A NON-ORTHOGONAL ELLIPTICAL TOROIDAL WORM GEAR PAIR

Abstract

The present invention belongs to the field of mechanical transmission technology, and proposes a non-orthogonal elliptical toroidal worm gear pair, including an involute cylindrical gear and an elliptical toroidal worm formed by a primary envelope of the involute cylindrical gear; both adopt spatial non-orthogonal transmission, and the shaft angle satisfies the self-locking condition and the restriction condition of minimum tooth top width; the toroidal generatrix of the elliptical toroidal worm is elliptical, which can increase the number of meshing teeth and the total length of instantaneous contact line. A non-orthogonal elliptical toroidal worm gear pair proposed in this invention has the characteristics of toroidal worm drive and can realize the whole facewidth of the gear to participate in the meshing drive. It can be used in the fields of precision continuous indexing transmission, continuous lapping of cylindrical gear tooth flank, and has good popularization application value and industrialization prospect.

Claims

1. A non-orthogonal elliptical toroidal worm gear pair, which is characterized in that the non-orthogonal elliptical toroidal worm gear pair comprising an involute cylindrical gear and an elliptical toroidal worm spreading from a single-envelope of the involute cylindrical gear; the involute cylindrical gear includes involute spur gear and involute helical gear; the tooth flank of the involute spur cylindrical gear is the involute cylindrical surface formed by the involute stretching along the axial direction, the tooth flank of the involute helical gear is the involute helicoid formed by the involute doing spiral movement along the axial direction, the involute is generated by the generating line doing pure rolling on the base circle; the involute cylindrical gear is ground and shaped by the hard tooth flank wear-resistant material; the equations of the left flank of the involute cylindrical gear are as follows: ${\begin{matrix} x_{1 L} = r_{b} \cos (u - ?_{0} - ?_{1} ?) + r_{b} u \sin (u - ?_{0} - ?_{1} ?) \\ y_{1 L} = - r_{b} \sin (u - ?_{0} - ?_{1} ?) + r_{b} u \cos (u - ?_{0} - ?_{1} ?) \\ z_{1 L} = ?_{1} ?_{1} ? + ?_{2} h_{L} \end{matrix}$ where, x.sub.1L is the x coordinate of each point on the left tooth surface, y.sub.1L is the y coordinate of each point on the left tooth surface, z.sub.1L is the z coordinate of each point on the left tooth surface; r.sub.b is the radius of the base circle of the involute gear; u is the rolling angle formed by the dominant involute tooth profile; a ?.sub.0 is half of the base circular angle corresponding to the transverse base thickness of the involute cylindrical gear; h.sub.L is the axial parameter of the tooth flank; ?.sub.1 is the helical parameter; ? is the angle at which the involute makes a spiral motion along the axial direction; ?.sub.1 is the parameter taking the value of 0 or 1, ?.sub.2 is the parameter taking the value of 0 or 1; the equations of the right flank of the involute cylindrical gear are as follows: ${\begin{matrix} x_{1 R} = r_{b} \cos (u - ?_{0} + ?_{1} ?) + r_{b} u \sin (u - ?_{0} + ?_{1} ?) \\ y_{1 R} = r_{b} \sin (u - ?_{0} + ?_{1} ?) - r_{b} u \cos (u - ?_{0} + ?_{1} ?) \\ z_{1 R} = ?_{1} ?_{1} ? + ?_{2} h_{R} \end{matrix}$ where, x.sub.1R is the x coordinate of each point on the right flank, y.sub.1R is the y coordinate of each point on the right flank, z.sub.1R is the z coordinate of each point on the right flank; h.sub.R is the right flank axial parameter; when the tooth flank equations of the above involute cylindrical gear satisfies ?.sub.1=0 and ?.sub.2=1, the corresponding left and right tooth flank equation is the tooth flank equation of the involute spur cylindrical gear; similarly, when ?.sub.1=1 and ?.sub.2=0, the corresponding left and right tooth flank equation is the tooth flank equations of the involute helical cylindrical gear; the reference surface of the elliptical toroidal worm described in the present invention is an elliptical toroidal surface, the generatrix of the elliptical toroidal surface is the intersection line between the inclined section and the reference cylinder within the working length of the worm, the inclined section passes through the axis of rotation of the elliptical toroidal worm and the angle with the horizontal plane is the shaft angle ?; the equation satisfied by the generatrix of the elliptical toroidal surface is as follows: $\frac{y^{2}}{{(r \csc ?)}^{2}} + \frac{x^{2}}{r^{2}} = 1$ where, r is the radius of the base circle of the involute indexing cylinder, x is the x-coordinate of any point on the generatrix, y is the y-coordinate of any point on the generatrix; the elliptical toroidal worm and involute cylindrical gear adopt spatially non-orthogonal transmission, and the shaft angle is determined according to the self-locking condition; the facewidth of the involute cylindrical gear is related to the working length of the elliptical toroidal worm and the shaft angle, and in order to achieve the full facewidth of the involute cylindrical gear to participate in the mesh, the following relationship should be satisfied:
b=L sin ? where, b is the facewidth of the involute cylindrical gear, L is the working length of the elliptical toroidal worm gear; the tooth flank of the elliptical toroidal worm is made of the involute spur gear's involute cylindrical surface or involute helical gear's involute helicoid as the tool generatrix surface according to the envelope method, and the corresponding transmission coordinate system is established according to the position relationship between the involute cylindrical gear and the elliptical toroidal worm mesh transmission; the details are as follows: the equation of the tooth flank of the involute cylindrical gear is obtained by the coordinate transformation and the principle of tooth conjugate meshing of the elliptical toroidal worm; the transmission subsets are ground and shaped with hard tooth material, so the equations of the tooth surface on the upper side of the elliptical toroidal worm are as follows: ${\begin{matrix} \begin{matrix} x_{2 L} = (x_{1 L} \cos ?_{1} - y_{1 L} \sin ?_{1} - a) \cos ?_{2} + \\ [(x_{1 L} \sin ?_{1} + y_{1 L} \cos ?_{1}) \cos ? + z_{1 L} \sin ?] \sin ?_{2} \end{matrix} \\ \begin{matrix} y_{2 L} = [(x_{1 L} \sin ?_{1} + y_{1 L} \cos ?_{1}) \cos ? + z_{1 L} \sin ?] \cos ?_{2} - \\ (x_{1 L} \cos ?_{1} - y_{1 L} \sin ?_{1} - a) \sin ?_{2} \end{matrix} \\ z_{2 L} = - (x_{1 L} \sin ?_{1} + y_{1 L} \cos ?_{1}) \sin ? + z_{1 L} \cos ? \\ x_{1 L} = r_{b} \cos ? + r_{b} u \sin ? \\ y_{1 L} = - r_{b} \sin ? + r_{b} u \cos ? \\ z_{1 L} = ?_{1} ?_{1} ? + ?_{2} h_{L} \\ ? = u - ?_{0} - ?_{1} ? \\ h_{L} = \frac{r_{b} (i_{12} - \cos ?) + a \cos (u - ?_{0} - ?_{1}) \cos ?}{- \sin (u - ?_{0} - ?_{1}) \sin ?} \\ u = \frac{\begin{matrix} - r_{b}^{2} \cos (? - ?_{1}) \sin ? + ?_{1}^{2} (? + ?_{0}) \sin (? - ?_{1}) \sin ? - \\ ?_{1} a \cos (? - ?_{1}) \cos ? - r_{b} ?_{1} (i_{12} - \cos ?) + r_{b} a \sin ? \end{matrix}}{(r_{b}^{2} + ?_{1}^{2}) \sin (? - ?_{1}) \sin ?} \end{matrix}$ where, a is the spread center distance, ?.sub.1 is the rotation angle of the involute cylindrical gear, ?.sub.2 is the rotation angle of elliptical toroidal worm, i.sub.12 is the reciprocal of the transmission ratio of the worm gear pair, ?.sub.0 is half of the base circular angle corresponding to the transverse base thickness of the involute cylindrical gear, u is the rolling angle formed by the dominant involute tooth profile; similarly, the equations of the lower side tooth surface of the elliptical toroidal worm are as follows: ${\begin{matrix} \begin{matrix} x_{2 R} = (x_{1 R} \cos ?_{1} - y_{1 R} \sin ?_{1} - a) \cos ?_{2} + \\ [(x_{1 R} \sin ?_{1} + y_{1 R} \cos ?_{1}) \cos ? + z_{1 R} \sin ?] \sin ?_{2} \end{matrix} \\ \begin{matrix} y_{2 R} = [(x_{1 R} \sin ?_{1} + y_{1 R} \cos ?_{1}) \cos ? + z_{1 R} \sin ?] \cos ?_{2} - \\ (x_{1 R} \cos ?_{1} - y_{1 R} \sin ?_{1} - a) \sin ?_{2} \end{matrix} \\ z_{2 R} = - (x_{1 R} \sin ?_{1} + y_{1 R} \cos ?_{1}) \sin ? + z_{1 R} \cos ? \\ x_{1 R} = r_{b} \cos (?) + r_{b} u \sin (?) \\ y_{1 R} = r_{b} \sin (?) - r_{b} u \cos (?) \\ z_{1 R} = ?_{1} ?_{1} ? + ?_{2} h_{R} \\ ? = u - ?_{0} + ?_{1} ? \\ h_{R} = \frac{r_{b} (i_{12} - \cos ?) + a \cos (u - ?_{0} + ?_{1}) \cos ?}{\sin (u - ?_{0} + ?_{1}) \sin ?} \\ u = \frac{\begin{matrix} - r_{b}^{2} \cos (? + ?_{1}) \sin ? + ?_{1}^{2} (? + ?_{0}) \sin (? + ?_{1}) \sin ? - \\ ?_{1} a \cos (? + ?_{1}) \cos ? - r_{b} ?_{1} (i_{12} - \cos ?) + r_{b} a \sin ? \end{matrix}}{(r_{b}^{2} + ?_{1}^{2}) \sin (? + ?_{1}) \sin ?} \end{matrix}$ the above tooth equations are determined by two parameters ?.sub.1 and u, other parameters are known, and upper tooth surface and lower tooth surface of elliptical toroidal gear vice can be obtained by MATLAB numerical analysis and 3D modeling software in the range of ?.sub.1 and u, and then they are stitched with the top toroidal surface and root toroidal surface of elliptical toroidal worm to generate the 3D solid model of non-orthogonal elliptical toroidal worm gear pair, and finally the non-orthogonal elliptical toroidal worm gear pair is obtained.

Description

BRIEF DESCRIPTIONS OF THE DRAWINGS

[0020] FIG. 1 Schematic diagram of the elliptical toroidal generatrix of an elliptical toroidal worm;

[0021] FIG. 2 Schematic diagram of the tooth flank structure of an involute spur cylindrical gear;

[0022] FIG. 3 Schematic diagram of the coordinate system of elliptical toroidal worm drive;

[0023] FIG. 4 Schematic diagram of elliptical toroidal worm gear tooth flank;

[0024] FIG. 5 Schematic diagram of involute spur cylindrical gear meshing with elliptical toroidal worm drive;

[0025] wherein, 1 illustrates inclined section; 2 illustrates the reference cylinder of involute cylindrical gear; 3 illustrates an elliptical toroidal generatrix; 4 illustrates an involute spur cylindrical gear; illustrates an elliptic toroidal worm; 5-1 illustrates an upper tooth surface of elliptic toroidal worm; 5-2 illustrates a lower tooth surface of elliptic toroidal worm.

DETAILED DESCRIPTION

[0026] Taking an involute spur cylindrical gear with a modulus m=2 mm, a tooth number z=120 and a pressure angle ?=20?, and an elliptical toroidal worm formed by a primary envelope of the involute cylindrical gear as an example, the technical solution is specified in conjunction with the attached drawings.

[0027] First, the reference surface of elliptical toroidal worm 5 is an elliptical toroidal surface, which is different from the conventional circular toroidal worm. The intersection line of the inclined section 1 and the gear reference cylinder surface 2 is the generatrix 3 of the elliptical toroidal surface, and the projection of the generatrix 3 on the cylindrical end face is a circular arc. The angle between the inclined section and the cylindrical end face is the shaft angle s of the transmission sub, and the radius of the cylindrical base circle is r. Then the equation of the cylindrical base circle is as follows:

x.sup.2+y.sup.2=r.sup.2

So, the equation satisfied by the generatrix 3 of the elliptical torus is as follows:

[00006] $\frac{x^{2}}{{(r \csc ?)}^{2}} + \frac{y^{2}}{r^{2}} = 1$

When r=118 mm and r=122.5 mm, the above equations of generatrix 3 correspond to the equations of the top generatrix and root generatrix of the elliptical toroidal worm gear respectively.

[0028] Further, the working length, axis of rotation and other parameters of the elliptical toroidal worm 5 are determined based on the parameters of the involute spur cylindrical gear 4. The facewidth of the involute spur cylindrical gear 4 is related to the working length of the elliptical toroidal worm 5 and the shaft angle, in order to achieve the whole facewidth of the involute spur cylindrical gear to participate in the meshing needs to meet the following relationship:

b=L sin ?

[0029] The working length L of the elliptical toroidal worm is 72 mm and the facewidth b of the involute cylindrical gear is 8 mm when the selected shaft angle is 5? under the constraints of the self-locking condition and the minimum tooth top width of the worm.

[0030] The involute spur cylindrical gear 4 is ground and shaped with hardened wear-resistant material, and the left flank equations of the involute spur cylindrical gear 4 is as follows:

[00007] ${\begin{matrix} x_{1 L} = r_{b} \cos (u - ?_{0}) + r_{b} u \sin (u - ?_{0}) \\ y_{1 L} = - r_{b} \sin (u - ?_{0}) + r_{b} u \cos (u - ?_{0}) \\ z_{1 L} = h_{L} \end{matrix}$

where, r.sub.b is the radius of the base circle of the involute cylindrical gear, the size of 112.7631 mm; u is the rolling angle formed by the dominant involute tooth profile, the value range is [0.2649,0.3850]; ?.sub.0 is half of the base circular angle corresponding to the transverse base thickness of the involute cylindrical gear, the size of 1.6043?; h is the tooth flank axial parameter, which is related to the rolling angle u and the angle ?.sub.1 of the elliptical toroidal worm gear.

[0031] Similarly, the equations of the right flank of the involute spur cylindrical gear 4 is as follows:

[00008] ${\begin{matrix} x_{1 R} = r_{b} \cos (u - ?_{0}) + r_{b} u \sin (u - ?_{0}) \\ y_{1 R} = r_{b} \sin (u - ?_{0}) - r_{b} u \cos (u - ?_{0}) \\ z_{1 R} = h_{R} \end{matrix}$

[0032] The tooth flank of the involute spur cylindrical gear 4 is used as the tool generatrix surface, and the tooth flank equations of the elliptical toroidal worm 5 are formed according to the generating method envelope. In the established elliptical toroidal worm space drive coordinate system, the coordinate systems ?(o; x, y, z) and ?.sub.p(o.sub.p; x.sub.p, y.sub.p, z.sub.p) represent the starting positions of elliptical toroidal worm 5 and worm wheelinvolute straight cylindrical gear 4, respectively, which are fixed coordinate systems. z and z.sub.p are the elliptical toroidal worm 5 and the two axes are non-orthogonal in space and the shaft angle is ?. The x.sub.p-axis and x-axis are in the same line and in the same direction. ?.sub.1(o.sub.1; x.sub.1, y.sub.1, z.sub.1) and ?.sub.2(o.sub.2; x.sub.2, y.sub.2, z.sub.2) denote the dynamic coordinate systems fixed to the spur cylindrical gear 4 and the elliptical toroidal worm 5, respectively, the spur cylindrical gear 4 and the elliptical toroidal worm 5 are rotated with angular velocities w.sub.1 and w.sub.2 rotate around z.sub.1 and z.sub.2 axes respectively, and the angle of rotation is ?.sub.1 and ?.sub.2 respectively, and the starting position is at ?.sub.1=0 and ?.sub.2=0. The shortest distance between z.sub.1 and z.sub.2 axes is a, which is the center distance of spur gear 4 and elliptical toroidal worm 5, and its value is 135 mm.

[0033] The tooth flank equations of elliptical toroidal worm 5 is obtained from the tooth equations of involute spur cylindrical gear 4 by transformation of the spatial coordinate system, and the coordinate transformation matrix M.sub.12 of the spatial coordinate system is as follows:

[00009] $M_{12} = [\begin{matrix} \cos ?_{2} & \sin ?_{2} & 0 & 0 \\ - \sin ?_{2} & \cos ?_{2} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}]$ $[\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & \sin ? & - \cos ? & 0 \\ 0 & \cos ? & \sin ? & 0 \\ 0 & 0 & 0 & 1 \end{matrix}] [\begin{matrix} 1 & 0 & 0 & - ? \\ 0 & 0 & 1 & 0 \\ 0 & - 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}] [\begin{matrix} \cos ?_{1} & - \sin ?_{1} & 0 & 0 \\ \sin ?_{1} & \cos ?_{1} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}]$

[0034] The angle ?.sub.1 of the elliptical toroidal worm is calculated to be in the range of [?17.25?,17.25?], which is related to its working half angle.

[0035] In the process of spatial conjugate meshing, the two tooth flanks participating in the mesh are in tangential contact at any instant, and there is always a common tangent plane at the tangent point, there is the same normal n. The equation of meshing satisfied at the contact is as follows:

v?n=0

where v is the relative velocity of the conjugate tooth flank at the meshing point.

[0036] This ensures that the two meshing tooth flanks can slide into contact continuously without interfering with each other. After satisfying the above requirements, the equations of the upper side tooth surface of elliptical toroidal worm 5 are as follows:

[00010] ${\begin{matrix} \begin{matrix} x_{2 L} = (x_{1 L} \cos ?_{1} - y_{1 L} \sin ?_{1} - a) \cos ?_{2} + \\ [(x_{1 L} \sin ?_{1} + y_{1 L} \cos ?_{1}) \cos ? + z_{1 L} \sin ?] \sin ?_{2} \end{matrix} \\ \begin{matrix} y_{2 L} = [(x_{1 L} \sin ?_{1} + y_{1 L} \cos ?_{1}) \cos ? + z_{1 L} \sin ?] \cos ?_{2} - \\ (x_{1 L} \cos ?_{1} - y_{1 L} \sin ?_{1} - a) \sin ?_{2} \end{matrix} \\ z_{2 L} = - (x_{1 L} \sin ?_{1} + y_{1 L} \cos ?_{1}) \sin ? + z_{1 L} \cos ? \\ x_{1 L} = r_{b} \cos (u - ?_{0}) + r_{b} u \sin (u - ?_{0}) \\ y_{1 L} = - r_{b} \sin (u - ?_{0}) + r_{b} u \cos (u - ?_{0}) \\ z_{1 L} = h_{L} \\ h_{L} = \frac{r_{b} (i_{12} - \cos ?) + a \cos (u - ?_{0} - ?_{1}) \cos ?}{- \sin (u - ?_{0} - ?_{1}) \sin ?} \end{matrix}$

[0037] Similarly, the equations of the lower side tooth surface of the elliptical toroidal worm 5 is as follows:

[00011] ${\begin{matrix} \begin{matrix} x_{2 R} = (x_{1 R} \cos ?_{1} - y_{1 R} \sin ?_{1} - a) \cos ?_{2} + \\ [(x_{1 R} \sin ?_{1} + y_{1 R} \cos ?_{1}) \cos ? + z_{1 R} \sin ?] \sin ?_{2} \end{matrix} \\ \begin{matrix} y_{2 R} = [(x_{1 R} \sin ?_{1} + y_{1 R} \cos ?_{1}) \cos ? + z_{1 R} \sin ?] \cos ?_{2} - \\ (x_{1 R} \cos ?_{1} - y_{1 R} \sin ?_{1} - a) \sin ?_{2} \end{matrix} \\ x_{2 R} = - (x_{1 R} \sin ?_{1} + y_{1 R} \cos ?_{1}) \sin ? + z_{1 R} \cos ? \\ x_{1 R} = r_{b} \cos (u - ?_{0}) + r_{b} u \sin (u - ?_{0}) \\ y_{1 R} = r_{b} \sin (u - ?_{0}) - r_{b} u \cos (u - ?_{0}) \\ z_{1 R} = h_{R} \\ h_{R} = \frac{r_{b} (i_{12} - \cos ?) + a \cos (u - ?_{0} + ?_{1}) \cos ?}{\sin (u - ?_{0} + ?_{1}) \sin ?} \end{matrix}$

[0038] The above tooth equations are determined by two parameters ?.sub.1 and u, other parameters are known, and upper tooth surface 5-1 and lower tooth surface 5-2 of elliptical toroidal gear pair can be obtained by MATLAB numerical analysis and 3D modeling software such as UG or Pro/E in the range of ?.sub.1 and u, and then they are stitched with the top toroidal surface and root toroidal surface of elliptical toroidal worm to generate the 3D solid model of non-orthogonal elliptical toroidal worm gear pair; after the 3D solid model is assembled with the involute cylindrical spur gear, the upper tooth surface 5-1 and lower tooth surface 5-2 of elliptical toroidal worm are in contact with the gear tooth flank and there is no tooth flank interference, which verifies the feasibility of this transmission form.

[0039] The transmission ratio adopted in this example is 120, and the number of teeth of involute cylindrical gear 4 involved in meshing at the same time is 12, so it has a good error equalization effect; when the shaft angle takes the maximum value of 7.7?, the facewidth of whole tooth contact of involute cylindrical gear can reach 9.65 mm, and the gear tooth flank wear is uniform and the accuracy retention is good, which can be used in precision continuous indexing transmission, comprehensive deviation measurement of elliptical toroidal worm, continuous grinding and processing of cylindrical gear tooth flank, etc. So, it has good popularization application value and industrialization prospect.

[0040] Finally, the above embodiments are only used to illustrate the technical solution of the present invention and not to limit it. To a person of ordinary skill in the art, equivalent substitutions or changes can be made according to the technical solution of the patent of the present invention and its specific embodiments, and all such changes or substitutions shall fall within the scope of protection of the patent of the present invention.

A NON-ORTHOGONAL ELLIPTICAL TOROIDAL WORM GEAR PAIR

Inventors

Cpc classification

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F16H1/16

MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING

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F16H55/22

MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING

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F16H55/0806

MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING

Classification Explorer

F16H2057/0087

MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING

International classification

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F16H1/16

MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING

Classification Explorer

F16H55/08

MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING

Classification Explorer

F16H55/22

MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING

Abstract

Claims

Description