LINE GEAR MECHANISM HAVING VARIABLE TRANSMISSION RATIO

Abstract

The present invention is a variable-ratio line gear mechanism. The mechanism forms a transmission pair consisting of a driving line gear and a driven line gear, of which axes intersect at an arbitrary angle. Transmission is generated by point contact meshing movement of line teeth between the driving line gear and the driven line gear. A contact curve of the line tooth is designed in accordance with space conjugate curve meshing theory, and the designing equation is divided into an equal transmission ratio part and a variable transmission ratio part. The equal transmission ratio part provides a uniform transmission, and the variable transmission ratio part makes the transmission ratio smoothly transit. The line gear mechanism is able to provide periodically transmission with variable transmission ratio, to provide a plurality of transmission ratios during a movement period of the driven line gear, and to enable smooth transitions between respective transmission ratios in accordance with movement rules.

Description

TECHNICAL FIELD

The present invention relates to the field of mechanical transmission, in particular, to a line gear mechanism meshing on a basis of space conjugate curve meshing theory and able to provide periodically variable transmission ratio.

BACKGROUND

In gearing-up or gearing-down between two axes, a gear able to realize a variable transmission velocity ratio is called the variable-ratio gear. The variable-ratio gear is mainly used in transmission occasions with special requirements, for example, in aspects of stepless speed regulation in automobile, oval gear flowmeter, and manipulator with variable transmission ratio.

A gear often used for transmission with variable transmission ratio is mainly a non-circular gear, since a non-circular gear mechanism may realize a special movement and function operation, such as swing, indexing, and varying speed, and in the meantime, the non-circular gear may be designed according to transmission ratio function. At present, the non-circular gear used for transmission with variable speed ratio refers to a non-cylindrical gear, and a non-bevel gear. Elliptic bevel gear is a typical non-bevel gear, in named for its large end node being a spherical ellipse. Orthogonal non-circular surface gear pair is a new-style variable-ratio gear transmission, and it has advantages of the non-circular gear, non-bevel gear and face gear, with its design and processing being more simple than the non-circular gear with variable transmission ratio. In addition, variable transmission ratio may be realized by a planetary gear set. Continuous variable magnetic gear comprises a three-phase winding stator and three concentric rotors. By controlling the speed of a central rotor, the transmission ratio between an input rotor and an output rotor can vary.

Line gear is a new-type gear utilizing a space curve meshing theory in the place of a traditional space curve surface meshing theory, which is mainly applied to micro transmission field, and has advantages such as small dimension, large transmission ratio, and easy manufacturing. At present, the line gear pair can be applied to transmissions on two vertical axes, two intersecting axes and two skewed axes, and its researches such as design equation, contact ratio, strength criterion, micro transmission and manufacturing field have tended to be perfect.

In a micro mechanical device with limited dimension, sometimes there is complicated transmission requirement of periodically variable transmission ratio, which is hard to realize by controlling, while traditional non-circular gear is not suitable for micro dimension.

SUMMARY OF THE INVENTION

Based on the theory of a line gear, the present invention provides a solution for designing a variable-ratio line gear mechanism. A line gear pair mechanism designed by this solution may provide a transmission with periodically variable transmission ratio.

The technical solution adopted in the present invention is as follows.

A variable-ratio line gear mechanism forms a transmission pair consisting of a driving line gear and a driven line gear, of which axes intersect at an arbitrary angle. The driving line gear consists of a wheel body and line teeth, and the driven line gear consists of a wheel body and line teeth. Contact curves of the line tooth of the driving line gear and the line tooth of the driven line gear mesh in accordance with a pair of spatial conjugate curves. The driving line gear is connected with an actuator to provide an input. There are one or more line teeth on the driving line gear. The line tooth of the driving line gear meshes with the line tooth of the driven line gear by point contact. The driven line gear is connected with an output end to provide a movement or an output of force. The line tooth on the driven line gear is the one with property of variable transmission ratio. In one movement period, a plurality of transmission ratios exist, and a smooth transition among different transmission ratios can be made, resulting in a transmission with periodically variable transmission ratio.

In the above-described variable-ratio line gear mechanism, the contact curve on the line tooth of the driving line gear for meshing is a circular helix. The line tooth of the driven line gear is divided into an equal transmission ratio part and a variable transmission ratio part. There are two equations for the contact curves on the line tooth for meshing: one is an equal transmission ratio equation for realizing equal transmission ratio, and the other one is a variable transmission ratio equation for realizing variable transmission ratio.

During the transmission of the above-described variable-ratio line gear mechanism, the variable transmission ratio equation can make the transmission ratio of the line gear smoothly change from one value to another, i.e. a derivative value of transmission ratio function increases or decreases from 0 to a certain value, and then smoothly back to 0.

During the transmission of the above-described variable-ratio line gear mechanism, in order to reduce the rotation and jumpiness of the operation of said mechanism, derivatives of both the variable transmission ratio equation and the variable transmission ratio equation can transit smoothly, i.e. derivative and second derivative of the variable transmission ratio equation both increase or decrease from 0 to a certain value, and then smoothly back to 0.

Accordingly, in the above-described variable-ratio line gear mechanism, O−xyz is a Cartesian coordinate system that is arbitrarily fixed in space, wherein O is an origin of the coordinate system O−xyz, and x, y, z are three coordinate axes of the coordinate system O−xyz. A Cartesian coordinate system O_p−x_py_pz_p is determined in accordance with a position of the coordinate system O−xyz, wherein a plane x_pO_pz_p is in a same plane as a plane xOz, a distance from a coordinate origin O_p to the axis z is a, a distance from O_p to the axis x is b, and an angle between the axis z and an axis z_p is (π−θ). θ is an angle between angular velocity vectors of the driving and driven line gears, with 0°≦θ≦180°. Coordinate systems O₁−x₁y₁z_i and O₂−x₂y₂z₂ are respectively the coordinate systems fixed on the driving line gear and the driven line gear. During transmission, the driving line gear and driven line gear rotate around the axis z and the axis z_p, respectively, and an initial meshing place of the driving line gear and the driven line gear is an initial position. At the initial position, coordinate systems O−x₁y₁z₁ and O₂−x₂y₂z₂ coincide with the coordinate systems O−xyz and O_p−x_py_pz_p respectively. At any time, the origin O₁ coincides with O, an axis z₁ coincides with the axis z, the origin O₂ coincides with O_p, an axis z₂ coincides with the axis z_p, the driving line gear rotates around the axis z at a uniform angular velocity ω₁, angular velocity direction of the driving line gear is a negative direction of the axis z, and an angle that the driving line gear rotates through around the axis z is φ₁; the driven line gear rotates around the axis z_p at a uniform angular velocity ω₂, angular velocity direction of the driven line gear is a negative direction of the axis z_p, an angle that the driven line gear rotates through around the axis z_p is φ₂, then an equation of the contact curve for the driving line gear is as follows in the coordinate system O−x₁y₁z₁

$\{\begin{array}{c}{x}_M^{\left(1\right)}=m\cos t\\ y_M^{\left(1\right)}=m\sin t\\ z_M^{\left(1\right)}=n\pi +nt\end{array}\left(-\pi \le t\le -\frac{\pi }{2}\right),$

and then said variable transmission ratio equation is determined by the following two methods:

1) The variable transmission ratio equation smoothly transits, with an equation for its variable transmission ratio contact curve being as follows:

$\{\begin{array}{c}{x}_M^{\left(2\right)}=\left[\left(m-a\right)\cos \theta -\left(n\pi +nt-b\right)\sin \theta \right]\cos {\varphi }_2\\ y_M^{\left(2\right)}=-\left[\left(m-a\right)\cos \theta -\left(n\pi +nt-b\right)\sin \theta \right]\sin {\varphi }_2\\ z_M^{\left(2\right)}=-\left(m-a\right)\sin \theta -\left(n\pi +nt-b\right)\cos \theta \\ {\varphi }_2=-\frac{{A\left({\varphi }_b-{\varphi }_a\right)}^2}{{\pi }^2}\sin \frac{\pi \left({\varphi }_1-{\varphi }_a\right)}{{\varphi }_b-{\varphi }_a}+C{\varphi }_1\end{array}\hspace{1em}$

wherein, A and C are determined by equations

$\frac{d{\varphi }_2\left({\varphi }_a\right)}{d{\varphi }_1}=\frac{1}{i_a}\mathrm{and}\frac{d{\varphi }_2\left({\varphi }_b\right)}{d{\varphi }_1}=\frac{1}{i_b},$

m is a helical radius of the contact curve of the driving line gear, n is a parameter of the contact curve of the driving line gear related to a pitch, with

$n=\frac{p}{2\pi }$

being defined if the pitch is p, and t is a parameter, with

$-\pi \le t\le -\frac{\pi }{2}$

representing that the contact curve of one line tooth of the driving line gear is a helix of ¼ circumference. When t=−π, the line teeth of the driving line gear and driven line gear begin to mesh. When

$t=-\frac{\pi }{2},$

the driving line gear rotates through ¼ circumference, and the line teeth of the driving line gear and driven line gear mesh to an end and begin to demesh. i_a and i_b are respectively the transmission ratio before and after some period of change process; φ_a and φ_b are respectively a start angle and an end angle of some line tooth of the driving line gear during transmission in a variable transmission ratio process, for example, when

$-\pi <t<-\frac{\pi }{2},{\varphi }_b-{\varphi }_a=\frac{\pi }{2}.$

2) Derivatives of the variable transmission ratio equation and variable transmission equation both can smoothly transit, i.e. equation for the variable transmission ratio contact curve is as follows:

$\{\begin{array}{c}{x}_M^{\left(2\right)}=\left[\left(m-l_1\right)\cos \theta -\left(n\pi +nt-l_2\right)\sin \theta \right]\cos {\varphi }_2\\ y_M^{\left(2\right)}=-\left[\left(m-l_1\right)\cos \theta -\left(n\pi +nt-l_2\right)\sin \theta \right]\sin {\varphi }_2\\ z_M^{\left(2\right)}=-\left(m-l_1\right)\sin \theta -\left(n\pi +nt-l_2\right)\cos \theta \\ {\varphi }_2=\frac{{E\left({\varphi }_b-{\varphi }_a\right)}^3}{8{\pi }^3}\cos \frac{\pi \left({\omega }_1t_1-{\varphi }_a\right)}{{\varphi }_b-{\varphi }_a}+\frac{F}{2}{\left({\omega }_1t_1\right)}^2+G{\omega }_1t_1\\ {\varphi }_1=t+\pi \end{array}\hspace{1em}$

Wherein, E, F and G are determined by equations

$\frac{d^2{\varphi }_2\left({\varphi }_b\right)}{d{\varphi }_1^2}=0,\frac{d{\varphi }_2\left({\varphi }_a\right)}{d{\varphi }_1}=1/i_a\mathrm{and}\frac{d{\varphi }_2\left({\varphi }_b\right)}{d{\varphi }_1}=1/i_b,$

m is a helical radius of the contact curve of the driving line gear, n is a parameter of the contact curve of the driving line gear related to a pitch, with

$n=\frac{p}{2\pi }$

being defined if the pitch is p, and t is a parameter, with

$-\pi \le t\le -\frac{\pi }{2}$

representing that the contact curve of one line tooth of the driving line gear is a helix of ¼ circumference. When t=−π, the line teeth of the driving line gear and driven line gear begin to mesh. When

$t=-\frac{\pi }{2},$

the driving line gear rotates through ¼ circumference, and the line teeth of the driving line gear and driven line gear mesh to an end and begin to demesh. i_a and i_b are respectively the transmission ratio before and after some period of change process; φ_a and φ_b are respectively a start angle and an end angle of some line tooth of the driving line gear during transmission in a variable transmission ratio process, for example, when

$-\pi <t<-\frac{\pi }{2},{\varphi }_b-{\varphi }_a=\frac{\pi }{2}.$

The variable transmission ratio equations created in accordance with claims 5 and 6 may be used in the variable-ratio line gear mechanism.

The line tooth on the driving line gear obtained in the present mechanism is designed on the basis of a circular helix, and there may be one or more line teeth. The line tooth of the driving line gear meshes with the line tooth of the driven line gear by point contact. The driven line gear is connected with an output end to provide a movement or an output of force. Line tooth on the driven line gear includes multiple designs, and may be designed as the line tooth with property of variable transmission ratio according to different equations for the contact curve on the line gear. In one movement period, a plurality of transmission ratios may exist, and a smooth transition among different transmission ratios can be made.

Principle of the present invention is that: in accordance with space conjugate curve meshing theory of the line gear, contact curve equation of the line tooth of the driven line gear required by transmission with equal transmission ratio can be designed; the contact curve equation of the line tooth of the driven line gear is improved, so that the transmission ratio is a certain value when it begins to enter into meshing, and the transmission ratio becomes another value when demeshing, with the transmission ratio smoothly changing in this period, i.e. a derivative value of transmission ratio function increases or decreases from 0 to a certain value, and then smoothly back to 0, or the derivative value of the variable transmission ratio function and second derivative of the variable transmission ratio function both increase or decrease from 0 to a certain value, and then smoothly back to 0, thereby the contact curve equation of the line tooth of the driven line gear required by the transmission with variable transmission ratio being obtained.

Compared with the prior arts, the present invention has the following advantages:

1. It is able to provide a plurality of transmission ratios during a movement period of the driven line gear, and to enable smooth transitions between respective transmission ratios in accordance with movement rules.

2. The transmission only depends on the point contact between the line tooth of the driving line gear and the line tooth of the driven line gear, so as long as the accuracy of the contact curve is guaranteed, with the design being simple, the processing being convenient and the weight and bulk being smaller than that of a traditional gear with variable transmission ratio, which is more suitable for microminiature mechanical and electrical products.

3. Line tooth number of the driving line gear is at least 1, being able to provide a transmission with greater transmission ratio, and able to design the line gear pair under a condition of axes intersecting at an arbitrary angle.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is the coordinate system of the mechanism according to the present invention.

FIG. 2 is two embodiments according to the present invention, including the driving line gear and driven line gear.

FIG. 3 is the embodiment of the driven line gear according to the present invention.

FIG. 4 is the establishment method of the line tooth entity.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT

The present invention is further described below in combination with the drawings, but the implementations of the present invention are not limited hereto.

1. A variable-ratio line gear mechanism described in the present invention comprises a driving line gear and a driven line gear. Coordinate systems of the driving line gear and driven line gear, which are shown in FIG. 1, are used for establishing a contact curve equation of a line tooth of the line gear.

As shown in FIG. 1, O−xyz and O_p−x_py_pz_p are fixed Cartesian coordinate systems, and O_p−x_py_pz_p is determined in accordance with a position of O−xyz: a plane x_pO_pz_p is in a same plane as a plane xOz, a distance from O_p to the axis z is a, a distance from O_p to the axis x is b, and an angle between the axis z and an axis z_p is (π−θ). θ is an angle between angular velocity vectors of the driving and driven line gears.

O₁−x₁y₁z₁ and O₂−x₂y₂z₂ are respectively the coordinate systems fixed on the driving line gear and the driven line gear. During transmission, the driving line gear and driven line gear rotate around the axis z and the axis z_p, respectively.

2. The variable-ratio line gear mechanism is shown in FIG. 2a and FIG. 2b. The left one is the driving line gear 1. On the driving line gear, a line tooth 2 of the driving line gear is provided. The right one is the driven line gear 3.

As shown in FIG. 3, line teeth 4 and 5 with equal transmission ratio and line teeth 6 and 7 with variable transmission ratio are provided on the driven line gear. When the driving line gear and driven line gear mesh to the line teeth 4 and 5 with equal transmission ratio, the transmission ratio is i_a and i_b. When they mesh to the line teeth 6 and 7 with variable transmission ratio, the transmission ratios smoothly transits from i_a to i_b, and smoothly transits from i_b to i_a, respectively. However, the implementations of the present invention are not limited hereto.

An equation designing the above-described line teeth with equal transmission ratio and variable transmission ratio is determined by the following equations.

The line tooth of the driving line gear is determined by its contact curve equation. The equation of the contact curve in O₁−x₁y₁z₁ is:

$\begin{array}{cc}\{\begin{array}{c}{x}_M^{\left(1\right)}=m\cos t\\ y_M^{\left(1\right)}=m\sin t\\ z_M^{\left(1\right)}=n\pi +nt\end{array}\left(-\pi \le t\le -\frac{\pi }{2}\right)& \left(1\right)\end{array}$

The line tooth of the driven line gear is determined by its contact curve equation. The driven contact curve is calculated by the contact curve of the line tooth of the driving line gear and a space curve meshing theory, and its equation in O₂−x₂y₂z₂ is as follows:

When the transmission ratio is i, the contact curve equation on the line tooth with equal transmission ratio is:

$\begin{array}{cc}\{\begin{array}{c}{x}_M^{\left(2\right)}=\left[\left(m-a\right)\cos \theta -\left(n\pi +nt-b\right)\sin \theta \right]\cos \left(\frac{t+\pi }{i}\right)\\ y_M^{\left(2\right)}=-\left[\left(m-a\right)\cos \theta -\left(n\pi +nt-b\right)\sin \theta \right]\sin \left(\frac{t+\pi }{i}\right)\\ z_M^{\left(2\right)}=-\left(m-a\right)\sin \theta -\left(n\pi +nt-b\right)\cos \theta \end{array}& \left(2\right)\end{array}$

When the transmission ratio is i_a to i_b, there may be two forms of the contact curve equation on the variable-ratio line gear mechanism, one is:

$\begin{array}{cc}\{\begin{array}{c}{x}_M^{\left(2\right)}=\left[\left(m-a\right)\cos \theta -\left(n\pi +nt-b\right)\sin \theta \right]\cos {\varphi }_2\\ y_M^{\left(2\right)}=-\left[\left(m-a\right)\cos \theta -\left(n\pi +nt-b\right)\sin \theta \right]\sin {\varphi }_2\\ z_M^{\left(2\right)}=-\left(m-a\right)\sin \theta -\left(n\pi +nt-b\right)\cos \theta \\ {\varphi }_2=\frac{{A\left({\varphi }_b-{\varphi }_a\right)}^2}{{\pi }^2}\sin \frac{\pi \left({\varphi }_1-{\varphi }_a\right)}{{\varphi }_b-{\varphi }_a}+C{\varphi }_1\end{array}& \left(3\right)\end{array}$

wherein, A and C is determined by:

$\begin{array}{cc}\frac{d{\varphi }_2\left({\varphi }_a\right)}{d{\varphi }_1}=\frac{1}{i_a}& \left(4\right)\\ \frac{d{\varphi }_2\left({\varphi }_b\right)}{d{\varphi }_1}=\frac{1}{i_b}.& \left(5\right)\end{array}$

The other one is:

$\begin{array}{cc}\{\begin{array}{c}{x}_M^{\left(2\right)}=\left[\left(m-l_1\right)\cos \theta -\left(n\pi +nt-l_2\right)\sin \theta \right]\cos {\varphi }_2\\ y_M^{\left(2\right)}=-\left[\left(m-l_1\right)\cos \theta -\left(n\pi +nt-l_2\right)\sin \theta \right]\sin {\varphi }_2\\ z_M^{\left(2\right)}=-\left(m-l_1\right)\sin \theta -\left(n\pi +nt-l_2\right)\cos \theta \\ {\varphi }_2=\frac{{E\left({\varphi }_b-{\varphi }_a\right)}^3}{8{\pi }^3}\cos \frac{\pi \left({\omega }_1t_1-{\varphi }_a\right)}{{\varphi }_b-{\varphi }_a}+\frac{F}{2}{\left({\omega }_1t_1\right)}^2+G{\omega }_1t_1\\ {\varphi }_1=t+\pi \end{array}& \left(6\right)\end{array}$

wherein, E, F and G are determined by:

$\begin{array}{cc}\frac{d^2{\varphi }_2\left({\varphi }_b\right)}{d{\varphi }_1^2}=0& \left(7\right)\\ \frac{d{\varphi }_2\left({\varphi }_a\right)}{d{\varphi }_1}=\frac{1}{i_a}& \left(8\right)\\ \frac{d{\varphi }_2\left({\varphi }_b\right)}{d{\varphi }_1}=\frac{1}{i_b}& \left(9\right)\end{array}$

The physical meaning of each parameter in the equations is as follows:

m is a helical radius of the contact curve of the driving line gear;

n is a parameter of the contact curve of the driving line gear related to a pitch, with

$n=\frac{p}{2\pi }$

being defined if the pitch is p;

t is a parameter, with

$-\pi \le t\le -\frac{\pi }{2}$

representing that the contact curve of one line tooth of the driving line gear is a helix of ¼ circumference. When t=−π, the line teeth of the driving line gear and driven line gear begin to mesh; when

$t=-\frac{\pi }{2},$

the driving line gear rotates through ¼ circumference, and the line teeth of the driving line gear and driven line gear mesh to an end and begin to demesh;

a and b are position parameters of the driving line gear and driven line gear, as shown in FIG. 1;

θ is an angle parameter of the driving line gear and driven line gear, as shown in FIG. 1;

i_a and i_b are two transmission ratios which are required;

φ₁ is a rotation angle of the driving line gear;

φ₂ is a rotation angle of the driven line gear;

φ_a and φ_b are respectively a start angle and an end angle of some line tooth of the driving line gear during transmission in a variable transmission ratio process, for example, when

$-\pi <t<-\frac{\pi }{2},{\varphi }_b-{\varphi }_a=\frac{\pi }{2}.$

3. In accordance with equations (1) to (3), it is possible to establish a line tooth entity. The line tooth entity only needs to meet the strength requirement, and line tooth entity itself does not have any specific shape requirement. As shown in FIG. 4, at each meshing point, a certain bulk is inversely extended respectively at both side (−γ₁ and γ₁ in FIG. 4) of the contact direction of the driving and driven line teeth, that the required line teeth can be generated. A wheel body is used to fix and connect the line teeth.

Claims

1. A variable-ratio line gear mechanism, characterized in that, the mechanism forms a transmission pair consisting of a driving line gear and a driven line gear, of which axes intersect at an arbitrary angle, the driving line gear consisting of a wheel body and line teeth, the driven line gear consisting of a wheel body and line teeth, contact curves of the line tooth of the driving line gear and the line tooth of the driven line gear meshing in accordance with a pair of space conjugate curves, the driving line gear being connected with an actuator to provide an input, there being one or more line teeth on the driving line gear, the line tooth of the driving line gear being meshed with the line tooth of the driven line gear by point contact, the driven line gear being connected with an output end to provide a movement or an output of force, the line tooth on the driven line gear being the one with property of variable transmission ratio, so that in one movement period, a plurality of transmission ratios exist, and a smooth transition among different transmission ratios can be made, resulting in a transmission with periodically variable transmission ratio.

2. The variable-ratio line gear mechanism according to claim 1, wherein the contact curve on the line tooth of the driving line gear for meshing is a circular helix, the line tooth of the driven line gear is divided into an equal transmission ratio part and a variable transmission ratio part, and there are two equations for the contact curve on the line tooth for meshing: one is an equal transmission ratio equation for realizing equal transmission ratio, and the other one is a variable transmission ratio equation for realizing variable transmission ratio.

3. The variable-ratio line gear mechanism according to claim 2, wherein during the transmission of said mechanism, the variable transmission ratio equation can make the transmission ratio of the line gear smoothly change from one value to another, i.e. a derivative value of transmission ratio function increases or decreases from 0 to a certain value, and then smoothly back to 0.

4. The variable-ratio line gear mechanism according to claim 3, wherein said variable transmission ratio equation is determined as follows: O−xyz is a Cartesian coordinate system that is arbitrarily fixed in space, O is an origin of the coordinate system O−xyz x, y, z are three coordinate axes of the coordinate system O−xyz, a Cartesian coordinate system Op−xpypzp is determined in accordance with a position of the coordinate system O−xyz, a plane xpOpzp is in a same plane as a plane xOz, a distance from a coordinate origin Op to the axis z is a, a distance from Op to the axis X is b, an angle between the axis z and an axis zp is (π−θ), θ is an angle between angular velocity vectors of the driving and driven line gears, with 0°≦θ≦180°, and coordinate systems O1−x1y1z1 and O2−x2y2z2 are respectively the coordinate systems fixed on the driving line gear and the driven line gear; during transmission, the driving line gear and driven line gear rotate around the axis z and the axis zp respectively, and an initial meshing place of the driving line gear and the driven line gear is an initial position; at the initial position, coordinate systems O1−x1y1z1 and O2−x2y2z2 coincide with the coordinate systems O−xyz and Op−xpypzp, respectively; at any time, the origin O1 coincides with O, an axis z1 coincides with the axis z, the origin O2 coincides with Op, an axis z2 coincides with the axis zp, the driving line gear rotates around the axis z at a uniform angular velocity ω1, angular velocity direction of the driving line gear is a negative direction of the axis z, and an angle that the driving line gear rotates through around the axis z is φ1; the driven line gear rotates around the axis zp at a uniform angular velocity zp, angular velocity direction of the driven line gear is a negative zp direction of the axis zp, an angle that the driven line gear rotates through around the axis zp is φ2, then an equation of the driving contact curve is as follows in the coordinate system O1−x1y1z1:

5. The variable-ratio line gear mechanism according to claim 2, wherein during the transmission of said mechanism, in order to reduce the rotation and jumpiness of the operation of said mechanism, derivative of the variable transmission ratio equation can also transit smoothly, i.e. second derivative of the variable transmission ratio equation increases or decreases from 0 to a certain value, and then smoothly back to 0.

6. The variable-ratio line gear mechanism according to claim 5, wherein said equation for variable transmission ratio contact curve can be expressed as: