CN110704975A - Design method of non-circular line gear generating periodic motion - Google Patents

Design method of non-circular line gear generating periodic motion Download PDF

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CN110704975A
CN110704975A CN201910938678.7A CN201910938678A CN110704975A CN 110704975 A CN110704975 A CN 110704975A CN 201910938678 A CN201910938678 A CN 201910938678A CN 110704975 A CN110704975 A CN 110704975A
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contact line
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CN110704975B (en
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陈扬枝
肖小平
邵琰杰
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South China University of Technology SCUT
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    • F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
    • F16—ENGINEERING ELEMENTS AND UNITS; GENERAL MEASURES FOR PRODUCING AND MAINTAINING EFFECTIVE FUNCTIONING OF MACHINES OR INSTALLATIONS; THERMAL INSULATION IN GENERAL
    • F16H—GEARING
    • F16H55/00—Elements with teeth or friction surfaces for conveying motion; Worms, pulleys or sheaves for gearing mechanisms
    • F16H55/02—Toothed members; Worms
    • F16H55/08—Profiling
    • F16H55/084—Non-circular rigid toothed member, e.g. elliptic gear
    • F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
    • F16—ENGINEERING ELEMENTS AND UNITS; GENERAL MEASURES FOR PRODUCING AND MAINTAINING EFFECTIVE FUNCTIONING OF MACHINES OR INSTALLATIONS; THERMAL INSULATION IN GENERAL
    • F16H—GEARING
    • F16H55/00—Elements with teeth or friction surfaces for conveying motion; Worms, pulleys or sheaves for gearing mechanisms
    • F16H55/02—Toothed members; Worms
    • F16H55/17—Toothed wheels
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
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Abstract

The invention discloses a design method of a non-circular gear generating periodic motion, wherein the non-circular gear comprises a driving wheel and a driven wheel, the driving wheel comprises a driving wheel body and a plurality of herringbone driving line teeth, the herringbone driving line teeth are uniformly distributed and fixed on the driving wheel body, and the tooth surface of each driving line tooth is provided with a driving contact line; the driven wheel comprises a driven wheel body and a plurality of herringbone driven line teeth, the herringbone driven line teeth are uniformly distributed and fixed on the driven wheel body, and driven contact lines are arranged on the tooth surfaces of the driven line teeth; the driving contact line is meshed with the driven contact line; according to the design method, a contact line equation is established according to the angle between the axis of the driving wheel and the axis of the driven wheel, and then a solid model of the non-circular line gear is established based on the contact line equation and the like. The non-circular line gear obtained by the design method can realize the non-sliding output periodic variation speed ratio, and can avoid the sudden change of angular acceleration in the transmission process, thereby avoiding the occurrence of impact.

Description

Design method of non-circular line gear generating periodic motion
Technical Field
The invention relates to the technical field of gears, in particular to a design method of a non-circular line gear generating periodic motion.
Background
Non-circular gears, which may achieve precise and non-uniform transmission between the two shafts, are commonly used and include non-cylindrical gears, non-conical gears, and face-toothed non-circular gears. Non-cylindrical gears are used for transmission between parallel shafts, while non-conical gears are used for transmission between intersecting shafts.
The line gear is a novel gear designed according to the space conjugate curve meshing principle, and a pair of space conjugate curves (main and auxiliary line tooth contact lines) always keep a point contact meshing state in the transmission process. The linear gear has the characteristics of small size, light weight, large transmission ratio and the like, and can realize transmission of parallel shafts, crossed shafts with any angle or staggered shafts in a tiny space, so that the linear gear is particularly suitable for light-weight and tiny machines.
However, the current non-circular line gear has the following defects: the speed ratio change can only realize the trapezoidal change rule, and cannot realize the rule that the speed ratio change meets the similar sinusoidal change; the problem of slip rate can also be caused in the transmission process of the existing non-circular gear, so that the transmission efficiency is reduced; meanwhile, in the transmission process, the speed changes in a jumping mode, and the speed ratio cannot be changed continuously.
Disclosure of Invention
The invention aims to overcome the defects of the prior art and provides a design method of a non-circular line gear generating periodic motion. The non-circular gear of the design method of the non-circular gear generating the periodic motion can reduce the slip rate and the impact in the transmission process, and can ensure that the output angular speed of the driven wheel meets the change rule similar to a sine curve.
The purpose of the invention is realized by the following technical scheme: the design method of the non-circular gear generating the periodic motion comprises a driving wheel and a driven wheel, wherein the driving wheel comprises a driving wheel body and a plurality of herringbone driving line teeth, the plurality of herringbone driving line teeth are uniformly distributed and fixed on the driving wheel body, and the tooth surface of each driving line tooth is provided with a driving contact line; the driven wheel comprises a driven wheel body and a plurality of herringbone driven line teeth, the herringbone driven line teeth are uniformly distributed and fixed on the driven wheel body, and driven contact lines are arranged on the tooth surfaces of the driven line teeth; the driving contact line is meshed with the driven contact line; the design method comprises the following steps:
s1: establishing a pair of reference coordinate systems of relative engagement, the reference coordinate systems being o-xyz and o, respectivelyp-xpypzpThe active contact line is fixedly connected with the active coordinate system o1-x1y1z1And the active coordinate system o1-x1y1z1At an angular velocity ω1Rotate around the o-xyz coordinate system by a rotation angle phi1(ii) a The driven contact line is fixed on the driven coordinate system o2-x2y2z2This slave coordinate system o2-x2y2z2At an angular velocity ω2Around op-xpypzpThe coordinate system rotates by a rotation angle phi2Wherein o ispDistance a, o to z axispDistance to x-axis b, theta x-axis and xpAngle of intersection between the axes, θ ∈ [0, π ∈ ]]；
S2: selecting an equation of the active contact line and an equation of the passive contact line according to the value of theta:
s2-1: when θ is 0 or θ is pi, the equation for the active contact line is:
Figure BDA0002222289100000021
this is represented by the formula (1),
the equation for the driven wheel contact line is:
Figure BDA0002222289100000022
this is formula (2);
s2-2: when theta is more than 0 and less than pi, the equation of the active contact line is as follows:
Figure BDA0002222289100000031
this is represented by the formula (3),
the equation for the driven wheel contact line is:
this is formula (4);
m in the formulae (1), (2), (3) and (4)(1)And M(2)The contact points of the driving wheel and the driven wheel are respectively, and n is a pitch parameter of the driving contact line and the driven contact line; i is the speed ratio of the driving wheel to the driven wheel, phi2For the driven coordinate system o2-x2y2z2Around op-xpypzpA rotation angle of the coordinate system; phi is a1As a main coordinate system o1-x1y1z1A rotation angle around an o-xyz coordinate system; p is a herringbone tooth contact line selection parameter.
S3: at ω, depending on the angular velocity of the driven wheel during a cycle1And ω2The number of the periodic conversion is obtained to obtain the upper limit of the rotation angle of the single-line tooth meshing area between the driving wheel and the driven wheel
Figure BDA0002222289100000034
Solving the tooth number of the driving wheel and the driven wheel;
s4: suppose phi2Satisfying a piecewise function relation with the time t and establishing an equation relation to solve the speed ratio control parameter;
s5: according to phi2And t, determining a relationship between the speed ratio i and the time t, the relationship between the speed ratio i and the time t being:
Figure BDA0002222289100000033
this is formula (5);
and establishing a solid model of the non-circular line gear by combining the formulas (1) to (4).
Preferably, step S3 includes the steps of:
s3-1: four meshing points are arranged on the driving wheel or the driven wheel, the four meshing points are respectively a meshing point A, a meshing point B, a meshing point C and a meshing point D, and the meshing point A, the meshing point B, the meshing point C and the meshing point D are sequentially distributed along the rotation direction
S3-2: if the driving wheel rotates anticlockwise, the driven wheel rotates clockwise; therefore, when the meshing point between the driving wheel and the driven wheel moves from the meshing point A to the meshing point B, the clockwise rotation angle of the driven wheel is
Figure BDA0002222289100000041
At the moment, the rotation angle of the driving wheel is
Figure BDA0002222289100000042
And is
Figure BDA0002222289100000043
This is the formula (6) in which the coefficient α ∈ (i)1,i2)；
When the meshing point moves from the point C to the point D, the transmission ratio is changed from i2Become i1The clockwise rotation angle of the driven wheel is
Figure BDA0002222289100000044
At the moment, the rotation angle of the driving wheel is
Figure BDA0002222289100000045
And is
Figure BDA0002222289100000046
This is the formula (7) in which the coefficient β ∈ (i)1,i2)；
In each of the transmission cycles, the transmission is,
Figure BDA0002222289100000047
this is the formula (8),
wherein the number of teeth of the driving wheel is 2N, and the number of teeth of the driven wheel is 2Nd。
Preferably, step S4 includes the steps of:
s4-1: suppose that: at t1At time, the speed ratio is i1(ii) a At t8At time, the speed ratio is i2(ii) a From t1To t8The rotation angle of the driving wheel is
Figure BDA0002222289100000048
The angle of rotation of the driven wheel is
Figure BDA0002222289100000049
During the speed ratio change, the angular acceleration and the angular jerk of the driven wheel cannot have sudden change, 1/i (t) must be conducted in the second order, and the angular acceleration and the angular jerk are t1Time t and8the values of the moments are all equal to 0, based on which the conditional equation is established:
Figure BDA00022222891000000410
this is formula (9);
Figure BDA00022222891000000411
this is formula (10);
Figure BDA00022222891000000412
this is formula (11);
Figure BDA00022222891000000413
this is formula (12);
Figure BDA00022222891000000414
this is formula (13);
Figure BDA0002222289100000051
this is formula (14);
Figure BDA0002222289100000052
this is formula (15);
Figure BDA0002222289100000053
it is characterized in thatFormula (16);
s4-2: according to the formulae (9) to (16), establishment
Figure BDA0002222289100000054
The equation of (c):
Figure BDA0002222289100000055
this is formula (17);
in the formula (17), psmn，pjmn，pαmn，pimn，
Figure BDA0002222289100000056
As a transmission ratio control parameter, psmnRepresents tmTo tnThe instantaneous spasm parameter, pjmnRepresents tmTo tnJerk parameter of time, pαmnRepresents tmTo tnAngular acceleration parameter of time, pimnRepresents tmTo tnThe speed ratio parameter at the time of day,
Figure BDA0002222289100000057
represents tmTo tnAn angle parameter of a moment; m is equal to [1,2,3,4,5,6,7 ]]The corresponding n ∈ [2,3,4,5,6,7,8 ]]。
Preferably, step S5 includes the steps of:
obtained according to formula (17)
Figure BDA0002222289100000058
According to formula (5) pair
Figure BDA0002222289100000059
Obtaining i (t) by derivation
Figure BDA00022222891000000510
And i (t) substituting equations in the formulas (1) to (4) to obtain equations of the driving contact line and the driven contact line, and establishing a solid model of the non-circular line gear based on the equations of the driving contact line and the driven contact line.
Compared with the prior art, the invention has the following advantages:
1. the design method of the non-circular line gear generating the periodic motion is characterized in that a driving contact line and a driven contact line are designed according to an angle theta between parallel shafts or crossed shafts so as to realize the speed ratio of non-sliding output periodic change, herringbone line teeth can be adopted as the driving line teeth and the driven line teeth, and the contact strength can be improved.
2. The design method of the non-circular line gear generating the periodic motion is characterized in that a driving contact line and a driven contact line are designed according to an angle theta between parallel shafts or crossed shafts so as to ensure that the slip rate is zero in the transmission process, avoid reduction of the transmission efficiency, and ensure that the speed ratio of the non-circular line gear does not jump in the transmission process, thereby ensuring that the speed ratio is periodically and continuously changed (as shown in figure 1).
3. The design method of the non-circular gear generating the periodic motion adopts a segmented quartic curve to design the change rule of the rotation angle of the non-circular gear along with time so as to ensure that the angular speed and the angular acceleration do not generate sudden change in the transmission process.
Drawings
FIG. 1 is a graphical representation of the ratio cycling profile of the non-circular wire gears of the present invention that produce cyclic motion.
Fig. 2 is a schematic view of the driven wheel in the non-circular line gear of the present invention that produces periodic motion.
Fig. 3 is a meshing coordinate system of the non-circular wire gears of the present invention that produce periodic motion.
FIG. 4 is a graph of the angle of rotation of a single tooth of a non-circular wire gear of the present invention as a function of time.
Fig. 5 is a schematic view of the meshing state of the non-circular gear generating periodic motion of the present invention when the axes of the driving pulley and the driven pulley are parallel.
Fig. 6 is a schematic view of the meshing state of the non-circular gear generating the periodic motion of the present invention when the axis of the driving pulley and the axis of the driven pulley intersect.
Wherein, 1 is the action wheel, 2 is driven the driving wheel, 3 is the action wheel body, 4 is the initiative line tooth, 41 and 42 are first initiative line tooth, 43 and 44 equal second initiative line tooth, 5 is the driven wheel body, 6 is the driven line tooth, 61 and 62 are first driven line tooth, 63 and 64 are the second driven line tooth.
Detailed Description
The invention is further illustrated by the following figures and examples.
The design method of the non-circular gear generating the periodic motion as shown in fig. 2 includes a driving wheel and a driven wheel, the driving wheel includes a driving wheel body and a plurality of herringbone driving teeth, the herringbone driving teeth are uniformly distributed and fixed on the driving wheel body, and a driving contact line is arranged on the tooth surface of the driving teeth; the driven wheel comprises a driven wheel body and a plurality of herringbone driven line teeth, the herringbone driven line teeth are uniformly distributed and fixed on the driven wheel body, and driven contact lines are arranged on the tooth surfaces of the driven line teeth; the driving contact line is meshed with the driven contact line; the design method comprises the following steps:
s1: establishing a pair of reference coordinate systems of relative engagement, the reference coordinate systems being o-xyz and o, respectivelyp-xpypzpThe active contact line is fixedly connected with the active coordinate system o1-x1y1z1And the active coordinate system o1-x1y1z1At an angular velocity ω1Rotate around the o-xyz coordinate system by a rotation angle phi1(ii) a The driven contact line is fixed on the driven coordinate system o2-x2y2z2This slave coordinate system o2-x2y2z2At an angular velocity ω2Around op-xpypzpThe coordinate system rotates by a rotation angle phi2Wherein o ispDistance a, o to z axispDistance to x-axis b, theta x-axis and xpAngle of intersection between the axes, θ ∈ [0, π ∈ ]]；
S2: selecting an equation of the active contact line and an equation of the passive contact line according to the value of theta:
s2-1: when θ is 0 or θ is pi, the equation for the active contact line is:
Figure BDA0002222289100000071
this is represented by the formula (1),
the expression for the driven wheel contact line is:
Figure BDA0002222289100000072
this is formula (2);
s2-2: when theta is more than 0 and less than pi, the equation of the active contact line is as follows:
Figure BDA0002222289100000073
this is represented by the formula (3),
the contact line of the driven wheel is as follows:
Figure BDA0002222289100000074
this is formula (4);
m in the formulae (1), (2), (3) and (4)(1)And M(2)The contact points of the driving wheel and the driven wheel are respectively, and n is a pitch parameter of the driving contact line and the driven contact line; i is the speed ratio of the driving wheel to the driven wheel, phi2For the driven coordinate system o2-x2y2z2Around op-xpypzpA rotation angle of the coordinate system; phi is a1As a main coordinate system o1-x1y1z1A rotation angle around an o-xyz coordinate system; p is a herringbone tooth contact line selection parameter. Specifically, the herringbone line tooth contact line selection parameters include p-1 and p-1, and are mainly used for establishing equations corresponding to the driving contact line and the driven contact line. As shown in fig. 1, 5 and 6, the driving wheel 1 includes a driving wheel body 3 and a plurality of herringbone driving teeth 4, and the herringbone driving teeth are uniformly distributed and fixed on the contour of the driving wheel body 3; the driving line teeth 4 are composed of first driving line teeth 41, 42 and second driving line teeth 43, 44, and the corresponding first driving line teeth 41, 42 and second driving line teeth 43, 44 are distributed in a herringbone shape. The driven wheel 2 comprises a driven wheel body 5 and a plurality of herringbone driven line teeth 6, and the herringbone driven line teeth are uniformly distributed and fixed on the outline of the driven wheel body 5; wherein the driven wire teeth are composed of first driven wire teeth 61, 62 and a second driven wire teethTwo driven line teeth 63, 64, the corresponding first 61, 62 and second 63, 64 driven line teeth being arranged in a chevron pattern.
When p is 1, equations (1) and (3) are equations of the driving contact lines included in the first driving wire teeth 41 and 42 (i.e., the first driving wire tooth located above), and equations (2) and (4) are equations of the driven contact lines included in the first driven wire teeth 61 and 62 (i.e., the first driven wire tooth located above). When p is-1, equations (1) and (3) are equations of the driving contact line contained in the second driving wire teeth 43, 44 (i.e., the second driving wire tooth located below), and equations (2) and (4) are equations of the driven contact line contained in the second driven wire teeth 63, 64 (i.e., the second driven wire tooth located below).
S3: at ω, depending on the angular velocity of the driven wheel during a cycle1And ω2The number of the periodic conversion is obtained to obtain the upper limit of the rotation angle of the single-line tooth meshing area between the driving wheel and the driven wheel
Figure BDA0002222289100000081
Solving the tooth number of the driving wheel and the driven wheel;
step S3 includes the following steps:
s3-1: four meshing points are arranged on the driving wheel or the driven wheel, the four meshing points are respectively a meshing point A, a meshing point B, a meshing point C and a meshing point D, and the meshing point A, the meshing point B, the meshing point C and the meshing point D are sequentially distributed along the rotation direction
S3-2: if the driving wheel rotates anticlockwise, the driven wheel rotates clockwise; therefore, when the meshing point between the driving wheel and the driven wheel moves from the meshing point A to the meshing point B, the clockwise rotation angle of the driven wheel is
Figure BDA0002222289100000091
At the moment, the rotation angle of the driving wheel is
Figure BDA0002222289100000092
And is
Figure BDA0002222289100000093
This is equation (6) where the transmission ratio coefficient α ∈ (i)1,i2)；
When the meshing point moves from the point C to the point D, the transmission ratio is changed from i2Become i1The clockwise rotation angle of the driven wheel is
Figure BDA0002222289100000094
At the moment, the rotation angle of the driving wheel is
Figure BDA0002222289100000095
And is
Figure BDA0002222289100000096
This is the formula (7) in which the transmission ratio coefficient β ∈ (i)1,i2)；
In each of the transmission cycles, the transmission is,
Figure BDA0002222289100000097
this is the formula (8),
wherein the number of teeth of the driving wheel is 2N, and the number of teeth of the driven wheel is 2Nd。
S4: suppose phi2Satisfying a piecewise function relation with the time t and establishing an equation relation to solve the speed ratio control parameter;
step S4 includes the following steps:
s4-1: suppose that: at t1At time, the speed ratio is i1(ii) a At t8At time, the speed ratio is i2(ii) a From t1To t8The rotation angle of the driving wheel is
Figure BDA0002222289100000098
The angle of rotation of the driven wheel isDuring the speed ratio change, the angular acceleration and the angular jerk of the driven wheel cannot have sudden change, 1/i (t) must be conducted in the second order, and the angular acceleration and the angular jerk are t1Time t and8the values of the moments are all equal to 0, based on which the conditional equation is established:
Figure BDA00022222891000000910
this is formula (9);
Figure BDA00022222891000000911
this is formula (10);
Figure BDA00022222891000000912
this is formula (11);
Figure BDA00022222891000000913
this is formula (12);
Figure BDA00022222891000000914
this is formula (13);
Figure BDA0002222289100000101
this is formula (14);
Figure BDA0002222289100000102
this is formula (15);
Figure BDA0002222289100000103
this is formula (16);
s4-2: according to the formulae (9) to (16), establishment
Figure BDA0002222289100000104
The equation of (c):
Figure BDA0002222289100000105
this is formula (17);
in the formula (17), psmn，pjmn，pαmn，pimn，
Figure BDA0002222289100000106
As a transmission ratio control parameter, psmnRepresents tmTo tnThe instantaneous spasm parameter, pjmnRepresents tmTo tnJerk parameter of time, pαmnRepresents tmTo tnAngular acceleration parameter of time, pimnRepresents tmTo tnThe speed ratio parameter at the time of day,
Figure BDA0002222289100000107
represents tmTo tnAn angle parameter of a moment; m is equal to [1,2,3,4,5,6,7 ]]The corresponding n ∈ [2,3,4,5,6,7,8 ]]. Further, if during the transmission, at t1Speed ratio at time i2(ii) a At t8Speed ratio at time i1(ii) a The rotation angle of the driving wheel is taken as the rotation angle of the driving wheel in the formula (9) - (16)The angle of rotation of the driven wheel is
S5: according to phi2And t, determining a relationship between the speed ratio i and the time t, the relationship between the speed ratio i and the time t being:
Figure BDA00022222891000001010
this is formula (5);
and establishing a solid model of the non-circular line gear by combining the formulas (1) to (4). In S3-S5, the upper limit of the angle is determined mainly according to the number of transitions, that is, the upper limit of the rotation angle of a single linear tooth meshing area is obtained by dividing 360 degrees by the number of transitions (namely the number of teeth), and the whole non-circular linear gear is formed by a plurality of linear tooth arrays, and then phi is solved2. Specifically, step S5 includes the following steps:
obtained according to formula (17)
Figure BDA00022222891000001011
According to formula (5) pair
Figure BDA00022222891000001012
Obtaining i (t) by derivation
Figure BDA0002222289100000111
And i (t) substituting the equations (1) to (4) to obtain an equation of the driving contact line and the driven contact line, establishing a solid model of the non-circular gear based on the equation of the driving contact line and the driven contact line, specifically, establishing a solid of the line teeth by reversely establishing a certain volume in the main normal vector direction of the driving contact line and the driven contact line, and further establishing the solid model of the non-circular gear according to the number of teeth.
In this embodiment:
speed ratio i 11, speed ratio i21.5; rotation angle
Figure BDA0002222289100000112
Rotation angle
Figure BDA0002222289100000113
The transmission ratio coefficient α ═ β ═ 1.2;
then the following equations can be obtained according to equations (6), (7), (8):
Figure BDA0002222289100000114
Figure BDA0002222289100000115
Figure BDA0002222289100000116
therefore, it is not only easy to use
Figure BDA0002222289100000117
So that the number of non-circular driving gears is2N is 12, and the number of the non-circular driven gears is 2Nd＝10。
The contact lines are now solved separately:
when x axis and xpWhen the intersection angle θ between the shafts is 180 °, the driving contact line equation and the driven contact line equation have the following two cases:
A. when p is 1, the equation for the active contact line is as follows:
Figure BDA0002222289100000118
this is formula (1');
the equation for the corresponding driven contact line is as follows:
Figure BDA0002222289100000121
this is formula (2');
B. when p is-1, the equation for the active contact line is as follows:
Figure BDA0002222289100000122
this is formula (3');
the equation for the corresponding driven contact line is as follows:
this is formula (4');
a solid model of the non-circular gear is established according to equations (1 ') - (4'), in which the axis of the driving wheel and the axis of the driven wheel are parallel, as shown in FIG. 5:
the driving wheel 1 comprises a driving wheel body 3 and a plurality of herringbone driving line teeth 4, and the driving line teeth are uniformly distributed and fixed on the outline of the driving wheel body 3; the driving linear teeth 4 are composed of a first driving linear tooth and a second driving linear tooth, and the corresponding first driving linear tooth and the corresponding second driving linear tooth are distributed in a herringbone shape.
In the transmission process, of two adjacent first driving linear teeth 41 and 42, the first driving linear tooth 41 corresponds to a non-circular linear gear ratio of i1Become intoi2And the first driving linear gear 42 corresponds to the non-circular linear gear ratio of i2Become i1(ii) a And of the two adjacent second driving linear teeth 43 and 44, the second driving linear tooth 43 corresponds to the non-circular linear gear ratio of i1Become i2The second driving gear 44 corresponds to a non-circular gear ratio of i2Become i1。
The driven wheel 2 parallel to the driving wheel 1 comprises a driven wheel body 5 and a plurality of herringbone driven line teeth 6, and the herringbone driven line teeth are uniformly distributed and fixed on the profile of the driven wheel body; the driven linear teeth are composed of first driven linear teeth and second driven linear teeth, and the corresponding first driven linear teeth and the corresponding second driven linear teeth are distributed in a herringbone mode.
In the transmission process, of two adjacent first driven linear teeth 61 and 62, the first driven linear tooth 61 corresponds to a non-circular linear gear ratio and is formed by i1Become i2And the first driven linear gear 62 corresponds to a non-circular linear gear ratio of i2Become i1(ii) a And of the two adjacent second driven linear teeth 63 and 64, the second driven linear tooth 63 corresponds to the non-circular linear gear ratio of i1Become i2The second driven linear gear 64 corresponds to a non-circular linear gear ratio of i2Become i1。
(II) when the x-axis and xpWhen the intersection angle θ between the shafts is 120 °, the driving contact line equation and the driven contact line equation have the following two cases:
A. when q is 1, the equation for the active contact line is as follows:
Figure BDA0002222289100000131
this is formula (1 ");
the corresponding equation for the driven contact line is:
Figure BDA0002222289100000132
this is formula (2 ");
B. when p is-1, the equation for the active contact line is as follows:
Figure BDA0002222289100000141
this is formula (3 ");
the equation for the corresponding driven contact line is as follows:
Figure BDA0002222289100000142
this is formula (4 ");
a physical model of the non-circular line gear is established according to the contact line equations (1 ") - (4") when the axis of the driven wheel intersects the axis of the driving wheel, as shown in fig. 6:
the driving wheel 1 comprises a driving wheel 3 and a plurality of herringbone driving line teeth 4 which are uniformly distributed and fixed on the outline of the driving wheel body; the driving linear teeth are composed of first driving linear teeth and second driving linear teeth, and the corresponding first driving linear teeth and the corresponding second driving linear teeth are distributed in a herringbone mode.
In the transmission process, of two adjacent first driving linear teeth 41 and 42, the first driving linear tooth 41 corresponds to a non-circular linear gear ratio of i1Become i2And the first driving linear gear 42 corresponds to the non-circular linear gear ratio of i2Become i1(ii) a And of the two adjacent second driving linear teeth 43 and 44, the second driving linear tooth 43 corresponds to the non-circular linear gear ratio of i1Become i2The second driving gear 44 corresponds to a non-circular gear ratio of i2Become i1。
The driven wheel 2 crossed with the axis of the driving wheel 1 comprises a driven wheel body 5 and a plurality of herringbone driven line teeth 6, and the herringbone driven line teeth are uniformly distributed and fixed on the profile of the driven wheel body; the driven linear teeth are composed of first driven linear teeth and second driven linear teeth, and the corresponding first driven linear teeth and the corresponding second driven linear teeth are distributed in a herringbone mode.
In the transmission process, of two adjacent first driven linear teeth 61 and 62, the first driven linear tooth 61 corresponds to a non-circular linear gear ratio and is formed by i1Become i2And the first driven linear gear 62 corresponds to a non-circular linear gear ratio of i2Become i1(ii) a And two adjacent second driven linesOf the teeth 63, 64, the second driven-line tooth 63 corresponds to a non-circular-line gear ratio of i1Become i2The second driven linear gear 64 corresponds to a non-circular linear gear ratio of i2Become i1。
The above-mentioned embodiments are preferred embodiments of the present invention, and the present invention is not limited thereto, and any other modifications or equivalent substitutions that do not depart from the technical spirit of the present invention are included in the scope of the present invention.

Claims (4)

1. A design method of a non-circular line gear generating periodic motion is characterized in that: the non-circular line gear comprises a driving wheel and a driven wheel, the driving wheel comprises a driving wheel body and a plurality of herringbone driving line teeth, the plurality of herringbone driving line teeth are uniformly distributed and fixed on the driving wheel body, and driving contact lines are arranged on the tooth surfaces of the driving line teeth; the driven wheel comprises a driven wheel body and a plurality of herringbone driven line teeth, the herringbone driven line teeth are uniformly distributed and fixed on the driven wheel body, and driven contact lines are arranged on the tooth surfaces of the driven line teeth; the driving contact line is meshed with the driven contact line; the design method comprises the following steps:
s1: establishing a pair of reference coordinate systems of relative engagement, the reference coordinate systems being o-xyz and o, respectivelyp-xpypzpThe active contact line is fixedly connected with the active coordinate system o1-x1y1z1And the active coordinate system o1-x1y1z1At an angular velocity ω1Rotate around the o-xyz coordinate system by a rotation angle phi1(ii) a The driven contact line is fixed on the driven coordinate system o2-x2y2z2This slave coordinate system o2-x2y2z2At an angular velocity ω2Around op-xpypzpThe coordinate system rotates by a rotation angle phi2Wherein o ispDistance a, o to z axispDistance to x-axis b, theta x-axis and xpAngle of intersection between the axes, θ ∈ [0, π ∈ ]]；
S2: selecting an equation of the active contact line and an equation of the passive contact line according to the value of theta:
s2-1: when θ is 0 or θ is pi, the equation for the active contact line is:
Figure FDA0002222289090000011
the equation for the driven wheel contact line is:
Figure FDA0002222289090000012
s2-2: when theta is more than 0 and less than pi, the equation of the active contact line is as follows:
Figure FDA0002222289090000021
the equation for the driven wheel contact line is:
m in the formulae (1), (2), (3) and (4)(1)And M(2)The contact points of the driving wheel and the driven wheel are respectively, and n is a pitch parameter of the driving contact line and the driven contact line; i is the speed ratio of the driving wheel to the driven wheel, phi2For the driven coordinate system o2-x2y2z2Around op-xpypzpA rotation angle of the coordinate system; phi is a1As a main coordinate system o1-x1y1z1A rotation angle around an o-xyz coordinate system; p is a herringbone tooth contact line selection parameter.
S3: at ω, depending on the angular velocity of the driven wheel during a cycle1And ω2The number of the periodic conversion is obtained to obtain the upper limit of the rotation angle of the single-line tooth meshing area between the driving wheel and the driven wheel
Figure FDA0002222289090000024
Solving the tooth number of the driving wheel and the driven wheel;
s4: suppose phi2Satisfying a piecewise function relation with the time t and establishing an equation relation to solve the speed ratio control parameter;
s5: according to phi2And t, determining a relationship between the speed ratio i and the time t, the relationship between the speed ratio i and the time t being:
Figure FDA0002222289090000023
and establishing a solid model of the non-circular line gear by combining the formulas (1) to (4).
2. The method of designing a non-circular wire gear that generates a periodic motion of claim 1, wherein: step S3 includes the following steps:
s3-1: four meshing points are arranged on the driving wheel or the driven wheel, the four meshing points are respectively a meshing point A, a meshing point B, a meshing point C and a meshing point D, and the meshing point A, the meshing point B, the meshing point C and the meshing point D are sequentially distributed along the rotation direction
S3-2: if the driving wheel rotates anticlockwise, the driven wheel rotates clockwise; therefore, when the meshing point between the driving wheel and the driven wheel moves from the meshing point A to the meshing point B, the clockwise rotation angle of the driven wheel is
Figure FDA0002222289090000031
At the moment, the rotation angle of the driving wheel is
Figure FDA0002222289090000032
And is
Figure FDA0002222289090000033
This is equation (6) where the transmission ratio coefficient α ∈ (i)1,i2)；
When the meshing point moves from the point C to the point D, the transmission ratio is changed from i2Become i1At the time ofThe clockwise rotation angle of the wheel being
Figure FDA0002222289090000034
At the moment, the rotation angle of the driving wheel is
Figure FDA0002222289090000035
And is
This is the formula (7) in which the transmission ratio coefficient β ∈ (i)1,i2)；
In each of the transmission cycles, the transmission is,
wherein the number of teeth of the driving wheel is 2N, and the number of teeth of the driven wheel is 2Nd。
3. The method of designing a non-circular wire gear that generates a periodic motion according to claim 2, characterized in that: step S4 includes the following steps:
s4-1: suppose that: at t1At time, the speed ratio is i1(ii) a At t8At time, the speed ratio is i2(ii) a From t1To t8The rotation angle of the driving wheel isThe angle of rotation of the driven wheel is
Figure FDA0002222289090000039
During the speed ratio change, the angular acceleration and the angular jerk of the driven wheel cannot have sudden change, 1/i (t) must be conducted in the second order, and the angular acceleration and the angular jerk are t1Time t and8the values of the moments are all equal to 0, based on which the conditional equation is established:
Figure FDA00022222890900000310
Figure FDA00022222890900000311
Figure FDA0002222289090000041
Figure FDA0002222289090000042
Figure FDA0002222289090000043
Figure FDA0002222289090000044
s4-2: according to the formulae (9) to (16), establishment
Figure FDA0002222289090000045
The equation of (c):
Figure FDA0002222289090000046
in the formula (17), psmn，pjmn，pαmn，pimn，
Figure FDA00022222890900000410
As a transmission ratio control parameter, psmnRepresents tmTo tnThe instantaneous spasm parameter, pjmnRepresents tmTo tnJerk parameter of time, pαmnRepresents tmTo tnAngular acceleration parameter of time, pimnRepresents tmTo tnThe speed ratio parameter at the time of day,
Figure FDA0002222289090000047
represents tmTo tnAn angle parameter of a moment; m is equal to [1,2,3,4,5,6,7 ]]The corresponding n ∈ [2,3,4,5,6,7,8 ]]。
4. The method of designing a non-circular wire gear that generates a periodic motion of claim 3, wherein: step S5 includes the following steps:
obtained according to formula (17)
Figure FDA0002222289090000048
According to formula (5) pair
Figure FDA00022222890900000411
Obtaining i (t) by derivation
Figure FDA0002222289090000049
And i (t) substituting equations in the formulas (1) to (4) to obtain equations of the driving contact line and the driven contact line, and establishing a solid model of the non-circular line gear based on the equations of the driving contact line and the driven contact line.
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