CN111625758B - Planetary gear time-varying meshing stiffness calculation method based on tooth profile correction method - Google Patents

Planetary gear time-varying meshing stiffness calculation method based on tooth profile correction method Download PDF

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CN111625758B
CN111625758B CN202010584803.1A CN202010584803A CN111625758B CN 111625758 B CN111625758 B CN 111625758B CN 202010584803 A CN202010584803 A CN 202010584803A CN 111625758 B CN111625758 B CN 111625758B
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尚志武
高茂生
庞海玉
俞燕
李万祥
周士琦
刘飞
张宝仁
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Tianjin Polytechnic University
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Abstract

The invention discloses a time-varying meshing stiffness calculation method of a planetary gear based on a tooth profile correction method, which belongs to the technical field of mechanical dynamics, fully considers the influence of tooth profile on the calculation of comprehensive meshing stiffness, and improves the calculation accuracy of the time-varying meshing stiffness of the planetary gear, and comprises the following specific steps of: firstly, basic parameters of the sun wheel, the planet wheel and the inner gear ring are determined, and then the radius r of the sun wheel and the planet wheel is determinedfThe method has the beneficial effects that the principle is simple, the calculation is easy, and the practicability is strong.

Description

Planetary gear time-varying meshing stiffness calculation method based on tooth profile correction method
Technical Field
The invention belongs to the technical field of mechanical dynamics, and relates to a gear meshing time-varying meshing stiffness calculation method, in particular to a gear meshing time-varying meshing stiffness calculation method based on a tooth profile correction method.
Background
The planetary gear transmission system mainly comprises an inner gear ring, a plurality of planet wheels, a sun gear and a planet carrier, and is more complex in structure than a common dead axle gear train, the planet wheels are simultaneously meshed with the sun gear and the inner gear ring, and a plurality of pairs of sun wheels, the planet wheels and the inner gear ring are simultaneously meshed.
The planetary gear transmission system is widely applied to various power transmission processes due to the advantages of compact structure, strong bearing capacity, easy realization of large transmission ratio and the like, is often in severe working environments such as low-speed heavy load and the like, and is very easy to induce faults. The dynamic characteristics of the planetary transmission system can be researched to deepen the fault mechanism research of the planetary transmission system, and a simulation experiment basis is provided for fault diagnosis of the planetary transmission system. The most important thing for studying the dynamics of the planetary transmission system is to accurately calculate the meshing stiffness of the gear teeth. However, the existing gear tooth meshing stiffness calculation method has the defects of large calculation amount, complex formula derivation process and calculation process, inaccurate calculation result and the like. The actual tooth profile of the gear needs to consider the influence of basic parameters of the gear, such as a deflection coefficient, a tooth number, a modulus and the like, so that the calculation methods of the gear tooth meshing stiffness of the gears with different basic parameters are different. If the gear teeth are directly simplified into the cantilever beam model with the trapezoidal section, the influence of the involute tooth profile of the gear can be ignored, and meanwhile, the excessive circular arc of the tooth root can also have an important influence on the accuracy of the meshing rigidity of the gear teeth, so that the most accurate method is to calculate the meshing rigidity according to the actual shape of the gear teeth.
The calculation method of the time-varying meshing stiffness of the planetary gear has important influence and value on the research of the dynamic characteristics and the failure mechanism of the planetary transmission system.
Disclosure of Invention
The invention aims to solve the problems and designs a planetary gear time-varying meshing stiffness calculation method based on a tooth profile correction method.
The technical scheme of the invention is that a planetary gear time-varying meshing stiffness calculation method based on a tooth profile correction method comprises the following specific steps:
the method comprises the following steps: the method is characterized in that basic parameters of the sun gear, the planet gear and the inner gear ring are defined, and the basic parameters comprise: number of teeth z, modulus m, tooth width B, tooth crest height coefficient haTop clearance coefficient c, pitch circle pressure angle α, displacement coefficient x (positive value for positive displacement and negative value for negative displacement), pitch circle pressure angle αwShear modulus G of the material, elastic modulus E of the material, Poisson ratio mu, rotation speed n and transmission power P;
step two: the radiuses of the sun wheel and the planet wheel are determined to be rfThe relative position relationship between the + c m circle and the base circle defines the relative position relationship between the addendum circle and the base circle of the inner gear ring, and specifically comprises the following steps:
respectively verifying the tooth number of the sun wheel and the planet wheel when the conditions are met
Figure BSA0000212286140000011
The gear base radius rbLess than rf+ c + m when the condition is satisfied
Figure BSA0000212286140000012
The gear base radius rbGreater than rf+ c × m, for the inner ring gear, the addendum diameter d of the inner ring gearaAlways larger than the base circle diameter.
Step three: calculating the meshing stiffness of the single gear teeth of the sun gear, the planet gear and the inner gear ring, and comprising the following steps: shear stiffness ksAxial tension and compression stiffness kaBending stiffness kbContact stiffness khAnd gear base flexible deformation rigidity kfThe specific calculation method is as follows:
1) for sun gear and rowStar wheel with base radius less than rf+ c m, including the case where the radius of the base circle is smaller than the radius of the root circle, i.e.
Figure BSA0000212286140000021
Time, shear stiffness ksAxial tension and compression stiffness kaAnd bending stiffness kbThe calculation formula of (2) is as follows:
Figure BSA0000212286140000022
Figure BSA0000212286140000023
Figure BSA0000212286140000024
rcxrepresents the distance from the current meshing point of the gear teeth (the meshing point always changes between the tooth profile working sections) to the axle center of the gear, McxThe bending moment of the meshing force at the meshing point to the tooth root part is represented by the following calculation formula:
Figure BSA0000212286140000025
Figure BSA0000212286140000026
hfdenotes the chord tooth thickness, r, on the root circlefDenotes the root circle radius, rxRepresenting radius, alpha, at any position on the tooth profilecxIndicating the pressure angle at the current point of engagement,
Figure BSA0000212286140000027
hixchord tooth thickness, h, representing any position of involute profile portionxThe chord tooth thickness at any position on the tooth profile is represented by the following calculation formula:
Figure BSA0000212286140000028
wherein alpha isxIndicating the pressure angle at any position on the tooth profile,
Figure BSA0000212286140000029
Δ h represents the half-chord tooth thickness at the root transition arc, hixAnd Δ h are calculated as:
Figure BSA00002122861400000210
wherein, the calculation modes of a and b are as follows:
b2＝(0.5m(z-2ha*-2c*+2x)+0.38m)2-a2 (8)
Figure BSA00002122861400000211
invα＝tanα-α (10)
contact stiffness khThe calculation formula of (2) is as follows:
Figure BSA00002122861400000212
wherein the content of the first and second substances,
Figure BSA00002122861400000213
ρ1、ρ2radius of curvature of tooth profiles of two gear teeth at contact point positions, respectively, E1、E2Respectively, the modulus of elasticity, μ1、μ2Respectively representing the Poisson's ratio of the material of two teeth, bcShowing the length of the meshing contact line of the two gear teeth. Rho1、ρ2The calculation formula of (2) is as follows:
Figure BSA0000212286140000031
wherein r isx1、rx2Respectively represents the radius r of any point on the involute profiles of the two gearsb1、rb2Respectively, the base radii of the two gears.
Gear matrix flexible deformation rigidity kfThe calculation formula of (2) is as follows:
Figure BSA0000212286140000032
wherein u isfxThe shortest distance from the intersection point of the meshing force extension line and the gear tooth radial symmetry line to the tooth root circle is represented by the following calculation formula:
Figure BSA0000212286140000033
the coefficients L, M, P and Q are represented by XiExpressed, the calculation formula is:
Figure BSA0000212286140000034
Figure BSA0000212286140000035
hfi＝rf/rin (17)
wherein, thetafRepresents the central angle corresponding to the upper half tooth thickness of the tooth root circle of the gear tooth,
Figure BSA0000212286140000036
hfi＝rf/rin。
2) for sun and planet gears, the base radius is greater than rf+ c x m, i.e.
Figure BSA0000212286140000037
And the meshing point varies between the base circle and the addendum circle, the shear stiffness ksAxial tension and compression stiffness kaAnd bending stiffness kbThe calculation formula of (2) is as follows:
Figure BSA0000212286140000038
Figure BSA0000212286140000039
Figure BSA00002122861400000310
the meshing point is between the base circle and rfShear stiffness k when varying between + c m circlessAxial tension and compression stiffness kaAnd bending stiffness kbThe calculation formula of (2) is as follows:
Figure BSA00002122861400000311
Figure BSA00002122861400000312
Figure BSA00002122861400000313
Mcxrepresenting the bending moment of the meshing force at the meshing point on the root portion, hbThe chordal tooth thickness on the gear base circle is expressed by the following calculation formula:
Figure BSA0000212286140000041
Figure BSA0000212286140000042
Figure BSA0000212286140000043
Figure BSA0000212286140000044
Figure BSA0000212286140000045
Figure BSA0000212286140000046
Figure BSA0000212286140000047
Figure BSA0000212286140000048
b2＝(0.5m(z-2ha*-2c*+2x)+0.38m)2-a2 (32)
contact stiffness khThe calculation formula of (2) is as follows:
Figure BSA0000212286140000049
Figure BSA00002122861400000410
Figure BSA00002122861400000411
gear matrix flexible deformation rigidity kfThe calculation formula of (2) is as follows:
Figure BSA00002122861400000412
Figure BSA00002122861400000413
the coefficients L, M, P and Q are represented by XiExpressed, the calculation formula is:
Figure BSA0000212286140000051
Figure BSA0000212286140000052
hfi＝rf/rin (40)
3) for an annulus gear, shear stiffness ksAxial tension and compression stiffness kaAnd bending stiffness kbThe calculation formula of (2) is as follows:
Figure BSA0000212286140000053
Figure BSA0000212286140000054
Figure BSA0000212286140000055
Figure BSA0000212286140000056
Figure BSA0000212286140000057
Figure BSA0000212286140000058
Figure BSA0000212286140000059
contact stiffness khThe calculation formula of (2) is as follows:
Figure BSA00002122861400000510
Figure BSA00002122861400000511
Figure BSA00002122861400000512
gear matrix flexible deformation rigidity kfThe calculation formula of (c) is:
Figure BSA00002122861400000513
Figure BSA00002122861400000514
hfi＝rf/rg (53)
rgrepresenting the radius of the annular gear base body, the coefficients L, M, P and Q being represented by XiExpressed, the calculation formula is:
Figure BSA0000212286140000061
Figure BSA0000212286140000062
hfi＝rf/rin (56)
step four: calculating comprehensive time-varying meshing rigidity comprising external meshing time-varying meshing rigidity k of the sun wheel and the planet wheelspnInner gearing time-varying meshing rigidity k meshed with inner gear ring and planet gearrpn。
The technical scheme of the invention is utilized to manufacture a planetary gear time-varying meshing stiffness calculation method based on a tooth profile correction method. The method firstly defines each basic parameter of the sun wheel, the planet wheel and the inner gear ring, and comprises the following steps: number of teeth z, modulus m, crest height factor haTop clearance coefficient c, pitch circle pressure angle α, displacement coefficient x (positive value for positive displacement and negative value for negative displacement), pitch circle pressure angle αwShear modulus G of the material, elastic modulus E of the material, Poisson's ratio mu, rotation speed n, transmission power P, etc.; then the radiuses of the sun wheel and the planet wheel are determined to be rfThe relative position relation between the circle of +0.25m and the base circle defines the relative position relation between the addendum circle of the inner gear ring and the base circle; and then calculating the meshing rigidity of the single gear teeth of the sun gear, the planet gear and the inner gear ring, wherein the method comprises the following steps: shear stiffness ksAxial tension and compression stiffness kaBending stiffness kbContact stiffness khAnd gear base flexible deformation rigidity kf(ii) a Finally, calculating the comprehensive time-varying meshing stiffness k including the external meshing time-varying meshing stiffness k of the meshing of the sun wheel and the planet wheelspnInner gearing time-varying meshing rigidity k meshed with inner gear ring and planet gearrpn. By the method, different gear types can be reasonably divided according to the tooth number and the meshing mode, and the gear tooth meshing rigidity of the gear in the fixed tooth number interval is reasonably calculated, so that the calculation result is more accurate, the assumption is reduced as much as possible, and the calculation is carried out close to the actual gear tooth shape as much as possible.
The invention has the beneficial effects that:
(1) according to the method, the gear tooth number is accurately divided according to the relation among the tooth number, the displacement coefficient and the pressure angle, so that the tooth profile of the gear is accurately divided, and a solid foundation is provided for accurately calculating the time-varying meshing rigidity of the gear;
(2) the invention carries out necessary modification on the tooth profile, so that the tooth profile is more in line with the actual tooth profile, the traditional method of simplifying the tooth root into a section of straight line is changed, and the accuracy of the calculation of the meshing stiffness is improved;
(3) the influence of the gear deflection coefficient on the meshing rigidity of the gear teeth is fully considered, the application range of the existing deflection gear is quite wide, the shape of the gear teeth can be changed due to the existence of the gear deflection coefficient (the gear teeth can be thickened by positive deflection, and the gear teeth can be thinned by negative deflection), so that the bearing capacity of the gear teeth is increased or reduced, the meshing rigidity of the gear teeth is changed, and the time-varying meshing rigidity of the deflection or non-deflection planetary gear can be accurately calculated;
(4) according to the invention, the calculation formula of the gear time-varying meshing stiffness is deduced from the angle of the chord tooth thickness, so that the calculation complexity is reduced, and the calculation principle is simpler and more intuitive and is easy to understand.
Drawings
FIG. 1 is a work flow chart of a planetary gear time-varying meshing stiffness calculation method based on a tooth profile correction method.
FIG. 2 is a schematic diagram of an experimental planetary transmission system.
FIG. 3 is a schematic diagram of a calculation of the thickness of the teeth of the planet gears.
FIG. 4 is a schematic diagram of the calculation of chordal thickness at the transition arc of the sun and planet gear tooth roots.
Fig. 5 is a schematic representation of parameters of the flexural deformation stiffness of the gear base body of the planet gear and the sun gear.
FIG. 6 is a schematic view of a chordal tooth thickness calculation for a sun gear tooth.
Fig. 7 is a schematic diagram of calculating the chordal thickness of the gear teeth of the inner gear ring.
Fig. 8 is a schematic diagram of the calculation of the chordal thickness at the transition arc of the ring gear tooth root.
FIG. 9 is a representation diagram of various parameters of the flexible deformation rigidity of the gear base body of the inner gear ring.
Fig. 10 is a schematic view of the initial position of each planet in the planetary transmission system.
FIG. 11 is a time varying mesh stiffness k for sun and planet meshingspnA stiffness value map of (2).
FIG. 12 is a time varying meshing stiffness k of the ring gear meshing with the sun gearrpnA stiffness value map of (2).
Detailed Description
The invention is described in detail below with reference to the accompanying drawings, and as shown in fig. 1, a method for calculating time-varying meshing stiffness of a planetary gear based on a tooth profile correction method includes the following specific steps:
the method comprises the following steps: the method is characterized in that basic parameters of the sun gear, the planet gear and the inner gear ring are defined, and the basic parameters comprise: number of teeth z, modulus m, tooth width B, tooth crest height coefficient haTop clearance coefficient c, pitch circle pressure angle α, displacement coefficient x (positive value for positive displacement and negative value for negative displacement), pitch circle pressure angle αwShear modulus G of the material, elastic modulus E of the material, Poisson ratio mu, rotation speed n and transmission power P;
TABLE 1 Gear basic parameters
Figure BSA0000212286140000071
Step two: the radiuses of the sun wheel and the planet wheel are determined to be rfThe relative position relation between the + c x m circle and the base circle defines the relative position relation between the addendum circle of the inner gear ring and the base circle;
step three: calculating the meshing stiffness of the single gear teeth of the sun gear, the planet gear and the inner gear ring, and comprising the following steps: shear stiffness ksAxial tension and compression stiffness kaBending stiffness kbContact stiffness khAnd gear base flexible deformation rigidity kf；
Step four: calculating comprehensive time-varying meshing stiffness including external meshing time-varying meshing stiffness k of the sun wheel and the planet wheelspnInner gearing time-varying meshing rigidity k meshed with inner gear ring and planet gearrpn。
The invention is specifically illustrated by the following specific examples, which are specific implementation processes:
the method comprises the following steps: the method is characterized in that basic parameters of the sun gear, the planet gear and the inner gear ring are defined, and the basic parameters comprise: number of teeth z, modulus m, tooth width B, tooth crest height coefficient haTop clearance coefficient c, pitch circle pressure angle α, displacement coefficient x (positive value for positive displacement and negative value for negative displacement), pitch circle pressure angle αwShear modulus G of the material, elastic modulus E of the material, Poisson ratio mu, rotation speed n and transmission power P;
the basic structure of the planetary transmission system used in this embodiment is as shown in fig. 2, 3 planetary wheels are uniformly distributed, basic parameters of each gear are as shown in table 1, the input rotation speed n of the sun wheel is 1445r/min, and the transmission power is 4000W.
Step two: the radiuses of the sun wheel and the planet wheel are determined to be rfThe relative position relationship between the + c m circle and the base circle defines the relative position relationship between the addendum circle and the base circle of the inner gear ring, and specifically comprises the following steps:
respectively verifying the tooth number of the sun wheel and the planet wheel when the conditions are met
Figure BSA0000212286140000072
The gear base radius rbLess than rf+ c + m when the condition is satisfied
Figure BSA0000212286140000073
The gear base radius rbGreater than rf+ c × m, for the inner gear ring, the addendum circle diameter d of the inner gear ringaAlways larger than the base circle diameter.
Number of sun gear teeth zsNumber of teeth z of planet gear 12pn31, ring gear tooth number zrSun gear shift factor x of 75s0.4152, planetary gear deflection coefficient xpn0.2724, ring gear shift coefficient xrWhen the pressure angle is 0.4 and all the three pressure angles are 20 degrees, the following steps are performed:
Figure BSA0000212286140000081
so that each gear has a radius rfThe relative position relationship between + c m circle and base circle is as follows:
a sun gear: radius of base circle is greater than rf+c*m；
Planet wheel: radius of base circle less than rf+c*m；
An inner gear ring: radius of base circle less than rf+c*m。
The following was verified by direct method:
the calculation formula of the tooth root circle radius of the external gear is as follows:
rf＝0.5m(z-2ha*-2c*+2x) (2)
the calculation formula of the base circle radius of the external gear is as follows:
rb＝0.5mz cosα (3)
the calculation formula of the radius of the tooth root circle of the inner gear ring is as follows:
rf＝0.5m(z+2ha*+2c*+2x) (4)
the calculation formula of the base circle radius of the inner gear ring is as follows:
rb＝0.5mz cosα (5)
the root radius r of the sun gearfs＝0.0103，rfs+0.25 m-0.0108, base radius rbs0.0113; radius r of tooth root of planet gearfpn＝0.0290，rfpn+0.25 m-0.0295, radius of base circle rbpn0.0291; radius r of tooth root circle of inner gear ringfr＝0.0783，rfr+0.25 m-0.0788, radius of base circle rbr＝0.0705。
The verification calculation result is consistent with the conclusion obtained according to the tooth number verification condition.
Step three: calculating the gear tooth meshing rigidity of the sun gear, the planet gear and the inner gear ring, and comprising the following steps of: shear stiffness ksAxial tension and compression stiffness kaBending stiffness kbContact stiffness khAnd gear base flexible deformation rigidity kf；
According to the conclusion in the step two, the planet wheel is calculated according to the following calculation formula, and the gear tooth shear stiffness k of the planet wheelsAxial tension and compression stiffness kaAnd bending stiffness kbThe calculation formula of (2) is as follows:
Figure BSA0000212286140000082
Figure BSA0000212286140000083
Figure BSA0000212286140000084
wherein h isfDenotes the chord tooth thickness, r, on the root circlefDenotes the root circle radius, rcxRepresenting the distance, r, from the current tooth meshing point (which always varies between the tooth profile working sections) to the gear axisxRepresenting radius, alpha, at any position on the tooth profilecxThe pressure angle at the point of engagement is indicated,
Figure BSA0000212286140000085
Mcxthe bending moment of the meshing force at the meshing point to the tooth root part is represented by the following calculation formula:
Figure BSA0000212286140000086
Figure BSA0000212286140000091
hxthe chord tooth thickness at any position on the tooth profile is represented, the calculation schematic diagram is shown in the attached figure 3, and the calculation formula is as follows:
Figure BSA0000212286140000092
wherein alpha isxIndicating the pressure angle at any position on the tooth profile,
Figure BSA0000212286140000093
Δ h represents the half chord tooth thickness at the root transition arc, hixAnd Δ h are calculated as:
Figure BSA0000212286140000094
wherein, the specific representation of a and b is shown in the attached figure 4(a), and the calculation formula is as follows:
b2＝(0.5m(z-2ha*-2c*+2x)+0.38m)2-a2 (13)
Figure BSA0000212286140000095
invα＝tanα-α (15)
contact stiffness khThe calculation formula of (2) is as follows:
Figure BSA0000212286140000096
wherein the content of the first and second substances,
Figure BSA0000212286140000097
ρ1、ρ2radius of curvature of two tooth profiles at the contact point, E1、E2Respectively, the modulus of elasticity, μ1、μ2Respectively representing the Poisson's ratio of the material of two teeth, bcShowing the length of the meshing contact line of the two gear teeth. Rho1、ρ2The calculation formula of (2) is as follows:
Figure BSA0000212286140000098
wherein r isx1、rx2Respectively represents the radius r of any point on the involute profiles of the two gearsb1、rb2Respectively, the base radii of the two gears.
Gear matrix flexible deformation rigidity kfThe calculation formula of (2) is as follows:
Figure BSA0000212286140000099
wherein u isfxThe shortest distance from the intersection point of the meshing force extension line and the gear tooth radial symmetry line to the tooth root circle is represented by the following calculation formula:
Figure BSA00002122861400000910
the coefficients L, M, P and Q are represented by XiExpressed, the calculation formula is:
Figure BSA00002122861400000911
wherein, thetafRepresents the central angle corresponding to the upper half tooth width of the tooth root circle of the gear teeth,
Figure BSA0000212286140000101
hfi＝rf/rin，Ai、Bi、Ci、Di、Ei、Fiis a constant coefficient, and the value is shown in table 2:
values of coefficients in the formula of Table 2
Figure BSA0000212286140000102
The specific representation of each parameter in the flexible deformation rigidity of the gear base body of the planet gear is shown in the attached figure 5.
According to the conclusion in the second step, the sun wheel is calculated according to the following formula, and when the meshing point changes between the base circle and the addendum circle, the shearing rigidity ksAxial tension and compression stiffness kaAnd bending stiffness kbThe calculation formula of (2) is as follows:
Figure BSA0000212286140000103
Figure BSA0000212286140000104
Figure BSA0000212286140000105
the meshing point is between the base circle and rfShear stiffness k when varying between + c m circlessAxial tension and compression stiffness kaAnd bending stiffness kbThe calculation formula of (2) is as follows:
Figure BSA0000212286140000106
Figure BSA0000212286140000107
Figure BSA0000212286140000108
wherein, Δ h, hx、a、b2、MxAnd hbThe isoparametric calculation is as follows, the chordal tooth thickness hxIs shown in FIG. 6, hbRepresenting the chordal tooth thickness on the gear base circle.
Figure BSA0000212286140000109
Figure BSA00002122861400001010
Figure BSA00002122861400001011
Figure BSA00002122861400001012
Figure BSA0000212286140000111
Figure BSA0000212286140000112
Figure BSA0000212286140000113
The concrete representation of a and b in the calculation of the meshing stiffness of the single gear teeth of the sun gear is shown in fig. 4 (b). Contact stiffness khAnd gear base flexible deformation rigidity kfThe formula of (c) is the same as the planetary gear, and is not repeated here. A schematic diagram of the calculation of the gear matrix compliance deflection stiffness of the sun gear teeth is shown in FIG. 5.
According to the conclusion in the step two, the shearing rigidity k of the gear teeth of the inner gear ringsAxial tension and compression stiffness kaAnd bending stiffness kbThe calculation formula of (2) is as follows:
Figure BSA0000212286140000114
Figure BSA0000212286140000115
Figure BSA0000212286140000116
Figure BSA0000212286140000117
Figure BSA0000212286140000118
Figure BSA0000212286140000119
a schematic diagram of the calculation of the chordal thickness of the inner gear ring is shown in the attached FIG. 7, and the concrete representation of a and b in the calculation of the meshing rigidity of the single gear teeth is shown in the attached FIG. 8.
Gear matrix flexible deformation rigidity kfThe calculation formula of (a) requires correction of ufxAnd hfiComprises the following steps:
Figure BSA00002122861400001110
hfi＝rf/rg (41)
wherein r isgThe radius of the ring gear base is indicated. The concrete representation of each parameter in the gear matrix flexible deformation rigidity of the inner gear ring is shown in the attached figure 9.
Contact rigidity k of gear teeth of inner gear ringhAnd other parameters are calculated in the same manner as the planets and will not be repeated here.
Step four: calculating comprehensive time-varying meshing stiffness including external meshing time-varying meshing stiffness k of the sun wheel and the planet wheelspnInner gearing time-varying meshing rigidity k meshed with inner gear ring and planet gearrpn. The specific calculation process is as follows:
firstly, the contact ratio of the two gears is calculated, and the calculation method is as follows:
Figure BSA0000212286140000121
in the formula, alphaaRepresenting a tooth tip circle pressure angle of the gear; when the two gears are externally engaged, plus or minus is selected as a plus sign; when the two gears are in internal engagement, plus or minus signs are taken, and 2 represents the inner gear ring.
The gear pair is always alternately engaged between the single pair of gears and the double pair of gears, and the interval of the single gear engagement interval and the interval of the double gear engagement interval of the gear engagement are calculated in the following way, wherein the interval of the single gear engagement interval is expressed as follows:
Figure BSA0000212286140000122
the double tooth meshing interval is represented as follows:
Figure BSA0000212286140000123
wherein n represents the nth period of alternation of single and double teeth of the gear pair, epsilonrpn、εspnRespectively representing the degree of overlap of the inner and outer meshes, singlespn、singlekrpnRespectively represents a single pair of gear tooth meshing intervals, doublek, where the nth planet gear is respectively meshed with the sun gear and the inner gear ringspn、doublekrpnAnd the meshing intervals of the double pairs of gear teeth, in which the nth planet gear is respectively meshed with the sun gear and the inner gear ring, are respectively shown.
In this embodiment, since the planetary transmission system has three planetary wheels, when each planetary wheel is engaged with the sun gear and each planetary wheel is engaged with the inner gear ring, a certain engagement phase difference exists, and therefore, it is further necessary to determine a phase difference between the three planetary wheels, and the phase difference between each external engagement and the phase difference between each internal engagement are calculated by using the following formula:
Figure BSA0000212286140000124
wherein phi isspne、φrpneRespectively representing the phase difference between the nth external and internal meshing rigidities and the initial phase.
The phase difference between the inner mesh and the outer mesh of the same planet wheel is calculated by adopting the following formula:
Figure BSA0000212286140000125
wherein phispn、φrpnRespectively represents the initial phase of the comprehensive engagement transmission error of the engagement of the nth planet wheel with the sun wheel and the inner gear ringspnrIndicating the phase difference between the outer and inner meshing.
Figure BSA0000212286140000131
ψpn＝ψpn0±ωct (48)
ωcIndicating the rotational speed of the planet carrier, #pn0And the initial position angle of the nth planet wheel is shown, N is the number of the planet wheels, and N is the number of the planet wheels. When the sun gear rotates clockwise, the minus sign is taken, and when the sun gear rotates counterclockwise, the plus sign is taken, and in the embodiment, the sun gear rotates clockwise, and then the minus sign is taken here.
The initial positions of the planet wheels of the planetary transmission system in the embodiment are shown in figure 10.
TABLE 3 planetary gear system engagement phase difference
Figure BSA0000212286140000132
Finally, the variable meshing rigidity k of external meshing is obtainedspnAnd inner meshing time-varying meshing stiffness krpn. Since gear meshing is always performed alternately between a single tooth and a double tooth, the calculation formula of the time-varying meshing stiffness when a pair of gear teeth mesh is as follows:
Figure BSA0000212286140000133
the calculation formula of the time-varying meshing stiffness of the meshing of the two pairs of gear teeth is as follows:
Figure BSA0000212286140000134
kspnthe results are shown in FIG. 11, where k issp1Representing the gear time-varying meshing stiffness, k, of the planet 1 meshing with the sunsp2Representing the gear time-varying meshing stiffness, k, of the planet 2 meshing with the sunsp3Representing the time-varying meshing stiffness of the gear in which the planet wheels 3 mesh with the sun wheel. k is a radical ofrpnThe results are shown in FIG. 12, where k isrp1Representing the time-varying meshing stiffness, k, of the gear in which the planet wheel 1 meshes with the inner gear ringrp2Representing the time-varying meshing stiffness, k, of the gear in which the planet 2 meshes with the inner gear ringrp3Representing the gear time-varying meshing stiffness with which the planet gears 3 mesh with the annulus gear.
The technical solutions described above only represent the preferred technical solutions of the present invention, and some possible modifications to some parts of the technical solutions by those skilled in the art all represent the principles of the present invention, and fall within the protection scope of the present invention.

Claims (1)

1. A planetary gear time-varying meshing stiffness calculation method based on a tooth profile correction method is characterized by comprising the following specific steps:
the method comprises the following steps: the method is characterized in that basic parameters of the sun gear, the planet gear and the inner gear ring are defined, and the basic parameters comprise: number of teeth z, modulus m, tooth width B, tooth crest height coefficient haTop clearance coefficient c, pitch circle pressure angle α, displacement coefficient x (positive value for positive displacement and negative value for negative displacement), pitch circle pressure angle αwShear modulus G of the material, elastic modulus E of the material, Poisson ratio mu, rotation speed n and transmission power P;
step two: the radiuses of the sun wheel and the planet wheel are determined to be rfThe relative position relation between the + c m circle and the base circle, the relative position relation between the addendum circle and the base circle of the inner gear ring, and rfRepresenting the root circle radius;
step three: computingThe single tooth meshing rigidity of sun gear, planet wheel and ring gear includes: shear stiffness ksAxial tension and compression stiffness kaBending stiffness kbContact stiffness khAnd gear base flexible deformation rigidity kf；
Step four: calculating comprehensive time-varying meshing stiffness including external meshing time-varying meshing stiffness k of the sun wheel and the planet wheelspnInner gearing time-varying meshing rigidity k meshed with inner gear ring and planet gearrpn；
In the second step, the radiuses of the sun wheel and the planet wheel are determined to be rfThe relative position relationship between the + c m circle and the base circle defines the relative position relationship between the addendum circle and the base circle of the inner gear ring, and specifically comprises the following steps:
respectively verifying the tooth number of the sun wheel and the planet wheel when the conditions are met
Figure FSB0000198327600000011
The gear base radius rbLess than rf+ c + m when the condition is satisfied
Figure FSB0000198327600000012
The gear base radius rbGreater than rf+ c × m, for the inner gear ring, the addendum circle diameter d of the inner gear ringaIs always larger than the diameter of the base circle;
in the third step, calculating the meshing stiffness of the single gear teeth of the sun gear, the planet gear and the inner gear ring comprises the following steps: shear stiffness ksAxial tension and compression stiffness kaBending stiffness kbContact stiffness khAnd gear base flexible deformation rigidity kfThe specific calculation method is as follows:
1) for sun and planet gears, the base radius is less than rf+ c m, including the case where the radius of the base circle is smaller than the radius of the root circle, i.e.
Figure FSB0000198327600000013
Time, shear stiffness ksAxial tension and compression stiffness kaAnd bending stiffness kbIs calculated by the formula：
Figure FSB0000198327600000014
Figure FSB0000198327600000015
Figure FSB0000198327600000016
Δ h denotes the half chord tooth thickness at the root transition arc, αxIndicating the pressure angle at any position on the tooth profile,
Figure FSB0000198327600000017
hixrepresenting the chordal tooth thickness, r, of any position of the involute profile partxRepresenting radius at any position on the tooth profile, rcxRepresenting the distance, M, from the current tooth meshing point (which always varies between the tooth profile working sections) to the gear axiscxThe bending moment of the meshing force at the meshing point to the tooth root part is represented by the following calculation formula:
Figure FSB0000198327600000018
Figure FSB0000198327600000021
hfdenotes the chord tooth thickness, alpha, on the root circlecxIndicating the pressure angle at the current point of engagement,
Figure FSB0000198327600000022
hxthe chord tooth thickness of any position on the tooth profile is represented by the following calculation formula:
Figure FSB0000198327600000023
wherein h isixAnd Δ h are calculated as:
Figure FSB0000198327600000024
wherein, the calculation modes of a and b are as follows:
b2＝(0.5m(z-2ha*-2c*+2x)+0.38m)2-a2 (8)
Figure FSB0000198327600000025
invα＝tanα-α (10)
contact stiffness khThe calculation formula of (2) is as follows:
Figure FSB0000198327600000026
wherein the content of the first and second substances,
Figure FSB0000198327600000027
ρ1、ρ2radius of curvature of two tooth profiles at the contact point, E1、E2Respectively, the modulus of elasticity, μ1、μ2Respectively representing the Poisson's ratio of the material of two teeth, bcThe length of the meshing contact line of the gear teeth of the two gears is shown by rx1、rx2Respectively represents the radius of any point on the involute profiles of the two gears by rb1、rb2Respectively representing the base radii of the two gears, then ρ1、ρ2The calculation formula of (2) is as follows:
Figure FSB0000198327600000028
gear matrix flexible deformation rigidity kfThe calculation formula of (2) is as follows:
Figure FSB0000198327600000029
wherein u isfxThe shortest distance from the intersection point of the meshing force extension line and the gear tooth radial symmetry line to the tooth root circle is represented by the following calculation formula:
Figure FSB00001983276000000210
the coefficients L, M, P and Q are represented by XiExpressed, the calculation formula is:
Figure FSB0000198327600000031
wherein A isi、Bi、Ci、Di、Ei、FiIs a constant coefficient, θfThe central angle corresponding to the upper half tooth thickness of the tooth root circle of the gear tooth is represented by the following specific calculation formula:
Figure FSB0000198327600000032
hfi＝rf/rin (17)
2) for sun and planet gears, the base radius is greater than rf+ c x m, i.e.
Figure FSB0000198327600000033
And the meshing point varies between the base circle and the addendum circle, the shear stiffness ksAxial tension and compression stiffness kaAnd bending stiffness kbThe calculation formula of (2) is as follows:
Figure FSB0000198327600000034
Figure FSB0000198327600000035
Figure FSB0000198327600000036
hbrepresenting the chordal thickness on the gear base circle with the mesh point between the base circle and rfShear stiffness k when varying between + c m circlessAxial tension and compression stiffness kaAnd bending stiffness kbThe calculation formula of (2) is as follows:
Figure FSB0000198327600000037
Figure FSB0000198327600000038
Figure FSB0000198327600000039
Mcxthe bending moment of the meshing force at the meshing point to the tooth root part is represented by the following calculation formula:
Figure FSB00001983276000000310
Figure FSB00001983276000000311
Figure FSB00001983276000000312
Figure FSB00001983276000000313
Figure FSB0000198327600000041
Figure FSB0000198327600000042
Figure FSB0000198327600000043
Figure FSB0000198327600000044
b2＝(0.5m(z-2ha*-2c*+2x)+0.38m)2-a2 (32)
contact stiffness khThe calculation formula of (2) is as follows:
Figure FSB0000198327600000045
Figure FSB0000198327600000046
Figure FSB0000198327600000047
gear matrix flexible deformation rigidity kfThe calculation formula of (2) is as follows:
Figure FSB0000198327600000048
Figure FSB0000198327600000049
the coefficients L, M, P and Q are represented by XiExpressed, the calculation formula is:
Figure FSB00001983276000000410
Figure FSB00001983276000000411
hfi＝rf/rin (40)
3) for ring gear, shear stiffness ksAxial tension and compression stiffness kaAnd bending stiffness kbThe calculation formula of (c) is:
Figure FSB00001983276000000412
Figure FSB0000198327600000051
Figure FSB0000198327600000052
Figure FSB0000198327600000053
Figure FSB0000198327600000054
Figure FSB0000198327600000055
Figure FSB0000198327600000056
contact stiffness khThe calculation formula of (2) is as follows:
Figure FSB0000198327600000057
Figure FSB0000198327600000058
Figure FSB0000198327600000059
gear matrix flexible deformation rigidity kfThe calculation formula of (2) is as follows:
Figure FSB00001983276000000510
Figure FSB00001983276000000511
hfi＝rf/fg (53)
rgrepresenting the radius of the annular gear base body, the coefficients L, M, P and Q being represented by XiExpressed, the calculation formula is:
Figure FSB00001983276000000512
Figure FSB00001983276000000513
hfi＝rf/rin (56)
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