CN111079300B - Straight gear meshing rigidity calculation method considering tooth direction error - Google Patents

Straight gear meshing rigidity calculation method considering tooth direction error Download PDF

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CN111079300B
CN111079300B CN201911334013.1A CN201911334013A CN111079300B CN 111079300 B CN111079300 B CN 111079300B CN 201911334013 A CN201911334013 A CN 201911334013A CN 111079300 B CN111079300 B CN 111079300B
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tooth
teeth
gear
meshing
stiffness
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CN111079300A (en
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王奇斌
孔宪光
怀天澍
徐锟
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Xidian University
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Abstract

The invention provides a straight gear meshing stiffness calculation method considering a tooth direction error, which improves the calculation accuracy of the straight gear meshing stiffness while ensuring the calculation efficiency. The method comprises the following implementation steps: setting parameters of a straight gear pair; calculating the tooth base rigidity of the straight gear pair; acquiring a plurality of slicing teeth of each straight gear in the straight gear pair along the tooth width direction; calculating the total error of the slicing teeth, the total rigidity of each slicing tooth and the coupling rigidity between adjacent slicing teeth in each straight gear; constructing a slice coupling model of a pair of teeth which are meshed with each other in the gear pair; derivation series spring group
Figure DDA0002330459140000012
The relation between the medium deformation amounts; derivation of
Figure DDA0002330459140000011
A formula for balancing the force and deformation of (a); calculating the deformation of the slicing teeth; calculating the meshing force of the gear pair; calculating the time-varying meshing stiffness of the gear pair. The invention considers the axial modification and misalignment errors of the straight gear and the coupling effect between the slicing teeth, has high calculation precision and can be used for the dynamic performance analysis and the optimized design of the straight gear.

Description

Straight gear meshing rigidity calculation method considering tooth direction error
Technical Field
The invention belongs to the technical field of mechanical dynamics, and relates to a straight gear meshing stiffness calculation method considering a tooth direction error, which can be used for dynamic performance analysis and optimization design of a straight gear with a large tooth direction error.
Background
Gears are one of the most widely used rotating machine components, and the performance of a gear system directly determines the performance of the rotating machine. In recent years, high-end numerical control equipment is continuously developing towards high speed, high precision and long service life, and higher requirements are put on the dynamic characteristics of a key part, namely a straight gear. Periodic changes in the meshing stiffness, which are one of the main forms of excitation of the transmission system, are referred to as time-varying meshing stiffness, and directly affect the dynamics of the transmission. When a gear has a tooth direction error, the meshing rigidity of the gear is greatly influenced, so that a method for calculating the meshing rigidity of the spur gear by considering the tooth direction error is necessary.
At present, analytical methods, finite element methods and analytical finite element methods are mainly used for calculating the time-varying meshing stiffness of a gear transmission system. The analytic method applies the principle of material mechanics, has the advantages of high calculation efficiency, convenience in use and the like, is widely applied to calculation of time-varying meshing stiffness of the gear pair, but usually ignores many factors such as tooth direction modification and misalignment, and therefore is influenced in calculation accuracy. The finite element method can simulate an actual tooth profile, including tooth profile modification, machining errors, installation errors and the like, and has higher precision in the time-varying meshing stiffness analysis of the gear pair, however, the finite element method needs repeated modeling of different gears and needs a lot of computing resources. The analytic finite element method is a calculation method combining the analytic method and the finite element method, and can quickly calculate the time-varying meshing stiffness under various tooth shapes, such as tooth profile modification, a thin web tooth base, a tooth base with holes, cracks and the like.
From the published data, for helical gears and straight gears with modified tooth direction and non-centering, the meshing stiffness of the gear pair is usually calculated by using an analytical method and a slicing theory. However, the current method generally considers the slicing teeth to be independent from each other, and neglects the coupling between the slicing teeth. For example, chinese patent application publication No. CN 107153736 a, entitled "a modified gear pair meshing characteristic analysis method considering drum-wise modification", discloses a modified gear pair meshing characteristic analysis method considering drum-wise modification, which is a method for obtaining basic parameters and drum-wise modification parameters of a gear pair, decomposing gear teeth of the gear pair into N independent and uniform thin plate gears along a tooth width direction, and calculating a time-varying meshing stiffness of each thin plate gear pair based on a tooth profile error of a drum-wise modification gear pair by using a gear pair meshing characteristic analysis method considering an influence of prolonged meshing; and establishing a three-dimensional model of the meshing gear pair with the drum-direction modification through three-dimensional drawing simulation software, importing the three-dimensional model into ANSYS software to establish a three-dimensional finite element contact model, and solving time-varying meshing rigidity data in the whole meshing process of the gear. The method considers the influences of nonlinear contact rigidity, finite element correction matrix rigidity and prolonged meshing action, improves the calculation accuracy of the gear meshing rigidity to a certain extent, but does not consider the influence of the coupling action between the sheet gears on the time-varying meshing rigidity and does not consider the condition of non-centering of the axis, so that the calculation accuracy is influenced to a certain extent.
Disclosure of Invention
The invention aims to overcome the defects in the prior art, provides a method for calculating the meshing stiffness of a straight gear by considering a tooth direction error, and aims to improve the calculation accuracy of the meshing stiffness of the straight gear with the tooth direction error while ensuring the calculation efficiency.
In order to achieve the purpose, the technical scheme adopted by the invention comprises the following steps:
(1) setting parameters of a straight gear pair:
the tooth widths of a driving gear p and a driven gear g in a straight gear pair are both B, the modulus is m, and the elastic moduli of the driving gear p and the driven gear g are respectively E p And E g The number of teeth of the driving gear p and the driven gear g is z 1 And z 2 ；
(2) Calculating tooth base rigidity k of straight gear pair tf ：
Calculating the rigidity k of the single-tooth meshing tooth base of the driving gear p in the straight gear pair fp Rigidity k of single tooth meshing base of driven gear g fg And through k fp And k fg Calculating tooth base rigidity k of straight gear pair tf ；
(3) Acquiring a plurality of slicing teeth of each straight gear in the straight gear pair along the tooth width direction:
the driving gear p in the straight gear pair is equivalent to N slicing teeth with the width delta l along the tooth width direction
Figure BDA0002330459120000021
The driven gear g is simultaneously equivalent to N slicing teeth with the width delta l along the tooth width direction
Figure BDA0002330459120000022
Wherein the content of the first and second substances,
Figure BDA0002330459120000023
the ith slice tooth is shown,
Figure BDA0002330459120000024
the ith slice tooth is shown,
Figure BDA0002330459120000031
N≥2；
(4) calculating slicing tooth
Figure BDA0002330459120000032
And
Figure BDA0002330459120000033
total error of each chip tooth, total stiffness of each chip tooth and coupling stiffness between adjacent chip teeth in each spur gear:
(4a) calculating slicing tooth
Figure BDA0002330459120000034
And
Figure BDA0002330459120000035
total error E of i Including slicing teeth
Figure BDA0002330459120000036
Axial modification error e lp 、
Figure BDA0002330459120000037
Axial modification error e lg And
Figure BDA0002330459120000038
and
Figure BDA0002330459120000039
misalignment of e m ，E i ＝e lp +e lg +e m ；
(4b) By slicing teeth
Figure BDA00023304591200000310
Bending stiffness of
Figure BDA00023304591200000311
Shear stiffness
Figure BDA00023304591200000312
Radial compression stiffness
Figure BDA00023304591200000313
And Hertz contact stiffness Δ k h Calculating
Figure BDA00023304591200000314
Total rigidity of
Figure BDA00023304591200000315
By slicing teeth
Figure BDA00023304591200000316
Bending stiffness of
Figure BDA00023304591200000317
Shear stiffness
Figure BDA00023304591200000318
Radial compression stiffness
Figure BDA00023304591200000319
And Hertz contact stiffness Δ k h Calculating
Figure BDA00023304591200000320
Total rigidity of
Figure BDA00023304591200000321
(4c) Calculating slicing tooth
Figure BDA00023304591200000322
And its adjacent slicing teeth
Figure BDA00023304591200000323
Stiffness of coupling therebetween
Figure BDA00023304591200000324
Calculating slicing tooth
Figure BDA00023304591200000325
And its adjacent slicing teeth
Figure BDA00023304591200000326
Rigidity of coupling therebetween
Figure BDA00023304591200000327
Figure BDA00023304591200000328
Figure BDA00023304591200000329
Wherein C c Represents a coupling coefficient;
(5) constructing a section coupling model S of a pair of mutually meshed teeth in the gear pair:
(5a) to slice teeth
Figure BDA00023304591200000330
Simulated as the total stiffness with itself
Figure BDA00023304591200000331
Equal drive gear segment tooth compression spring
Figure BDA00023304591200000332
To obtainSet of N drive gear segment tooth compression springs arranged in parallel
Figure BDA00023304591200000333
To slice teeth
Figure BDA00023304591200000334
And its adjacent slicing teeth
Figure BDA00023304591200000335
The coupling between the two is simulated as the coupling stiffness
Figure BDA00023304591200000336
Equal driving gear coupling compression spring
Figure BDA00023304591200000337
The set consisting of N-1 driving gear coupling compression springs is obtained
Figure BDA00023304591200000338
And will be
Figure BDA00023304591200000339
And
Figure BDA00023304591200000340
by passing
Figure BDA00023304591200000341
Connecting to obtain a slice coupling model of the driving gear p;
(5b) to slice teeth
Figure BDA00023304591200000342
Simulated as the total stiffness with itself
Figure BDA00023304591200000343
Equal driven gear slicing tooth compression spring
Figure BDA0002330459120000041
To obtain a plurality of N parallel arrangementsDriven gear segment tooth compression spring assembly
Figure BDA0002330459120000042
To slice into teeth
Figure BDA0002330459120000043
And its adjacent slicing teeth
Figure BDA0002330459120000044
The coupling between the two is simulated as the coupling stiffness
Figure BDA0002330459120000045
Equal driven gear coupling compression spring
Figure BDA0002330459120000046
The set consisting of N-1 driven gear coupling compression springs is obtained
Figure BDA0002330459120000047
And will be
Figure BDA0002330459120000048
And with
Figure BDA0002330459120000049
By passing
Figure BDA00023304591200000410
Connecting to obtain a slice coupling model of the driven gear g;
(5c) corresponding cutting teeth of the driving gear p and the driven gear g
Figure BDA00023304591200000411
And with
Figure BDA00023304591200000412
Contact between the teeth is simulated as corresponding to the slicing teeth
Figure BDA00023304591200000413
And
Figure BDA00023304591200000414
rigidity ak of contact therebetween h Equal contact compression spring
Figure BDA00023304591200000415
Obtaining a set of N contact compression springs
Figure BDA00023304591200000416
(5d) Coupling slices of the driving gear p into a model
Figure BDA00023304591200000417
In a section coupling model with the driven gear g
Figure BDA00023304591200000418
By passing
Figure BDA00023304591200000419
Are connected to form a series spring group
Figure BDA00023304591200000420
And combining N series spring groups into a series spring group set
Figure BDA00023304591200000421
(5e) Will be provided with
Figure BDA00023304591200000422
And
Figure BDA00023304591200000423
and
Figure BDA00023304591200000424
combining into a slice coupling model of a pair of meshed teeth in the gear pair;
(6) series spring group derivation
Figure BDA00023304591200000425
The relation between the middle deformation amounts is as follows:
(6a) series spring set in hypothetical slice coupling model
Figure BDA00023304591200000426
Under force F i Is deformed by the action of
Figure BDA00023304591200000427
The sum of the deformation of the N series spring groups is the meshing deformation of the gear pair
Figure BDA00023304591200000428
Wherein delta pi Is composed of
Figure BDA00023304591200000429
Amount of deformation of delta gi Is composed of
Figure BDA00023304591200000430
Amount of deformation of delta hi Is composed of
Figure BDA00023304591200000431
The amount of deformation of (a);
(6b) by F i 、δ pi 、δ gi 、δ hi And series spring group
Figure BDA00023304591200000432
The balance equation constructed by the stiffness of each compression spring
Figure BDA00023304591200000433
And
Figure BDA00023304591200000434
series spring group derivation
Figure BDA00023304591200000435
The relation between the middle deformation amounts is as follows:
Figure BDA0002330459120000051
(7) derivation of
Figure BDA0002330459120000052
Force and deformation equation of (1):
(7a) by F i And delta pi And a driving gear segment tooth compression spring of the segment coupling model of the driving gear p
Figure BDA0002330459120000053
Compression spring coupled with driving gear
Figure BDA0002330459120000054
Rigidity of
Figure BDA0002330459120000055
And
Figure BDA0002330459120000056
compression spring for deriving cutting teeth of driving gear
Figure BDA0002330459120000057
Amount of deformation δ pi And F i Obtaining a balance equation of the deformation and force of the N driving gear blade tooth compression springs:
Figure BDA0002330459120000058
(7b) converting the balance equation of the deformation and force of the N drive gear blade tooth compression springs into a matrix form K c X is F, and it is defined as
Figure BDA0002330459120000059
Is a formula for balancing the force and deformation, wherein K c Representing a stiffness matrix, X and F representing a deformation vector and an applied force vector, respectively,
Figure BDA00023304591200000510
X＝[δ p1 ,δ p2 ,…,δ pi ,…,δ pN ] T
F＝[F 1 ,F 2 ,…,F i ,…,F N ] T
[·] T representation matrix [ ·]Transposing;
(8) calculating the time-varying meshing stiffness of the gear pair:
(8a) let F be an external force applied to a section coupling model S of a pair of teeth meshing with each other in a gear pair e Tolerance error of F ε And let δ be 0;
(8b) determine delta and slice teeth
Figure BDA0002330459120000061
And
Figure BDA0002330459120000062
total error E of i The relationship delta > E i If yes, slicing the teeth
Figure BDA0002330459120000063
And slicing teeth
Figure BDA0002330459120000064
Meshing, all the slicing teeth in the meshing state in the driving gear p are combined into a set
Figure BDA0002330459120000065
Otherwise, slicing the teeth
Figure BDA0002330459120000066
And slicing teeth
Figure BDA0002330459120000067
Non-meshing, namely slicing teeth in a non-meshing state in the driving gear p
Figure BDA0002330459120000068
Are combined into sets
Figure BDA0002330459120000069
Wherein, the first and the second end of the pipe are connected with each other,
Figure BDA00023304591200000610
the W-th slicing tooth in the meshing state in the driving gear p, W is the total number of the slicing teeth in the meshing state,
Figure BDA00023304591200000611
the number of the S-th slicing teeth in the driving gear p in the non-meshed state is S, the total number of the slicing teeth in the non-meshed state is S + W, and the sum of S + W is N;
(8c) when slicing teeth
Figure BDA00023304591200000612
And slicing teeth
Figure BDA00023304591200000613
When engaged, the deformation of W slice teeth in the engaged state in the driving gear p is all
Figure BDA00023304591200000614
According to the distance set A p” Section tooth with nearest area and in meshing state
Figure BDA00023304591200000615
The amount of deformation of the teeth is calculated to obtain the slicing teeth in a non-meshed state
Figure BDA00023304591200000616
Amount of deformation delta' ps And will be calculated
Figure BDA00023304591200000617
Amount of deformation δ pi Fill in X ═ delta p1 ,δ p2 ,…,δ pi ,…,δ pN ] T In which delta " ps The calculation formula of (c) is:
Figure BDA00023304591200000618
wherein, gamma is j(j+1) The deformation transfer coefficient between the jth slicing tooth and the jth +1 slicing tooth is obtained;
(8d) calculating the meshing force F of a gear pair m And judging | F e -F m |≤F ε If yes, obtaining the meshing deformation delta of the gear pair and the load distribution of the gear pair, and executing the step (8e), otherwise, enabling the gear pair to have the same load distribution
Figure BDA00023304591200000619
And performing step (8 b);
(8e) and calculating the meshing stiffness k of the gear pair according to the meshing deformation delta of the gear pair, and calculating the meshing stiffness of each meshing position in a meshing period, namely the time-varying meshing stiffness of the gear pair.
Compared with the prior art, the invention has the following advantages:
in the invention, in the process of acquiring the time-varying meshing stiffness of the gear pair by an analytic method and a slicing theory, slicing teeth are calculated
Figure BDA0002330459120000071
And
Figure BDA0002330459120000072
when the total error is larger than the total error, the tooth direction modification error and the misalignment error which have influence on the meshing rigidity of the straight gear are considered; when a slice coupling model is constructed, the coupling effect between the slice teeth and the adjacent slice teeth is equivalent to a spring, the coupling effect between the adjacent slice teeth of the driving gear and the coupling effect between the adjacent slice teeth of the driven gear are fully considered, and compared with the prior art, the calculation efficiency is ensured, and the calculation precision of the time-varying meshing stiffness of the straight gear in the presence of a tooth direction error is effectively improved.
Drawings
FIG. 1 is a flow chart of an implementation of the present invention;
FIG. 2 is a graph illustrating a parameter defining stiffness of a tooth base mesh according to an embodiment of the present invention;
FIG. 3 is a schematic view of a slicer tooth according to an embodiment of the present invention;
FIG. 4 is a diagram illustrating the definition of misalignment error parameters according to an embodiment of the present invention;
FIG. 5 is a schematic view of a slice coupling model of a driving gear p according to an embodiment of the present invention;
FIG. 6 is a schematic view of a slice coupling model of a driven gear g according to an embodiment of the present invention;
FIG. 7 is a schematic view of a sliced coupling model of a pair of intermeshing teeth in a gear pair according to an embodiment of the present invention;
Detailed Description
The invention is described in further detail below with reference to the figures and the specific embodiments.
Referring to fig. 1, the present invention includes the steps of:
step 1) setting parameters of a straight gear pair:
the tooth widths of a driving gear p and a driven gear g in a straight gear pair are both B, the modulus is m, and the elastic moduli of the driving gear p and the driven gear g are respectively E p And E g The number of teeth of the driving gear p and the driven gear g is z 1 And z 2 ；
The parameter values of this embodiment are shown in table 1:
TABLE 1 parameters of spur gear set
Figure BDA0002330459120000073
Figure BDA0002330459120000081
Step 2) calculating the tooth base rigidity k of the straight gear pair tf ：
Calculating the rigidity k of the single-tooth meshing tooth base of the driving gear p in the straight gear pair fp Rigidity k of single tooth meshing base of driven gear g fg And through k fp And k fg Calculating tooth base rigidity k of straight gear pair tf ；
Wherein the active straightRigidity k of single-tooth meshing tooth base of gear p fp Rigidity k of single-tooth meshing tooth base of driven straight gear g fg The calculation formula is as follows:
Figure BDA0002330459120000082
wherein, the parameters needed for calculating the tooth base rigidity of the spur gear pair are shown in fig. 2, u fq Distance of force to tooth root, S fq Is the width of the tooth root,
Figure BDA0002330459120000083
as angle of pressure at the contact point, coefficient L * 、M * 、P * And Q * Can be according to a polynomial function
Figure BDA0002330459120000084
To obtain, X * Represents the coefficient L * 、M * 、P * And Q * Wherein h is fq ＝r fq /r intq ；A i 、B i 、C i 、D i 、E i And F i The subscript q takes the values of p and g, representing the driving gear and the driven gear respectively;
and through k fp And k fg Calculating tooth base rigidity k of straight gear pair tf The calculation formula is as follows:
Figure BDA0002330459120000085
λ p is the tooth base stiffness correction factor, λ, of the driving gear p g Is a tooth base stiffness correction coefficient of the driven gear g.
TABLE 2A i ,B i ,C i ,D i ,E i ,F i Coefficient value of
Figure BDA0002330459120000091
Step 3) obtaining a plurality of slicing teeth of each straight gear in the straight gear pair along the tooth width direction:
the driving gear p in the straight gear pair is equivalent to N slicing teeth with the width delta l along the tooth width direction
Figure BDA00023304591200000920
The driven gear g is simultaneously equivalent to N slice teeth with the width delta l along the tooth width direction
Figure BDA0002330459120000092
Wherein the content of the first and second substances,
Figure BDA0002330459120000093
the ith slice tooth is shown and,
Figure BDA0002330459120000094
the ith slice tooth is shown and,
Figure BDA0002330459120000095
n is more than or equal to 2, and in the embodiment, N is 100;
each spur gear is equivalent to a chip tooth as shown in fig. 3, where Δ l is the width of each chip tooth and the z-axis direction is the tooth width direction.
Step 4) calculating the slicing teeth
Figure BDA0002330459120000096
And
Figure BDA0002330459120000097
total error of each chip tooth, total stiffness of each chip tooth and coupling stiffness between adjacent chip teeth in each spur gear:
step 4a) calculating the slicing teeth
Figure BDA0002330459120000098
And
Figure BDA0002330459120000099
total error E of i Including slicing teeth
Figure BDA00023304591200000910
Axial modification error e lp 、
Figure BDA00023304591200000911
Axial modification error e lg And
Figure BDA00023304591200000912
and
Figure BDA00023304591200000913
misalignment of e m ，E i ＝e lp +e lg +e m (ii) a Wherein the slicing teeth
Figure BDA00023304591200000914
Axial modification error e lp 、
Figure BDA00023304591200000915
Axial modification error e lg And
Figure BDA00023304591200000916
and
Figure BDA00023304591200000917
misalignment of e m The calculation formulas are respectively as follows:
Figure BDA00023304591200000918
Figure BDA00023304591200000919
Figure BDA0002330459120000101
wherein C is β Representing the tooth trimming amount; b 0 The crest of the axial modification curve in the tooth width directionPoint coordinates, b i Coordinates of each slice in the tooth width direction are represented; s represents a bending coefficient, β, which determines the shape of the profile of the axial modification b Representing the gear base helix angle, θ x1 、θ x2 、θ y1 And theta y2 The degrees of freedom of oscillation, psi, of the driving gear p and the driven gear, respectively, as shown in FIG. 4 12 Can be expressed as:
Figure BDA0002330459120000102
wherein alpha is 12 (0≤α 12 2 pi) is the relative position angle of the gear pair, as shown in fig. 4, the axial modification amount C is shown in the embodiment β 20 μm, misalignment angle θ y1 ＝0.05°；
Step 4b) by slicing teeth
Figure BDA0002330459120000103
Bending stiffness of
Figure BDA0002330459120000104
Shear stiffness
Figure BDA0002330459120000105
Radial compression stiffness
Figure BDA0002330459120000106
And Hertz contact stiffness Δ k hi Calculating
Figure BDA0002330459120000107
Total rigidity of
Figure BDA0002330459120000108
By slicing teeth
Figure BDA0002330459120000109
Bending stiffness of
Figure BDA00023304591200001010
Shear stiffness
Figure BDA00023304591200001011
Radial compression stiffness
Figure BDA00023304591200001012
And Hertz contact stiffness Δ k h Calculating
Figure BDA00023304591200001013
Total rigidity of
Figure BDA00023304591200001014
Bending rigidity of each segment tooth of the driving gear p and the driven gear g by potential energy method
Figure BDA00023304591200001015
Shear stiffness
Figure BDA00023304591200001016
Radial compression stiffness
Figure BDA00023304591200001017
And Hertz contact stiffness Δ k h The method has the same calculation formula, and the calculation formula is respectively as follows:
Figure BDA00023304591200001018
Figure BDA00023304591200001019
Figure BDA00023304591200001020
Figure BDA00023304591200001021
Figure BDA0002330459120000111
wherein
Figure BDA0002330459120000112
Figure BDA0002330459120000113
In order to index the circular pressure angle,
Figure BDA0002330459120000114
is the pressure angle at the point of contact, r bq Is the base radius, θ bq Is half of the base tooth angle,
Figure BDA0002330459120000115
the crest coefficient, gamma and tau are integral ranges, y 1q Is the horizontal coordinate, y, at any point on the transition curve 2q Is a horizontal coordinate at any point on the involute, I y1q Is the section moment of inertia, I, at any position on the transition curve y2q Is the section moment of inertia, G, at any position on the involute q For shear modulus, A y1q Is the cross-sectional area at any position on the transition curve, A y2q The cross-sectional area at any position on the involute is shown, E is the synthetic elastic modulus, and subscript q takes the values of p and g and respectively represents a driving gear and a driven gear.
Step 4c) calculating the slicing teeth
Figure BDA0002330459120000116
And its adjacent slicing teeth
Figure BDA0002330459120000117
Rigidity of coupling therebetween
Figure BDA0002330459120000118
Calculating slicing tooth
Figure BDA0002330459120000119
And its adjacent slicing teeth
Figure BDA00023304591200001110
Rigidity of coupling therebetween
Figure BDA00023304591200001111
Figure BDA00023304591200001112
Figure BDA00023304591200001113
Wherein C is c Represents a coupling coefficient;
step 5), constructing a slice coupling model S of a pair of meshed teeth in the gear pair:
step 5a) slicing teeth
Figure BDA00023304591200001114
Simulated as the total stiffness with itself
Figure BDA00023304591200001115
Equal drive gear segment tooth compression spring
Figure BDA00023304591200001116
Obtaining a set consisting of N driving gear cutting teeth compression springs which are arranged in parallel
Figure BDA00023304591200001117
To slice teeth
Figure BDA00023304591200001118
And its adjacent slicing teeth
Figure BDA00023304591200001119
The coupling between the two is simulated as the coupling stiffness
Figure BDA00023304591200001120
Equal mainMovable gear coupling compression spring
Figure BDA00023304591200001121
The set consisting of N-1 driving gear coupling compression springs is obtained
Figure BDA00023304591200001122
And will be
Figure BDA00023304591200001123
And
Figure BDA00023304591200001124
by passing
Figure BDA00023304591200001125
Connecting to obtain a slice coupling model of the driving gear p shown in FIG. 5;
step 5b) slicing the teeth
Figure BDA00023304591200001126
Simulated as the total stiffness with itself
Figure BDA00023304591200001127
Equal driven gear slicing tooth compression spring
Figure BDA0002330459120000121
Obtaining a set consisting of N driven gear slicing tooth compression springs arranged in parallel
Figure BDA0002330459120000122
To slice into teeth
Figure BDA0002330459120000123
And its adjacent slicing teeth
Figure BDA0002330459120000124
Is simulated as a stiffness of the coupling
Figure BDA0002330459120000125
Equal driven teethWheel coupling compression spring
Figure BDA0002330459120000126
The set consisting of N-1 driven gear coupling compression springs is obtained
Figure BDA0002330459120000127
And will be
Figure BDA0002330459120000128
And
Figure BDA0002330459120000129
by passing
Figure BDA00023304591200001210
Connecting to obtain a slice coupling model of the driven gear g shown in FIG. 6;
step 5c) corresponding slicing teeth of the driving gear p and the driven gear g
Figure BDA00023304591200001211
And
Figure BDA00023304591200001212
contact between the teeth is simulated as corresponding to the slicing teeth
Figure BDA00023304591200001213
And
Figure BDA00023304591200001214
contact rigidity Δ k therebetween h Equal contact compression spring
Figure BDA00023304591200001215
Obtaining a set of N contact compression springs
Figure BDA00023304591200001216
Step 5d) coupling the slices of the driving gear p into a model
Figure BDA00023304591200001217
In a section coupling model with the driven gear g
Figure BDA00023304591200001218
By passing
Figure BDA00023304591200001219
Are connected to form a series spring group
Figure BDA00023304591200001220
And combining N series spring group sets into a series spring group set
Figure BDA00023304591200001221
Step 5e) will
Figure BDA00023304591200001222
And
Figure BDA00023304591200001223
and
Figure BDA00023304591200001224
combining into a slice coupling model of a pair of intermeshing teeth in a gear pair as shown in figure 7;
step 6) deducing the series spring group
Figure BDA00023304591200001225
The relation between the middle deformation amounts is as follows:
step 6a) assuming series spring groups in the slice coupling model
Figure BDA00023304591200001226
Under force F i Is deformed by the action of
Figure BDA00023304591200001227
The sum of the deformation of N series spring sets is the meshing deformation of the gear pair
Figure BDA00023304591200001228
Wherein delta pi Is composed of
Figure BDA00023304591200001229
Amount of deformation of (d), δ gi Is composed of
Figure BDA00023304591200001230
Amount of deformation of delta hi Is composed of
Figure BDA00023304591200001231
The amount of deformation of (a);
step 6b) by F i 、δ pi 、δ gi 、δ hi And series spring group
Figure BDA00023304591200001232
The balance equation constructed by the stiffness of each compression spring
Figure BDA00023304591200001233
And
Figure BDA00023304591200001234
series spring group derivation
Figure BDA00023304591200001235
Relation between the medium deformation amounts:
Figure BDA0002330459120000131
step 7) derivation
Figure BDA0002330459120000132
A formula for balancing the force and deformation of (a);
step 7a) by F i And delta pi And a driving gear segment tooth compression spring of the segment coupling model of the driving gear p
Figure BDA0002330459120000133
Compression spring coupled with driving gear
Figure BDA0002330459120000134
Rigidity of
Figure BDA0002330459120000135
And
Figure BDA0002330459120000136
compression spring for deriving cutting teeth of driving gear
Figure BDA0002330459120000137
Amount of deformation δ pi And F i Obtaining a balance equation of the deformation and force of the N driving gear blade tooth compression springs:
Figure BDA0002330459120000138
step 7b) converting the force of the deformation of the N drive gear blade tooth compression springs into a matrix form K according to a balance formula of the deformation and a balance equation of the force c X is F, and it is defined as
Figure BDA0002330459120000139
Is a formula for balancing the force and deformation, wherein K c Representing a stiffness matrix, X and F representing a deformation vector and an applied force vector, respectively,
Figure BDA00023304591200001310
X＝[δ p1 ,δ p2 ,…,δ pi ,…,δ pN ] T
F＝[F 1 ,F 2 ,…,F i ,…,F N ] T
[·] T representation matrix [ ·]Transposing;
step 8) calculating the time-varying meshing stiffness of the gear pair:
step 8a) providing cuts applied to a pair of intermeshing teeth in a gear pairThe external force on the sheet coupling model S is F e Tolerance error of F ε And let δ be 0;
step 8b) determining the delta and the slicing teeth
Figure BDA0002330459120000141
And
Figure BDA0002330459120000142
total error E of i The relationship delta > E i If true, the slicing teeth are determined
Figure BDA0002330459120000143
And slicing teeth
Figure BDA0002330459120000144
Meshing, all the slicing teeth in the meshing state in the driving gear p are combined into a set
Figure BDA0002330459120000145
Otherwise, slicing the teeth
Figure BDA0002330459120000146
And slicing teeth
Figure BDA0002330459120000147
Non-meshing, namely all the slicing teeth in the driving gear p in a non-meshing state
Figure BDA0002330459120000148
Are combined into sets
Figure BDA0002330459120000149
Wherein the content of the first and second substances,
Figure BDA00023304591200001410
is the W-th slicing tooth in the meshing state in the driving gear p, W is the total number of the slicing teeth in the meshing state,
Figure BDA00023304591200001411
is the s-th one of the driving gear pThe slicing teeth are in a non-meshed state, S is the total number of the slicing teeth in the non-meshed state, and S + W is equal to N;
step 8c) when slicing teeth
Figure BDA00023304591200001412
And slicing teeth
Figure BDA00023304591200001413
When engaged, the deformation of W slice teeth in the engaged state in the driving gear p is all
Figure BDA00023304591200001414
According to the distance set A p” Section tooth with nearest area and in meshing state
Figure BDA00023304591200001415
The amount of deformation of (2) calculating the slicing teeth in a non-meshing state
Figure BDA00023304591200001416
Amount of deformation delta' ps And will be calculated
Figure BDA00023304591200001417
Amount of deformation δ pi Fill in X ═ delta p1 ,δ p2 ,…,δ pi ,…,δ pN ] T In which delta " ps The calculation formula of (2) is as follows:
Figure BDA00023304591200001418
wherein, gamma is j(j+1) The deformation transfer coefficient between the jth slicing tooth and the (j + 1) th slicing tooth is obtained;
when the slicing teeth are meshed, the equivalent spring is in a contact state, the deformation of the spring in a non-contact state can be represented by the deformation of the spring in the contact state, the deformation transmission coefficient represents the relation between the deformation amounts of the adjacent driving gear slicing teeth arranged in parallel and compressing the spring, and the deformation transmission coefficient gamma is j(j+1) The calculation formula of (2) is as follows:
Figure BDA00023304591200001419
when the subscript is greater than N, the,
Figure BDA00023304591200001420
step 8d) calculating the meshing force F of the gear pair m And judging | F e -F m |≤F ε If yes, obtaining the meshing deformation delta of the gear pair and the load distribution of the gear pair, and executing the step (8e), otherwise, enabling the gear pair to be in meshing deformation delta and load distribution of the gear pair
Figure BDA0002330459120000151
And performing step (8 b);
wherein the engaging force F m As each component F in the force vector i And the calculation formula is as follows:
F＝KX
Figure BDA0002330459120000152
where K is a stiffness matrix, K being K when a single tooth is engaged c When the two teeth are engaged
Figure BDA0002330459120000153
Step 8e) calculating the meshing stiffness k of the gear pair according to the meshing deformation delta of the gear pair, and calculating the meshing stiffness of each meshing position in a meshing period, namely the time-varying meshing stiffness of the gear pair;
tooth pair meshing stiffness k tt And the meshing rigidity k of the gear pair, and the calculation formula is respectively as follows:
Figure BDA0002330459120000154
Figure BDA0002330459120000155
the technical effects of the invention are explained by combining simulation experiments as follows:
1. simulation conditions and contents:
the calculation method of the present invention and the corrected straight gear meshing stiffness calculation accuracy of the prior art spur gear meshing stiffness analysis method considering the drum direction modification were compared and simulated using Matlab software, and the results are shown in table 3.
2. And (3) simulation result analysis:
as can be seen from table 3, the calculation accuracy of the meshing stiffness of the spur gear is significantly improved compared to the prior art in consideration of axial modification and misalignment.
TABLE 3 simulation comparison results
Calculation method Accuracy at 20 μm flank modification Accuracy at 0.05 deg. non-centering angle
The invention discloses a computing method 98.7％ 98％
Prior art calculation method 82.9％ 94.9％

Claims (7)

1. A spur gear meshing rigidity calculation method considering a tooth direction error is characterized by comprising the following steps:
(1) setting parameters of a straight gear pair:
the tooth widths of a driving gear p and a driven gear g in a straight gear pair are both B, the modulus is m, and the elastic moduli of the driving gear p and the driven gear g are respectively E p And E g The number of teeth of the driving gear p and the driven gear g is z 1 And z 2 ；
(2) Calculating tooth base rigidity k of straight gear pair tf ：
Calculating the rigidity k of the single-tooth meshing tooth base of the driving gear p in the straight gear pair fp Rigidity k of base of single tooth meshing with driven gear g fg And through k fp And k fg Calculating tooth base rigidity k of straight gear pair tf ；
(3) Acquiring a plurality of slicing teeth of each straight gear in the straight gear pair along the tooth width direction:
the driving gear p in the straight gear pair is equivalent to N slicing teeth with the width delta l along the tooth width direction
Figure FDA0002330459110000011
The driven gear g is simultaneously equivalent to N slicing teeth with the width delta l along the tooth width direction
Figure FDA0002330459110000012
Wherein the content of the first and second substances,
Figure FDA0002330459110000013
the ith slice tooth is shown,
Figure FDA0002330459110000014
the ith slice tooth is shown,
Figure FDA0002330459110000015
N≥2；
(4) calculating slicing tooth
Figure FDA0002330459110000016
And
Figure FDA0002330459110000017
total error of each chip tooth, total stiffness of each chip tooth and coupling stiffness between adjacent chip teeth in each spur gear:
(4a) calculating slicing tooth
Figure FDA0002330459110000018
And
Figure FDA0002330459110000019
total error E of i Including slicing teeth
Figure FDA00023304591100000110
Axial modification error e lp 、
Figure FDA00023304591100000111
Axial modification error e lg And
Figure FDA00023304591100000112
and with
Figure FDA00023304591100000113
Misalignment of e m ，E i ＝e lp +e lg +e m ；
(4b) By slicing teeth
Figure FDA00023304591100000114
Bending stiffness of
Figure FDA00023304591100000115
Shear stiffness
Figure FDA00023304591100000116
Radial compression stiffness
Figure FDA00023304591100000117
And Hertz contact stiffness Δ k h Calculating
Figure FDA00023304591100000118
Total rigidity of
Figure FDA00023304591100000119
By slicing teeth
Figure FDA00023304591100000120
Bending stiffness of
Figure FDA00023304591100000121
Shear stiffness
Figure FDA00023304591100000122
Radial compression stiffness
Figure FDA00023304591100000123
And Hertz contact stiffness Δ k h Calculating
Figure FDA00023304591100000124
Total rigidity of
Figure FDA00023304591100000125
(4c) Calculating slicing tooth
Figure FDA0002330459110000021
And its adjacent slicing teeth
Figure FDA0002330459110000022
Stiffness of coupling therebetween
Figure FDA0002330459110000023
Calculating slicing tooth
Figure FDA0002330459110000024
And its adjacent slicing teeth
Figure FDA0002330459110000025
Rigidity of coupling therebetween
Figure FDA0002330459110000026
Figure FDA0002330459110000027
Figure FDA0002330459110000028
Wherein C is c Represents a coupling coefficient;
(5) constructing a section coupling model S of a pair of mutually meshed teeth in the gear pair:
(5a) to slice teeth
Figure FDA0002330459110000029
Simulated as the total stiffness with itself
Figure FDA00023304591100000210
Equal drive gear segment tooth compression spring
Figure FDA00023304591100000211
Obtaining a set consisting of N driving gear cutting teeth compression springs which are arranged in parallel
Figure FDA00023304591100000212
To slice teeth
Figure FDA00023304591100000213
And its adjacent slicing teeth
Figure FDA00023304591100000214
The coupling between the two is simulated as the coupling stiffness
Figure FDA00023304591100000215
Equal driving gear coupling compression spring
Figure FDA00023304591100000216
The set consisting of N-1 driving gear coupling compression springs is obtained
Figure FDA00023304591100000217
And will be
Figure FDA00023304591100000218
And
Figure FDA00023304591100000219
by passing
Figure FDA00023304591100000220
Connecting to obtain a slice coupling model of the driving gear p;
(5b) to slice teeth
Figure FDA00023304591100000221
Simulated as the total stiffness with itself
Figure FDA00023304591100000222
Equal driven gear slicing tooth compression spring
Figure FDA00023304591100000223
Obtaining a set consisting of N driven gear slicing tooth compression springs which are arranged in parallel
Figure FDA00023304591100000224
To slice teeth
Figure FDA00023304591100000225
And adjacent slices thereofTooth
Figure FDA00023304591100000226
Is simulated as a stiffness of the coupling
Figure FDA00023304591100000227
Equal driven gear coupling compression spring
Figure FDA00023304591100000228
The set consisting of N-1 driven gear coupling compression springs is obtained
Figure FDA00023304591100000229
And will be
Figure FDA00023304591100000230
And
Figure FDA00023304591100000231
by passing
Figure FDA00023304591100000232
Connecting to obtain a slice coupling model of the driven gear g;
(5c) corresponding the driving gear p and the driven gear g to the slicing teeth
Figure FDA00023304591100000233
And with
Figure FDA00023304591100000234
Contact between the teeth is simulated as corresponding to the slicing teeth
Figure FDA00023304591100000235
And
Figure FDA00023304591100000236
contact rigidity Δ k therebetween h Equal contact compression spring
Figure FDA00023304591100000237
Obtaining a set of N contact compression springs
Figure FDA00023304591100000238
(5d) Coupling the slices of the driving gear p in a model
Figure FDA0002330459110000031
In a section coupling model with the driven gear g
Figure FDA0002330459110000032
By passing
Figure FDA0002330459110000033
Connected to form a series spring set
Figure FDA0002330459110000034
And combining N series spring group sets into a series spring group set
Figure FDA0002330459110000035
(5e) Will be provided with
Figure FDA0002330459110000036
And
Figure FDA0002330459110000037
and
Figure FDA0002330459110000038
combining into a slice coupling model of a pair of meshed teeth in the gear pair;
(6) series spring group derivation
Figure FDA0002330459110000039
The relation between the middle deformation amounts is as follows:
(6a) series spring set in hypothetical slice coupling model
Figure FDA00023304591100000310
Under force F i Is deformed by delta i t ＝δ pi +δ gi +δ hi The sum of the deformation of the N series spring groups is the meshing deformation of the gear pair
Figure FDA00023304591100000311
Wherein delta pi Is composed of
Figure FDA00023304591100000312
Amount of deformation of (d), δ gi Is composed of
Figure FDA00023304591100000313
Amount of deformation of delta hi Is composed of
Figure FDA00023304591100000314
The amount of deformation of (2);
(6b) by F i 、δ pi 、δ gi 、δ hi And series spring group
Figure FDA00023304591100000315
The balance equation constructed by the stiffness of each compression spring
Figure FDA00023304591100000316
And delta i t ＝δ pi +δ gi +δ hi Deducing series spring group
Figure FDA00023304591100000317
Relation between the medium deformation amounts:
Figure FDA00023304591100000318
(7) derivation of
Figure FDA00023304591100000319
Force and deformation equation of (1):
(7a) by F i And delta pi And a driving gear segment tooth compression spring of the segment coupling model of the driving gear p
Figure FDA00023304591100000320
Compression spring coupled with driving gear
Figure FDA00023304591100000321
Rigidity of
Figure FDA00023304591100000322
And
Figure FDA00023304591100000323
compression spring for deriving cutting teeth of driving gear
Figure FDA00023304591100000324
Amount of deformation δ pi And F i Obtaining a balance equation of the deformation and force of the N driving gear blade tooth compression springs:
Figure FDA0002330459110000041
(7b) converting the balance equation of the deformation and force of the N drive gear blade tooth compression springs into a matrix form K c X is equal to F, and is taken as
Figure FDA0002330459110000042
Is a formula for balancing the force and deformation, wherein K c Representing a stiffness matrix, X and F representing a deformation vector and an applied force vector, respectively,
Figure FDA0002330459110000043
X＝[δ p1 ,δ p2 ,…,δ pi ,…,δ pN ] T
F＝[F 1 ,F 2 ,…,F i ,…,F N ] T
[·] T representation matrix [ ·]Transposing;
(8) calculating the time-varying meshing stiffness of the gear pair:
(8a) let F be an external force applied to a section coupling model S of a pair of teeth meshing with each other in a gear pair e Tolerance error of F ε And let δ be 0;
(8b) determine delta and slice teeth
Figure FDA0002330459110000044
And
Figure FDA0002330459110000045
total error E of i The relationship delta > E i If true, the slicing teeth are determined
Figure FDA0002330459110000046
And slicing teeth
Figure FDA0002330459110000047
Meshing, all the slicing teeth in the meshing state in the driving gear p are combined into a set
Figure FDA0002330459110000048
Otherwise, slicing the teeth
Figure FDA0002330459110000049
And slicing teeth
Figure FDA00023304591100000410
Non-meshing, namely cutting all the driving gears p in a non-meshing stateSheet tooth
Figure FDA00023304591100000411
Are combined into sets
Figure FDA00023304591100000412
Wherein the content of the first and second substances,
Figure FDA00023304591100000413
is the W-th slicing tooth in the meshing state in the driving gear p, W is the total number of the slicing teeth in the meshing state,
Figure FDA0002330459110000051
the number of the S-th slicing teeth in the driving gear p in the non-meshed state is S, the total number of the slicing teeth in the non-meshed state is S + W, and the sum of S + W is N;
(8c) when slicing teeth
Figure FDA0002330459110000052
And slicing teeth
Figure FDA0002330459110000053
When engaged, the deformation of W slice teeth in the engaged state in the driving gear p is all
Figure FDA00023304591100000512
According to the distance set A p” Section tooth with nearest area and in meshing state
Figure FDA0002330459110000055
The amount of deformation of the teeth is calculated to obtain the slicing teeth in a non-meshed state
Figure FDA0002330459110000056
Amount of deformation δ ″) ps And will be calculated
Figure FDA0002330459110000057
Amount of deformation δ pi Fill in X ═ delta p1 ,δ p2 ,…,δ pi ,…,δ pN ] T In which δ ″) ps The calculation formula of (2) is as follows:
Figure FDA0002330459110000058
wherein, gamma is j(j+1) The deformation transfer coefficient between the jth slicing tooth and the (j + 1) th slicing tooth is obtained;
(8d) calculating the meshing force F of a gear pair m And judging | F e -F m |≤F ε If yes, obtaining the meshing deformation delta of the gear pair and the load distribution of the gear pair, and executing the step (8e), otherwise, enabling the gear pair to have the meshing deformation delta and the load distribution of the gear pair
Figure FDA0002330459110000059
And performing step (8 b);
(8e) and calculating the meshing stiffness k of the gear pair according to the meshing deformation delta of the gear pair, and calculating the meshing stiffness of each meshing position in a meshing period, namely the time-varying meshing stiffness of the gear pair.
2. A spur gear meshing stiffness calculating method considering a tooth direction error according to claim 1, wherein the single-tooth meshing tooth base stiffness k of the driving spur gear p in the step (2) fp Rigidity k of single-tooth meshing tooth base of driven straight gear g fg And tooth base rigidity k of spur gear pair tf The calculation formulas are respectively as follows:
Figure FDA00023304591100000510
Figure FDA00023304591100000511
wherein the content of the first and second substances,
Figure FDA0002330459110000061
is the pressure angle at the contact point u fq Distance of force to tooth root, S fq Root width, L * ,M * ,P * ,Q * The subscript q takes the values p and g, representing the driving and driven gears, λ, respectively, for the coefficients of the formula p Is the tooth base stiffness correction factor, λ, of the driving gear p g Is a tooth base stiffness correction coefficient of the driven gear g.
3. A method for calculating a tooth-meshing stiffness of a spur gear in consideration of a tooth-direction error according to claim 1, wherein the sliced tooth in the step (4a) is
Figure FDA0002330459110000062
Axial modification error e lp 、
Figure FDA0002330459110000063
Axial modification error e lg And
Figure FDA0002330459110000064
and
Figure FDA0002330459110000065
misalignment of e m The calculation formulas are respectively as follows:
Figure FDA0002330459110000066
Figure FDA0002330459110000067
Figure FDA0002330459110000068
wherein C is β Representing the tooth trimming amount; b 0 Representing the vertex coordinates of the relief curve in the tooth-width direction, b i Coordinates representing each slice in the tooth width direction; s represents a bending coefficient, β, which determines the shape of the relief curve b Representing the gear base helix angle, θ x1 、θ x2 、θ y1 And theta y2 The degrees of freedom of oscillation, psi, of the driving gear p and the driven gear g, respectively 12 Is a parameter related to the direction of rotation of the drive gear.
4. A method for calculating stiffness of spur gear meshing considering tooth flank error according to claim 1, wherein said step (4b) is performed in step (4b)
Figure FDA0002330459110000069
Total rigidity of
Figure FDA00023304591100000610
And
Figure FDA00023304591100000611
total rigidity of
Figure FDA00023304591100000612
The calculation formulas are respectively as follows:
Figure FDA00023304591100000613
Figure FDA00023304591100000614
5. a method of calculating stiffness of spur gear meshing considering a tooth flank error according to claim 1, wherein a distortion transfer coefficient between the jth tooth and the (j + 1) th tooth in the step (8c) is calculated by:
Figure FDA0002330459110000071
wherein, when the subscript is greater than N,
Figure FDA0002330459110000072
6. the method for calculating the meshing stiffness of a spur gear considering the tooth orientation error as set forth in claim 1, wherein the meshing force F in the step (8d) m The calculation formula is as follows:
F＝KX
Figure FDA0002330459110000073
wherein K is a stiffness matrix, K-K when the single teeth mesh c When two teeth are engaged
Figure FDA0002330459110000074
7. The method for calculating meshing stiffness of a spur gear considering a tooth flank error as set forth in claim 1, wherein the tooth-to-tooth meshing stiffness k in the step (8e) tt And the gear pair meshing rigidity k, the calculation formulas are respectively as follows:
Figure FDA0002330459110000075
Figure FDA0002330459110000076
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