CN107654585B - Non-circular gear planetary gear system design method based on kinematics mapping - Google Patents
Non-circular gear planetary gear system design method based on kinematics mapping Download PDFInfo
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- CN107654585B CN107654585B CN201710760318.3A CN201710760318A CN107654585B CN 107654585 B CN107654585 B CN 107654585B CN 201710760318 A CN201710760318 A CN 201710760318A CN 107654585 B CN107654585 B CN 107654585B
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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
- F16H1/00—Toothed gearings for conveying rotary motion
- F16H1/28—Toothed gearings for conveying rotary motion with gears having orbital motion
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- A—HUMAN NECESSITIES
- A01—AGRICULTURE; FORESTRY; ANIMAL HUSBANDRY; HUNTING; TRAPPING; FISHING
- A01C—PLANTING; SOWING; FERTILISING
- A01C11/00—Transplanting machines
- A01C11/006—Other parts or details or planting machines
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- A—HUMAN NECESSITIES
- A01—AGRICULTURE; FORESTRY; ANIMAL HUSBANDRY; HUNTING; TRAPPING; FISHING
- A01C—PLANTING; SOWING; FERTILISING
- A01C11/00—Transplanting machines
- A01C11/02—Transplanting machines for seedlings
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- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F30/00—Computer-aided design [CAD]
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Abstract
The invention discloses the non-circular gear planetary gear system design methods mapped based on kinematics.The method difficulty of existing reverse transplanting mechanism is big, and optimization process is complicated.The method that the present invention is mapped by kinematics, two closed tracks can be formed by acquiring two sets of four-bar mechanisms by five given posture points, then it is taken two closed tracks and obtains angular displacement curve a little and with uniform B-Spline interpolation three times, resultant gear ratio curve is found out by angular displacement curve, carries out transmission ratio distribution；The pitch curve of two pairs of non-circular gears is found out by two stage gear ratios.The present invention optimizes adjustment by the point on two closing tracks respectively, and the coupling degree of association is small, and adjustability is stronger, and the process of optimization transplanting track is easier.
Description
Technical Field
The invention belongs to the technical field of machinery, relates to a transplanter, and particularly relates to a non-circular gear planetary gear train design method based on kinematic mapping.
Background
The transplanting of the pot seedlings is an important way for improving the agricultural economic benefit. The use of the transplanter greatly improves the labor productivity of agriculture and planting industry, particularly seedlings, reduces the labor intensity, improves the overall level of the agriculture machine, and pushes the agricultural production to a new era.
The slideway mechanism is easy to realize special tracks, flexible and convenient in design, complex in structure, high in cost and low in efficiency. The rotary mechanism is difficult to design, but simple in structure, low in cost and high in efficiency. The rotary mechanism in the market is used as a substitute product to gradually eliminate the connecting rod type transplanting mechanism.
Disclosure of Invention
The invention aims to solve the problem of high design difficulty of the existing rotating mechanism, and provides a non-circular gear planetary gear train design method based on kinematic mapping, two closed tracks can be formed by solving two sets of four-bar mechanisms through five given attitude points through the kinematic mapping method, then point is taken on the two closed tracks, an angular displacement curve is obtained by three-time non-uniform B spline interpolation, a total transmission ratio curve is solved through the angular displacement curve, and transmission ratio distribution is carried out; and solving pitch curves of the two pairs of non-circular gears through two-stage transmission ratio.
In order to achieve the purpose, the invention adopts the technical scheme that:
the method comprises the following specific steps:
the method comprises the following steps that firstly, a non-circular gear planetary gear train is constructed, and the non-circular gear planetary gear train comprises a planetary carrier, a transplanting arm, a first-stage driving wheel, a first-stage driven wheel, a second-stage driving wheel and a second-stage driven wheel which are arranged in the planetary carrier; the first-stage driving wheel is fixedly connected to the frame; one end of the planet carrier is hinged with the first-stage driving wheel, the other end of the planet carrier is hinged with the second-stage driven wheel, and the middle part of the planet carrier is hinged with the first-stage driven wheel; the first-stage driven wheel is fixedly connected with the second-stage driving wheel; the hinge point of the first-stage driving wheel is defined as a fixed hinge point, and the hinge point of the second-stage driven wheel is defined as a movable hinge point; the first-stage driving wheel is meshed with the first-stage driven wheel; the second-stage driving wheel is meshed with the second-stage driven wheel; the shell of the transplanting arm is fixedly connected with the second-stage driven wheel; and a cam of the transplanting arm is fixedly connected with the planet carrier.
And step two, reversely solving two sets of four-bar mechanisms based on a kinematic mapping method.
The coordinate (x, y) of the dynamic hinge point in the dynamic coordinate system XOY is converted into the coordinate expression form in the static coordinate system XOY as follows:
wherein, the distance from the origin of the movable coordinate system xoy to the X axis is d1, the distance from the origin of the movable coordinate system xoy to the Y axis is d2, and the included angle between the X axis and the X axis is
Order to
Will be provided withd1And d2By Z1、Z2、Z3And Z4Expressing to obtain
Because the movable hinge point is bound to be on a circle with the fixed hinge point as the center of the circle, namely the movable hinge point meets the circular equation:
2a1X+2a2Y+a3＝a0(X2+Y2) (3)
wherein, a0、a1、a2And a3Are all coefficients.
Substituting equation (2) into equation (3) yields:
wherein p is1＝-a0，p2＝a0x，p3＝a0y，p4＝a1，p5＝a2，p6＝-a1y+a2x，p7＝-(a1x+a2y)/2，p8＝(a3-a0(x2+y2))/4；
Eight coefficients p1、p2、p3、p4、p5、p6、p7And p8Are not independent, but must satisfy the following two equations
p1p6+p2p5-p3p4＝0 (5)
2p1p7-p2p4-p3p5＝0 (6)
Expressing the posture pointsIn the form of three-dimensional coordinatesGet five attitude points j is 1,2,3,4,5, and is substituted into the formula (1) to obtain five groups ZiSolution, i ═ 1,2,3,4, five groups ZiIs solved as Zji. In order to meet the requirement of the transplanting track, three attitude points are selected near the seedling taking point, the other two attitude points are selected at the seedling pushing point, and the five attitude points simultaneously restrict the overall height of the track.
Five groups ZiThe solutions are respectively substituted into the equations in equation (4) and written in matrix form as follows:
wherein the matrix coefficientsAj2＝Zj1Zj3-Zj2Zj4，Aj3＝Zj2Zj3+Zj1Zj4，Aj4＝Zj1Zj3+Zj2Zj4，Aj5＝Zj2Zj3-Zj1Zj4，Aj6＝Zj3Zj4， p＝[p1 p2 p3p4 p5 p6 p7 p8]T，
Let coefficient matrix
Matrix [ A ]]T[A]Three eigenvalues are zero, corresponding to three eigenvectors vα，vβAnd vγConstituting the basis of the null space.
Let α, γ be three real parameters, and the vector p be expressed as:
p＝αvα+βvβ+γvγ (7)
the vector p satisfies the formulas (5) and (6), and p in the formula (7)1、p2、p3、p4、p5、p6And p7Substituting into the formulas (5) and (6) to obtain
K10α2+K11β2+K12αβ+K13αγ+K14βγ+K15γ2＝0 (8)
K20α2+K21β2+K22αβ+K23αγ+K24βγ+K25γ2＝0 (9)
Wherein KmnEach of m 1,2, n 0,1,2,3,4, and 5 is expressed by an expression composed of three feature vectors. Setting gamma not equal to 0, and dividing gamma on both sides of formulas (8) and (9)2Is obtained aboutAndtwo sets of real solutions to the two binary quadratic equations, i.e., two sets of solutions to the vector p. Two groups of solutions of the vector p are obtained and are respectively substituted back to p1＝-a0，p2＝a0x，p3＝a0y，p4＝a1，p5＝a2，p6＝-a1y+a2x，p7＝-(a1x+a2y)/2，p8＝(a3-a0(x2+y2) B)/4, find two groups a0，a1，a2，a3X, y. Two groups a0、a1、a2、a3Substituting into formula (3) and converting the circular equation into a circle center radius formula. The coordinates of the centers of the two circular equations are the two fixed hinge points.
One attitude point is taken from three attitude points near the seedling point, and two groups of x and y values are substituted intoAndand calculating coordinates of the two movable hinge points corresponding to the attitude point. The connecting line of the two fixed hinge points is used as a frame, the connecting line of the two fixed hinge points and the corresponding movable hinge point is used as a crank or a swing rod, and the connection of the two movable hinge points is used as a connecting rod to form a first set of four-bar mechanism.
One attitude point is taken from two attitude points near the seedling pushing point, and two groups of x and y values are substituted intoAndand calculating coordinates of the two movable hinge points corresponding to the attitude point. The connecting line of the two fixed hinge points is used as a frame, the connecting line of the two fixed hinge points and the corresponding movable hinge point is used as a crank or a swing rod, and the two movable hingesThe connection of the chain points is a connecting rod to form a second set of four-bar mechanism.
And step three, fitting an angular displacement curve to obtain a total transmission ratio curve.
Taking 36 value points from the two closed tracks respectively, wherein the value points are taken by taking one point every 10 degrees of the crank rotation; then, sequentially taking seven points on the first closed track close to the seedling taking position according to the crank steering, and calculating the angular displacement of the planet carrier corresponding to the seven points and the angular displacement difference value of the planet carrier and the transplanting arm; and sequentially taking three points on the second closed track close to the seedling pushing point according to the crank steering, and calculating the angular displacement of the planet carrier corresponding to the three points and the angular displacement difference value of the planet carrier and the transplanting arm.
The angular displacement of the planet carrier is used as an abscissa, the angular displacement difference value between the planet carrier and the transplanting arm is used as an ordinate, seven points are described according to the angular displacement of the planet carrier calculated from seven points taken from a first closed track and the angular displacement difference value between the planet carrier and the transplanting arm, three points are described according to the angular displacement of the planet carrier calculated from three points taken from a second closed track and the angular displacement difference value between the planet carrier and the transplanting arm, seven interpolation points are given, and the difference of the ordinate of the first point and the ordinate of the last point of the seventeen interpolation points is ensured to be 2 pi. And obtaining an angular displacement curve through three times of non-uniform B spline interpolation according to the seventeen interpolation points, and inserting fitting points with the number larger than 20 between every two adjacent interpolation points. The angular displacement curve needs to be monotonous, namely the non-circular gear does not rotate backwards.
Total gear ratioWherein w1Is the angular velocity, w, of the planet carrier2The angular speed of the transplanting arm is obtained, the reciprocal of the slope calculated by two adjacent points of the angular displacement curve is a negative value, namely the total transmission ratio corresponding to the two adjacent points, and then the whole total transmission ratio curve is obtained according to the angular displacement curve.
And step four, calculating the length of the transplanting arm.
In order to meet the transplanting requirement, a fixed hinge point outside a closed loop formed by connecting the broken lines of five attitude points is taken when the length of the transplanting arm is calculated, and the length of the transplanting arm is obtained according to the coordinate of one attitude point and a movable hinge point corresponding to the attitude point in a four-bar mechanism where the fixed hinge point is located.
And step five, distributing the total transmission ratio, and calculating to obtain pitch curves of the two pairs of non-circular gears.
And obtaining the angular displacement of the planet carrier and the transplanting arm according to the angular displacement curve, and obtaining the transplanting track of the sharp point of the transplanting arm by combining the lengths of the planet carrier and the transplanting arm. Optimizing a transplanting track by adjusting the vertical coordinate of an interpolation point on the angular displacement curve; and translating three points, which are described by the angular displacement difference value of the planet carrier and the transplanting arm and are calculated according to the three points on the second closed track, so that the curves of the peak section and the valley section of the total transmission ratio curve are changed, and finally the purpose of improving the concave of the gear pitch curve is achieved.
According to the total gear ratio curve, two-stage gear ratios are distributed. The first stage transmission ratio isThe second stage transmission ratio is
The pitch curve of the non-circular gears is represented by polar coordinates, the angular displacement of the planet carrier is set to be theta, and the center distances of the two non-circular gears are all a. The first stage driving wheel has a pole diameter ofPolar angle ofThe first stage driven wheel has a pole diameter r2＝a-r1Polar angle isWill be provided withSubstitution intoBecause the first stage driving wheel rotates for one circle, the first stage driven wheel also rotates for one circle, namely the momentAnd obtaining xs. The diameter of the second stage driving wheel isPolar angle ofThe second stage driven wheel has a pole diameter r4＝a-r3Polar angle is
The invention has the beneficial effects that:
1. the invention designs the transmission of the non-circular planetary gear train based on the kinematic mapping, so that the track meets the transplanting requirement.
Drawings
FIG. 1 is a schematic diagram of the mechanism of the present invention;
FIG. 2 is a diagrammatic view of the movement of the mechanism of the present invention;
FIG. 3 is a schematic view of two sets of four-bar linkages calculated by way of example;
FIG. 4 is a graph of angular displacement calculated by example;
FIG. 5 is a total ratio curve and ratio distribution curve calculated by example;
FIG. 6 is a schematic diagram of two closed tracks obtained by example calculation;
FIG. 7 is a pitch curve of the primary drive pulley calculated by example;
FIG. 8 is a pitch curve of a first stage driven wheel calculated by an example;
FIG. 9 is a pitch curve of the secondary drive pulley calculated by example;
fig. 10 is a pitch curve of the second stage driven wheel obtained by example calculation.
Detailed Description
The invention is further explained below with reference to the drawings and examples.
As shown in fig. 1 and 2, a kinematic mapping-based non-circular planetary gear train design method is used for solving two sets of four-bar mechanisms by using a kinematic mapping method; two sections of closed tracks formed by the two sets of four-bar mechanisms are used for taking points on the two sections of closed tracks and obtaining a section of complete angular displacement curve through three times of non-uniform B-spline interpolation; calculating a total transmission ratio curve through an angular displacement curve, and distributing the transmission ratios; the pitch curves of the two pairs of non-circular gears are obtained through two stages of transmission ratios, and the specific steps are as follows:
step one, constructing a non-circular gear planetary gear train, which comprises a planet carrier 3, a transplanting arm 6, a first-stage driving wheel 4, a first-stage driven wheel 2, a second-stage driving wheel 1 and a second-stage driven wheel 5, wherein the first-stage driving wheel 4, the first-stage driven wheel 2, the second-stage driving wheel 1 and the second-stage driven wheel 5 are arranged in the planet carrier 3; the first-stage driving wheel 4 is fixedly connected to the frame; one end of the planet carrier 3 is hinged with the first-stage driving wheel 4, the other end is hinged with the second-stage driven wheel 5, and the middle part is hinged with the first-stage driven wheel 2; the first-stage driven wheel 2 is fixedly connected with the second-stage driving wheel 1; the hinge point of the first-stage driving wheel 4 is defined as a fixed hinge point, and the hinge point of the second-stage driven wheel 5 is defined as a movable hinge point; the first-stage driving wheel 4 is meshed with the first-stage driven wheel 2; the second-stage driving wheel 1 is meshed with the second-stage driven wheel 5; the shell of the transplanting arm 6 is fixedly connected with the second-stage driven wheel 5; the cam of the transplanting arm 6 is fixedly connected with the planet carrier 3; the transplanting arm 6 is constructed by using the transplanting arm disclosed in the patent application No. 201110164729.9. The planet carrier 3 drives the first-stage driven wheel 2 to rotate; the first-stage driven wheel 2 is fixedly connected with the second-stage driving wheel 1, so that the first-stage driven wheel 2 drives the second-stage driving wheel 1 to rotate, and the second-stage driving wheel 1 drives the second-stage driven wheel 5 to rotate; the transplanting arm 6 rotates along with the second-stage driven wheel 5, and the tail end of the transplanting arm forms a transplanting track.
And step two, reversely solving two sets of four-bar mechanisms based on a kinematic mapping method.
The coordinate (x, y) of the dynamic hinge point in the dynamic coordinate system XOY is converted into the coordinate expression form in the static coordinate system XOY as follows:
wherein, the distance from the origin of the movable coordinate system xoy to the X axis is d1, the distance from the origin of the movable coordinate system xoy to the Y axis is d2, and the included angle between the X axis and the X axis is
Order to
Will be provided withd1And d2By Z1、Z2、Z3And Z4Expressing to obtain
Because the movable hinge point is bound to be on a circle with the fixed hinge point as the center of the circle, namely the movable hinge point meets the circular equation:
2a1X+2a2Y+a3＝a0(X2+Y2) (3)
wherein, a0、a1、a2And a3Are all coefficients.
Substituting equation (2) into equation (3) yields:
wherein p is1＝-a0，p2＝a0x，p3＝a0y，p4＝a1，p5＝a2，p6＝-a1y+a2x，p7＝-(a1x+a2y)/2，p8＝(a3-a0(x2+y2))/4；
Eight coefficients p1、p2、p3、p4、p5、p6、p7And p8Are not independent, but must satisfy the following two equations
p1p6+p2p5-p3p4＝0 (5)
2p1p7-p2p4-p3p5＝0 (6)
Expressing the attitude point into a three-dimensional coordinate formGet five attitude points j ═ 1,2,3,4,5, (235, 120, 10), (252, 112, 0), (237, 121, 12), (80, -270, -85), and (58, -258, -75), respectively; five attitude point coordinates are respectively substituted into the formula (1) to obtain five groups of ZiSolution, i ═ 1,2,3,4, five groups ZiIs solved as Zji. A first group: z11＝-70.0125，Z12＝111.8235，Z13＝-0.0872，Z140.9962; second group: z21＝-56，Z22＝126，Z23＝0，Z241. Third group: z31＝-47.782，Z32＝124.1748，Z33＝0.1045，Z340.9945. And a fourth group: z41＝72.5088，Z42＝120.6958，Z43＝-0.6756，Z440.7373. And a fifth group: z51＝84.6885，Z52＝101.5375，Z53＝-0.6088，Z54＝0.7934。
Five groups ZiThe solutions are respectively substituted into the equations in equation (4) and written in matrix form as follows:
wherein the matrix coefficientsAj2＝Zj1Zj3-Zj2Zj4，Aj3＝Zj2Zj3+Zj1Zj4，Aj4＝Zj1Zj3+Zj2Zj4，Aj5＝Zj2Zj3-Zj1Zj4，Aj6＝Zj3Zj4， p＝[p1 p2 p3p4 p5 p6 p7 p8]T，
Let coefficient matrix
Matrix [ A ]]T[A]Three eigenvalues are zero, corresponding to three eigenvectors vα，vβAnd vγConstituting the basis of the null space. Wherein,
vα＝[0.0004，0.0320，0.0101，-0.0337，0.0138，0.9988，-0.0026，0.0017]T，
vβ＝[0.0008，0.0545，0.0237，-0.0655，0.0378，-0.0021，0.9954，0.0032]T，
vγ＝[-0.0005，-0.0223，-0.0091，0.0545，-0.0266，0.0013，0.0029，0.9979]T。
let α, γ be three real parameters, any vector can be represented in null space by the following equation, so vector p is expressed as:
p＝αvα+βvβ+γvγ (7)
the vector p satisfies the formulas (5) and (6), and p in the formula (7)1、p2、p3、p4、p5、p6And p7Substituting into the formulas (5) and (6) to obtainTo
K10α2+K11β2+K12αβ+K13αγ+K14βγ+K15γ2＝0 (8)
K20α2+K21β2+K22αβ+K23αγ+K24βγ+K25γ2＝0 (9)
Wherein KmnEach of m 1,2, n 0,1,2,3,4, and 5 is expressed by an expression composed of three feature vectors. Setting gamma not equal to 0, and dividing gamma on both sides of formulas (8) and (9)2Is obtained aboutAndtwo sets of real solutions to the two binary quadratic equations, i.e., two sets of solutions to the vector p. A first group: get vector p and then back-substitute to p1＝-a0，p2＝a0x，p3＝a0y，p4＝a1，p5＝a2，p6＝-a1y+a2x，p7＝-(a1x+a2y)/2，p8＝(a3-a0(x2+y2) B) 4, corresponding to a0＝-8.52×10-5，a1＝0.0037，a2＝-0.0013，a3-0.6627, x-199.9259, and y-69.1470. Second group:the same back substitution is carried out to obtain the corresponding a0＝2.143×10-4，a1＝0.0295，a2＝-0.0151，a33.633, x-10.0829 and y-9.4963. Two groups a0、a1、a2、a3Substituting into formula (3) and converting the circular equation into a circle center radius formula, which is respectively: (X +43.835)2+(Y-15.533)2＝99.7242And (X-137.487)2+(Y+70.4616)2＝202.0462。
The fixed hinge points are the coordinates of the center of the circle of the equation, the obtained fixed hinge points are (-43.835, 15.533) and (137.487, -70.4616), respectively, and the track of the moving hinge point is a circle, namely the distance L1 between the moving hinge point and the fixed hinge point is 99.724 mm.
In order to meet the requirement of the transplanting track, three attitude points (235, 120, 10), (252, 112, 0), (237, 121, 12) are used for restraining the track of a sharp mouth (containing a small buckle of the seedling taking point) near the seedling taking point, the other two attitude points (80, -270, -85), (58, -258, -75) are used for restraining the track near the seedling pushing point, the five attitude points simultaneously restrain the overall height of the track, the rest middle sections have no requirement, and the black solid points in fig. 3 are the attitude points.
Taking a coordinate (235, 120, 10) of the attitude point, and substituting two groups of x and y values into Andthe coordinates of the attitude point (235, 120, 10) corresponding to the two moving hinge points are calculated as (26.1041, 86.6203) and (223.421, 112.399). The connecting line of the fixed hinge points (-43.835, 15.533) and (137.487, -70.4616) is taken as a frame, and the connecting line of the two fixed hinge points (-43.835, 15.533) and (137.487, -70.4616) and the corresponding movable hinge points (26.1041, 86.6203) and (223.421, 112.399) is taken as a crank or a swing rod (the crank with shorter rod length) so as to moveThe connection of the hinge points (26.1041, 86.6203) and (223.421, 112.399) is a linkage, forming a first set of four-bar linkages, such as the upper-side linkage shown in fig. 3.
Taking a posture point coordinate (80, -270, -85), and substituting two groups of x and y values into Andthe coordinates of the attitude point (80, -270, -85) are calculated to correspond to the coordinates of the two moving hinge points (-6.3085, -76.8614) and (69.6611, -260.7831). A connecting line of the fixed hinge points (-43.835, 15.533) and (137.487, -70.4616) is used as a frame, a connecting line of the two fixed hinge points (-43.835, 15.533) and (137.487, -70.4616) and the corresponding movable hinge points (-6.3085, -76.8614) and (69.6611, -260.7831) is used as a swing rod, and a connection of the movable hinge points (-6.3085, -76.8614) and (69.6611, -260.7831) is used as a connecting rod to form a second set of four-bar mechanism, such as a lower-side mechanism shown in fig. 3.
And step three, fitting an angular displacement curve to obtain a total transmission ratio curve.
The first set of four-bar mechanisms passes exactly three stance points (235, 120, 10), (252, 112, 0), (237, 121, 12) and forms a first closed trajectory, and the second set of four-bar mechanisms passes exactly two stance points (80, -270, -85), (58, -258, -75) and forms a second closed trajectory. Taking 36 value-taking points from the two closed tracks respectively, wherein the value-taking method is that one point is taken every 10 degrees of the crank rotation, then 7 points are sequentially taken at the position close to the seedling-taking point on the first closed track according to the crank rotation direction, and the angular displacement of the planet carrier corresponding to the 7 points and the angular displacement difference value of the planet carrier and the transplanting arm are calculated; and sequentially taking 3 points on the second closed track close to the seedling pushing point according to the crank steering, and calculating the angular displacement of the planet carrier corresponding to the 3 points and the angular displacement difference value of the planet carrier and the transplanting arm.
As shown in fig. 4, the angular displacement of the planet carrier is used as the abscissa, the angular displacement difference between the planet carrier and the transplanting arm is used as the ordinate, seven points (the section from point a to point b in fig. 4) are described according to the angular displacement of the planet carrier calculated from 7 points taken on the first closed trajectory and the angular displacement difference between the planet carrier and the transplanting arm, three points (the section from point c to point d in fig. 4) are described according to the angular displacement of the planet carrier calculated from 3 points taken on the second closed trajectory and the angular displacement difference between the planet carrier and the transplanting arm, and seven interpolation points are given, so that the difference between the ordinate of the first point and the ordinate of the last interpolation point of the seventeen interpolation points. And obtaining an angular displacement curve through cubic non-uniform B spline interpolation according to the seventeen interpolation points, wherein the precision can be set as 30 fitting points inserted between two adjacent interpolation points. The angular displacement curve needs to be monotonous, namely the non-circular gear does not rotate backwards.
Total gear ratioWherein w1Is the angular velocity, w, of the planet carrier 32The angular velocity of the transplanting arm 6 is obtained, the reciprocal of the slope calculated by two adjacent points of the angular displacement curve is a negative value, the total transmission ratio corresponding to the two adjacent points is obtained, and then the whole total transmission ratio curve is obtained according to the angular displacement curve, as shown in fig. 5.
And step four, calculating the length of the transplanting arm 6.
In order to meet the transplanting requirements, when the length of the transplanting arm 6 is calculated, a fixed hinge point (-43.835, 15.533) outside a closed loop formed by connecting broken lines of five posture points is taken, the coordinates of a moving hinge point corresponding to the posture point coordinates (235, 120, 10) are (26.1041, 86.6203), and further the length L2 of the transplanting arm 6 is obtained to be 211.5459 mm. At this time, the locus of the moving hinge point is a circle having a circular point (-43.835, 15.533) and a radius of 99.724 mm.
And step five, distributing the total transmission ratio, and calculating to obtain pitch curves of the two pairs of non-circular gears.
Obtaining the planet carrier 3 according to the angular displacement curve and transplantingThe angular displacement of the arm 6 is combined with the lengths of the planet carrier 3 and the transplanting arm 6 to obtain the transplanting track at the tail end of the transplanting arm 6. The height of the transplanting track, the length and the width of the sharp mouth and the angular displacement (namely the angle) of the transplanting arm at the seedling taking point and the seedling pushing point are required to be ensured) The transplanting requirement is met, the transplanting requirement can be optimized by adjusting the ordinate of the interpolation point on the angular displacement curve, and the optimized transplanting track is shown in fig. 6. The point c can be translated to three points on the section d, so that the curve of the peak section and the valley section of the total transmission ratio curve is changed, and the purpose of improving the concave of the pitch curve is finally achieved.
According to the total gear ratio curve, two-stage gear ratios are distributed. The first stage transmission ratio isThe second stage transmission ratio isAs shown in fig. 5, the sub-ratio profile becomes smoother after the total ratio profile split is complete.
The pitch curve of the non-circular gears is represented by polar coordinates, the angular displacement of the planet carrier is set to be theta, and the center distances of the two non-circular gears are all a. The first stage driving wheel 4 has a pole diameter ofPolar angle ofThe first stage driven wheel 2 has a pole diameter r2＝a-r1Polar angle isWill be provided withSubstitution into Because the first-stage driving wheel 4 rotates for one circle, the first-stage driven wheel 2 also rotates for one circle, namely the momentObtaining xs as 0.968; thus, the second-stage driving wheel 1 can be ensured to rotate one circle and the second-stage driven wheel 5 can also rotate one circle without processing the second-stage transmission ratio. The diameter of the second-stage driving wheel 1 isPolar angle ofThe second stage driven wheel 5 has a pole diameter r4＝a-r3Polar angle isWhen a is 70mm, the pitch curve of the primary driving pulley 4, the primary driven pulley 2, the secondary driving pulley 1, and the secondary driven pulley 5 are calculated as shown in fig. 7, 8, 9, and 10, respectively.
Claims (1)
1. A non-circular gear planetary gear train design method based on kinematic mapping is characterized in that: the method comprises the following specific steps:
the method comprises the following steps that firstly, a non-circular gear planetary gear train is constructed, and the non-circular gear planetary gear train comprises a planetary carrier, a transplanting arm, a first-stage driving wheel, a first-stage driven wheel, a second-stage driving wheel and a second-stage driven wheel which are arranged in the planetary carrier; the first-stage driving wheel is fixedly connected to the frame; one end of the planet carrier is hinged with the first-stage driving wheel, the other end of the planet carrier is hinged with the second-stage driven wheel, and the middle part of the planet carrier is hinged with the first-stage driven wheel; the first-stage driven wheel is fixedly connected with the second-stage driving wheel; the hinge point of the first-stage driving wheel is defined as a fixed hinge point, and the hinge point of the second-stage driven wheel is defined as a movable hinge point; the first-stage driving wheel is meshed with the first-stage driven wheel; the second-stage driving wheel is meshed with the second-stage driven wheel; the shell of the transplanting arm is fixedly connected with the second-stage driven wheel; the cam of the transplanting arm is fixedly connected with the planet carrier;
step two, reversely solving two sets of four-bar mechanisms based on a kinematic mapping method;
the coordinate (x, y) of the dynamic hinge point in the dynamic coordinate system XOY is converted into the coordinate expression form in the static coordinate system XOY as follows:
wherein, the distance from the origin of the movable coordinate system xoy to the X axis is d1, the distance from the origin of the movable coordinate system xoy to the Y axis is d2, and the included angle between the X axis and the X axis is
Order to
Will be provided withd1And d2By Z1、Z2、Z3And Z4Expressing to obtain
Because the movable hinge point is bound to be on a circle with the fixed hinge point as the center of the circle, namely the movable hinge point meets the circular equation:
2a1X+2a2Y+a3＝a0(X2+Y2) (3)
wherein, a0、a1、a2And a3Are all tiedCounting;
substituting equation (2) into equation (3) yields:
wherein p is1＝-a0，p2＝a0x，p3＝a0y，p4＝a1，p5＝a2，p6＝-a1y+a2x，p7＝-(a1x+a2y)/2，p8＝(a3-a0(x2+y2))/4；
Eight coefficients p1、p2、p3、p4、p5、p6、p7And p8Are not independent, but must satisfy the following two equations
p1p6+p2p5-p3p4＝0 (5)
2p1p7-p2p4-p3p5＝0 (6)
Expressing the attitude point into a three-dimensional coordinate formGet five attitude points j is 1,2,3,4,5, and is substituted into the formula (1) to obtain five groups ZiSolution, i ═ 1,2,3,4, five groups ZiIs solved as Zji(ii) a In order to meet the requirement of a transplanting track, three attitude points are selected near a seedling taking point, the other two attitude points are selected at a seedling pushing point, and the five attitude points simultaneously restrict the overall height of the track;
five groups ZiThe solutions are respectively substituted into the formula (4)) And written in matrix form as follows:
wherein the matrix coefficientsAj2＝Zj1Zj3-Zj2Zj4，Aj3＝Zj2Zj3+Zj1Zj4，Aj4＝Zj1Zj3+Zj2Zj4，Aj5＝Zj2Zj3-Zj1Zj4，Aj6＝Zj3Zj4， p＝[p1 p2 p3 p4p5 p6 p7 p8]T，
Let coefficient matrix
Matrix [ A ]]T[A]Three eigenvalues are zero, corresponding to three eigenvectors vα，vβAnd vγA base constituting a null space;
let α, γ be three real parameters, and the vector p be expressed as:
p＝αvα+βvβ+γvγ (7)
the vector p satisfies the formulas (5) and (6), and p in the formula (7)1、p2、p3、p4、p5、p6And p7Substituting into the formulas (5) and (6) to obtain
K10α2+K11β2+K12αβ+K13αγ+K14βγ+K15γ2＝0 (8)
K20α2+K21β2+K22αβ+K23αγ+K24βγ+K25γ2＝0 (9)
Wherein KmnEach of m 1,2, n 0,1,2,3,4,5 is expressed by an expression composed of three feature vectors; setting gamma not equal to 0, and dividing gamma on both sides of formulas (8) and (9)2Is obtained aboutAndthe two binary quadratic equations have two groups of real number solutions, namely two groups of solutions of the vector p; two groups of solutions of the vector p are obtained and are respectively substituted back to p1＝-a0，p2＝a0x，p3＝a0y，p4＝a1，p5＝a2，p6＝-a1y+a2x，p7＝-(a1x+a2y)/2，p8＝(a3-a0(x2+y2) B)/4, find two groups a0，a1，a2，a3X, y; two groups a0、a1、a2、a3Substituting the equation into a formula (3) and converting the circular equation into a circle center radius formula; the coordinates of the centers of the circles of the two circular equations are two fixed hinge points;
one attitude point is taken from three attitude points near the seedling point, and two groups of x and y values are substituted intoAndcalculating coordinates of two movable hinge points corresponding to the attitude point; the connecting line of the two fixed hinge points is taken as a frame, the connecting line of the two fixed hinge points and the corresponding movable hinge point is taken as a crank or a swing rod, and the connection of the two movable hinge points is taken as a connecting rod to form a first set of four-bar mechanism;
one attitude point is taken from two attitude points near the seedling pushing point, and two groups of x and y values are substituted intoAndcalculating coordinates of two movable hinge points corresponding to the attitude point; the connecting line of the two fixed hinge points is taken as a frame, the connecting line of the two fixed hinge points and the corresponding movable hinge point is taken as a crank or a swing rod, and the connection of the two movable hinge points is taken as a connecting rod to form a second set of four-bar mechanism;
step three, fitting an angular displacement curve to obtain a total transmission ratio curve;
taking 36 value points from the two closed tracks respectively, wherein the value points are taken by taking one point every 10 degrees of the crank rotation; then sequentially taking seven interpolation points on the first closed track close to the seedling taking position according to the crank steering, and calculating the angular displacement of the planet carrier corresponding to the seven interpolation points and the angular displacement difference value of the planet carrier and the transplanting arm; sequentially taking three interpolation points at the position close to the seedling pushing point on the second closed track according to the crank steering, and calculating the angular displacement of the planet carrier corresponding to the three interpolation points and the angular displacement difference value of the planet carrier and the transplanting arm;
the angular displacement of the planet carrier is used as an abscissa, the angular displacement difference value of the planet carrier and the transplanting arm is used as an ordinate, the seven interpolation points are described according to the angular displacement of the planet carrier calculated by the seven interpolation points taken on the first closed track and the angular displacement difference value of the planet carrier and the transplanting arm, the three interpolation points are described according to the angular displacement of the planet carrier calculated by the three interpolation points taken on the second closed track and the angular displacement difference value of the planet carrier and the transplanting arm, and in addition, the seven interpolation points are given to ensure that the first and last point ordinate of the seventeen interpolation points has a 2 pi difference; obtaining an angular displacement curve through three times of non-uniform B spline interpolation according to seventeen interpolation points, and inserting fitting points with the number larger than 20 between every two adjacent interpolation points; the angular displacement curve needs to be ensured to be monotonous, namely, the non-circular gear does not rotate backwards;
total gear ratioWherein w1Is the angular velocity, w, of the planet carrier2The angular speed of the transplanting arm is adopted, the reciprocal of the slope calculated by two adjacent points of the angular displacement curve is a negative value, namely the total transmission ratio corresponding to the two adjacent points is obtained, and then the whole total transmission ratio curve is obtained according to the angular displacement curve;
step four, calculating the length of the transplanting arm;
in order to meet the transplanting requirement, a fixed hinge point outside a closed loop formed by connecting line of broken lines of five attitude points is taken when the length of the transplanting arm is calculated, and the length of the transplanting arm is obtained according to the coordinate of one attitude point and a movable hinge point corresponding to the attitude point in a four-bar mechanism;
step five, distributing the total transmission ratio, and calculating to obtain pitch curves of the two pairs of non-circular gears;
obtaining the angular displacement of the planet carrier and the transplanting arm according to the angular displacement curve, and obtaining the transplanting track of the tip point of the transplanting arm by combining the lengths of the planet carrier and the transplanting arm; optimizing a transplanting track by adjusting the vertical coordinate of an interpolation point on the angular displacement curve; translating three interpolation points which are described by the angular displacement difference value of the planet carrier and the transplanting arm and are calculated according to the three interpolation points which are taken on the second closed track, so that the curves of the peak section and the valley section of the total transmission ratio curve are changed, and finally the purpose of improving the concave of the gear pitch curve is achieved;
distributing two-stage transmission ratio according to the total transmission ratio curve; the first stage transmission ratio isThe second stage transmission ratio is
The pitch curve of the non-circular gears is represented by polar coordinates, the angular displacement of the planet carrier is set to be theta, and the center distances of the two non-circular gears are all a; the first stage driving wheel has a pole diameter ofPolar angle ofThe first stage driven wheel has a pole diameter r2＝a-r1Polar angle isWill be provided withSubstitution intoBecause the first stage driving wheel rotates for one circle, the first stage driven wheel also rotates for one circle, namely the momentObtaining xs; the diameter of the second stage driving wheel isPolar angle ofThe second stage driven wheel has a pole diameter r4＝a-r3Polar angle is
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