profile parameter equation;

[0127] P F4 - Radius of tooth root circle and point P F3 The intersection of the oblique lines with a slope of 1;

[0128] P G3-The intersection point of the rack transition fillet starting length and the rack working tooth profile parameter equation;

[0129] P G4 -The length of the rack tooth root and the point P G3 The intersection of the lines with a slope of 1;

[0130] T P1 -Unit tangent vector of point P1;

[0131] T P2 -Unit tangent vector of point P2;

[0132] T G1 -Unit tangent vector of point G1;

[0133] T G2 -Unit tangent vector of point G2;

[0134] x P (P F3 )-Point P F3 The x coordinate of

[0135] y p (P F3 )-Point P F3 The y coordinate of

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

[0137] x p (P F4 )-Point P F4 The x coordinate of

[0138] y p (P F4 )-Point P F4 The y coordinate of

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

[0140] x G (P G3 )-Point P G3 The x coordinate of

[0141] y G (P G3 )-Point P G3 The y coordinate of

[0142] z G (P G3 )-Point P G3The z coordinate of

[0143] x G (P G4 )-Point P G4 The x coordinate of

[0144] y G (P G4 )-Point P G4 The y coordinate of

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

[0146] Δd-face width coefficient;

[0147] b-gear tooth width, b = Δd × 2R1; (21)

[0148] α t - End pressure angle, α t =20°;

[0149] - tooth top height coefficient,

[0150] -Headspace coefficient,

[0151] R1- is the pitch radius of the gear, R1 = m t Z1 / 2; (22)

[0152] R2- is the pitch line of the rack;

[0153] a-center distance between rack and gear, a=R1+R2; (23)

[0154] h a - Tooth top height,

[0155] h f -Tooth root height,

[0156] R a1 - Gear tooth tip radius, R a1 =R1+h a ; (26)

[0157] R f1 -Gear tooth root radius, R f1 =R1-h f ; (27)

[0158] R h1 - Gear transition fillet start radius, Rh1 =R1-h a ; (28)

[0159] R a2 - Length of rack tooth tip, R a2 =R2+h a ; (29)

[0160] R f2 - Length of rack tooth root, R f2 =R2-h f ; (30)

[0161] R h2 - Rack transition fillet start length, R h2 =R2-h a ; (31)

[0162] ε-coincidence,

[0163] p t - Face pitch, p t =πm t ; (33).

[0164] The relevant parameters are Z1=16, i=1, m t =2, b=32mm,α t =20°, Δσ=0.2, R1=16mm, R2=38mm;

[0165] Then, substituting the above values ​​into equations (1) to (33), we can obtain the gear contact line 10, rack contact line 11 parameter equations and the end face tooth profile parameter equations of the gear and the rack in this example, and obtain the gear and rack tooth surface structure, which can be assembled according to the correct center distance.

[0166] Example 2

[0167] like Figure 7As shown, this embodiment also provides another pure rolling rack-and-pinion mechanism based on hyperbolic cosine tooth profile and inverse tangent 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 overlap ε=2 of the pure rolling rack-and-pinion mechanism based on the hyperbolic cosine tooth profile and inverse tangent tooth line is pre-set, 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 the hyperbolic cosine tooth profile and inverse tangent tooth line in the rotational motion.

[0168] 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;

[0169] Substituting the above values ​​into equations (1) to (33) can obtain the contact line parameter equation and the end tooth profile parameter equation of the gear 5 and the rack 6 in this example. Then, according to the motion law of the meshing point, the structure of the rack 6 and the gear 5 can be obtained, and they can be assembled according to the correct center distance.

[0170] 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.

[0171] 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 hyperbolic cosine tooth profile and inverse tangent 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 working tooth profile curves 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 hyperbolic cosine curves; the tooth surfaces of the gear and the rack have an inverse tangent curve 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 moving trajectories of the meshing points of the gear and the rack form a meshing line, and two contact lines are formed on the tooth surfaces of the gear and the rack respectively.

2. A pure rolling gear rack mechanism based on hyperbolic cosine tooth profile and inverse tangent tooth line as claimed in claim 1, characterized in that: The tooth surface structures of the gear and the rack are respectively formed by the movement of their respective end face working tooth profile curves along the contact line with the contact point, and the contact line is an inverse tangent function curve symmetrical about the z-axis after being unfolded along the pitch cylindrical surface of the gear, and the rack is an inverse tangent function curve unfolded along the rack pitch surface.

3. A pure rolling gear rack mechanism based on hyperbolic cosine tooth profile and inverse tangent tooth line as claimed in claim 1, characterized in that: The right side end surface working tooth profile curves of the gear and the rack are both formed by hyperbolic cosine function curves, and the right side tooth root transition curves of the gear and the rack are composed of Hermite curves; the shape of the end surface working tooth profile curve is controlled by the intersection point of the tooth addendum circle and the hyperbolic 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 hyperbolic cosine function curve. G4 control, and 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.

4. A pure rolling gear rack mechanism based on hyperbolic cosine tooth profile and inverse tangent tooth line as claimed in claim 3, characterized in that: The contact line between the gear and the rack is determined as follows: Construct 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 two meshing points move along the meshing line, contact lines are formed on the tooth surfaces of the gear 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 parametric equation of the contact line of the gear tooth surface is obtained from equations (1) and (5): The parametric equation of the contact line of the rack tooth surface is obtained from equations (1) and (4):

5. A pure rolling gear rack mechanism based on hyperbolic cosine tooth profile and inverse tangent tooth line as claimed in claim 4, characterized in that: The end surface working tooth profile curves of the gear and the rack are determined by the following method: The gear is at O k -x k ,y k ,z k The parametric equation of the working tooth profile curve on the right side: The rack is in the coordinate system O k -x k ,y k ,z k The parametric equation of the working tooth profile curve on the right side: The rack is in the coordinate system O k -x k ,y k ,z k The parametric equation of the working tooth profile curve on the left side: The parametric equation of the working tooth profile curve of the gear on the right side in the coordinate system O1-x1, y1, z1 is: The parametric equation of the working tooth profile curve of the gear on the left side in the coordinate system O1-x1, y1, z1 is:

6. A pure rolling gear rack mechanism based on hyperbolic cosine tooth profile and inverse tangent tooth line as claimed in claim 5, characterized in that: 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 by movement along 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:

7. A pure rolling rack and pinion mechanism based on hyperbolic cosine tooth profile and inverse tangent tooth line as claimed in claim 6, characterized in that: The right tooth root of the end face of the gear 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 right working tooth profile curve of the gear and the radius R of the tooth 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 end face of the rack uses a Hermite curve as a transition curve, wherein the Hermite curve is from point P G3 and P G4 , and P G3 and P G4 The tangent vectors are T G3 and T G4 Decision, P G3 The point is the right working tooth profile curve of the gear and the radius R of the tooth 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 - parameter of the hyperbolic cosine function equation; p - set the hyperbolic 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; i-gear ratio 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).

Images