CN112434432A - Modeling method for meshing stiffness under external meshing straight gear abrasion - Google Patents

Modeling method for meshing stiffness under external meshing straight gear abrasion Download PDF

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CN112434432A
CN112434432A CN202011368519.7A CN202011368519A CN112434432A CN 112434432 A CN112434432 A CN 112434432A CN 202011368519 A CN202011368519 A CN 202011368519A CN 112434432 A CN112434432 A CN 112434432A
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meshing
tooth
gear
stiffness
wear
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CN112434432B (en
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陈雪峰
沈智宪
乔百杰
杨来浩
罗巍
曹宏瑞
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Xian Jiaotong University
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Abstract

The invention discloses a modeling method of meshing stiffness under the wearing of an external meshing straight gear, which comprises the following steps of acquiring the wearing capacity distribution of the wearing of the external meshing straight gear on a tooth profile; establishing a single-tooth meshing model based on the wear loss distribution, and solving the single-tooth meshing model to obtain a single-tooth meshing relation under the wear of the external meshing straight gear; deducing a gear meshing relationship under multi-tooth meshing based on the single-tooth meshing relationship; and calculating the gear meshing stiffness under the abrasion of the external meshing straight gear based on the gear meshing relation.

Description

Modeling method for meshing stiffness under external meshing straight gear abrasion
Technical Field
The invention belongs to the technical field of gear measurement, and particularly relates to a modeling method for meshing stiffness under the condition of external meshing straight gears.
Background
Gear transmission is widely applied to transmission systems of mechanical equipment, such as helicopters, wind driven generators, gear fan engines and the like, and gear failure is a key factor causing breakdown of the transmission systems. Gear abrasion is a common early failure mode, causes vibration and noise increase of a transmission system, and is a key cause of serious failures such as gear breakage, cracks and the like. In order to monitor the wear state of the gear by means of vibration signals and the like, the dynamic analysis of a transmission system after the gear is worn needs to be carried out, the vibration response characteristics of the gear wear are researched, and the monitoring basis of the gear wear is provided. The gear meshing stiffness is used as important internal excitation in a gear transmission system, and whether the meshing stiffness after the gear is abraded can be accurately obtained is the key for researching the gear abrasion failure dynamics.
The above information disclosed in this background section is only for enhancement of understanding of the background of the invention and therefore it may contain information that does not form the prior art that is already known in this country to a person of ordinary skill in the art.
Disclosure of Invention
Aiming at the problems in the prior art, the invention provides a modeling method of the meshing stiffness under the wearing of an external meshing straight gear, and the accurate calculation of the meshing stiffness after the tooth surface wearing fault of the gear occurs is realized through a geometric modeling method.
The invention aims to realize the technical scheme that a method for modeling the meshing stiffness under the abrasion of an external meshing straight gear comprises the following steps:
in the first step, the wear loss distribution of the external meshing spur gear on the tooth profile is obtained;
in the second step, a single-tooth meshing model is established based on the wear loss distribution, and the single-tooth meshing model is solved to obtain a single-tooth meshing relation under the wear of the external meshing straight gear;
in the third step, based on the single-tooth meshing relation, deducing the gear meshing relation under multi-tooth meshing;
and in the fourth step, calculating the gear meshing rigidity under the abrasion of the external meshing straight gear based on the gear meshing relation.
In the method, the firstIn the step, the wearing capacity of the external meshing straight gear on the tooth profile of a pair of meshing gears is distributed as
Figure BDA0002804207040000021
And
Figure BDA0002804207040000022
wherein the content of the first and second substances,
Figure BDA0002804207040000023
showing the profile of the driving wheel1Point at coordinate x1The amount of wear of the parts is reduced,
Figure BDA0002804207040000024
showing the profile of the driving wheel2Point at coordinate x2The amount of wear at the point.
In the method, in the second step, the single-tooth meshing model is:
γ1perfect＝arctan(α2+α1)-α1，
Figure BDA0002804207040000025
Figure BDA0002804207040000026
Figure BDA0002804207040000027
γ2perfect＝arctan(α2′+α1′)-α1′，
Figure BDA0002804207040000028
Figure BDA0002804207040000029
Figure BDA00028042070400000210
Figure BDA00028042070400000211
Figure BDA00028042070400000212
θ1＝θ2，
θ1＝δ1-ψ1，
θ2＝δ2+ψ2，
Figure BDA0002804207040000031
Figure BDA0002804207040000032
δ1＝λ1+γ1，
δ2＝λ2+γ2，
Figure BDA0002804207040000033
Figure BDA0002804207040000034
wherein A is1perfect、A1Respectively represents the meshing points on the perfect tooth profile and the worn tooth profile of the driving wheel,
Figure BDA0002804207040000035
the radius of the base circle of the driving wheel is shown,
Figure BDA0002804207040000036
respectively represent the engagement points A on the driving wheel1perfectAnd A1Radius of (a)2、α1Respectively represents half of the angle occupied by the base circle on the single tooth of the driving wheel and the meshing point A1perfectAngle of (a) gamma1perfect、γ1Respectively represent the engagement points A on the driving wheel1perfectAnd A1Angle of (A)2perfect、A2The mesh points on the perfect tooth profile and the worn tooth profile of the driven wheel are respectively shown,
Figure BDA0002804207040000037
representing the base radius of the driven wheel,
Figure BDA0002804207040000038
each representing a mesh point on the driven wheel2perfectAnd A2Radius, α2′、α1' represents half of the angle occupied by the base circle on the single tooth of the driven wheel and the meshing point A respectively2perfectAngle of (a) gamma2perfect、γ2Respectively representing the engagement points A on the driven wheels2perfectAnd A2Angle of (a)ωThe center distance between the driving wheel and the driven wheel is shown,
Figure BDA0002804207040000039
representing the rotation angles, theta, of the driving and driven wheels, respectively1、θ2Respectively showing the meshing points A of the driving wheel and the driven wheel1And A2Is at an angle delta to the centre line1、δ2Respectively showing the meshing points A of the driving wheel and the driven wheel1And A2Is at an angle psi to the corresponding radius1、ψ2Respectively showing the meshing points A of the driving wheel and the driven wheel1And A2Is included angle of radius and center line, lambda1、λ2Respectively showing the meshing points A of the driving wheel and the driven wheel1And A2Tangent to and coordinate axis X1And X2The angle of,
Figure BDA0002804207040000041
respectively representing the amount of wear of the driven wheels
Figure BDA0002804207040000042
And amount of driven wheel wear
Figure BDA0002804207040000043
At the coordinate x1And x2The slope of (d).
In the method, in the third step, the gear engagement relation under multi-tooth engagement is derived through a multi-tooth engagement criterion based on the single-tooth engagement relation, wherein the multi-tooth engagement criterion comprises the rotation angle based on the driving wheel and the driven wheel
Figure BDA0002804207040000044
Calculating the rotation angle of each pair of gears in multi-tooth engagement
Figure BDA0002804207040000045
Figure BDA0002804207040000046
Wherein z is1And z2Number of teeth of driving and driven wheels, 1st、2nd、3rd、nthThe 1 st, 2 nd, 3 rd and n th pairs of meshing gears are shown, and the gear meshing relationship under multi-tooth meshing is obtained based on the rotation angle of each pair of gears in multi-tooth meshing.
In the method, the fourth step of calculating the gear mesh stiffness under the wear of the external-mesh spur gear based on the gear mesh relationship includes,
the gear stiffness of the drive wheel includes: hertz contact stiffness khBending stiffness kbShear deformation stiffness ksAxial compression stiffness kaAnd elastic matrix deformation stiffness kf，
Figure BDA0002804207040000047
Figure BDA0002804207040000048
Figure BDA0002804207040000049
Figure BDA00028042070400000410
Figure BDA0002804207040000051
And is
Figure BDA0002804207040000052
Figure BDA0002804207040000053
Wherein b represents the tooth width, v represents the Poisson's ratio, E, G represents the modulus and shear modulus, respectively,
Figure BDA0002804207040000054
Figure BDA0002804207040000055
respectively show a cross section d1The moment of inertia and the cross-sectional moment of the (c),
Figure BDA0002804207040000056
indicating the coordinate X of the driving wheel1Upper wear amount, D, H represents the algebraic number of simplified stiffness calculations, ufThe distance between the intersection point of the meshing line and the symmetric line of the gear teeth and the base circle is shown,Sfrepresenting arc length, L, of single tooth profile*、M*、P*、Q*Four parameters relating to the number of gear teeth and the module are shown. The rigidity calculation process of the driven wheel is completely consistent with that of the driving wheel.
L*、M*、P*、Q*The values of (a) can be obtained by polynomial fitting:
Figure BDA0002804207040000057
Ai、Bi、Ci、Di、Ei、Fithe values of (b) are shown in Table 1, hfi＝rf/rint，rfDenotes the root circle radius, rintIndicating the gear shaft bore diameter, thetafRepresenting the angle occupied by the single tooth profile.
TABLE 1Ai、Bi、Ci、Di、Ei、FiParameter table
Ai Bi Ci Di Ei Fi
L* -5.754×10-5 -1.999×10-3 -2.302×10-4 -4.77×10-3 0.027 6.805
M* -60.11×10-5 28.1×10-3 -83.43×10-4 -9.926×10-3 0.162 0.909
P* -50.95×10-5 185.5×10-3 0.0538×10-4 53.3×10-3 0.29 0.924
Q* -6.204×10-5 9.0889×10-3 -4.096×10-4 7.8297×10-3 -0.15 0.59
The single and multi-tooth meshing stiffness was as follows:
single tooth meshing stiffness:
Figure BDA0002804207040000061
multi-tooth meshing stiffness:
Figure BDA0002804207040000062
wherein k isb1、ks1、ka1、kf1Representing gear stiffness, k, of the driving wheelb2、ks2、ka2、kf2The index i is 1, and 2 indicates the 1 st and 2 nd pairs of meshing gears in multiple tooth meshing.
Compared with the prior art, the beneficial effect that this disclosure brought does:
according to the modeling method for the meshing stiffness under the abrasion fault of the external meshing straight gear, the meshing stiffness under the abrasion of the gear is calculated by using a geometric modeling method, and a key basis is provided for dynamics and diagnosis research of the abrasion fault of the gear. And establishing and solving a single-tooth meshing relation under the abrasion fault by adopting a geometric modeling method, deducing a new gear meshing relation under the abrasion fault during multi-tooth meshing by utilizing a multi-tooth meshing criterion, and finally calculating the meshing rigidity under the abrasion of the gear according to a potential energy method. The method has the advantages of being accurate in calculation and high in efficiency. The meshing stiffness solved by the method can provide a key basis for the dynamic and diagnostic research of the gear wear fault.
The above description is only an overview of the technical solutions of the present invention, and in order to make the technical means of the present invention more clearly apparent, and to make the implementation of the content of the description possible for those skilled in the art, and to make the above and other objects, features and advantages of the present invention more obvious, the following description is given by way of example of the specific embodiments of the present invention.
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Various other advantages and benefits of the present invention will become apparent to those of ordinary skill in the art upon reading the following detailed description of the preferred embodiments. The drawings are only for purposes of illustrating the preferred embodiments and are not to be construed as limiting the invention. It is obvious that the drawings described below are only some embodiments of the invention, and that for a person skilled in the art, other drawings can be derived from them without inventive effort. Also, like parts are designated by like reference numerals throughout the drawings.
In the drawings:
FIG. 1 is a flow chart of a method for modeling mesh stiffness under wear of an external meshing spur gear according to one embodiment of the present disclosure;
fig. 2(a) to 2(c) are wear amount distributions on tooth profiles of gears provided by another embodiment of the present disclosure, where 2(a) is a schematic diagram of the wear amount distribution on tooth profile, 2(b) is the gear wear amount on the tooth profile of the driving gear, and 2(c) is the gear wear amount on the tooth profile of the driven gear;
fig. 3(a) to 3(c) are a single tooth model and a single tooth meshing model under gear wear provided by another embodiment of the present disclosure, where 3(a) is a driving gear wear single tooth model, 3(b) is a driven gear wear single tooth model, and 3(c) is a gear wear single tooth meshing model;
4(a) -4 (b) are a single tooth meshing relationship and a multiple tooth meshing relationship under wear of a gear provided by another embodiment of the present disclosure, wherein 4(a) is a single tooth meshing relationship and 4(b) is a multiple tooth meshing relationship;
FIG. 5 is a graph of mesh stiffness provided by another embodiment of the present disclosure, normal and under gear wear;
fig. 6(a) to 6(b) are finite element models of gear wear provided by another embodiment of the present disclosure, wherein 6(a) is the finite element model and 6(b) is the result of the finite element model under normal and gear wear.
The invention is further explained below with reference to the figures and examples.
Detailed Description
Specific embodiments of the present invention will be described in more detail below with reference to fig. 1 to 6 (b). While specific embodiments of the invention are shown in the drawings, it should be understood that the invention may be embodied in various forms and should not be construed as limited to the embodiments set forth herein. Rather, these embodiments are provided so that this disclosure will be thorough and complete, and will fully convey the scope of the invention to those skilled in the art.
It should be noted that certain terms are used throughout the description and claims to refer to particular components. As one skilled in the art will appreciate, various names may be used to refer to a component. This specification and claims do not intend to distinguish between components that differ in name but not function. In the following description and in the claims, the terms "include" and "comprise" are used in an open-ended fashion, and thus should be interpreted to mean "include, but not limited to. The description which follows is a preferred embodiment of the invention, but is made for the purpose of illustrating the general principles of the invention and not for the purpose of limiting the scope of the invention. The scope of the present invention is defined by the appended claims.
For the purpose of facilitating understanding of the embodiments of the present invention, the following description will be made by taking specific embodiments as examples with reference to the accompanying drawings, and the drawings are not to be construed as limiting the embodiments of the present invention.
The method for modeling the meshing stiffness under the wearing of the external meshing spur gear comprises the following steps,
1) acquiring the wear amount distribution of gear wear on tooth profile;
2) according to the distribution of the gear wear amount on the tooth profile, a single-tooth meshing model after the gear is worn is established, and then the model is solved to obtain a single-tooth meshing relation under the gear wear;
3) according to the single-tooth meshing relationship under the gear abrasion, a multi-tooth meshing rule is adopted to deduce a new gear meshing relationship under multi-tooth meshing;
4) and analytically calculating the gear meshing stiffness under the gear wear by adopting a potential energy method according to the new gear meshing relationship.
Preferably, in step 1), the abrasion loss of the gear abraded on the tooth profile of the pair of meshed gear is distributed as
Figure BDA0002804207040000081
And
Figure BDA0002804207040000082
wherein the content of the first and second substances,
Figure BDA0002804207040000083
showing the profile of the driving wheel1Point at coordinate x1The amount of wear of the parts is reduced,
Figure BDA0002804207040000084
showing the profile of the driving wheel2Point at coordinate x2The amount of wear at the point.
Preferably, in step 2), the worn single-tooth meshing model of the gear is as follows:
γ1perfect＝arctan(α2+α1)-α1
Figure BDA0002804207040000085
Figure BDA0002804207040000086
Figure BDA0002804207040000087
γ2perfect＝arctan(α2′+α1′)-α1′
Figure BDA0002804207040000091
Figure BDA0002804207040000092
Figure BDA0002804207040000093
Figure BDA0002804207040000094
Figure BDA0002804207040000095
θ1＝θ2
θ1＝δ1-ψ1
θ2＝δ2+ψ2
Figure BDA0002804207040000096
Figure BDA0002804207040000097
δ1＝λ1+γ1
δ2＝λ2+γ2
Figure BDA0002804207040000098
Figure BDA0002804207040000099
wherein A is1perfect、A1Respectively represents the meshing points on the perfect tooth profile and the worn tooth profile of the driving wheel,
Figure BDA00028042070400000910
the radius of the base circle of the driving wheel is shown,
Figure BDA00028042070400000911
respectively represent the engagement points A on the driving wheel1perfectAnd A1Radius of (a)2、α1Respectively represents half of the angle occupied by the base circle on the single tooth of the driving wheel and the meshing point A1perfectAngle of (a) gamma1perfectAnd gamma 1 respectively represent an engagement point A on the driving wheel1perfectAnd A1Angle of (A)2perfect、A2The mesh points on the perfect tooth profile and the worn tooth profile of the driven wheel are respectively shown,
Figure BDA0002804207040000101
representing the base radius of the driven wheel,
Figure BDA0002804207040000102
each representing a mesh point on the driven wheel2perfectAnd A2Radius, α2′、α1' represents half of the angle occupied by the base circle on the single tooth of the driven wheel and the meshing point A respectively2perfectAngle of (a) gamma2perfect、γ2Respectively representing the engagement points A on the driven wheels2perfectAnd A2Angle of (a)ωThe center distance between the driving wheel and the driven wheel is shown,
Figure BDA0002804207040000103
representing the rotation angles, theta, of the driving and driven wheels, respectively1、θ2Respectively showing the meshing points A of the driving wheel and the driven wheel1And A2Is at an angle delta to the centre line1、δ2Respectively showing the meshing points A of the driving wheel and the driven wheel1And A2Is at an angle psi to the corresponding radius1、ψ2Respectively showing the meshing points A of the driving wheel and the driven wheel1And A2Is included angle of radius and center line, lambda1、λ2Respectively showing the meshing points A of the driving wheel and the driven wheel1And A2Tangent to and coordinate axis X1And X2The angle of,
Figure BDA0002804207040000104
respectively representing the amount of wear of the driven wheels
Figure BDA0002804207040000105
And amount of driven wheel wear
Figure BDA0002804207040000106
At the coordinate x1And x2The slope of (d).
And directly solving to obtain a new single-tooth meshing relation after the gear is abraded according to the established single-tooth meshing model under the gear abrasion.
Preferably, in step 3), the multidentate meshing criterion is as follows:
solving the rotation angle of the driving wheel and the driven wheel after the gear is worn according to the step 2)
Figure BDA0002804207040000107
The rotation angle of each pair of gears in the multi-tooth meshing is calculated:
Figure BDA0002804207040000108
wherein z is1And z2Number of teeth of driving and driven wheels, 1st、2nd、3rd、nthRespectively showing a 1 st pair, a 2 nd pair, a 3 rd pair and an nth pair of meshed gears.
The new gear meshing relation after the gear is worn under the multi-tooth meshing can be directly obtained according to the principle.
Preferably, in step 4), the process of calculating the meshing stiffness by using a potential energy method is as follows:
the stiffness of the gears includes: hertz contact stiffness khBending stiffness kbShear deformation stiffness ksAxial compression stiffness kaAnd elastic matrix deformation stiffness kf。
Figure BDA0002804207040000111
Figure BDA0002804207040000112
Figure BDA0002804207040000113
Figure BDA0002804207040000114
Figure BDA0002804207040000115
And is
Figure BDA0002804207040000116
Figure BDA0002804207040000117
Wherein b represents the tooth width, v represents the Poisson's ratio, E, G represents the modulus and shear modulus, respectively,
Figure BDA0002804207040000118
Figure BDA0002804207040000119
respectively show a cross section d1The moment of inertia and the cross-sectional moment of the (c),
Figure BDA00028042070400001110
indicating the coordinate X of the driving wheel1Upper wear amount, D, H represents the algebraic number of simplified stiffness calculations, ufRepresenting the distance between the intersection point of the meshing line and the symmetrical line of the gear teeth and the base circle, SfRepresenting arc length, L, of single tooth profile*、M*、P*、Q*Four parameters relating to the number of gear teeth and the module are shown. The rigidity calculation process of the driven wheel is completely consistent with that of the driving wheel.
The following meshing stiffnesses for single and double teeth are thus obtained:
single tooth meshing stiffness:
Figure BDA0002804207040000121
multi-tooth meshing stiffness:
Figure BDA0002804207040000122
wherein k isb1、ks1、ka1、kf1Representing gear stiffness, k, of the driving wheelb2、ks2、ka2、kf2The index i is 1, and 2 indicates the 1 st and 2 nd pairs of meshing gears in multiple tooth meshing.
In one embodiment, as shown in fig. 1, a method for modeling meshing stiffness under an external meshing spur gear wear failure includes the following steps:
1) acquiring the wear amount distribution of gear wear on tooth profile;
in this step, the distribution of the gear wear amount is as shown in fig. 2(a) to 2(c), and the wear amount of the gear wear on the pair of tooth profiles of the meshing gear is distributed as
Figure BDA0002804207040000123
And
Figure BDA0002804207040000124
wherein the content of the first and second substances,
Figure BDA0002804207040000125
showing the profile of the driving wheel1Point at coordinate x1The amount of wear of the parts is reduced,
Figure BDA0002804207040000126
showing the profile of the driving wheel2Point at coordinate x2The amount of wear at the point.
2) According to the distribution of the gear wear amount on the tooth profile, a single-tooth meshing model after the gear is worn is established, and then the model is solved to obtain a single-tooth meshing relation under the gear wear;
in this step, taking a pair of external meshing spur gears as an example, the basic parameters are shown in table 1:
TABLE 1 meshing parameters for a pair of external meshing spur gears
Figure BDA0002804207040000127
Figure BDA0002804207040000131
Single tooth model and single tooth meshing model under gear wear established by adopting geometric modeling method are shown in figures 3(a) to (c), and coordinate systems XOY and X are established1OY1、X2O2Y2Specifically, it is represented as:
γ1perfect＝arctan(α2+α1)-α1
Figure BDA0002804207040000132
Figure BDA0002804207040000133
Figure BDA0002804207040000134
γ2perfect＝arctan(α2′+α1′)-α1′
Figure BDA0002804207040000135
Figure BDA0002804207040000136
Figure BDA0002804207040000137
Figure BDA0002804207040000138
Figure BDA0002804207040000139
θ1＝θ2
θ1＝δ1-ψ1
θ2＝δ2+ψ2
Figure BDA0002804207040000141
Figure BDA0002804207040000142
δ1＝λ1+γ1
δ2＝λ2+γ2
Figure BDA0002804207040000143
Figure BDA0002804207040000144
wherein A is1perfect、A1Respectively represents the meshing points on the perfect tooth profile and the worn tooth profile of the driving wheel,
Figure BDA0002804207040000145
the radius of the base circle of the driving wheel is shown,
Figure BDA0002804207040000146
respectively represent the engagement points A on the driving wheel1perfectAnd A1Radius of (a)2、α1Respectively represents half of the angle occupied by the base circle on the single tooth of the driving wheel and the meshing point A1perfectAngle of (a) gamma1perfect、γ1Respectively represent the engagement points A on the driving wheel1perfectAnd A1Angle of (A)2perfect、A2The mesh points on the perfect tooth profile and the worn tooth profile of the driven wheel are respectively shown,
Figure BDA0002804207040000147
representing the base radius of the driven wheel,
Figure BDA0002804207040000148
each representing a mesh point on the driven wheel2perfectAnd A2Radius, α2′、α1' represents half of the angle occupied by the base circle on the single tooth of the driven wheel and the meshing point A respectively2perfectAngle of (a) gamma2perfect、γ2Respectively representing the engagement points A on the driven wheels2perfectAnd A2Angle of (a)ωThe center distance between the driving wheel and the driven wheel is shown,
Figure BDA0002804207040000149
representing the rotation angles, theta, of the driving and driven wheels, respectively1、θ2Respectively showing the meshing points A of the driving wheel and the driven wheel1And A2Is at an angle delta to the centre line1、δ2Respectively showing the meshing points A of the driving wheel and the driven wheel1And A2Is at an angle psi to the corresponding radius1、ψ2Respectively showing the meshing points A of the driving wheel and the driven wheel1And A2Is included angle of radius and center line, lambda1、λ2Respectively showing the meshing points A of the driving wheel and the driven wheel1And A2Tangent to and coordinate axis X1And X2The angle of,
Figure BDA0002804207040000151
respectively representing the amount of wear of the driven wheels
Figure BDA0002804207040000152
And amount of driven wheel wear
Figure BDA0002804207040000153
At the coordinate x1And x2The slope of (d).
And directly solving to obtain a new single-tooth meshing relation after the gear is abraded according to the established single-tooth meshing model under the gear abrasion.
3) According to the single-tooth meshing relationship under the gear abrasion, a multi-tooth meshing rule is adopted to deduce a new gear meshing relationship under multi-tooth meshing;
the multi-tooth meshing criteria are as follows:
solving the rotation angle of the driving wheel and the driven wheel after the gear is worn according to the step 2)
Figure BDA0002804207040000154
The rotation angle of each pair of gears in the multi-tooth meshing is calculated:
Figure BDA0002804207040000155
wherein z is1And z2Number of teeth of driving and driven wheels, 1st、2nd、3rd、nthRespectively showing a 1 st pair, a 2 nd pair, a 3 rd pair and an nth pair of meshed gears.
Fig. 4(a) to 4(b) show multi-tooth meshing relationships obtained from a single-tooth meshing relationship and a multi-tooth meshing criterion under wear of the gears.
4) According to the new gear meshing relation, analytically calculating gear meshing rigidity under gear wear by adopting a potential energy method;
the stiffness of the gears includes: hertz contact stiffness khBending stiffness kbShear deformation stiffness ksAxial compression stiffness kaAnd elastic matrix deformation stiffness kf。
Figure BDA0002804207040000156
Figure BDA0002804207040000161
Figure BDA0002804207040000162
Figure BDA0002804207040000163
Figure BDA0002804207040000164
And is
Figure BDA0002804207040000165
Figure BDA0002804207040000166
Wherein b represents the tooth width, v represents the Poisson's ratio, E, G represents the modulus and shear modulus, respectively,
Figure BDA0002804207040000167
Figure BDA0002804207040000168
respectively show a cross section d1The moment of inertia and the cross-sectional moment of the (c),
Figure BDA0002804207040000169
indicating the coordinate X of the driving wheel1Upper wear amount, D, H represents the algebraic number of simplified stiffness calculations, ufRepresenting the distance between the intersection point of the meshing line and the symmetrical line of the gear teeth and the base circle, SfRepresenting arc length, L, of single tooth profile*、M*、P*、Q*Four parameters relating to the number of gear teeth and the module are shown. The rigidity calculation process of the driven wheel is completely consistent with that of the driving wheel.
L*、M*、P*、Q*The values of (a) can be obtained by polynomial fitting:
Figure BDA00028042070400001610
Ai、Bi、Ci、Di、Ei、Fithe values of (b) are shown in Table 2, hfi＝rf/rint，rfDenotes the root circle radius, rintIndicating the gear shaft bore diameter, thetafRepresenting the angle occupied by the single tooth profile.
TABLE 2Ai、Bi、Ci、Di、Ei、FiParameter table
Ai Bi Ci Di Ei Fi
L* -5.754×10-5 -1.999×10-3 -2.302×10-4 -4.77×10-3 0.027 6.805
M* -60.11×10-5 28.1×10-3 -83.43×10-4 -9.926×10-3 0.162 0.909
P* -50.95×10-5 185.5×10-3 0.0538×10-4 53.3×10-3 0.29 0.924
Q* -6.204×10-5 9.0889×10-3 -4.096×10-4 7.8297×10-3 -0.15 0.59
The following meshing stiffnesses for single and double teeth are thus obtained:
single tooth meshing stiffness:
Figure BDA0002804207040000171
multi-tooth meshing stiffness:
Figure BDA0002804207040000172
wherein k isb1、ks1、ka1、kf1Representing gear stiffness, k, of the driving wheelb2、ks2、ka2、kf2The index i is 1, and 2 indicates the 1 st and 2 nd pairs of meshing gears in multiple tooth meshing.
Fig. 5 shows the meshing stiffness under normal and gear wear. FIG. 6(a) shows a finite element model, and FIG. 6(b) shows the results of the finite element model, as shown in Table 3:
TABLE 3 comparison of the results of the present method with finite element models
Item Method for producing a composite material Finite element model
Time consuming About 1 minute About 150 minutes
Average stiffness (Normal) 9.255×108N/m 8.914×108N/m
Average stiffness (wear) 9.307×108N/m 8.645×108N/m
As can be seen from Table 3, the meshing stiffness obtained by solving the two methods is very close, and the accuracy of the method disclosed by the invention is verified. In addition, the analysis model provided by the invention has extremely short time consumption and high calculation efficiency.
The method comprises the steps of 1) obtaining depth distribution of gear abrasion on tooth profile; 2) establishing a geometric model of single-tooth meshing after the gear is worn, and solving the model to obtain a single-tooth meshing relation after the gear is worn; 3) deducing a new gear meshing relation under gear wear during multi-tooth meshing by adopting a multi-tooth meshing rule; 4) the meshing stiffness is analytically calculated by adopting a potential energy method, the gear meshing stiffness under the gear wear failure is obtained, a new gear meshing relationship under the gear tooth surface wear is established, and the meshing stiffness under the gear wear is calculated according to the new gear meshing relationship, so that the method has the advantages of accurate calculation and high efficiency. The meshing stiffness obtained by the method can be applied to the vibration response characteristic research of the gear wear failure.
Although the embodiments of the present invention have been described above with reference to the accompanying drawings, the present invention is not limited to the above-described embodiments and application fields, and the above-described embodiments are illustrative, instructive, and not restrictive. Those skilled in the art, having the benefit of this disclosure, may effect numerous modifications thereto without departing from the scope of the invention as defined by the appended claims.

Claims (5)

1. A method of modeling mesh stiffness under outer mesh spur gear wear, the method comprising the steps of:
in a first step (S1), acquiring a wear amount distribution of the wear of an external meshing spur gear on a tooth profile;
in the second step (S2), a single-tooth meshing model is established based on the wear loss distribution, and the single-tooth meshing model is solved to obtain a single-tooth meshing relation under the wear of the external meshing straight gear;
in a third step (S3), a gear meshing relationship under multi-tooth meshing is derived based on the single-tooth meshing relationship;
in the fourth step (S4), gear mesh stiffness under wear of the external-mesh spur gear is calculated based on the gear mesh relationship.
2. The method according to claim 1, wherein preferably, in the first step (S1), the wear amount of the outer meshing spur gear wear on the pair of meshing gear tooth profiles is distributed as
Figure FDA0002804207030000011
And
Figure FDA0002804207030000012
wherein the content of the first and second substances,
Figure FDA0002804207030000013
showing the profile of the driving wheel1Point at coordinate x1The amount of wear of the parts is reduced,
Figure FDA0002804207030000014
showing the profile of the driving wheel2Point at coordinate x2The amount of wear at the point.
3. The method according to claim 2, wherein in a second step (S2), the single-tooth meshing pattern is:
γ1perfect＝arctan(α2+α1)-α1，
Figure FDA0002804207030000015
Figure FDA0002804207030000016
Figure FDA0002804207030000017
γ2perfect＝arctan(α2′+α1′)-α1′，
Figure FDA0002804207030000021
Figure FDA0002804207030000022
Figure FDA0002804207030000023
Figure FDA0002804207030000024
Figure FDA0002804207030000025
θ1＝θ2，
θ1＝δ1-ψ1，
θ2＝δ2+ψ2，
Figure FDA0002804207030000026
Figure FDA0002804207030000027
δ1＝λ1+γ1，
δ2＝λ2+γ2，
Figure FDA0002804207030000028
Figure FDA0002804207030000029
wherein A is1perfect、A1Respectively represents the meshing points on the perfect tooth profile and the worn tooth profile of the driving wheel,
Figure FDA00028042070300000210
the radius of the base circle of the driving wheel is shown,
Figure FDA0002804207030000031
respectively represent the engagement points A on the driving wheel1perfectAnd A1Radius of (a)2、α1Respectively represents half of the angle occupied by the base circle on the single tooth of the driving wheel and the meshing point A1perfectAngle of (a) gamma1perfect、γ1Respectively represent the engagement points A on the driving wheel1perfectAnd A1Angle of (A)2perfect、A2The mesh points on the perfect tooth profile and the worn tooth profile of the driven wheel are respectively shown,
Figure FDA0002804207030000032
representing the base radius of the driven wheel,
Figure FDA0002804207030000033
each representing a mesh point on the driven wheel2perfectAnd A2Radius, α2′、α1' represents half of the angle occupied by the base circle on the single tooth of the driven wheel and the meshing point A respectively2perfectAngle of (a) gamma2perfect、γ2Respectively representing the engagement points A on the driven wheels2perfectAnd A2Angle of (a)ωThe center distance between the driving wheel and the driven wheel is shown,
Figure FDA0002804207030000034
representing the rotation angles, theta, of the driving and driven wheels, respectively1、θ2Respectively showing the meshing points A of the driving wheel and the driven wheel1And A2Is at an angle delta to the centre line1、δ2Respectively showing the meshing points A of the driving wheel and the driven wheel1And A2Is at an angle psi to the corresponding radius1、ψ2Respectively showing the meshing points A of the driving wheel and the driven wheel1And A2Is included angle of radius and center line, lambda1、λ2Respectively showing the meshing points A of the driving wheel and the driven wheel1And A2Tangent to and coordinate axis X1And X2The angle of,
Figure FDA0002804207030000035
respectively representing the amount of wear of the driven wheels
Figure FDA0002804207030000036
And a slaveAmount of wheel wear
Figure FDA0002804207030000037
At the coordinate x1And x2The slope of (d).
4. A method according to claim 3, wherein in a third step (S3), gear engagement relationships under multiple tooth engagement are derived via multiple tooth engagement criteria based on the single tooth engagement relationships, the multiple tooth engagement criteria including based on the rotational angles of the drive and driven wheels
Figure FDA0002804207030000038
Calculating the rotation angle of each pair of gears in multi-tooth engagement
Figure FDA0002804207030000039
Wherein z is1And z2Number of teeth of driving and driven wheels, 1st、2nd、3rd、nthThe 1 st, 2 nd, 3 rd and n th pairs of meshing gears are shown, and the gear meshing relationship under multi-tooth meshing is obtained based on the rotation angle of each pair of gears in multi-tooth meshing.
5. The method according to claim 1, wherein the fourth step (S4) of calculating gear mesh stiffness under wear of an outer mesh spur gear based on the gear mesh relationship includes,
the gear stiffness of the drive wheel includes: hertz contact stiffness khBending stiffness kbShear deformation stiffness ksAxial compression stiffness kaAnd elastic matrix deformation stiffness kf，
Figure FDA0002804207030000041
Figure FDA0002804207030000042
Figure FDA0002804207030000043
Figure FDA0002804207030000044
Figure FDA0002804207030000045
And is
Figure FDA0002804207030000046
Figure FDA0002804207030000047
Wherein b represents the tooth width, v represents the Poisson's ratio, E, G represents the modulus and shear modulus, respectively,
Figure FDA0002804207030000048
Figure FDA0002804207030000049
respectively show a cross section d1The moment of inertia and the cross-sectional moment of the (c),
Figure FDA00028042070300000410
indicating the coordinate X of the driving wheel1Upper wear amount, D, H represents the algebraic number of simplified stiffness calculations, ufRepresenting the distance between the intersection point of the meshing line and the symmetrical line of the gear teeth and the base circle, SfRepresenting arc length, L, of single tooth profile*、M*、P*、Q*Showing four and gear teethThe number and the modulus are related, the rigidity calculation process of the driven wheel is consistent with that of the driving wheel,
the single and multi-tooth meshing stiffness was as follows:
single tooth meshing stiffness:
Figure FDA0002804207030000051
multi-tooth meshing stiffness:
Figure FDA0002804207030000052
wherein k isb1、ks1、ka1、kf1Representing gear stiffness, k, of the driving wheelb2、ks2、ka2、kf2The index i is 1, and 2 indicates the 1 st and 2 nd pairs of meshing gears in multiple tooth meshing.
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