CN112464481B

CN112464481B - Dynamic transmission precision numerical calculation method of cycloidal pin gear speed reducer for robot

Info

Publication number: CN112464481B
Application number: CN202011392955.8A
Authority: CN
Inventors: 韩林山; 范素香; 王迎佳; 上官林建; 雷冀南
Original assignee: North China University of Water Resources and Electric Power
Current assignee: North China University of Water Resources and Electric Power
Priority date: 2020-12-02
Filing date: 2020-12-02
Publication date: 2024-04-16
Anticipated expiration: 2040-12-02
Also published as: CN112464481A

Abstract

A dynamic transmission precision numerical calculation method of cycloidal pin gear speed reducer for robot comprises defining key errors of each part, measuring and calculating to determine the key errors, establishing a dynamic model of dynamic transmission errors, calculating displacement generated at part contact positions according to the dynamic model of errors, further determining displacement generated at shaft or bearing and gear tooth engagement positions of micro-displacement of each part, and determining acting force of each part at each part contact position according to the errors and the displacements; establishing a dynamic transmission error mathematical model; the actual rotation angle theta of the output shaft at any moment can be determined by calculation _c Dynamic transmission error delta theta _c According to the method, various errors, non-linear factors such as contact gaps, part contact deformation and inertial load of rotating parts are comprehensively considered, the situation that the speed reducer is in a live state is more similar to the situation that the transmission error is obtained is more similar to the actual situation, and the production and assembly processes of parts of the speed reducer can be guided more accurately.

Description

Dynamic transmission precision numerical calculation method of cycloidal pin gear speed reducer for robot

Technical Field

The invention relates to the technical field of cycloidal pin gear speed reducers for robots, in particular to a dynamic transmission precision numerical calculation method of a cycloidal pin gear speed reducer for a robot.

Background

The cycloidal pin gear transmission is a small tooth difference planetary transmission adopting cycloidal gears and pin gears for meshing, the transmission mechanism is first developed and succeeded in Germany in the beginning of thirty of the twentieth century, lorenzBraren established Sai Gule Co., ltd in Munich in 1931, and the manufacture and sale of cycloidal pin gear speed reducers are started. The transmission mechanism has the characteristics of large transmission ratio, high transmission efficiency, small volume, light weight, large bearing capacity, stable operation, low noise, long service life and the like.

Because of the manufacturing errors and assembly errors of parts of the cycloidal pin gear speed reducer and the existence of temperature deformation and elastic deformation in the transmission process, the input and output transmission errors are unavoidable. The transmission error is the deviation value between the actual rotation angle and the theoretical rotation angle of the output shaft, and is an important index for evaluating the transmission precision of the cycloidal pin gear speed reducer.

The transmission error means: when the input shaft rotates unidirectionally, the difference between the actual value and the theoretical value of the rotation angle of the output shaft is calculated. For a cycloidal pin gear speed reducer with a planet carrier as an output shaft, if the input end of an input shaft (namely a sun gear) rotates at a constant speed and the rotation angle of the input end is theta _s The actual rotation angle of the output shaft (i.e. the planet carrier) is theta _c The transmission ratio of the system is i _z The transmission error of the system is: Δθ _c ＝θ _c -θ _s /i _z . The transmission accuracy is the aforementioned transmission error, and fig. 1 is a waveform schematic diagram of the transmission error.

The determination of the transmission precision has practical guiding significance for the production and assembly processes of parts of the cycloidal pin gear speed reducer for the robot, can determine the optimal matching relationship of the parts in advance, and has great significance for matching parts with good complementarity into the speed reducer with better performance.

Disclosure of Invention

The invention aims to solve the problems, and provides a dynamic transmission precision numerical calculation method of a cycloidal pin gear speed reducer for a robot, in particular to a dynamic transmission precision numerical calculation method of a three-crank cycloidal pin gear speed reducer for a robot.

The technical scheme of the invention is as follows: the basic principle of the method is as follows: the actual mass center position of the part and the rotation angle of the part deviate from ideal positions due to nonlinear factors such as machining errors, mounting errors, gear tooth meshing gaps, bearing gaps, part contact deformation, inertial load of the rotating part and the like of all parts in a transmission system, the deviation is described by linear displacement and angular displacement, and the linear displacement and the angular displacement are collectively called micro displacement; at the same time, the contact positions of the bearing or the shaft support, the gear tooth meshing and the like among the parts are equivalent by springs, And is described in terms of physical bearing or shaft stiffness, gear tooth engagement stiffness. By analyzing the stress condition of each part in the transmission system at the ideal position, a dynamic differential equation set of the system, namely a numerical calculation mathematical model of dynamic transmission errors, is established according to the D' Alemmbert principle; solving a differential equation set by adopting a nonlinear direct integration method such as a Wilson-theta method, a Newmark method and the like; then programming and calculating by using a computer to determine the actual rotation angle theta of the output shaft of the speed reducer at any moment _c Further determine the dynamic transmission error delta theta _c 。

The method specifically comprises the following steps:

s1, determining basic structural parameters of the three-crank cycloidal pin gear speed reducer according to design parameters of the speed reducer to be simulated and calculated,

the method specifically comprises the following parameters: sun gear tooth number Z _s Tooth number Z of planet wheel _p Cycloidal gear tooth number Z _b Number of teeth Z of needle wheel _r Involute gear modulus m (mm), involute gear pressure angle α (°), sun gear tooth displacement coefficient x _cs Shift coefficient x of planet gear teeth _cp Planet gear tooth width B _z (mm), cycloidal gear tooth width B _c (mm), crank shaft eccentricity e (mm), pin tooth pin radius r _d (mm), theoretical radius r of pin or pin housing tooth space distribution circle _z (mm), 2 cycloidal gears, 3 planetary gears, 3 crankshafts;

s2, defining key errors of a sun gear, 3 planet gears, 2 cycloid gears, 3 crankshafts, a planet carrier, a pin gear pin, a pin gear shell and a roller bearing in the speed reducer, measuring the key errors through a high-precision measuring instrument, and determining the key errors according to calculation of a measuring result;

the key errors of the parts are defined as follows:

1) Definition of critical error of sun gear

The key errors of the sun gear are as follows: base circle eccentricity error of sun gear (E _s ,β _s ) The assembly error of the sun gear is (A) _s ,γ _s ) The method comprises the steps of carrying out a first treatment on the surface of the Wherein: e (E) _s 、A _s Respectively show the base circle eccentric error,Size, dimension (um) of assembly error; beta _s 、γ _s The directions of the base circle eccentric error and the assembly error are respectively represented, and the dimension (degree) is shown;

2) Critical error definition for planet gears

The key errors of the planet wheel are as follows: the base circle eccentric errors of the 3 planet wheels are (E) _pi ,β _pi ) (i=1, 2, 3); wherein: e (E) _pi (i=1, 2, 3) represents the base circle eccentricity error size, dimension (um) of the planet wheels 1,2,3, respectively; beta _pi (i=1, 2, 3) represents the base circle eccentric error direction of the planetary gears 1,2,3, respectively, dimension (°;

3) Definition of key errors for a three crank cycloidal gear

The key errors of the three-crank cycloidal gear are as follows: eccentric errors (E) of crank shaft holes 1,2,3 in the cycloid gears 1,2 _hji ,β _hji ) (j=1, 2; i=1, 2, 3); tooth space deviation R of gear teeth on cycloidal gears 1 and 2 _jk (j=1, 2), pitch deviation P _jk (j=1, 2); wherein: e (E) _hji The magnitude and dimension (um) of the eccentric error of the crank shaft hole are shown; beta _hji Representing the eccentric error direction and dimension (°) of the crank shaft hole; r is R _jk 、P _jk Respectively representing the tooth socket deviation and the tooth pitch deviation of cycloid gear teeth, and dimension (um);

4) Definition of critical errors of pin gear shells

The key errors of the pin gear housing are: z is Z _r Average radius error delta of individual pin tooth slots _cr (um) on needle gear case Z _r Tooth space deviation (abbreviated as pin tooth space deviation) R of circle centers of individual tooth spaces _k (k＝1,2,…,Z _r ) Pitch deviation (abbreviated as pin wheel pitch deviation) P _k (k＝1,2,…,Z _r ) The method comprises the steps of carrying out a first treatment on the surface of the Wherein: r is R _k 、P _k The tooth space deviation and the tooth pitch deviation of the kth needle tooth are respectively represented, and the dimension (um) is shown;

5) Definition of key errors of a crankshaft

The key errors of the crankshaft are: eccentric errors (E) of the eccentric cams 1,2 on the crankshafts 1,2,3 _cji ,β _cji ) (j=1, 2; i=1, 2, 3); wherein: e (E) _cji Representing the eccentric error of an eccentric camSize, dimension (um); beta _cji Representing the eccentric error direction, dimension (°) of the eccentric cam;

6) Definition of critical errors for a planet carrier

The key errors of the planet carrier are as follows: eccentric errors (E) of the crank shaft holes 1,2,3 on the carrier _cai ,β _cai ) (i=1, 2, 3), the eccentricity error (a _c ,γ _c ) The method comprises the steps of carrying out a first treatment on the surface of the Wherein: e (E) _cai 、A _c The eccentric error of the crank shaft hole and the journal is shown, and the dimension (um) is shown; beta _cai 、γ _c Representing the eccentric error direction of the crank shaft hole and the shaft neck, and dimension (°);

7) Definition of key errors for pin teeth

The key errors of the pin are: z is Z _r Average diameter error delta of individual pin teeth _zr Dimension (um);

8) Definition of critical errors for bearings

In the three-crank cycloidal pin gear speed reducer, the bearings between the cycloidal gears 1,2 and the crankshafts 1,2,3 are roller bearings (fig. 12), and the bearing clearances are delta respectively _bji (j=1, 2; i=1, 2, 3), dimension (um);

the bearings between the planet carrier and the crankshafts 1,2 and 3 adopt tapered roller bearings, and the bearing clearances are delta respectively _xi (i=1, 2, 3), dimension (um);

the bearing between the planet carrier and the needle gear shell adopts an angular contact ball bearing, and the bearing clearance is delta _ca Dimension (um);

s3, establishing a mechanical model for calculating dynamic transmission errors of the three-crank cycloidal pin gear speed reducer, wherein the mechanical model is specifically as follows

1) In the mechanical model, the contact positions of the bearing or the shaft support, the gear tooth engagement and the like among the parts are equivalent by springs, and the physical quantity of the rigidity of the bearing or the shaft and the gear tooth engagement is used for describing; gear tooth engagement stiffness k between sun gear and planet wheel i (i=1, 2, 3) _i (N/um) (i=1, 2, 3), bearing support rigidity k between crank shaft i (i=1, 2, 3) and cycloid gear j (j=1, 2) shaft hole _ji (N/um) (j=1, 2; i=1, 2, 3), the bearing support rigidity k between the crank shaft i (i=1, 2, 3) and the carrier shaft hole _bi (N/um) (i=1, 2, 3), bearing support stiffness k between planet carrier and needle gear housing _ca (N/um), the meshing stiffness k between the cycloidal gear and the needle tooth _jk (N/um)(k＝1,2,…,Z _r ) Bearing stiffness k of sun gear shaft _s (N/um); the determination method of the rigidity comprises the following steps: k (k) _i Determining by adopting bending deformation of gear teeth; k (k) _s Calculating the bending deformation of the sun gear shaft; and k is _ji 、k _bi 、k _ca 、k _jk The contact stiffness is determined using the Palmgren formula.

The Palmgren formula is:

wherein,

middle v ₁ ,ν ₂ Poisson ratio of the elastomer 1,2;

E ₁ ,E ₂ -elastic modulus (MPa) of the elastomer 1,2;

delta-the amount of contact deformation (mm) between two elastomers;

l-the contact length (mm) between two elastomers;

f-load (N) between the two elastomers.

If use k ₀ Indicating the contact stiffness (N/mm) between two elastomers

F＝k ₀ δ (2)

The contact stiffness k between the two elastomers can be deduced from the formulae (1), (2) ₀ The method comprises the following steps:

from the formula (3), the contact stiffness k between the two elastomers ₀ Mainly depends on the materials of the two elastic bodies, the contact length and the acting load of the two elastic bodies;

2) Determining a static and dynamic coordinate system of the mechanical model; the theoretical center O of a sun gear shaft (or a planet carrier and a pin gear housing) is taken as an origin, and the section perpendicular to the axis of the speed reducer is a plane static coordinate system xoy. Starting from the input end, namely the sun gear end, the cycloid gear number j=1 near the sun gear, and the other cycloid gear number j=2; by theoretical centroid O of cycloidal gear _j As origin, let the eccentric direction of cycloid wheel j (j=1, 2) be eta _j The shaft, assuming the sun gear shaft to rotate anticlockwise, will be perpendicular to eta _j The shaft rotates 90 degrees along the revolution direction of the shaft to be zeta _j Shaft, establishing dynamic coordinate system eta of cycloid wheel j _j O _j ξ _j The method comprises the steps of carrying out a first treatment on the surface of the In the dynamic modeling process, taking the cycloid wheel eta with j=1 _j The axis is consistent with the x-axis direction of the static coordinate system when in the initial position;

3) In the mechanical model, each spring is arranged at the contact position of the positive direction of each part coordinate axis or the meshing position of gear teeth, and the tension of the spring is set to be positive and the compression of the spring is set to be negative; phi (phi) _i Representing the relative position of the crankshaft holes on the cycloidal gears (or planet carrier), taking phi _i ＝2π(i-1)/3(i＝1,2,3)，ψ _j Represents the theoretical centroid O of 2 cycloidal gears _j Is taken as ψ _j ＝(j-1)π(j＝1,2)；

S4, determining the displacement of the transmission system at the bearing position of the shaft or the bearing and the meshing position of the gear teeth according to the manufacturing errors and gaps of all parts in the three-crank cycloidal pin gear speed reducer determined in the S2 and combining the mechanical model established in the S3; the method comprises the following steps:

1) And determining the displacement of the base circle eccentric error and the assembly error of the sun gear at the meshing position of the gear teeth and the shaft bearing position of the sun gear.

If the base circle eccentricity error of the sun gear is (E _s ,β _s ) The displacement at the meshing of the teeth is:

e _si ＝E _s cos(θ _s +β _s -A _i ) (4)

In which A _i ＝θ _c +φ _i +π/2-α′，θ _c ＝θ _s /i _z

α' —the angle of engagement of the sun and planet (°);

θ _c -theoretical rotation angle (°) of the planet carrier.

If the sun gear assembly error is (A) _s ,γ _s ) The displacements thereof in the x, y directions at the sun gear shaft support are respectively:

e _sx ＝A _s cosγ _s (5)

e _sy ＝A _s sinγ _s (6)

2) Determining the displacement generated by the base circle eccentric error of the planet wheel at the meshing position of the gear teeth; base circle eccentric error of 3 planet wheels (E _pi ,β _pi ) (i=1, 2, 3) the displacement at the tooth engagement is:

e _pi ＝-E _pi cos(β _pi -θ _p -A _i )(i＝1,2,3) (7)

in theta _p -theoretical self-rotation angle (°) of the planet;

3) Determining the displacement generated by the eccentric error of the crank shaft hole on the cycloid wheel at the bearing support position of the cycloid wheel, and the displacement generated by the tooth socket deviation and the tooth pitch deviation at the gear tooth meshing position;

definition O _ji Represents the center of a crank shaft hole i (i=1, 2, 3) on a cycloid wheel j (j=1, 2), R _dc The radius of the circumference of the crank shaft hole on the cycloid wheel is represented.

If the eccentric error of the cycloid gear crank shaft hole is: (E) _hji ,β _hji ) (j=1, 2; i=1, 2, 3), the displacements thereof in the x, y directions of the static coordinate system at the bearing support are respectively:

e _hxji ＝-E _hji cos(θ _c +φ _i +β _hji ) (8)

e _hyji ＝-E _hji sin(θ _c +φ _i +β _hji ) (9)

defining the tooth space deviation of a cycloidal gearR _jk Deviation of pitch P _jk O represents the center of the pinwheel, O _j Represents the center of the cycloid gear j (j=1, 2), and P represents the node between the cycloid gear j and the pin gear;

if the tooth space deviation on the cycloid wheel j is R _jk (j=1, 2), pitch deviation P _jk (j=1, 2), then the displacements thereof at the tooth mesh are respectively:

alpha in the formula _jk The kth needle tooth center is connected with the jth cycloidal gear node and eta _j Positive axis angle (°), where k=1, 2, …, Z _r ；

φ _djk The kth needle tooth center and the jth cycloidal gear center O _j Connection line and eta _j Included angle (°) in positive direction of axis

4) Determining the tooth socket deviation of the circle center of the tooth socket on the needle gear shell and the displacement generated by the tooth pitch deviation at the meshing position of cycloid gear teeth;

a tooth space deviation (abbreviated as a needle wheel tooth space deviation) R for defining the circle center of the tooth space on the needle gear shell _k Pitch deviation (abbreviated as pin wheel tooth space deviation) P _k Wherein: o represents the center of the pinwheel, O _j The center of the cycloid gear j (j=1, 2) is represented, and P represents the node between the cycloid gear j and the pin gear.

If the tooth socket deviation of the pinwheel is R _k (k＝1,2,…,Z _r ) The pitch deviation is P _k (k＝1,2,…,Z _r ) The displacements at the meshing of the teeth are respectively:

e _Rk ＝-R _k cos(α _jk -φ _jk ) (12)

phi in _jk The kth needle tooth radial direction and the jth cycloidal gear eta _j Included angle (degree) between axial directions;

5) Determining the displacement of the eccentric error of the eccentric cam of the crankshaft at the bearing support thereof;

define the eccentric cam j eccentric error, O, of the crankshaft i _ji Represents the center of a crank shaft hole i (i=1, 2, 3) on a cycloid wheel j (j=1, 2), O _pj The center of rotation of the crankshaft corresponding to the cycloid gear j (j=1, 2) on the crankshaft i (i=1, 2, 3);

if the eccentric error of the eccentric cam j on the crankshaft i is (E) _cji ,β _cji ) (j=1, 2; i=1, 2, 3), then the displacements in x, y directions are respectively:

e _cxji ＝E _cji cos(θ _p +ψ _j +β _cji ) (14)

e _cyji ＝-E _cji sin(θ _p +ψ _j +β _cji ) (15)

6) Determining the displacement generated by the eccentric errors of a crank shaft hole and a shaft neck on the planet carrier at a bearing supporting position;

defining the eccentricity error of the planet carrier crank shaft hole i, wherein O _ci Represents the center of a crank shaft hole i (i=1, 2, 3) on the planet carrier, T _out Representing the load torque acting on the planet carrier;

if the eccentric error of the crank shaft hole i of the planet carrier is (E) _cai ,β _cai ) (i=1, 2, 3), then its displacement in the x, y direction at the support is respectively:

e _caxi ＝-E _cai cos(θ _c +φ _i +β _cai ) (16)

e _cayi ＝-E _cai sin(θ _c +φ _i +β _cai ) (17)

if the eccentricity of the journal of the planet carrier is assumed to be (A) _c ,γ _c ) It is generated in the x, y direction at the supportThe displacement of (2) is respectively:

e _cx ＝A _c cosγ _c (18)

e _cy ＝A _c sinγ _c (19)

7) Determining a diameter error of the pin, a gap formed by a tooth groove radius error on the pin housing at the meshing position of cycloid gear teeth, and determining displacement of the gap at the meshing position of the cycloid gear teeth;

definition O represents the theoretical distribution circle center of the needle tooth pin or the distribution circle center of the theoretical circle center of the tooth slot on the needle tooth shell, delta _jk Representing the meshing gap of cycloidal gear teeth;

if the average diameter of the pin is wrong _zr (um), average radius error of pin tooth slot delta _cr (um) then its resulting gap delta at the cycloidal tooth engagement _jk And the displacement e produced thereby _jk The method comprises the following steps of:

δ _jk ＝δ _zr +δ _cr (20)

e _jk ＝-δ _jk (21)

8) Displacement of the bearing gaps at their support

If the clearance of the roller bearing at the crank shaft hole i of the cycloid wheel j is delta _bji (j=1, 2; i=1, 2, 3), then its displacement at the support is:

e _bji ＝-δ _bji (22)

if the clearance of the conical roller bearing of the planet carrier at the crank shaft hole i is delta _xi (i=1, 2, 3), then its displacement at the support is:

e _xi ＝-δ _xi (23)

if the clearance between the planet carrier and the pin gear shell is delta _ca The displacement it produces at the support is:

e _ca ＝-δ _ca (24)

s5, determining displacement generated by micro displacement of each part in the three-crank cycloidal pin gear speed reducer at the meshing position of the shaft or the bearing and the gear teeth; the method comprises the following steps:

when the displacement generated by the micro displacement of each part in the three-crank cycloidal pin gear speed reducer at the contact position is analyzed, the pin gear housing is assumed to be fixed, the planet gears are fixedly connected with the crank shaft, the linear displacement directions of the sun gear, the planet gears with the crank shaft, the cycloidal gears and the planet carrier are the same as the positive direction of the coordinate axes of a static coordinate system or a dynamic coordinate system, and the angular displacement directions are the same as the design planning directions;

1) The micro-displacement of the sun gear generates displacement at the meshing position of the gear teeth and the supporting position of the sun gear shaft;

if the micro-displacement of the sun gear is x _s 、y _s 、θ _sa -θ _s The displacement at the meshing of the teeth is:

s _i ＝x _s cosA _i +y _s sinA _i +R _bs (θ _sa -θ _s ) (25)

wherein R is _bs -represents the base radius (mm) of the sun wheel;

the displacements generated in the x and y directions at the shaft support are:

s _x ＝x _s (26)

s _y ＝y _s (27)

2) The micro-displacement of the planet wheel and the crank shaft generates displacement at the meshing position of the gear teeth, the supporting position of the cycloidal gear roller bearing and the supporting position of the planet carrier tapered roller bearing;

let the micro displacement of the planet i (i=1, 2, 3) be x _pi 、y _pi And theta _pi -θ _p The displacement at the meshing of the teeth is:

s _pi ＝-x _pi cosA _i -y _pi sinA _i -R _bp (θ _pi -θ _p ) (28)

wherein R is _bp -a base radius (mm) of the planet wheel;

θ _pi -the actual self-rotation angle (°) of the planet wheel.

The displacements generated in the x and y directions at the cycloidal gear roller bearing support are as follows:

s _jix ＝x _pi -e(θ _pi -θ _p )sin(θ _p +ψ _j ) (29)

s _jiy ＝y _pi -e(θ _pi -θ _p )cos(θ _p +ψ _j ) (30)

where e-the eccentricity (mm) of the crankshaft.

The displacement generated in the x and y directions at the support position of the conical roller bearing of the planet carrier is as follows:

s _ix ＝x _pi (31)

s _iy ＝y _pi (32)

3) The micro displacement of the cycloidal gear generates displacement at the gear tooth meshing position and the crank shaft roller bearing supporting position;

let the micro-displacement of cycloid wheel j (j=1, 2) be η _dj 、θ _dj -θ _c And theta _Oj -θ _p Wherein eta _dj Represents the linear micro displacement of the cycloidal gear, theta _dj -θ _c Represents the rotation angle micro-displacement of the cycloid wheel, theta _Oj -θ _p Representing the male angular micro-displacement of the cycloidal gears;

the micro-displacement of the cycloidal gear j (j=1, 2) produces a displacement at its tooth engagement of:

s _jk ＝η _dj cosα _jk -R _d (θ _dj -θ _c )sinα _jk +e(θ _Oj -θ _p )sinα _jk (33)

wherein R is _d -pitch radius (mm) of the cycloid wheel;

the displacements generated in the x and y directions at the crank shaft roller bearing support are respectively:

s _cjix ＝R _dc (θ _dj -θ _c )sin(θ _c +φ _i )+e(θ _Oj -θ _p )sin(θ _p +ψ _j )-η _dj cos(θ _p +ψ _j ) (34)

s _cjiy ＝-R _dc (θ _dj -θ _c )cos(θ _c +φ _i )+η _dj sin(θ _p +ψ _j )+e(θ _Oj -θ _p )cos(θ _p +ψ _j ) (35)

wherein R is _dc -the centre distance (mm) between the sun wheel and the planet wheel;

4) The micro-displacement of the planet carrier is generated at the bearing position of the crank shaft tapered roller bearing and the bearing position of the needle gear shell angular contact ball bearing;

let three micro-displacements of the planet carrier be x _ca 、y _ca And theta _ca -θ _c The displacements generated in the x and y directions of the crankshaft tapered roller bearing support are respectively as follows:

s _bix ＝-x _ca +R _dc (θ _ca -θ _c )sin(θ _c +φ _i ) (36)

s _biy ＝-y _ca -R _dc (θ _ca -θ _c )cos(θ _c +φ _i ) (37)

the displacements (refer to fig. 23) generated in the x and y directions at the needle gear housing angular contact ball bearing support are respectively:

s _cax ＝x _ca (38)

s _cay ＝y _ca (39)

s6, determining acting forces of all parts at a bearing position of a shaft or a bearing and a gear tooth meshing position in the three-crank cycloidal pin gear speed reducer;

according to the steps S4 and S5, the determined manufacturing errors, gaps and micro-displacements of each part generate displacements at the positions of the support and the gear teeth, and simultaneously, the theoretical load transmitted by each part in an ideal state is considered, so that the acting force of each part at the positions of the shaft, the bearing and the gear teeth is obtained. If the acting force is positive, the contact force between two objects is pressed, otherwise pulled

1) The acting forces of the sun gear in the x and y directions at the supporting position are F respectively _sx (N)、F _sy (N) as shown in formulas (40), (41);

F _sx ＝k _s (s _x -e _sx )＝k _s (x _s -e _sx ) (40)

F _sy ＝k _s (s _y -e _sy )＝k _s (y _s -e _sy ) (41)

2) The tooth meshing force between the sun gear and the planet wheel i (i=1, 2, 3) is F _i (N) as shown in formula (42);

f in the formula _i0 -representing the theoretical tooth meshing force (N) between the sun and the planet;

if in equation (42) []If the internal value is less than or equal to 0, the acting force is represented as a pulling force, which indicates that a gap exists between the sun gear and the planet gear without contact, and F is taken at the moment _i Is zero;

3) Forces in x and y directions at the contact point of the cycloid wheel j (j=1, 2) and the crank shaft i (i=1, 2, 3) are respectively F _jix (N)、F _jiy (N) as shown in formulas (43), (44);

f in the formula _jix0 When representing an ideal state, cycloid wheel j (j=1, 2) and crank shaft i (i=1, 2, 3)

A load (N) applied to the roller bearing in the x direction;

f in the formula _jiy0 When representing an ideal state, cycloid wheel j (j=1, 2) and crank shaft i (i=1, 2, 3)

A load (N) applied to the roller bearing in the y direction;

the + -numbers in the formulas (43) and (44) are determined according to the deformation of the contact part of the cycloid gear and the crank shaft;

4) Cycloidal gear j (j=1, 2) teeth and kth (k=1, 2, …, Z) _r ) The acting force between the needle teeth is F _jk (N) as shown in formula (45).

F in the formula _jk0 -representing the tooth meshing force (N) of the cycloid gear j (j=1, 2) in an ideal state;

if F _jk If the gear tooth meshing force is less than or equal to 0, the gear tooth meshing force is a pulling force, which indicates that gaps exist between the cycloidal gear teeth and the pin teeth without contact, and F is taken _ijk ＝0；

5) Forces in x and y directions at the contact of the planet carrier and the crankshaft i (i=1, 2, 3) are respectively F _cix (N)、F _ciy (N) as shown in formulas (46), (47);

f in the formula _ci0 -the force (N) between the planet carrier and the crankshaft i (i=1, 2, 3) in ideal conditions.

The + -sign in the formulas (46), (47) is determined according to the magnitude of the deformation of the contact portion of the planet carrier and the crank shaft.

6) Planet carrier and needle gear shellThe acting forces in the x and y directions at the support are F _cax (N)、F _cay (N) as shown in formulas (48), (49).

F _cax ＝k _ca (s _cax -e _cx ±e _ca )＝k _ca (x _ca -e _cx ±e _ca ) (48)

F _cay ＝k _ca (s _cay -e _cy ±e _ca )＝k _ca (y _ca -e _cy ±e _ca ) (49)

The + -sign in the formulas (48), (49) is determined according to the deformation of the planet carrier and the needle gear shell.

S7, establishing a mathematical model for calculating dynamic transmission errors of the three-crank cycloidal pin gear speed reducer;

by analyzing the stress condition of each part in the three-crank cycloidal pin gear speed reducer at any position, a mathematical model of the dynamic transmission error of the system is established according to the D' Alemmbert principle; in the establishment of the mathematical model, the Coriolis acceleration of the cycloid gear, the planet gear and the crankshaft is considered, and the frictional resistance in the transmission system is ignored. The mathematical model of the dynamic transmission error of the three-crank cycloidal pin gear speed reducer is obtained by arrangement as follows:

M is in _s -mass of sun gear (kg);

m _sp -mass sum of planet and crankshaft (kg);

m _bx -mass of cycloidal gear (kg);

J _op moment of inertia (kg.m) of the planet and crankshaft ² )；

J _oj Moment of inertia of cycloidal gear (kg.m) ² )；

ω _c -the theoretical angular speed (rad/s) of the planet carrier;

ω _p -planet wheel rotation theoretical angleSpeed (rad/s).

The mathematical model can be organized into the following matrix form:

m, C, K in the formula, a mass array, a damping array and a rigidity array are respectively adopted, and the matrix orders are 21 multiplied by 21;

x-displacement vector, where x= (X) _s ,y _s ,θ _sa ,x _p1 ,y _p1 ,θ _p1 ,x _p2 ,y _p2 ,θ _p2 ,x _p3 ,y _p3 ,

θ _p3 ,η _d1 ,θ _o1 ,θ _d1 ,η _d2 ,θ _o2 ,θ _d2 ,x _ca ,y _ca ,θ _ca ) ^T ；

Q-generalized force vector.

Due to the stiffness coefficient k in the transmission system _ji 、k _bi 、k _ca 、k _jk The mathematical model of the dynamic transmission error of the three-crank cycloidal pin gear speed reducer is a function of displacement, and therefore, the established mathematical model belongs to a nonlinear dynamics model;

s8, solving a mathematical model of the dynamic transmission error of the established three-crank cycloidal pin gear speed reducer by adopting a nonlinear Newmark method and a direct integration method of a Wilson-theta method; the actual rotation angle theta of the output shaft of the speed reducer at any moment can be determined by programming and utilizing a computer to carry out numerical calculation _c Dynamic transmission error delta theta _c 。

Preferably, the error measuring instrument adopted in the step S2 has the functions of intelligent mode identification, automatic scanning and the like, and has the characteristics of high measuring precision, high sensitivity and high efficiency; the main technical parameters are as follows: the minimum reading unit is less than or equal to 0.1um, the measurement precision is not lower than 1.2+4L/1000 (um) (L is the length of the measured workpiece, the dimension is mm), and the plane dimension of the workbench is not smaller than 450mm multiplied by 400mm.

Preferably, in step S2, a high-precision three-coordinate measuring apparatus or other part error measuring apparatus is used; the calculation method adopts the basic theory and method of least square theory, mathematical transformation and filtering theory and optimization method, and the key errors of the parts are determined by the computer processing technology or method of the measured data.

The beneficial effects of the invention are as follows: firstly, defining key errors of all parts of a speed reducer, measuring and calculating to determine the key errors, then establishing a dynamic transmission error mechanical model, calculating displacement generated at a shaft or bearing support position and a gear tooth meshing position according to the error mechanical model, further determining displacement generated at the shaft or bearing and the gear tooth meshing position by micro-displacement of all parts, and determining acting force of all parts at all part contact positions according to the errors and the displacements; establishing a dynamic transmission error mathematical model of the speed reducer; the actual rotation angle theta of the output shaft of the speed reducer at any moment can be determined through the numerical calculation of a computer _c Dynamic transmission error delta theta _c According to the method, various errors, non-linear factors such as contact gaps, part contact deformation and inertial load of rotating parts are comprehensively considered, the situation that the speed reducer is in a live state is more similar to the situation that the transmission error is obtained is more similar to the actual situation, and the production and assembly processes of parts of the speed reducer can be guided more accurately.

Drawings

FIG. 1 is a waveform schematic diagram of a transmission error of a three-crank cycloidal pin gear speed reducer for a robot;

FIG. 2 is a schematic diagram of a three crank cycloidal pin gear speed reducer transmission system for a robot;

FIG. 3 is a disassembled view of parts of a three crank cycloidal pin gear speed reducer for a robot;

FIG. 4 is a schematic view of a sun gear shaft therein;

FIG. 5 is a schematic view of a planetary gear therein;

FIG. 6 is a schematic illustration of a three crank cycloidal gear;

FIG. 7 is a schematic view of an angular contact ball bearing at the planet carrier and the needle gear housing;

FIG. 8 is a schematic view of a needle gear housing therein;

FIG. 9 is a schematic view of a crankshaft therein;

FIGS. 10-11 are schematic diagrams of a planet carrier therein;

FIG. 12 is a schematic view of a pin tooth;

FIG. 13 is a schematic view of a cycloidal gear and roller bearing at a crankshaft;

FIG. 14 is a schematic view of tapered roller bearings at the crank shaft and planet carrier;

FIG. 15 is a schematic diagram of a mechanical model of the dynamic transmission error of the three crank cycloidal pin gear speed reducer;

FIG. 16 is a schematic illustration of base circle eccentricity errors of a sun gear and an ith planet;

FIG. 17 is a schematic illustration of assembly errors of a sun gear;

FIG. 18 is a schematic illustration of the eccentricity error of a cycloid crank shaft bore;

FIG. 19 is a schematic illustration of the tooth socket deviation and pitch deviation of a cycloid gear;

FIG. 20 is a schematic illustration of tooth slot deviation and pitch deviation of the center of a tooth slot on a pin housing;

FIG. 21 is a schematic illustration of the eccentric error of the eccentric cam on the crankshaft;

FIG. 22 is a schematic illustration of the eccentricity of a planet carrier crank shaft bore;

FIG. 23 is a schematic illustration of the journal eccentricity error of a planet carrier;

FIG. 24 is a schematic view of cycloidal tooth meshing clearances between pin and tooth slots;

Detailed Description

Embodiment one: referring to fig. 1-24, a dynamic transmission accuracy value calculation method of a cycloidal pin gear speed reducer for a robot comprises the following steps:

the method specifically comprises the following parameters: sun gear tooth number Z _s Tooth number Z of planet wheel _p Cycloidal gear tooth number Z _b Number of teeth Z of needle wheel _r Involute gear modulus m (mm), involute gear pressure angle α (°), sun gear tooth displacement coefficient x _cs Line (row)Star wheel tooth deflection coefficient x _cp Planet gear tooth width B _z (mm), cycloidal gear tooth width B _c (mm), crank shaft eccentricity e (mm), pin tooth pin radius r _d (mm), theoretical radius r of pin or pin housing tooth space distribution circle _z (mm), 2 cycloidal gears, 3 planetary gears, 3 crankshafts; as shown in fig. 2 to 3.

the key errors of the parts are defined as follows:

1) Definition of critical error of sun gear

Referring to fig. 4, the key errors of the sun gear are: base circle eccentricity error of sun gear (E _s ,β _s ) The assembly error of the sun gear is (A) _s ,γ _s ) The method comprises the steps of carrying out a first treatment on the surface of the Wherein: e (E) _s 、A _s The magnitude, dimension (um) of the base circle eccentricity error and the assembly error are respectively shown; beta _s 、γ _s The directions of the base circle eccentric error and the assembly error are respectively represented, and the dimension (degree) is shown;

2) Critical error definition for planet gears

Referring to fig. 5, the key errors of the planet are: the base circle eccentric errors of the 3 planet wheels are (E) _pi ,β _pi ) (i=1, 2, 3); wherein: e (E) _pi (i=1, 2, 3) represents the base circle eccentricity error size, dimension (um) of the planet wheels 1,2,3, respectively; beta _pi (i=1, 2, 3) represents the base circle eccentric error direction of the planetary gears 1,2,3, respectively, dimension (°;

3) Definition of key errors for a three crank cycloidal gear

Referring to fig. 6, the key errors of the three crank cycloidal gear are: eccentric errors (E) of crank shaft holes 1,2,3 in the cycloid gears 1,2 _hji ,β _hji ) (j=1, 2; i=1, 2, 3); tooth space deviation R of gear teeth on cycloidal gears 1 and 2 _jk (j=1, 2), pitch deviation P _jk (j=1, 2); wherein: e (E) _hji Representation yeastEccentric error of the shaft hole of the handle, dimension (um); beta _hji Representing the eccentric error direction and dimension (°) of the crank shaft hole; r is R _jk 、P _jk Respectively representing the tooth socket deviation and the tooth pitch deviation of cycloid gear teeth, and dimension (um);

4) Definition of critical errors of pin gear shells

Referring to fig. 8, the key errors of the pin housing are: z is Z _r Average radius error delta of individual pin tooth slots _cr (um) on needle gear case Z _r Tooth space deviation (abbreviated as pin tooth space deviation) R of circle centers of individual tooth spaces _k (k＝1,2,…,Z _r ) Pitch deviation (abbreviated as pin wheel pitch deviation) P _k (k＝1,2,…,Z _r ) The method comprises the steps of carrying out a first treatment on the surface of the Wherein: r is R _k 、P _k The tooth space deviation and the tooth pitch deviation of the kth needle tooth are respectively represented, and the dimension (um) is shown;

5) Definition of key errors of a crankshaft

Referring to fig. 9, the key errors of the crank shaft are: eccentric errors (E) of the eccentric cams 1,2 on the crankshafts 1,2,3 _cji ,β _cji ) (j=1, 2; i=1, 2, 3); wherein: e (E) _cji The magnitude and dimension (um) of the eccentric error of the eccentric cam are shown; beta _cji Representing the eccentric error direction, dimension (°) of the eccentric cam;

6) Definition of critical errors for a planet carrier

Referring to fig. 10-11, the key errors of the planet carrier are: eccentric errors (E) of the crank shaft holes 1,2,3 on the carrier _cai ,β _cai ) (i=1, 2, 3), the eccentricity error (a _c ,γ _c ) The method comprises the steps of carrying out a first treatment on the surface of the Wherein: e (E) _cai 、A _c The eccentric error of the crank shaft hole and the journal is shown, and the dimension (um) is shown; beta _cai 、γ _c Representing the eccentric error direction of the crank shaft hole and the shaft neck, and dimension (°);

7) Definition of key errors for pin teeth

Referring to fig. 12, the key errors of the pin are: z is Z _r Average diameter error delta of individual pin teeth _zr Dimension (um);

8) Definition of critical errors for bearings

In the three-crank cycloidal pin gear speed reducer, the bearings between the cycloidal gears 1,2 and the crankshafts 1,2,3 are roller bearings, see fig. 13, and the bearing clearances are delta respectively _bji (j=1, 2; i=1, 2, 3), dimension (um);

the bearings between the planet carrier and the crankshafts 1,2,3 are tapered roller bearings, see fig. 14, with bearing clearances delta respectively _xi (i=1, 2, 3), dimension (um);

the bearing between the planet carrier and the needle gear shell adopts an angular contact ball bearing, see figure 7, and the bearing clearance is delta _ca Dimension (um);

1) Referring to fig. 15, in the mechanical model, the contact points of the bearing or the shaft support, the gear tooth engagement and the like among the parts are equivalent by springs, and the physical quantities of the rigidity of the bearing or the shaft and the rigidity of the gear tooth engagement are described; gear tooth engagement stiffness k between sun gear and planet wheel i (i=1, 2, 3) _i (N/um) (i=1, 2, 3), bearing support rigidity k between crank shaft i (i=1, 2, 3) and cycloid gear j (j=1, 2) shaft hole _ji (N/um) (j=1, 2; i=1, 2, 3), the bearing support rigidity k between the crank shaft i (i=1, 2, 3) and the carrier shaft hole _bi (N/um) (i=1, 2, 3), bearing support stiffness k between planet carrier and needle gear housing _ca (N/um), the meshing stiffness k between the cycloidal gear and the needle tooth _jk (N/um)(k＝1,2,…,Z _r ) Bearing stiffness k of sun gear shaft _s (N/um); the determination method of the rigidity comprises the following steps: k (k) _i Determining by adopting bending deformation of gear teeth; k (k) _s Calculating the bending deformation of the sun gear shaft; and k is _ji 、k _bi 、k _ca 、k _jk The contact stiffness is determined using the Palmgren formula.

The Palmgren formula is:

wherein,

middle v ₁ ,ν ₂ Poisson ratio of the elastomer 1,2;

E ₁ ,E ₂ -elastic modulus (MPa) of the elastomer 1,2;

delta-the amount of contact deformation (mm) between two elastomers;

l-the contact length (mm) between two elastomers;

f-load (N) between the two elastomers.

If use k ₀ Indicating the contact stiffness (N/mm) between two elastomers

F＝k ₀ δ (2)

3) In the mechanical model, each spring is arranged at the positive contact position or gear tooth engagement position of each part coordinate axis, andsetting the tension and compression of the spring to be positive and negative; phi (phi) _i Representing the relative position of the crankshaft holes on the cycloidal gears (or planet carrier), taking phi _i ＝2π(i-1)/3(i＝1,2,3)，ψ _j Represents the theoretical centroid O of 2 cycloidal gears _j Is taken as ψ _j ＝(j-1)π(j＝1,2)；

S4, determining the displacement of the transmission system at the bearing position of the shaft or the bearing and the meshing position of the gear teeth according to the manufacturing errors and gaps of all parts in the three-crank cycloidal pin gear speed reducer determined in the S2 and combining the mechanical model established in the S3, wherein the displacement is shown in FIG. 15; the method comprises the following steps:

1) And determining the displacement of the base circle eccentric error and the assembly error of the sun gear at the meshing position of the gear teeth and the shaft bearing position of the sun gear. Fig. 16 is a diagram showing the base circle eccentricity error of the sun gear and the ith planetary gear, and fig. 17 is a diagram showing the assembly error of the sun gear.

e _si ＝E _s cos(θ _s +β _s -A _i ) (4)

in which A _i ＝θ _c +φ _i +π/2-α′，θ _c ＝θ _s /i _z

α' —the angle of engagement of the sun and planet (°);

θ _c -theoretical rotation angle (°) of the planet carrier.

e _sx ＝A _s cosγ _s (5)

e _sy ＝A _s sinγ _s (6)

2) Determining the displacement of the base circle eccentricity error of the planet at the meshing of its teeth, as shown in fig. 16; base circle eccentric error of 3 planet wheels (E _pi ,β _pi ) (i=1, 2, 3) the displacement at the tooth engagement is:

e _pi ＝-E _pi cos(β _pi -θ _p -A _i )(i＝1,2,3) (7)

in theta _p -theoretical self-rotation angle (°) of the planet;

3) Determining the displacement generated by the eccentric error of the crank shaft hole on the cycloid wheel at the bearing support position of the cycloid wheel, and the displacement generated by the tooth socket deviation and the tooth pitch deviation at the gear tooth meshing position; FIG. 18 is a schematic illustration of the eccentricity of a cycloid crank shaft bore;

definition wherein O _ji Represents the center of a crank shaft hole i (i=1, 2, 3) on a cycloid wheel j (j=1, 2), R _dc The radius of the circumference of the crank shaft hole on the cycloid wheel is represented.

e _hxji ＝-E _hji cos(θ _c +φ _i +β _hji ) (8)

e _hyji ＝-E _hji sin(θ _c +φ _i +β _hji ) (9)

Fig. 19 shows the tooth space deviation R of the cycloid gear _jk Deviation of pitch P _jk Schematic diagram, O represents the center of the pinwheel, O _j Represents the center of the cycloid gear j (j=1, 2), and P represents the node between the cycloid gear j and the pin gear;

FIG. 20 is a tooth space deviation R of the center of the tooth space on the pin housing (abbreviated as pin wheel tooth space deviation) _k Pitch deviation (abbreviated as pin wheel tooth space deviation) P _k Schematic diagram, wherein: o represents the center of the pinwheel, O _j The center of the cycloid gear j (j=1, 2) is represented, and P represents the node between the cycloid gear j and the pin gear.

e _Rk ＝-R _k cos(α _jk -φ _jk ) (12)

FIG. 21 is a schematic diagram of the eccentric error of eccentric cam j, O of crankshaft i _ji Represents the center of a crank shaft hole i (i=1, 2, 3) on a cycloid wheel j (j=1, 2), O _pj The center of rotation of the crankshaft corresponding to the cycloid gear j (j=1, 2) on the crankshaft i (i=1, 2, 3);

e _cxji ＝E _cji cos(θ _p +ψ _j +β _cji ) (14)

e _cyji ＝-E _cji sin(θ _p +ψ _j +β _cji ) (15)

FIG. 22 is a schematic diagram of the eccentricity error of the planet carrier crank shaft bore i, where O _ci Represents the center of a crank shaft hole i (i=1, 2, 3) on the planet carrier, T _out Representing the load torque acting on the planet carrier;

e _caxi ＝-E _cai cos(θ _c +φ _i +β _cai ) (16)

e _cayi ＝-E _cai sin(θ _c +φ _i +β _cai ) (17)

if the eccentricity of the journal of the planet carrier is assumed to be (A) _c ,γ _c ) The displacements in the x and y directions of the support are respectively:

e _cx ＝A _c cosγ _c (18)

e _cy ＝A _c sinγ _c (19)

FIG. 24 is a schematic view of the gap formed by the diameter error of the pin, the tooth space radius error on the pin housing at the cycloidal tooth mesh, ifAverage diameter error delta of pin _zr (um), average radius error of pin tooth slot delta _cr (um) then its resulting gap delta at the cycloidal tooth engagement _jk And the displacement e produced thereby _jk The method comprises the following steps of:

δ _jk ＝δ _zr +δ _cr (20)

e _jk ＝-δ _jk (21)

8) Displacement of the bearing gaps at their support

e _bji ＝-δ _bji (22)

e _xi ＝-δ _xi (23)

e _ca ＝-δ _ca (24)

when analyzing the displacement generated by the micro displacement of each part in the three-crank cycloidal pin gear speed reducer at the contact position, the pin gear shell is assumed to be fixed, the planet gears are fixedly connected with the crank shaft, the linear displacement directions of the sun gear, the planet gears with the crank shaft, the cycloidal gears and the planet carrier are the same as the positive direction of the coordinate axis of the static coordinate system or the dynamic coordinate system, and the angular displacement directions are the same as the design planning directions, namely the directions shown in fig. 15;

referring to FIG. 16, if the micro-displacement of the sun gear is x _s 、y _s 、θ _sa -θ _s The displacement at the meshing of the teeth is:

s _i ＝x _s cosA _i +y _s sinA _i +R _bs (θ _sa -θ _s ) (25)

wherein R is _bs -represents the base radius (mm) of the sun wheel;

the displacements generated in the x and y directions at the shaft support are:

s _x ＝x _s (26)

s _y ＝y _s (27)

referring to fig. 16, if the micro displacement of the planetary wheel i (i=1, 2, 3) is x _pi 、y _pi And theta _pi -θ _p The displacement at the meshing of the teeth is:

s _pi ＝-x _pi cosA _i -y _pi sinA _i -R _bp (θ _pi -θ _p ) (28)

wherein R is _bp -a base radius (mm) of the planet wheel;

θ _pi -the actual self-rotation angle (°) of the planet wheel.

s _jix ＝x _pi -e(θ _pi -θ _p )sin(θ _p +ψ _j ) (29)

s _jiy ＝y _pi -e(θ _pi -θ _p )cos(θ _p +ψ _j ) (30)

where e-the eccentricity (mm) of the crankshaft.

s _ix ＝x _pi (31)

s _iy ＝y _pi (32)

let the micro-displacements of the cycloid wheels j (j=1, 2) be η respectively with reference to fig. 15 _dj 、θ _dj -θ _c And theta _Oj -θ _p Wherein eta _dj Represents the linear micro displacement of the cycloidal gear, theta _dj -θ _c Represents the rotation angle micro-displacement of the cycloid wheel, theta _Oj -θ _p Representing the male angular micro-displacement of the cycloidal gears;

wherein R is _d -pitch radius (mm) of the cycloid wheel;

wherein R is _dc Sun gear and travellingCenter distance (mm) between star wheels;

referring to fig. 22, let three micro-displacements of the planet carrier be x _ca 、y _ca And theta _ca -θ _c The displacements generated in the x and y directions of the crankshaft tapered roller bearing support are respectively as follows:

s _bix ＝-x _ca +R _dc (θ _ca -θ _c )sin(θ _c +φ _i ) (36)

s _biy ＝-y _ca -R _dc (θ _ca -θ _c )cos(θ _c +φ _i ) (37)

s _cax ＝x _ca (38)

s _cay ＝y _ca (39)

F _sx ＝k _s (s _x -e _sx )＝k _s (x _s -e _sx ) (40)

F _sy ＝k _s (s _y -e _sy )＝k _s (y _s -e _sy ) (41)

A load (N) applied to the roller bearing in the x direction;

A load (N) applied to the roller bearing in the y direction;

6) The acting forces in the x and y directions of the supporting positions of the planet carrier and the needle gear shell are F respectively _cax (N)、F _cay (N) as shown in formulas (48), (49).

F _cax ＝k _ca (s _cax -e _cx ±e _ca )＝k _ca (x _ca -e _cx ±e _ca ) (48)

F _cay ＝k _ca (s _cay -e _cy ±e _ca )＝k _ca (y _ca -e _cy ±e _ca ) (49)

M is in _s -mass of sun gear (kg);

m _sp -mass sum of planet and crankshaft (kg);

m _bx -mass of cycloidal gear (kg);

J _op moment of inertia (kg.m) of the planet and crankshaft ² )；

J _oj Moment of inertia of cycloidal gear (kg.m) ² )；

ω _c -the theoretical angular speed (rad/s) of the planet carrier;

ω _p -the theoretical angular rotation speed (rad/s) of the planet wheel.

The mathematical model can be organized into the following matrix form:

Q-generalized force vector.

s8, solving a mathematical model of the dynamic transmission error of the established three-crank cycloidal pin gear speed reducer by adopting a nonlinear Newmark method and a direct integration method of a Wilson-theta method; the actual rotation angle theta of the output shaft of the speed reducer at any moment can be determined by programming and utilizing a computer to carry out numerical calculation _c Dynamic transmission error delta theta _c As shown in fig. 1.

The error measuring instrument adopted in the step S2 has the functions of intelligent mode identification, automatic scanning and the like, and has the characteristics of high measuring precision, high sensitivity and high efficiency; the main technical parameters are as follows: the minimum reading unit is less than or equal to 0.1um, the measurement precision is not lower than 1.2+4L/1000 (um) (L is the length of the measured workpiece, the dimension is mm), and the plane dimension of the workbench is not smaller than 450mm multiplied by 400mm.

In the step S2, a high-precision three-coordinate measuring instrument or other part error measuring instruments are adopted; the calculation method adopts the basic theory and method of least square theory, mathematical transformation and filtering theory and optimization method, and the key errors of the parts are determined by the computer processing technology or method of the measured data.

The method comprehensively considers nonlinear factors such as part machining errors, installation errors, gear tooth meshing gaps, bearing gaps, part contact deformation, inertial load of rotating parts and the like, and provides a dynamic transmission error numerical calculation method. The method not only can simulate and calculate transmission errors, namely transmission precision, but also can be particularly used for simulating and calculating the speed reducer which runs at high speed or with load, because the inertia wheels of the parts of the speed reducer and the force generated by torque between input and output can all influence the movement contact clearance and deformation of the parts, compared with simple static simulation, the method combines engineering practice to simulate, and is more suitable for guiding actual production.

Claims

1. A dynamic transmission precision numerical calculation method of cycloidal pin gear speed reducer for robot comprises the following steps:

the method specifically comprises the following parameters: sun gear tooth number Z _s Tooth number Z of planet wheel _p Cycloidal gear tooth number Z _b Number of teeth Z of needle wheel _r Involute gear modulus m, dimension mm, involute gear pressure angle alpha, dimension and sun gear tooth deflection coefficient x _cs Shift coefficient x of planet gear teeth _cp Planet gear tooth width B _z Dimension mm, cycloidal gear tooth width B _c Dimension mm, crank shaft eccentricity e, dimension mm, pin radius r _d Theoretical radius r of dimension mm, pin or pin housing tooth space distribution circle _z Dimension mm, 2 cycloid gears, 3 planetary gears and 3 crankshafts;

s2, defining key errors of a sun gear, 3 planet gears, 2 cycloid gears, 3 crankshafts, a planet carrier, a pin gear pin, a pin gear shell and a roller bearing in the speed reducer, measuring the key errors through a high-precision measuring instrument, and determining the key errors according to calculation of a measuring result; the key errors of each part are defined as follows:

1) Definition of critical error of sun gear

The key errors of the sun gear are as follows: base circle eccentricity error of sun gear (E _s ,β _s ) The assembly error of the sun gear is (A) _s ,γ _s ) The method comprises the steps of carrying out a first treatment on the surface of the Wherein: e (E) _s 、A _s The sizes of the base circle eccentric error and the assembly error are respectively shown, and the dimension um; beta _s 、γ _s Respectively represent the base circle eccentric error and the assembly errorDirection, dimension degree;

2) Critical error definition for planet gears

The key errors of the planet wheel are as follows: the base circle eccentric errors of the 3 planet wheels are (E) _pi ,β _pi ) I=1, 2,3; wherein: e (E) _pi I=1, 2,3 respectively represent the base circle eccentric error size, dimension um of the planet wheels 1,2,3; beta _pi I=1, 2,3 represents the base circle eccentric error direction of the planet wheels 1,2,3, respectively, dimension °;

3) Definition of key errors for a three crank cycloidal gear

The key errors of the three-crank cycloidal gear are as follows: eccentric errors (E) of crank shaft holes 1,2,3 in the cycloid gears 1,2 _hji ,β _hji ) J=1, 2; i=1, 2,3; tooth space deviation R of gear teeth on cycloidal gears 1 and 2 _jk J=1, 2, pitch deviation P _jk J=1, 2; wherein: e (E) _hji The eccentric error of the crank shaft hole is represented by dimension um; beta _hji Representing the eccentric error direction and dimension degree of the crank shaft hole; r is R _jk 、P _jk The tooth space deviation and the tooth pitch deviation of cycloid gear teeth are respectively represented, and the dimension um;

4) Definition of critical errors of pin gear shells

The key errors of the pin gear housing are: z is Z _r Average radius error delta of individual pin tooth slots _cr Dimension um, Z on needle gear shell _r Tooth groove deviation of circle centers of tooth grooves, namely tooth groove deviation R of needle teeth _k ,k＝1,2,…,Z _r Pitch deviation, abbreviated as pin wheel pitch deviation P _k ，k＝1,2,…,Z _r The method comprises the steps of carrying out a first treatment on the surface of the Wherein: r is R _k 、P _k The tooth space deviation and the tooth pitch deviation of the kth needle tooth are respectively represented, and the dimension um;

5) Definition of key errors of a crankshaft

The key errors of the crankshaft are: eccentric errors (E) of the eccentric cams 1,2 on the crankshafts 1,2,3 _cji ,β _cji ) J=1, 2; i=1, 2,3; wherein: e (E) _cji The magnitude of the eccentric error of the eccentric cam is shown, dimension um; beta _cji Representing the eccentric error direction and dimension degree of the eccentric cam;

6) Definition of critical errors for a planet carrier

The key errors of the planet carrier are as follows: eccentric errors (E) of the crank shaft holes 1,2,3 on the carrier _cai ,β _cai ) I=1, 2,3, the bearing-mounted journal eccentricity error (a _c ,γ _c ) The method comprises the steps of carrying out a first treatment on the surface of the Wherein: e (E) _cai 、A _c The eccentric error of the crank shaft hole and the shaft neck is represented, and the dimension um; beta _cai 、γ _c Representing the eccentric error direction of the crank shaft hole and the journal, and the dimension degree;

7) Definition of key errors for pin teeth

The key errors of the pin are: z is Z _r Average diameter error delta of individual pin teeth _zr Dimension um;

8) Definition of critical errors for bearings

In the three-crank cycloidal pin gear speed reducer, the bearings between the cycloidal gears 1 and 2 and the crankshafts 1,2 and 3 adopt roller bearings, and the bearing clearances are delta respectively _bji J=1, 2; i=1, 2,3, dimension um;

the bearings between the planet carrier and the crankshafts 1,2 and 3 adopt tapered roller bearings, and the bearing clearances are delta respectively _xi I=1, 2,3, dimension um;

the bearing between the planet carrier and the needle gear shell adopts an angular contact ball bearing, and the bearing clearance is delta _ca Dimension um;

1) In the mechanical model, the contact positions of the bearing or the shaft support, the gear tooth engagement and the like among the parts are equivalent by springs, and the physical quantity of the rigidity of the bearing or the shaft and the gear tooth engagement is used for describing; gear tooth engagement stiffness k between sun gear and planet gears i, i=1, 2,3 _i Bearing support rigidity k between dimension N/um, i=1, 2,3, crank shaft i, i=1, 2,3 and cycloid gear j, j=1, 2 shaft hole _ji Dimension N/um, j=1, 2; i=1, 2,3, bearing support stiffness k between crankshaft i, i=1, 2,3 and planet carrier shaft bore _bi Dimension N/um, i=1, 2,3, bearing support stiffness k between planet carrier and pin housing _ca N/um, meshing stiffness k between cycloidal gear and needle tooth _jk Dimension N/um, k=1,2,…,Z _r bearing stiffness k of sun gear shaft _s Dimension N/um; the determination method of the rigidity comprises the following steps: k (k) _i Determining by adopting bending deformation of gear teeth; k (k) _s Calculating the bending deformation of the sun gear shaft; and k is _ji 、k _bi 、k _ca 、k _jk The contact stiffness is determined by adopting a Palmgren formula;

the Palmgren formula is:

wherein,

middle v ₁ ,ν ₂ Poisson ratio of the elastomer 1, 2;

E ₁ ,E ₂ elastic modulus of the elastomer 1,2, dimensional MPa;

delta-contact deformation between two elastomers, dimension mm;

l-the contact length between two elastomers, dimension mm;

f, load between two elastic bodies, dimension N;

if use k ₀ Representing the contact stiffness between two elastomers, dimension N/mm, then

F＝k ₀ δ (2)

2) Determining a static and dynamic coordinate system of the mechanical model; the theoretical center O of the sun gear shaft or the planet carrier and the needle gear shell is taken as the originThe section perpendicular to the axis of the speed reducer is a plane static coordinate system xoy; starting from the input end, namely the sun gear end, the cycloid gear number j=1 near the sun gear, and the other cycloid gear number j=2; by theoretical centroid O of cycloidal gear _j As origin, let the eccentric direction of cycloid wheel j, j=1, 2 be eta _j The shaft, assuming the sun gear shaft to rotate anticlockwise, will be perpendicular to eta _j The shaft rotates 90 degrees along the revolution direction of the shaft to be zeta _j Shaft, establishing dynamic coordinate system eta of cycloid wheel j _j O _j ξ _j The method comprises the steps of carrying out a first treatment on the surface of the In the dynamic modeling process, taking the cycloid wheel eta with j=1 _j The axis is consistent with the x-axis direction of the static coordinate system when in the initial position;

3) In the mechanical model, each spring is arranged at the contact position of the positive direction of each part coordinate axis or the meshing position of gear teeth, and the tension of the spring is set to be positive and the compression of the spring is set to be negative; phi (phi) _i Representing the relative position of the crankshaft bores on the cycloidal gears or planet carriers, phi being taken _i ＝2π(i-1)/3，i＝1,2,3，ψ _j Represents the theoretical centroid O of 2 cycloidal gears _j Is taken as ψ _j ＝(j-1)π，j＝1,2；

S4, determining displacement of a transmission system at a shaft or bearing support position and a gear tooth meshing position according to manufacturing errors and gaps of all parts in the three-crank cycloidal pin gear speed reducer determined in the S2 and combining the mechanical model established in the S3; the method comprises the following steps:

1) Determining the displacement generated by the base circle eccentric error and the assembly error of the sun gear at the meshing position of gear teeth and the shaft bearing position of the sun gear respectively;

e _si ＝E _s cos(θ _s +β _s -A _i ) (4)

in which A _i ＝θ _c +φ _i +π/2-α′，θ _c ＝θ _s /i _z

Alpha' — the meshing angle of the sun gear and the planet gear, dimension degree;

θ _c -theoretical rotation angle, dimension °;

e _sx ＝A _s cosγ _s (5)

e _sy ＝A _s sinγ _s (6)

2) Determining the displacement generated by the base circle eccentric error of the planet wheel at the meshing position of the gear teeth; base circle eccentric error of 3 planet wheels (E _pi ,β _pi ) i=1, 2,3 at the tooth engagement:

e _pi ＝-E _pi cos(β _pi -θ _p -A _i ),i＝1,2,3 (7)

in theta _p -theoretical self-rotation angle, dimension degree of the planet wheel;

definition O _ji Represents the center of the crank shaft holes i, i=1, 2,3 on the cycloid gears j, j=1, 2, R _dc The radius of the circumference of the center of the crank shaft hole on the cycloid wheel is represented;

if the eccentric error of the cycloid gear crank shaft hole is: (E) _hji ,β _hji ) J=1, 2; i=1, 2,3, the displacements thereof in the x, y directions of the static coordinate system at the bearing support are respectively:

e _hxji ＝-E _hji cos(θ _c +φ _i +β _hji ) (8)

e _hyji ＝-E _hji sin(θ _c +φ _i +β _hji ) (9)

defining the tooth space deviation R of cycloidal gears _jk Deviation of pitch P _jk O represents the center of the pinwheel, O _j Represents the center of the cycloid gear j, j=1, 2, and P represents the node of the cycloid gear j and the pin gear;

if the tooth space deviation on the cycloid wheel j is R _jk J=1, 2, pitch deviation P _jk J=1, 2, then the displacements at the tooth mesh are respectively:

e _Rjk ＝R _jk cos(α _jk -φ _djk ) (10)

e _Pjk ＝P _jk sin(α _jk -φ _djk ) (11)

Alpha in the formula _jk The kth needle tooth center is connected with the jth cycloidal gear node and eta _j Positive axis angle, dimension, where k=1, 2, …, Z _r ；

φ _djk The kth needle tooth center and the jth cycloidal gear center O _j Connection line and eta _j Included angle of axial positive direction, dimension degree;

defining the tooth space deviation of the circle center of the tooth space on the needle gear shell, namely the tooth space deviation R of the needle gear _k Tooth space deviation, pin wheel tooth space deviation P for short _k Wherein: o represents the center of the pinwheel, O _j Represents the center of the cycloid gear j, j=1, 2, and P represents the node of the cycloid gear j and the pin gear;

if the tooth socket deviation of the pinwheel is R _k ,k＝1,2,…,Z _r The pitch deviation is P _k ,k＝1,2,…,Z _r The displacements at the meshing of the teeth are respectively:

e _Rk ＝-R _k cos(α _jk -φ _jk ) (12)

e _Pk ＝-P _k sin(α _jk -φ _jk ) (13)

phi in _jk The kth needle tooth radial direction and the jth cycloidal gear eta _j Included angle between axial positive direction, dimension degree;

define the eccentric cam j eccentric error, O, of the crankshaft i _ji Represents the center of the crank shaft holes i, i=1, 2,3 on the cycloid gears j, j=1, 2, O _pj The crank shaft rotation center corresponding to the cycloid gears j, j=1, 2 on the crank shafts i, i=1, 2, 3;

If the eccentric error of the eccentric cam j on the crankshaft i is (E) _cji ,β _cji ) J=1, 2; i=1, 2,3, then the displacements in x, y directions are respectively:

e _cxji ＝E _cji cos(θ _p +ψ _j +β _cji ) (14)

e _cyji ＝-E _cji sin(θ _p +ψ _j +β _cji ) (15)

defining the eccentricity error of the planet carrier crank shaft hole i, wherein O _ci Represents the center of a crank shaft hole i, i=1, 2,3 on the planet carrier, T _out Representing the load torque acting on the planet carrier;

if the eccentric error of the crank shaft hole i of the planet carrier is (E) _cai ,β _cai ) I=1, 2,3, then the displacements it produces in the x, y direction at the support are respectively:

e _caxi ＝-E _cai cos(θ _c +φ _i +β _cai ) (16)

e _cayi ＝-E _cai sin(θ _c +φ _i +β _cai ) (17)

e _cx ＝A _c cosγ _c (18)

e _cy ＝A _c sinγ _c (19)

definition O represents a pinThe theoretical distribution circle center of the tooth socket theoretical circle center on the pin gear shell or the distribution circle center delta _jk Representing the meshing gap of cycloidal gear teeth;

if the average diameter of the pin is wrong _zr Average radius error delta of dimension um and pin tooth slot _cr Dimension um, then its gap delta created at the cycloidal tooth engagement _jk And the displacement e produced thereby _jk The method comprises the following steps of:

δ _jk ＝δ _zr +δ _cr (20)

e _jk ＝-δ _jk (21)

8) Displacement of the bearing gaps at their support

If the clearance of the roller bearing at the crank shaft hole i of the cycloid wheel j is delta _bji J=1, 2; i=1, 2,3, then its displacement at the support is:

e _bji ＝-δ _bji (22)

if the clearance of the conical roller bearing of the planet carrier at the crank shaft hole i is delta _xi I=1, 2,3, then its displacement at the support is:

e _xi ＝-δ _xi (23)

e _ca ＝-δ _ca (24)

s _i ＝x _s cos A _i +y _s sin A _i +R _bs (θ _sa -θ _s ) (25)

wherein R is _bs -the base radius of the sun gear, dimension mm;

the displacements generated in the x and y directions at the shaft support are:

s _x ＝x _s (26)

s _y ＝y _s (27)

s _pi ＝-x _pi cos A _i -y _pi sin A _i -R _bp (θ _pi -θ _p ) (28)

wherein R is _bp -the radius of the base circle of the planet wheel, dimension mm;

θ _pi -the actual self-rotation angle of the planet wheel, dimension °;

s _jix ＝x _pi -e(θ _pi -θ _p )sin(θ _p +ψ _j ) (29)

s _jiy ＝y _pi -e(θ _pi -θ _p )cos(θ _p +ψ _j ) (30)

wherein e is the eccentricity of the crankshaft, and the dimension is mm;

s _ix ＝x _pi (31)

s _iy ＝y _pi (32)

let j, j=1, 2 the micro-displacement of cycloid gear be η _dj 、θ _dj -θ _c And theta _Oj -θ _p Wherein eta _dj Represents the linear micro displacement of the cycloidal gear, theta _dj -θ _c Represents the rotation angle micro-displacement of the cycloid wheel, theta _Oj -θ _p Representing the male angular micro-displacement of the cycloidal gears;

the micro-displacement of the cycloidal gear j, j=1, 2 produces a displacement at its tooth engagement of:

Wherein R is _d The pitch radius of the cycloid wheel is of a dimension mm;

wherein R is _dc -center distance between the sun gear and the planet gear, dimension mm;

s _bix ＝-x _ca +R _dc (θ _ca -θ _c )sin(θ _c +φ _i ) (36)

s _biy ＝-y _ca -R _dc (θ _ca -θ _c )cos(θ _c +φ _i ) (37)

the displacements generated in the x and y directions of the needle gear housing angular contact ball bearing support are respectively as follows:

s _cax ＝x _ca (38)

s _cay ＝y _ca (39)

according to the steps S4 and S5, the determined manufacturing errors, gaps and micro-displacements of the parts are generated at the positions of the support and the gear teeth, and meanwhile, the theoretical load transmitted by the parts in an ideal state is considered, so that the acting force of the parts at the positions of the shaft, the bearing and the gear teeth is obtained; if the acting force is positive, the contact force between two objects is pressed, otherwise pulled

1) The acting forces of the sun gear in the x and y directions at the supporting position are F respectively _sx Dimension N, F _sy Dimension N, as shown in formulas (40), (41);

F _sx ＝k _s (s _x -e _sx )＝k _s (x _s -e _sx ) (40)

F _sy ＝k _s (s _y -e _sy )＝k _s (y _s -e _sy ) (41)

2) The meshing force between the sun gear and the planetary gears i, i=1, 2 and 3 is F _i Dimension N, as shown in equation (42);

f in the formula _i0 -representing the theoretical tooth meshing force between the sun gear and the planet gears, dimension N;

3) Forces in x and y directions at contact points of the cycloid wheels j, j=1, 2 and the crank shafts i, i=1, 2 and 3 are respectively F _jix Dimension N, F _jiy Dimension N, as shown in formulas (43), (44);

f in the formula _jix0 When ideal, the cycloid gear j, j=1, 2 and the crank shaft i, i=1, 2,3

The load born by the roller bearing in the x direction, dimension N;

f in the formula _jiy0 When ideal, the cycloid gear j, j=1, 2 and the crank shaft i, i=1, 2,3

The load born by the roller bearing in the y direction, dimension N;

4) Cycloidal gear j, j=1, 2 teeth and kth, k=1, 2, …, Z _r The acting force between the needle teeth is F _jk Dimension N, as shown in equation (45);

f in the formula _jk0 -representing the tooth meshing force of the cycloid gear j, j=1, 2, dimension N, in an ideal state;

5) Forces in x and y directions at contact points of the planet carrier and the crank shafts i, i=1, 2 and 3 are respectively F _cix Dimension N, F _ciy Dimension N, as shown in formulas (46), (47);

f in the formula _ci0 -the force between the planet carrier and the crankshaft i, i=1, 2,3, dimension N, representing the ideal state;

the + -numbers in the formulas (46) and (47) are determined according to the deformation of the contact part of the planet carrier and the crank shaft;

6) The acting forces in the x and y directions of the supporting positions of the planet carrier and the needle gear shell are F respectively _cax Dimension N, F _cay Dimension N, as shown in formulas (48), (49);

F _cax ＝k _ca (s _cax -e _cx ±e _ca )＝k _ca (x _ca -e _cx ±e _ca ) (48)

F _cay ＝k _ca (s _cay -e _cy ±e _ca )＝k _ca (y _ca -e _cy ±e _ca ) (49)

the + -sign in the formulas (48) and (49) is determined according to the deformation of the planet carrier and the needle gear shell support;

by analyzing the stress condition of each part in the three-crank cycloidal pin gear speed reducer at any position, a mathematical model of the dynamic transmission error is established according to the D' Alemmbert principle; when the mathematical model is established, the Coriolis acceleration of the cycloid gear, the planet gear and the crank shaft is considered, the friction resistance in the transmission system is ignored, and the mathematical model of the dynamic transmission error of the three-crank cycloid pin gear speed reducer is obtained by arrangement as follows:

M is in _s -mass, dimension kg of sun gear;

m _sp -mass sum, dimension kg of planet wheel and crankshaft;

m _bx -mass, dimension kg of cycloidal gear;

J _op rotational inertia of planet wheel and crankshaft, dimension kg.m ² ；

J _oj Moment of inertia of cycloidal gear, dimension kg.m ² ；

ω _c -the theoretical angular velocity of the planet carrier, dimension rad/s;

ω _p -the rotation theoretical angular speed of the planet wheel, dimension rad/s;

the mathematical model can be organized into the following matrix form:

x-displacement vector, where x= (X) _s ,y _s ,θ _sa ,x _p1 ,y _p1 ,θ _p1 ,x _p2 ,y _p2 ,θ _p2 ,x _p3 ,y _p3 ,θ _p3 ,η _d1 ,θ _o1 ,θ _d1 ,η _d2 ,θ _o2 ,θ _d2 ,x _ca ,y _ca ,θ _ca ) ^T ；

Q—generalized force vector;

2. The method for calculating the dynamic transmission accuracy value of the cycloidal pin gear speed reducer for the robot according to claim 1, wherein the method comprises the following steps: the error measuring instrument adopted in the step S2 has the functions of intelligent mode identification, automatic scanning and the like, and has the characteristics of high measuring precision, high sensitivity and high efficiency; the main technical parameters are as follows: the minimum reading unit is less than or equal to 0.1um, the measurement precision is not lower than 1.2+4L/1000, the dimension um, L is the length of the measured workpiece, the dimension is mm, and the plane dimension of the workbench is not less than 450mm multiplied by 400mm.

3. The method for calculating the dynamic transmission accuracy value of the cycloidal pin gear speed reducer for the robot according to claim 1, wherein the method comprises the following steps: in the step S2, a high-precision three-coordinate measuring instrument or other part error measuring instruments are adopted; the calculation method adopts the basic theory and method of least square theory, mathematical transformation and filtering theory and optimization method, and the key errors of the parts are determined by the computer processing technology or method of the measured data.