Rotation-movement conversion linear gear mechanism

Abstract

The present invention discloses a line gear mechanism for rotation-movement conversion, comprising a driving line gear (1) and a driven line gear (2). A stagger angle between an axis of the driving line gear and an axis of the driven line gear is any value from 0 to 180. By a point contact meshing between a driving contact curve of a driving line tooth on the driving line gear (1) and a driven contact curve of a driven line tooth on the driven line gear (2), and by utilizing rotation of the driving line gear (1), it achieves that the driven line gear (2) rotates while moving smoothly. The line gear mechanism for rotation-movement conversion is simple in structure, easy to design, can achieve small displacement of movement, and is especially suitable for the conversion of small machinery from rotation to linear motion.

Claims

1. A line gear mechanism for rotation-movement conversion, characterized in that, the line gear mechanism comprises a driving line gear and a driven line gear, the driving line gear and the driven line gear forms a transmission pair, as the driving line gear rotates, the driven line gear moves smoothly while rotating, and a stagger angle between an axis of the driving line gear and an axis of the driven line gear is 0 to 180, wherein the transmission pair of the mechanism realizes transmission of the line gear mechanism by a point contact meshing between a driving contact curve on a driving line tooth and a driven contact curve on a driven line tooth, the driving contact curve and the driven contact curve being a pair of conjugate contact curves, and a space curve meshing equation of the line gear mechanism for rotation-movement conversion is determined as follows: oxyz and o.sub.qx.sub.qy.sub.qz.sub.q are two space Cartesian coordinate systems, o is an origin of the coordinate system oxyz with an arbitrary position, and x, y, z are three coordinate axes of the coordinate system oxyz; o.sub.q is an origin of the coordinate system o.sub.qx.sub.qy.sub.qz.sub.q, and x.sub.q, y.sub.q, z.sub.q are three coordinate axes of the coordinate system o.sub.qx.sub.qy.sub.qz.sub.q; a plane xoz is in a same plane as a plane x.sub.po.sub.pz.sub.p, a distance from the origin o.sub.q to a plane yoz is |a|, a distance from the origin o.sub.q to a plane xoy is |b|, a distance from the origin o.sub.q to the plane xoz is |b|, the y axis is parallel to the y.sub.q axis, and an angle between the Z axis and the z.sub.p axis is (), with 0180; a space Cartesian coordinate system o.sub.1x.sub.1y.sub.1z.sub.1 is fixedly connected with the driving line gear, o.sub.1 is an origin of the coordinate system o.sub.1x.sub.1y.sub.1z.sub.1, and x.sub.1, y.sub.1, z.sub.1 are three coordinate axes of a coordinate system o.sub.1x.sub.1y.sub.1z.sub.1; a space Cartesian coordinate system o.sub.3x.sub.3y.sub.3z.sub.3 is fixedly connected with the driven line gear, o.sub.3 is an origin of the coordinate system ox.sub.3y.sub.3z.sub.3, x.sub.3, y.sub.3, z.sub.3 are three coordinate axes of the coordinate system o.sub.3x.sub.3y.sub.3z.sub.3, and an initial meshing place of the driving line gear and the driven line gear is an initial position; at the initial position, the coordinate systems o.sub.1x.sub.1y.sub.1z.sub.1 and o.sub.3x.sub.3y.sub.3z.sub.3 coincide with the coordinate systems oxyz and o.sub.qx.sub.qy.sub.qz.sub.q, respectively; at any time, the origin o.sub.1 coincides with o, the z.sub.1 axis coincides with the z axis, the origin o.sub.3 coincides with o.sub.q, and the z.sub.3 axis coincides with the z.sub.q axis; when 0<90, the driving line gear rotates about the z axis at a uniform angular velocity .sub.1, a direction of the angular velocity of the driving line gear is a negative direction of the z axis, and an angle that the driving line gear rotates through about the z axis is .sub.1; the driven line gear rotates about the z.sub.q axis at a uniform angular velocity .sub.2, a direction of the angular velocity of the driven line gear is a negative direction of the z.sub.q axis, and an angle that the driven line gear rotates through about the z.sub.q axis is .sub.3; meanwhile, the driven line gear moves along the negative direction of the z.sub.q axis with a speed of A, a displacement of the driven line gear has a magnitude of s, and then a space curve meshing equation thereof is:
y.sub.M.sup.(1).sub.1.sub.x.sup.(1)x.sub.M.sup.(1).sub.1.sub.y.sup.(1)+y.sub.M.sup.(1).sub.2.sub.x.sup.(1)cos .sub.2x.sub.M.sup.(1).sub.y.sup.(1)cos (A sin c.sub.2 cos ).sub.x.sup.(1)cos .sub.1+(A sin c.sub.2 cos ).sub.y.sup.(1)sin .sub.1+(.sub.2 sin (z.sub.M.sup.(1)b+s)+.sub.2a cos )(.sub.x.sup.(1)sin .sub.1)(.sub.2 sin (z.sub.M.sup.(1)b+s)+.sub.2a cos )(.sub.y.sup.(1)cos .sub.1)+(A cos .sub.2 sin x.sub.M.sup.(1)sin .sub.1+.sub.2 sin y.sub.M.sup.(1)cos .sub.1c.sub.2 sin ).sub.z.sup.(1)=0 wherein, $\{\begin{array}{c}{x}_M^{\left(1\right)}=x_M^{\left(1\right)}\left(t\right)\\ y_M^{\left(1\right)}=y_M^{\left(1\right)}\left(t\right)\\ z_M^{\left(1\right)}=z_M^{\left(1\right)}\left(t\right)\end{array}$ is an equation of the driving contact curve of the driving line gear, t is a parameter, .sup.(1)=.sub.x.sup.(1)i.sup.(1)+.sub.y.sup.(1)j.sup.(1)+.sub.z.sup.(1)k.sup.(1) is a unit vector of a principal normal of the driving contact curve of the mechanism at a meshing point in the coordinate system o.sub.1x.sub.1y.sub.1z.sub.1, i.sup.(1), j.sup.(1) and l.sup.(1) are unit vectors of the x.sub.1, y.sub.1, z.sub.1 axes, an equation of the driven contact curve of the driven line gear in the coordinate system o.sub.3x.sub.3y.sub.3z.sub.3 is: $\{\begin{array}{c}{x}_M^{\left(3\right)}=\left(-\cos {}_1\cos {}_3\cos +\sin {}_1\sin {}_3\right)x_M^{\left(1\right)}+\\ \left(-\sin {}_1\cos {}_3\cos -\cos {}_1\sin {}_3\right)y_M^{\left(1\right)}-\\ \cos {}_3\sin z_M^{\left(1\right)}+\left(-a\cos {}_3\cos +b\cos {}_3\sin +c\sin {}_3\right)\\ y_M^{\left(3\right)}=\left(-\cos {}_1\sin {}_3\cos -\sin {}_1\cos {}_3\right)x_M^{\left(1\right)}+\\ \left(-\sin {}_1\sin {}_3\cos +\cos {}_1\cos {}_3\right)y_M^{\left(1\right)}-\\ \sin {}_3\sin z_M^{\left(1\right)}+\left(-a\sin {}_3\cos +b\sin {}_3\sin -c\cos {}_3\right)\\ z_M^{\left(3\right)}=\cos {}_1\sin x_M^{\left(1\right)}+\sin {}_1\sin y_M^{\left(1\right)}-\cos z_M^{\left(1\right)}+a\sin +\\ b\cos -s\end{array}$ wherein, .sub.2=i.sub.21.sub.1, .sub.3=i.sub.21.sub.1, i.sub.21 is a transmission ratio of the driving line gear to the driven line gear; when 90180, the driving line gear rotates about the z axis at the uniform angular velocity .sub.1, the direction of the angular velocity of the driving line gear is the negative direction of the z axis, at this moment, the driven line gear rotates about the z.sub.q axis at an angular velocity with a magnitude of .sub.2 and a direction being a positive direction of the z.sub.q axis; meanwhile, the driven line gear moves along the z.sub.q axis at a speed with a magnitude of A and a direction being the positive direction of the z.sub.q axis, an angle that the driving line gear rotates through about the z axis is .sub.1, an angle that the driven line gear rotates through about the z.sub.q axis is .sub.3, a displacement of moving along the z.sub.q axis is s, and then the space curve meshing equation of the mechanism is:
y.sub.M.sup.(1).sub.1.sub.x.sup.(1)+x.sub.M.sup.(1).sub.1.sub.y.sup.(1)+y.sub.M.sup.(1).sub.2.sub.x.sup.(1)cos .sub.2x.sub.M.sup.(1).sub.y.sup.(1)cos +(A sin c.sub.2 cos ).sub.x.sup.(1)cos .sub.1+(A sin c.sub.2 cos ).sub.y.sup.(1)sin .sub.1+(.sub.2 sin (z.sub.M.sup.(1)bs)+.sub.2a cos )(.sub.x.sup.(1)sin .sub.1)(.sub.2 sin (z.sub.M.sup.(1)bs)+.sub.2a cos )(.sub.y.sup.(1)cos .sub.1)+(A cos .sub.2 sin x.sub.M.sup.(1)sin .sub.1+.sub.2 sin y.sub.M.sup.(1)cos .sub.1c.sub.2 sin ).sub.z.sup.(1)=0 wherein, $\{\begin{array}{c}{x}_M^{\left(1\right)}=x_M^{\left(1\right)}\left(t\right)\\ y_M^{\left(1\right)}=y_M^{\left(1\right)}\left(t\right)\\ z_M^{\left(1\right)}=z_M^{\left(1\right)}\left(t\right)\end{array}$ is an equation of the driving contact curve of the driving line gear in the coordinate system o.sub.1x.sub.1y.sub.1z.sub.1, t is a parameter, .sup.(1)=.sub.x.sup.(1)i.sup.(1)+.sub.y.sup.(1)j.sup.(1)+.sub.z.sup.(1)k.sup.(1) is a unit vector of a principal normal of the driving contact curve of the mechanism at a meshing point in the coordinate system o.sub.1x.sub.1y.sub.1z.sub.1, i.sup.(1), j.sup.(1) and k.sup.(1) are unit vectors of the x.sub.1, y.sub.1, z.sub.1 axes, an equation of the driven contact curve of the driven line gear in the coordinate system o.sub.3x.sub.3y.sub.3z.sub.3 is: $\{\begin{array}{c}{x}_M^{\left(3\right)}=\left(-\cos {}_1\cos {}_3\cos -\sin {}_1\sin {}_3\right)x_M^{\left(1\right)}+\\ \left(-\sin {}_1\cos {}_3\cos +\cos {}_1\sin {}_3\right)y_M^{\left(1\right)}-\\ \cos {}_3\sin z_M^{\left(1\right)}+\left(-a\cos {}_3\cos +b\cos {}_3\sin -c\sin {}_3\right)\\ y_M^{\left(3\right)}=\left(\cos {}_1\sin {}_3\cos -\sin {}_1\cos {}_3\right)x_M^{\left(1\right)}+\\ \left(\sin {}_1\sin {}_3\cos +\cos {}_1\cos {}_3\right)y_M^{\left(1\right)}+\\ \sin {}_3\sin z_M^{\left(1\right)}+\left(a\sin {}_3\cos -b\sin {}_3\sin -c\cos {}_3\right)\\ z_M^{\left(3\right)}=\cos {}_1\sin x_M^{\left(1\right)}+\sin {}_1\sin y_M^{\left(1\right)}-\cos z_M^{\left(1\right)}+a\sin +\\ b\cos +s\end{array}$ wherein .sub.2=i.sub.21.sub.1, .sub.3=i.sub.21.sub.1, i.sub.21 is a transmission ratio of the driving line gear to the driven line gear.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) FIG. 1 is a schematic view of a spatial meshing coordinate system in an implementation.

(2) FIG. 2 is a schematic view of a driving line gear and its line tooth in an implementation.

(3) FIG. 3 is a schematic view of a driven line gear and its line tooth in an implementation.

(4) FIG. 4 is a schematic view of meshing of the driving line gear and driven line gear in an implementation.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT

(5) The implementation of the present invention will be further described below with reference to the accompanying drawings, and the present invention has been fully described for those skilled in the art, and the scope of protection of the present invention is not limited to the following content.

(6) A space curve meshing equation of a line gear mechanism for rotation-movement conversion of the driving contact curve and the driven contact curve in the present invention is in accordance with the space curve meshing theory.

(7) FIG. 1 illustrates a schematic view of a spatial meshing coordinate system of a line gear mechanism for rotation-movement conversion. oxyz, o.sub.px.sub.py.sub.pz.sub.p and o.sub.qx.sub.qy.sub.qz.sub.q are three space Cartesian coordinate system. o is an origin of the coordinate system oxyz, and x y z are three coordinate axes of the coordinate system oxyz. o.sub.p is an origin of the coordinate system o.sub.px.sub.py.sub.pz.sub.p and x.sub.p y.sub.p z.sub.p are three coordinate axes of the coordinate system o.sub.px.sub.py.sub.pz.sub.p. o.sub.q is an origin of the coordinate system o.sub.qx.sub.qy.sub.qz.sub.q, and x.sub.q y.sub.q z.sub.q are three coordinate axes of the coordinate system o.sub.qx.sub.qy.sub.qz.sub.q. A plane xoz is in a same plane as a plane x.sub.po.sub.pz.sub.p, a distance from the origin o.sub.p to the z axis is |a|, and a distance from the origin o.sub.p to the x axis is |b|. o.sub.qx.sub.qy.sub.qz.sub.q is obtained by translating by a distance |c| along the direction y.sub.p based on o.sub.px.sub.py.sub.pz.sub.p, an angle between the z axis and the z.sub.p axis has a supplementary angle of , 0180, and equals to the supplementary angle of the angle between the z axis and the z.sub.p axis. The space Cartesian coordinate system o.sub.1x.sub.1y.sub.1z.sub.1 is fixedly connected with the driving line gear, o.sub.1 is an origin of the coordinate system o.sub.1x.sub.1y.sub.1z.sub.1, and x.sub.1 y.sub.1 z.sub.1 are three coordinate axes of the coordinate system o.sub.1x.sub.1y.sub.1z.sub.1. The space Cartesian coordinate system o.sub.3x.sub.3y.sub.3z.sub.3 is fixedly connected with the driven line gear, o.sub.3 is an origin of the coordinate system o.sub.3x.sub.3y.sub.3z.sub.3 coordinate system, x.sub.3 y.sub.3 z.sub.3 are three coordinate axes of the coordinate system o.sub.3x.sub.3y.sub.3z.sub.3, and an initial meshing place of the driving line gear and the driven line gear is an initial position. At the initial position, the coordinate systems o.sub.1x.sub.1y.sub.1z.sub.1 and o.sub.3x.sub.3y.sub.3z.sub.3 coincide with the coordinate systems o.sub.qx.sub.q and o.sub.qx.sub.qy.sub.qz.sub.q, respectively. At any time, the origin o.sub.1 coincides with o, the z.sub.1 axis coincides with the z axis, the origin o.sub.3 coincides with o.sub.q, and the z.sub.3 axis coincides with the z.sub.q axis. The driving line gear rotates about the z axis at a uniform angular velocity .sub.1, a direction of the angular velocity of the driving line gear is a negative direction of the z axis, as shown in FIG. 1, and an angle that the driving line gear rotates through about the z axis is .sub.1. The driven line gear rotates about the z.sub.q axis at a uniform angular velocity .sub.2, a direction of the angular velocity of the driven line gear is as shown in FIG. 1, and an angle that the driven line gear rotates through about the z.sub.q axis is .sub.3. Meanwhile, the driven line gear moves along the negative direction of the z.sub.q axis with a speed of A, and a displacement of the driven line gear has a magnitude denoted by s.

(8) Using the knowledge of differential geometry and space curve meshing theory, an equation (1) may be obtained:

(9) $\begin{array}{cc}\{\begin{array}{c}-y_M^{\left(1\right)}{}_1{}_x^{\left(1\right)}+x_M^{\left(1\right)}{}_1{}_y^{\left(1\right)}+y_M^{\left(1\right)}{\stackrel{\_}{}}_2{}_x^{\left(1\right)}\cos -\\ {\stackrel{\_}{}}_2{x}_M^{\left(1\right)}{}_y^{\left(1\right)}\cos +\left(-A\sin -c{\stackrel{\_}{}}_2\cos \right){}_x^{\left(1\right)}\cos {}_1+\\ \left(-A\sin -c{\stackrel{\_}{}}_2\cos \right){}_y^{\left(1\right)}\sin {}_1+\\ \left({\stackrel{\_}{}}_2\sin \left(z_M^{\left(1\right)}-b-s\right)+{\stackrel{\_}{}}_{2}a\cos \right)\left({}_x^{\left(1\right)}\sin {}_1\right)-\\ \left({\stackrel{\_}{}}_2\sin \left(z_M^{\left(1\right)}-b-s\right)+{\stackrel{\_}{}}_{2}a\cos \right)\left({}_y^{\left(1\right)}\cos {}_1\right)+\\ (A\cos -{\stackrel{\_}{}}_2\sin x_M^{\left(1\right)}\sin {}_1+\\ {\stackrel{\_}{}}_2\sin y_M^{\left(1\right)}\cos {}_1-c{\stackrel{\_}{}}_2\sin ){}_z^{\left(1\right)}=0\\ x_M^{\left(1\right)}=x_M^{\left(1\right)}\left(t\right)\\ y_M^{\left(1\right)}=y_M^{\left(1\right)}\left(t\right)\\ z_M^{\left(1\right)}=z_M^{\left(1\right)}\left(t\right)\\ {}_2=i_{21}{}_1\\ {}_3=i_{21}{}_1\\ x_M^{\left(3\right)}=\left(-\cos {}_1\cos {}_3\cos -\sin {}_1\sin {}_3\right)x_M^{\left(1\right)}+\\ \left(-\sin {}_1\cos {}_3\cos +\cos {}_1\sin {}_3\right)y_M^{\left(1\right)}-\\ \cos {}_3\sin z_M^{\left(1\right)}+(-a\cos {}_3\cos +\\ b\cos {}_3\sin -c\sin {}_3)\\ y_M^{\left(3\right)}=\left(\cos {}_1\sin {}_3\cos -\sin {}_1\cos {}_3\right)x_M^{\left(1\right)}+\\ \left(\sin {}_1\sin {}_3\cos +\cos {}_1\cos {}_3\right)y_M^{\left(1\right)}+\\ \sin {}_3\sin z_M^{\left(1\right)}+(a\cos {}_3\cos -\\ b\sin {}_3\sin -c\cos {}_3)\\ z_M^{\left(3\right)}=\cos {}_1\sin x_M^{\left(1\right)}+\sin {}_1\sin y_M^{\left(1\right)}-\\ \cos z_M^{\left(1\right)}+a\sin +b\cos +s\end{array}\mathrm{wherein},& \left(1\right)\\ -y_M^{\left(1\right)}{}_1{}_x^{\left(1\right)}+x_M^{\left(1\right)}{}_1{}_y^{\left(1\right)}+y_M^{\left(1\right)}{\stackrel{\_}{}}_2{}_x^{\left(1\right)}\cos -{\stackrel{\_}{}}_2{x}_M^{\left(1\right)}{}_y^{\left(1\right)}\cos +\left(-A\sin -c{\stackrel{\_}{}}_2\cos \right){}_x^{\left(1\right)}\cos {}_1+\left(-A\sin -c{\stackrel{\_}{}}_2\cos \right){}_y^{\left(1\right)}\sin {}_1+\left({\stackrel{\_}{}}_2\sin \left(z_M^{\left(1\right)}-b-s\right)+{\stackrel{\_}{}}_{2}a\cos \right)\left({}_x^{\left(1\right)}\sin {}_1\right)-\left({\stackrel{\_}{}}_2\sin \left(z_M^{\left(1\right)}-b-s\right)+{\stackrel{\_}{}}_{2}a\cos \right)\left({}_y^{\left(1\right)}\cos {}_1\right)+\left(A\cos -{\stackrel{\_}{}}_2\sin x_M^{\left(1\right)}\sin {}_1+{\stackrel{\_}{}}_2\sin y_M^{\left(1\right)}\cos {}_1-c{\stackrel{\_}{}}_2\sin \right){}_z^{\left(1\right)}=0& \left(2\right)\end{array}$

(10) Equation (2) is the space curve meshing equation of the line gear mechanism for rotation-movement conversion.

(11) $\{\begin{array}{c}{x}_M^{\left(1\right)}=x_M^{\left(1\right)}\left(t\right)\\ y_M^{\left(1\right)}=y_M^{\left(1\right)}\left(t\right)\\ z_M^{\left(1\right)}=z_M^{\left(1\right)}\left(t\right)\end{array}$
is an equation for the driving contact curve in the coordinate system o.sub.1x.sub.1y.sub.1z.sub.1, and t is a parameter;

(12) .sup.(1) is a unit vector of a principal normal of the driving contact curve at a meshing point in the coordinate system o.sub.1x.sub.1y.sub.1z.sub.1, i.e., .sup.(1)=.sub.x.sup.(1)i.sup.(1)+.sub.y.sup.(1)j.sup.(1)+.sub.z.sup.(1)k.sup.(1), and i.sup.(1) j.sup.(1) k.sup.(1) are unit vectors of the z.sub.1 y.sub.1 z.sub.1 axes, respectively.

(13) In particular:

(14) ${}_x^{\left(1\right)}=\frac{\begin{array}{c}{x}_M^{{\left(1\right)}_{}}\left(t\right)\left[{x_M^{{\left(1\right)}_{}}\left(t\right)}^2+{y_M^{{\left(1\right)}_{}}\left(t\right)}^2+{z_M^{{\left(1\right)}_{}}\left(t\right)}^2\right]-\\ x_M^{{\left(1\right)}_{}}\left(t\right)\left[x_M^{{\left(t\right)}_{}}\left(t\right)x_M^{{\left(t\right)}_{}}\left(t\right)+y_M^{{\left(1\right)}_{}}\left(t\right)y_M^{{\left(1\right)}_{}}\left(t\right)+z_M^{{\left(1\right)}_{}}\left(t\right)z_M^{{\left(1\right)}_{}}\left(t\right)\right]\end{array}}{{\left[{x_M^{{\left(1\right)}_{}}\left(t\right)}^2+{y_M^{{\left(1\right)}_{}}\left(t\right)}^2+{z_M^{{\left(1\right)}_{}}\left(t\right)}^2\right]}^2}$${}_y^{\left(1\right)}=\frac{\begin{array}{c}{y}_M^{{\left(1\right)}_{}}\left(t\right)\left[{x_M^{{\left(1\right)}_{}}\left(t\right)}^2+{y_M^{{\left(1\right)}_{}}\left(t\right)}^2+{z_M^{{\left(1\right)}_{}}\left(t\right)}^2\right]-\\ y_M^{{\left(1\right)}_{}}\left(t\right)\left[x_M^{{\left(t\right)}_{}}\left(t\right)x_M^{{\left(t\right)}_{}}\left(t\right)+y_M^{{\left(1\right)}_{}}\left(t\right)y_M^{{\left(1\right)}_{}}\left(t\right)+z_M^{{\left(1\right)}_{}}\left(t\right)z_M^{{\left(1\right)}_{}}\left(t\right)\right]\end{array}}{{\left[{x_M^{{\left(1\right)}_{}}\left(t\right)}^2+{y_M^{{\left(1\right)}_{}}\left(t\right)}^2+{z_M^{{\left(1\right)}_{}}\left(t\right)}^2\right]}^2}$${}_z^{\left(1\right)}=\frac{\begin{array}{c}{z}_M^{{\left(1\right)}_{}}\left(t\right)\left[{x_M^{{\left(1\right)}_{}}\left(t\right)}^2+{y_M^{{\left(1\right)}_{}}\left(t\right)}^2+{z_M^{{\left(1\right)}_{}}\left(t\right)}^2\right]-\\ z_M^{{\left(1\right)}_{}}\left(t\right)\left[x_M^{{\left(t\right)}_{}}\left(t\right)x_M^{{\left(t\right)}_{}}\left(t\right)+y_M^{{\left(1\right)}_{}}\left(t\right)y_M^{{\left(1\right)}_{}}\left(t\right)+z_M^{{\left(1\right)}_{}}\left(t\right)z_M^{{\left(1\right)}_{}}\left(t\right)\right]\end{array}}{{\left[{x_M^{{\left(1\right)}_{}}\left(t\right)}^2+{y_M^{{\left(1\right)}_{}}\left(t\right)}^2+{z_M^{{\left(1\right)}_{}}\left(t\right)}^2\right]}^2}$$\begin{array}{cc}\{\begin{array}{c}{x}_M^{\left(3\right)}=\left(-\cos {}_1\cos {}_3\cos -\sin {}_1\sin {}_3\right)x_M^{\left(1\right)}+\\ \left(-\sin {}_1\cos {}_3\cos +\cos {}_1\sin {}_3\right)y_M^{\left(1\right)}-\\ \cos {}_3\sin z_M^{\left(1\right)}+(-a\cos {}_3\cos +\\ b\cos {}_3\sin -c\sin {}_3)\\ y_M^{\left(3\right)}=\left(\cos {}_1\sin {}_3\cos -\sin {}_1\cos {}_3\right)x_M^{\left(1\right)}+\\ \left(\sin {}_1\sin {}_3\cos +\cos {}_1\cos {}_3\right)y_M^{\left(1\right)}+\\ \sin {}_3\sin z_M^{\left(1\right)}+(a\cos {}_3\cos -\\ b\sin {}_3\sin -c\cos {}_3)\\ z_M^{\left(3\right)}=\cos {}_1\sin x_M^{\left(1\right)}+\sin {}_1\sin y_M^{\left(1\right)}-\\ \cos z_M^{\left(1\right)}+a\sin +b\cos +s\end{array}& \left(3\right)\end{array}$

(15) Equation (3) is an equation of the driven contact curve which is spatially conjugate with the driving contact curve in the coordinate system o.sub.3x.sub.3y.sub.3z.sub.3;

(16) in the equation: a, b, cthree coordinate scores of the origin o.sub.q in the coordinate system oxyz (as shown in FIG. 1);

(17) .sub.1,.sub.2the angular velocity of rotation of the driving wheel and the driven wheel;

(18) i.sub.21the transmission ratio of the driving wheel and the driven wheel.

(19) When 0<90, the direction of the driven wheel of the angular velocity .sub.2 is contrary to the direction shown in FIG. 1, and a direction of .sub.3 is also contrary to the direction shown in FIG. 1, while a direction of its moving velocity A is also contrary to the direction shown in FIG. 1, and the displacement s is also the negative direction of the z.sub.q axis. Therefore, .sub.2, .sub.3, A and s are inserted into the equations (2) and (3) to obtain the space curve meshing equation, a driving contact curve equation and a driven contact curve equation of the line gear mechanism for rotation-movement conversion at the angle , as shown in an equation (4):

(20) $\begin{array}{cc}\{\begin{array}{c}{y}_M^{\left(1\right)}{}_1{}_x^{\left(1\right)}-x_M^{\left(1\right)}{}_1{}_y^{\left(1\right)}+y_M^{\left(1\right)}{}_2{}_x^{\left(1\right)}\cos -\\ {}_2{x}_M^{\left(1\right)}{}_y^{\left(1\right)}\cos \\ \left(A\sin -c{\stackrel{\_}{}}_2\cos \right){}_x^{\left(1\right)}\cos {}_1+\\ \left(A\sin -c{\stackrel{\_}{}}_2\cos \right){}_y^{\left(1\right)}\sin {}_1+\\ \left({\stackrel{\_}{}}_2\sin \left(z_M^{\left(1\right)}-b-s\right)+{\stackrel{\_}{}}_{2}a\cos \right)\left({}_x^{\left(1\right)}\sin {}_1\right)-\\ \left({\stackrel{\_}{}}_2\sin \left(z_M^{\left(1\right)}-b+s\right)+{\stackrel{\_}{}}_{2}a\cos \right)\left({}_y^{\left(1\right)}\cos {}_1\right)+\\ (-A\cos -{\stackrel{\_}{}}_2\sin x_M^{\left(1\right)}\sin {}_1+\\ {\stackrel{\_}{}}_2\sin y_M^{\left(1\right)}\cos {}_1-c{\stackrel{\_}{}}_2\sin ){}_z^{\left(1\right)}=0\\ x_M^{\left(1\right)}=x_M^{\left(1\right)}\left(t\right)\\ y_M^{\left(1\right)}=y_M^{\left(1\right)}\left(t\right)\\ z_M^{\left(1\right)}=z_M^{\left(1\right)}\left(t\right)\\ {}_2=i_{21}{}_1\\ {}_3=i_{21}{}_1\\ x_M^{\left(3\right)}=\left(-\cos {}_1\cos {}_3\cos -\sin {}_1\sin {}_3\right)x_M^{\left(1\right)}+\\ \left(-\sin {}_1\cos {}_3\cos +\cos {}_1\sin {}_3\right)y_M^{\left(1\right)}-\\ \cos {}_3\sin z_M^{\left(1\right)}+(-a\cos {}_3\cos +\\ b\cos {}_3\sin +c\sin {}_3)\\ y_M^{\left(3\right)}=\left(-\cos {}_1\sin {}_3\cos -\sin {}_1\cos {}_3\right)x_M^{\left(1\right)}+\\ \left(-\sin {}_1\sin {}_3\cos +\cos {}_1\cos {}_3\right)y_M^{\left(1\right)}-\\ \sin {}_3\sin z_M^{\left(1\right)}+(-a\sin {}_3\cos +\\ b\sin {}_3\sin -c\cos {}_3)\\ z_M^{\left(3\right)}=\cos {}_1\sin x_M^{\left(1\right)}+\sin {}_1\sin y_M^{\left(1\right)}-\\ \cos z_M^{\left(1\right)}+a\sin +b\cos -s\end{array}& \left(4\right)\end{array}$

(21) When 908180, the direction of the driven wheel of the angular velocity .sub.2 is the same as the direction shown in FIG. 1, the direction of .sub.3 is also the same as the direction shown in FIG. 1, the direction of the moving velocity A is also shown as FIG. 1, and the direction of the displacement is the same as the direction of the z.sub.q axis. It may be obtained that the driving contact curve equation and the driven contact curve equation at the angle is shown in the equation (1).

(22) According to the curve meshing equation, choosing different angle and driving contact curve equation, the relation between .sub.1 and t may be obtained. Then according to the value, choosing the driven contact curve equation in the equation (3) or the equation (4), a conductor of the driving line tooth and a conductor of the driven line tooth may be obtained respectively. And a closed curve designed is the generatrix, and the generatrix moves along the two conductors respectively, and the obtained entities are the driving line tooth and the driven line tooth. According to actual requirements, the driving line gear wheel body and the driven line gear wheel body are designed, and thereby the driving line gear and the driven line gear are designed.

(23) If the driving line gear of the driving line gear is a space cylindrical spiral, it satisfies an equation (5) in the coordinate system o.sub.1x.sub.1y.sub.1z.sub.1:

(24) $\begin{array}{cc}\{\begin{array}{c}{x}_M^{\left(1\right)}=m\cos t\\ y_M^{\left(1\right)}=m\sin t\\ z_M^{\left(1\right)}=n+\mathrm{nt}\end{array}& \left(5\right)\end{array}$

(25) When 0<90, the equation (5) is inserted into the equation (3) to obtain the space curve meshing equation of the line gear mechanism with conversion from rotation to linear motion, as shown in an equation (6):
(A sin +c.sub.2 cos )cos(.sub.1t)(.sub.2 sin (n+ntb+s)+.sub.2a cos )sin(.sub.1t)=0(6)

(26) When 90<180, the equation (5) is inserted into the equation (4) to obtain the space curve meshing equation of the line gear mechanism with conversion from rotation to linear motion, as shown in an (7):
(A sin +c.sub.2 cos )cos(.sub.1t)(.sub.2 sin (n+ntbs)+.sub.2a cos )sin(.sub.1t)=0(7)

(27) Set A=k.sub.1, then s=k.sub.1. Substituting m=5 mm, n=8 mm, a=0 mm, b=0 mm, c=30 mm, =30, k=0.15, .sub.1=/2, t.sub.s=0.5, t.sub.e=0 and i.sub.21= into the equation of the driving contact curve in the coordinate system o.sub.1x.sub.1y.sub.1z.sub.1 can be obtained:

(28) 0$\begin{array}{cc}\{\begin{array}{c}{x}_M^{\left(1\right)}=5\cos t\\ y_M^{\left(1\right)}=5\sin t\\ z_M^{\left(1\right)}=8+8t\end{array}& \left(8\right)\end{array}$

(29) By the equation (6) and the equation (3), and by three fitting, an equation of the driven contact curve in the coordinate system o.sub.3x.sub.3y.sub.3z.sub.3 is:

(30) $\begin{array}{cc}\{\begin{array}{c}{x}_M^{\left(3\right)}=0.4189t^3+2.5757t^2-7.8227t-24.2499\\ y_M^{\left(3\right)}=0.1943t^3-2.1999t^2-11.1985t+13.2385\\ z_M^{\left(3\right)}=0.0019t^3+0.0297t^2-7.3796t-23.1574\end{array}& \left(9\right)\end{array}$

(31) According to the equations (8) and (9), and according to the required section, the line tooth entity can be established. The line tooth body only needs to meet the strength requirement, and the line tooth body and the line gear wheel body themselves have no special shape requirements. The number of the driving line teeth is set as N.sub.1=4, and according to the requirements to the transmission ratio and the moving displacement, the number of the driven line teeth is set as N.sub.2=23. It is worth mentioning that when the contact curve of the driven line gear is designed, the contact curve of the next line tooth moves on the z axis based on the contact curve of the previous line tooth by a moving distance of one line tooth, and rotates by an angle of rotation of one line tooth. Utilizing this method, it may obtain a schematic view of the driving line gear and its line tooth as shown in FIG. 2, a schematic view of the driven line gear and its line tooth as shown in FIG. 3, and a schematic view of meshing of the driving line gear and driven line gear as shown in FIG. 4. In FIG. 4, 1 represents the driving line gear, 2 represents the driving line tooth, 3 represents the driven line gear and 4 represents the driven line tooth.

(32) The present invention provides a method and a mechanism which are capable of converting rotation to movement for capable of providing rotation-movement conversion for a micro-mechanical device. This mechanism greatly simplifies the structure of the micromechanical transmission, realizes the rotation-movement conversion motion of the spatially staggered axes, reduces the geometry, reduces the mass, improves the flexibility of operation, is simple in production, has a low cost, and is easy to be applied in the micro electro mechanical field.