Pure rolling gear rack mechanism based on elliptic tooth profile combination function tooth trace

Abstract

The invention provides a tooth trace pure rolling gear rack mechanism based on an elliptic tooth profile combination function, and relates to the technical field of gear transmission, the tooth trace pure rolling gear rack mechanism comprises a gear and a rack, the end face tooth profile of the gear and the rack is composed of an end face working tooth profile curve and a tooth root transition curve, and the end face working tooth profile curve of the gear and the tooth root transition curve is an elliptic function. The tooth surfaces of the gear and the rack have arc tangent function and elliptic function combined tooth line structures, and control points are set for smooth connection; 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 respectively form two combined function contact lines on the tooth surfaces of the gear and the rack. The method has the beneficial effects that two functions are smoothly connected at a specific control point by setting the specific control point to form a combined tooth line, so that the axial force is eliminated, and the service life of the bearing is favorably prolonged.

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 tooth line of an elliptical tooth profile combination function. Background Art

[0002] In the field of mechanical engineering, gears and racks are key components in mechanical transmission systems and are widely used in various types of mechanical equipment, such as CNC machine tools, automation equipment, robotics, and various industrial production lines. For example, Chinese patent application number 201710016238.7 discloses "a concave-convex meshing circular arc gear rack mechanism without relative sliding", and Chinese patent 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, since 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 bearings and affects the accuracy and life of the transmission system. Summary of the invention

[0003] In view of this, in order to solve the 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 an elliptical tooth profile combination function tooth line.

[0004] An embodiment of the present invention provides a pure rolling gear and rack mechanism based on an elliptical tooth profile combined function 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 elliptical functions, the tooth surfaces of the gear and the rack have an inverse tangent function and an elliptical function combined tooth line structure, and the inverse tangent function and the elliptic function are smoothly connected by setting control points; 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 combined function contact lines on the tooth surfaces of the gear and the rack, respectively.

[0005] Furthermore, the two pure rolling contact points of the gear respectively have an elliptical function curve and an inverse tangent function curve after being unfolded along the pitch cylinder contact line, and the two pure rolling contact points of the rack respectively have an elliptical function curve and an inverse tangent function curve after being unfolded along the pitch surface, the two curves are smoothly connected at the set control point, and the tooth surfaces of the gear and the rack are formed by regular scanning of the end face tooth profile along the contact line.

[0006] Furthermore, the right working tooth profile curves of the gear and the rack are both formed by elliptic curves, and the tooth root transition curve is composed of Hermite curves; the value range of the end face working tooth profile curve is to control 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 elliptic curve, and the tooth root transition curve starting control point is formed by the intersection point P formed by the intersection of the elliptic curve and the tooth root transition starting 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 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 BB parameter equation of the meshing point motion of the gear and the rack is:

[0010]

[0011] In the coordinate system O k -x k ,y k ,z kIn the equation, the meshing line CC parameter equation of the meshing point motion of the gear and the rack is:

[0012]

[0013] The relationship between the gear rotation angle and the rack motion is:

[0014]

[0015] When the meshing point moves along the meshing line BB and the meshing line CC, two contact lines of different functions 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:

[0016] M 1k =M 10 ×M 0k (4)

[0017]

[0018] in,

[0019]

[0020] In formulas (4) and (6), R1 is the pitch radius of the gear, is the gear rotation angle;

[0021] The parameter equation of the contact line of the gear tooth surface is obtained by equations (1), (2) and (6):

[0022]

[0023] The parametric equation of the contact line of the rack tooth surface is obtained by equations (1), (2) and (5):

[0024]

[0025] Furthermore, the end face tooth profiles of the gear and the rack are determined by the following method:

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

[0027]

[0028] 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:

[0029]

[0030] 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:

[0031]

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

[0033]

[0034] The parametric equation of the left working tooth profile of the gear in the coordinate system O1-x1, y1, z1 is:

[0035]

[0036] Furthermore, the tooth surfaces of the gear and the rack are determined according to the following method:

[0037] One side of the gear tooth surface is swept along the contact line LL, and the parameter equation of the working left tooth surface of the gear is:

[0038]

[0039] The other side of the gear tooth surface is swept along the contact line KK. The parameter equation of the working left tooth surface of the gear is:

[0040]

[0041] One side of the gear tooth surface is swept along the contact line LL, and the parameter equation of the working right tooth surface of the gear is:

[0042]

[0043] The other side of the gear tooth surface is swept along the contact line KK. The parameter equation of the working right tooth surface of the gear is:

[0044]

[0045] The tooth surface on one side of the rack is realized by parallel motion along the rack contact line 11, and the parameter equation of the working left tooth surface of the rack is:

[0046]

[0047] The tooth surface on the other side of the rack is realized by parallel motion along the rack contact line kk. The parameter equation of the working left tooth surface of the rack is:

[0048]

[0049] The formation of the tooth surface on one side of the rack is achieved by parallel motion along the rack contact curve 11. The parameter equation of the working right tooth surface of the rack is:

[0050]

[0051] The formation of the tooth surface on one side of the rack is achieved by parallel motion along the rack contact curve kk. The parameter equation of the working right tooth surface of the rack is:

[0052]

[0053] Furthermore, the tooth root transition curves of the gear and the rack are determined according to the following method:

[0054] 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 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:

[0055]

[0056] 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 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 G3The Hermite curve parameter equation is determined by the slope of 1:

[0057]

[0058] in,

[0059]

[0060] The beneficial effects brought about by the technical solution provided by the embodiment of the present invention are:

[0061] 1. The present invention discloses a pure rolling gear rack mechanism based on a tooth line of an elliptical tooth profile combination function, 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 a tooth line composed of an inverse tangent function and an elliptical function after being unfolded on the pitch cylindrical surface. Similarly, the rack contact line is a tooth line composed of an inverse tangent function and an elliptical function at the pitch surface. By setting a specific control point, the two functions are smoothly connected at this point to form a combined tooth line to eliminate the axial force, which can significantly reduce the friction loss of the bearing and improve the transmission efficiency.

[0062] 2. The present invention is a pure rolling gear rack mechanism based on an elliptical tooth profile combined function tooth line. Both the gear tooth profile and the rack tooth profile are tooth profiles established based on elliptical functions. The gear contact line and the rack contact line are controlled by a set control point to smoothly connect the two curves at this point to prevent stress concentration at the control point during meshing. The tooth root transition curve uses the Hermite curve to effectively improve the tooth root bending strength. In addition, the gear rack is in pure rolling contact, so there is almost no sliding friction between the gear and the rack, and the friction loss and noise are significantly reduced, the transmission efficiency is greatly improved, and the occurrence of failure forms such as tooth surface wear and plastic deformation is reduced.

[0063] 3. The present invention is a pure rolling gear rack mechanism based on an elliptical tooth profile combined function 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

[0064] Figure 1 Schematic diagram of a pure rolling rack and pinion mechanism based on an elliptical tooth profile combined function tooth line in Example 1 of the present invention;

[0065] Figure 2 Schematic diagram of a spatial meshing coordinate system of a pure rolling gear rack mechanism based on an elliptical tooth profile combination function tooth line in Example 1;

[0066] Figure 3 It is a schematic end view of a pure rolling gear rack mechanism based on an elliptical tooth profile combination function tooth line in Example 1;

[0067] Figure 4 is a schematic diagram of the tooth profile of the gear and rack in Example 1;

[0068] Figure 5 is a schematic diagram of the gears in Example 1;

[0069] Figure 6 is a schematic diagram of the rack in Example 1;

[0070] Figure 7 It is a schematic diagram of a pure rolling gear rack mechanism based on an elliptical tooth profile combination function tooth line in Example 2.

[0071] In the figure: 1. driver; 2. input shaft; 3. coupling; 4. output shaft; 5. gear; 6. rack; 7. meshing line BB; 8. pitch cylinder; 9. rack pitch surface; 10. gear contact line KK; 11. meshing line CC; 12. rack contact line kk; 13. rack contact line ll; 14. gear contact line LL; 15. tooth root transition curve; 16. end face working tooth profile curve. DETAILED DESCRIPTION

[0072] 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.

[0073] 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.

[0074] 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.

[0075] 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.

[0076] 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.

[0077] 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.

[0078] The present invention provides a pure rolling gear rack mechanism based on an elliptical tooth profile combined function tooth line, comprising a gear 2 and a rack 6, wherein the end face tooth profiles of the gear 2 and the rack 6 are composed of an end face working tooth profile curve 16 and a tooth root transition curve 15, and the end face tooth profiles of the gear 2 and the rack 6 are symmetrical on both sides; the end face working tooth profile curve 16 of the gear 2 and the rack 6 is an elliptical function, and the tooth surfaces of the gear 2 and the rack 6 have an inverse tangent function and an elliptical function combined tooth line structure, and the inverse tangent function and the elliptic function are smoothly connected by setting control points; at least one pair of gear tooth meshing points of the gear 2 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 2 and the rack 6 form two combined function contact lines on the tooth surfaces of the gear 2 and the rack 6, respectively. The two pure rolling contact points of the gear 2 respectively have an elliptical function curve and an inverse tangent function curve after being unfolded along the contact line of the pitch cylinder 8, and the two pure rolling contact points of the rack 6 respectively have an elliptical function curve and an inverse tangent function curve after being unfolded along the rack pitch surface. The two curves are smoothly connected at the set control point, and the tooth surfaces of the gear 2 and the rack 6 are formed by regular scanning of the end face tooth profile along the contact line.

[0079] Please refer to Figure 1 In a gear rack mechanism with an end face inverse tangent function tooth profile and an elliptical 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.

[0080] See also 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 , one of the gear contact lines is an elliptical curve and the other is an inverse tangent curve after being unfolded along the pitch cylindrical surface of gear 5; similarly, one of the contact lines of rack 6 is an elliptical curve and the other is an inverse tangent curve after being unfolded along the rack pitch surface 9. The end face tooth profile of gear 5 is composed of an elliptical curve 11 and an end face tooth root transition curve 15, i.e., a Hermite curve, from the tooth top to the tooth root.

[0081] 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 15 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 tooth root transition curve 15 on the right side 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.

[0082] The gear 5 rotates under the drive of the driving motor 1, so that the rack 6 performs 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.

[0083] The contact line between the gear 5 and the rack 6 is determined by the following method:

[0084] 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 kThe 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;

[0085] In the coordinate system O k -x k ,y k ,z k In the figure, the parameter equation of the meshing line B-B7 of the meshing point motion of the gear 5 and the rack 6 is:

[0086]

[0087] In the coordinate system O k -x k ,y k ,z k In the figure, the meshing line C-C11 parameter equation of the meshing point motion of the gear 5 and the rack 6 is assumed to be:

[0088]

[0089] The relationship between the rotation angle of the gear 5 and the movement of the rack 6 is:

[0090]

[0091] When the meshing point moves along the meshing line B-B7 and the meshing line C-C11, two contact lines of different functions 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:

[0092] M 1k =M 10 ×M 0k (4)

[0093]

[0094] in,

[0095]

[0096] In formulas (4) and (6), R1 is the pitch circle radius of the gear 5, is the rotation angle of the gear 5;

[0097] The parametric equation of the contact line of the tooth surface of the gear 5 is obtained by equations (1), (2) and (6):

[0098]

[0099] The parametric equation of the contact line of the tooth surface of the rack 6 is obtained by equations (1), (2) and (5):

[0100]

[0101] The end face tooth profiles of the gear 5 and the rack 6 are determined by the following method:

[0102] 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:

[0103]

[0104] In the coordinate system O k -x k ,y k ,z k The parameter equation of the right working tooth profile of the rack 6 is:

[0105]

[0106] 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:

[0107]

[0108] The parametric equation of the right working tooth profile of the gear 5 in the coordinate system O1-x1, y1, z1 is:

[0109]

[0110] The parameter equation of the left working tooth profile of the gear 5 in the coordinate system O1-x1, y1, z1 is:

[0111]

[0112] The tooth surfaces of the gear 5 and the rack 6 are determined as follows:

[0113] One side of the tooth surface of the gear 5 is swept along the contact line L-L14. The parameter equation of the working left tooth surface of the gear 5 is:

[0114]

[0115] The other side of the tooth surface of the gear 5 is swept along the contact line K-K10. The parameter equation of the working left tooth surface of the gear 5 is:

[0116]

[0117] One side of the tooth surface of the gear 5 is swept along the contact line L-L14. The parameter equation of the working right tooth surface of the gear 5 is:

[0118]

[0119] The other side of the tooth surface of the gear 5 is swept along the contact line K-K10. The parameter equation of the working right tooth surface of the gear 5 is:

[0120]

[0121] The tooth surface on one side of the rack 6 is realized by parallel motion along the rack contact line l-l13. The parameter equation of the working left tooth surface of the rack 6 is:

[0122]

[0123] The tooth surface on the other side of the rack 6 is realized by parallel motion along the rack contact line k-k12. The parameter equation of the working left tooth surface of the rack 6 is:

[0124]

[0125] The formation of the tooth surface on one side of the rack 6 is achieved by parallel motion along the rack contact curve l-l13. The parameter equation of the working right tooth surface of the rack 6 is:

[0126]

[0127] The formation of the tooth surface on one side of the rack 6 is achieved by parallel motion along the rack contact curve k-k12. The parameter equation of the working right tooth surface of the rack 6 is:

[0128]

[0129] The tooth root transition curve 15 of the gear 5 and the rack 6 is determined as follows:

[0130] 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 F F4 The tangent vectors are T F3 and T F4Decision, 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:

[0131]

[0132] 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 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:

[0133]

[0134] in,

[0135]

[0136] In all the above formulas:

[0137] t-parameters of the elliptic function equation;

[0138] p-set elliptic curve parameter value;

[0139] i-coincidence

[0140] σ-motion parameter variable of meshing point M, and σ∈[0,Δσ];

[0141] Δσ-the maximum value of the motion parameter variable of the meshing point;

[0142] - is the linear proportional coefficient of the meshing point movement;

[0143] m t -end modulus;

[0144] Z1-number of gear teeth;

[0145] Z2-number of teeth on the rack after the gear rotates one circle;

[0146] T H is the Hermite type line parameter, 0.2≤T H ≤1.5;

[0147] t H is the value range of the Hermite type line, 0≤t H ≤1;

[0148] P F3 -The intersection point of the starting radius of the gear transition fillet and the gear working tooth profile parameter equation;

[0149] P F4 -Gear tooth root circle radius and passing point P F3 The intersection of the oblique lines with a slope of 1;

[0150] P G3 -The intersection point of the rack transition fillet starting length and the rack working tooth profile parameter equation;

[0151] P G4 -The length of the rack tooth root and the point P G3 The intersection of the lines with a slope of 1;

[0152] T P1 -Unit tangent vector of point P1;

[0153] T P2 -unit tangent vector of point P2;

[0154] T G1 -Unit tangent vector of point G1;

[0155] T G2 -Unit tangent vector of point G2;

[0156] x P (P F3 )-Point P F3 The x coordinate of

[0157] y p (P F3 )-Point P F3 The y coordinate of

[0158] z p (P F3 )-Point P F3 The z coordinate of

[0159] x p (P F4 )-Point P F4 The x coordinate of

[0160] yp (P F4 )-Point P F4 The y coordinate of

[0161] z p (P F4 )-Point P F4 The z coordinate of

[0162] x G (P G3 )-Point P G3 The x coordinate of

[0163] y G (P G3 )-Point P G3 The y coordinate of

[0164] z G (P G3 )-Point P G3 The z coordinate of

[0165] x G (P G4 )-Point P G4 The x coordinate of

[0166] y G (P G4 )-Point P G4 The y coordinate of

[0167] z G (P G4 )-Point P G4 The z coordinate of

[0168] Δd-face width coefficient;

[0169] b-gear tooth width, b = Δd × 2R1; (26)

[0170] α t - End pressure angle, α t =20°;

[0171] - tooth top height coefficient,

[0172] -Headspace coefficient,

[0173] R1- is the pitch radius of the gear, R1 = m t Z1 / 2; (27)

[0174] R2- is the pitch line of the rack;

[0175] a-center distance between rack and gear, a=R1+R2; (28)

[0176] h a - Tooth top height,

[0177] h f -Tooth root height,

[0178] R a1 - Gear tooth tip radius, R a1 =R1+h a ; (31)

[0179] R f1 -Gear tooth root radius, R f1 =R1-h f ; (32)

[0180] R h1 - Gear transition fillet start radius, R h1 =R1-h a ; (33)

[0181] R a2 - Length of rack tooth tip, R a2 =R2+h a ; (34)

[0182] R f2 - Length of rack tooth root, R f2 =R2-h f ; (35)

[0183] R h2 - Rack transition fillet start length, R h2 =R2-h a ; (36)

[0184] ε-coincidence,

[0185] p t - Face pitch, p t =πm t ; (38).

[0186] The relevant parameters are Z1=16, i=1, m t =2, b=32mm,α t =20°, Δσ=0.2, R1=16mm, R2=38mm;

[0187] Then, substituting the above values ​​into equations (1)-(38) 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.

[0188] Embodiment 2:

[0189] like Figure 7 As shown, embodiment 2 of the present invention also provides another pure rolling gear rack mechanism based on the tooth line of the elliptical tooth profile combination function, 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 gear rack mechanism based on the tooth line of the elliptical tooth profile combination function ensures that at each instant, at least two pairs of gear teeth participate in the meshing transmission at the same time, thereby realizing the continuous and stable meshing transmission of the pure rolling gear rack mechanism based on the tooth line of the elliptical tooth profile combination function in the rotational motion.

[0190] 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;

[0191] Substituting the above values ​​into equations (1)-(38) 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, the elliptical tooth profile combined tooth line pure rolling gear rack structure is obtained, and it can be assembled according to the correct center distance.

[0192] 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.

[0193] 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 an elliptical tooth profile combined function tooth line, characterized in that: It includes 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 an elliptic function, the tooth surface of the gear and the rack has an inverse tangent function and an elliptic function combined tooth line structure, and the inverse tangent function and the elliptic function are smoothly connected by setting control points; 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 combined function contact lines on the tooth surfaces of the gear and the rack respectively.

2. A pure rolling gear rack mechanism based on an elliptical tooth profile combined function tooth line as claimed in claim 1, characterized in that: The two pure rolling contact points of the gear respectively have an elliptic function curve and an inverse tangent function curve after being unfolded along the pitch cylinder contact line, and the two pure rolling contact points of the rack respectively have an elliptic function curve and an inverse tangent function curve after being unfolded along the pitch surface, the two curves are smoothly connected at the set control point, and the tooth surfaces of the gear and the rack are formed by regular scanning of the end face tooth profile along the contact line.

3. A pure rolling gear rack mechanism based on an elliptical tooth profile combined function tooth line as claimed in claim 1, characterized in that: The right working tooth profile curves of the gear and the rack are both formed by elliptical curves, and the tooth root transition curve is composed of Hermite curves; the value range of the end face working tooth profile curve is to control 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 elliptic curve, and the tooth root transition curve start control point is formed by the intersection point P formed by the intersection of the elliptic 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.

4. A pure rolling gear rack mechanism based on an elliptical tooth profile combined function tooth line 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 BB parameter equation of the meshing point motion of the gear and the rack is: In the coordinate system O k -x k ,y k ,z k In the equation, the meshing line CC 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 BB and the meshing line CC, two contact lines of different functions 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 (4) 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), (2) and (6): The parametric equation of the contact line of the rack tooth surface is obtained by equations (1), (2) and (5):

5. A pure rolling gear rack mechanism based on an elliptical tooth profile combined function tooth line 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 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:

6. A pure rolling gear rack mechanism based on an elliptical tooth profile combined function tooth line as claimed in claim 5, characterized in that: The tooth surfaces of the gear and the rack are determined as follows: One side of the gear tooth surface is swept along the contact line LL, and the parameter equation of the working left tooth surface of the gear is: The other side of the gear tooth surface is swept along the contact line KK. The parameter equation of the working left tooth surface of the gear is: One side of the gear tooth surface is swept along the contact line LL, and the parameter equation of the working right tooth surface of the gear is: The other side of the gear tooth surface is swept along the contact line KK. The parameter equation of the working right tooth surface of the gear is: The tooth surface on one side of the rack is realized by parallel motion along the rack contact line 11, and the parameter equation of the working left tooth surface of the rack is: The tooth surface on the other side of the rack is realized by parallel motion along the rack contact line kk. The parameter equation of the working left tooth surface of the rack is: The formation of the tooth surface on one side of the rack is achieved by parallel motion along the rack contact curve 11. The parameter equation of the working right tooth surface of the rack is: The formation of the tooth surface on one side of the rack is achieved by parallel motion along the rack contact curve kk. The parameter equation of the working right tooth surface of the rack is:

7. A pure rolling gear rack mechanism based on an elliptical tooth profile combined function tooth line 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 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 P 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 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 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-parameters of the elliptic function equation; p-set elliptic curve 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-coincidence m t End face 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; (26) 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; (27) R2- is the pitch line of the rack; a-center distance between rack and gear, a=R1+R2; (28) h a - Tooth top height, h f -Tooth root height, R a1 - Gear tooth tip radius, R a1 =R1+h a ; (31) R f1 -Gear tooth root radius, R f1 =R1-h f ; (32) R h1 - Gear transition fillet start radius, R h1 =R1-h a ; (33) R a2 - Length of rack tooth tip, R a2 =R2+h a ; (34) R f2 - Length of rack tooth root, R f2 =R2-h f ; (35) R h2 - Rack transition fillet start length, R h2 =R2-h a ; (36) ε-degree of polymerization, p t - Face pitch, p t =πm t ; (38).

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