CN111488682A - Involute helical gear pair tooth width modification dynamic model establishing method - Google Patents

Involute helical gear pair tooth width modification dynamic model establishing method Download PDF

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CN111488682A
CN111488682A CN202010274021.8A CN202010274021A CN111488682A CN 111488682 A CN111488682 A CN 111488682A CN 202010274021 A CN202010274021 A CN 202010274021A CN 111488682 A CN111488682 A CN 111488682A
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gear
tooth
gear pair
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CN111488682B (en
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刘辉
严鹏飞
张晨
高普
贝文瑾
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Beijing Institute of Technology BIT
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Abstract

The invention provides a method for establishing a tooth width modification dynamic model of an involute helical gear pair, which comprises the following steps of: modifying the gear tooth width of the bevel gear pair by adopting a polynomial function modification method; dispersing the helical gear pair into N sheet gear pairs with equal width along the tooth width direction; calculating the dynamic meshing force superposition of the sheet gear pair in a meshing state and obtaining the comprehensive dynamic meshing force of the bevel gear pair; calculating the deflection moment superposition of the sheet gear pair in the meshing state and obtaining the comprehensive swing direction moment of the bevel gear pair; and establishing a dynamic equation of the bevel gear pair, and substituting the comprehensive dynamic meshing force of the bevel gear pair and the comprehensive swinging moment of the bevel gear pair into the dynamic equation. The invention is calculated by dispersing the helical gear pair into N sheet gear pairs with equal width along the tooth width direction, thereby improving the accuracy of the dynamic characteristics of the gear pair with uneven tooth width direction.

Description

Involute helical gear pair tooth width modification dynamic model establishing method
Technical Field
The invention relates to the technical field of mechanical dynamics, in particular to a method for establishing an involute helical gear pair tooth width modification dynamic model.
Background
The helical gear is a high-power heavy-duty gear, the strength requirement on the gear is high, and for a gear pair with tooth-direction meshing deviation and uneven load distribution, the meshing center of the gear teeth is not positioned in the middle of the gear in the tooth surface and the tooth width direction, and even in the gear movement process, the meshing center varies left and right in the tooth width direction. Therefore, it is difficult to analyze the dynamic characteristic law of the gear pair caused by the variation in the tooth width direction.
Therefore, how to improve the accuracy of the dynamic characteristics of the gear pair with non-uniform tooth width direction is an urgent technical problem to be solved by those skilled in the art.
Disclosure of Invention
In view of the above, the present invention provides a method for establishing a tooth width modification dynamic model of an involute helical gear pair, which can improve the accuracy of the dynamic characteristics of the gear pair with non-uniform tooth width.
In order to achieve the purpose, the invention provides the following technical scheme:
a method for establishing an involute helical gear pair tooth width modification dynamic model comprises the following steps:
modifying the gear tooth width of the bevel gear pair by adopting a polynomial function modification method;
dispersing the helical gear pair into N sheet gear pairs with equal width along the tooth width direction;
calculating the dynamic meshing force superposition of the sheet gear pair in a meshing state and obtaining the comprehensive dynamic meshing force of the bevel gear pair;
calculating the deflection moment superposition of the sheet gear pair in a meshing state and obtaining the comprehensive swinging moment of the bevel gear pair;
and establishing a dynamic equation of the bevel gear pair, and substituting the comprehensive dynamic meshing force of the bevel gear pair and the comprehensive swinging moment of the bevel gear pair into the dynamic equation.
In one specific embodiment, the modification quantity formula of the polynomial function is:
Figure BDA0002444145210000021
in the formula,. DELTA.d,iRepresents the modification amount at any position in the tooth width direction, diIs the length of the tooth surface of the bevel gear from the center position, s is the bending index of a polynomial function, b is the tooth direction modification length, deltadThe tooth direction modification amount is d, the tooth width is d, and s is more than or equal to 1 and less than or equal to 5.
In another embodiment, the gear instantaneous pressure angle of the meshed gear pair is:
αL＜αt,i＜αU；
αLat the minimum pressure angle of the bevel gear, αt,iFor the instantaneous pressure angle of the gear of the thin-plate gear pair, i ═ 1,2 represents two bevel gears in the bevel gear pair, α, respectivelyUIs the maximum pressure angle of the helical gear.
In another specific embodiment, the calculation formula of the comprehensive dynamic meshing force of the bevel gear pair is as follows:
Figure BDA0002444145210000022
in the formula (I), the compound is shown in the specification,
Figure BDA0002444145210000023
is the combined dynamic engagement force of said bevel gear pair, Fm,iIs the dynamic meshing force of the ith pair of thin-plate gear pairs in the direction of the meshing line.
In another specific embodiment, the dynamic meshing force of each pair of thin-plate gear pairs in the meshing line direction is calculated by the formula:
Figure BDA0002444145210000024
Figure BDA0002444145210000031
in the formula (I), the compound is shown in the specification,i is 1, …, N, and represents the ith pair of sheet gears, keFor the time varying mesh stiffness of the ith pair of foil gears,
Figure BDA0002444145210000032
damping of the i-th pair of thin-plate gear pairs mesh, m1And m2Respectively the modulus, x, of the bevel gear pair1、x2、y1、y2、z1、z2The coordinate positions r of two bevel gears of the bevel gear pair on a coordinate systemb1And rb2The radiuses of two bevel gears of the bevel gear pair are respectively;
i(b,Δit) is the deformation of the meshing line of the ith pair of the thin plate gears,
Figure BDA0002444145210000033
is the first derivative, Delta, of the distortion of the meshing line of the ith pair of thin-plate gear pairsi(t) is the dynamic transmission error of the ith pair of slice gear pairs,
Figure BDA0002444145210000034
is the first derivative of the dynamic transmission error of the ith pair of the thin-plate gear pairs;
for the sign function, 1 stands for flank engagement, -1 stands for flank engagement, αiIs the dynamic meshing angle, gamma, of the ith pair of laminar gear pairsiIs the relative position angle of the ith pair of sheet gears at any time;
eithe tooth profile deviation caused by the combined error of the ith pair of sheet gear pairs and gear modification, es,iTooth profile error of ith pair of thin-plate gear pairs, ep,iFor the projected equivalent value of the assembly error of the ith pair of thin-plate gear pairs on the meshing line, em,iDeformation of bearing and housing parts to cause non-parallelism of drive shaft and errors in tooth width direction for manufacturing, mounting and wear, eβ,iThe tooth shape deviation caused by the tooth direction modification of the ith pair of the thin-plate gear pairs.
In another particular embodiment of the process of the present invention,
Figure BDA0002444145210000035
Figure BDA0002444145210000041
in the formula,. DELTA.i(t)≥bcosβbWhile being in tooth flank engagement, Δi(t)≤-bcosβbThe tooth back is engaged while the rest is disengaged.
In another specific embodiment, said es,iThe calculation formula of (2) is as follows: e.g. of the types,i＝e0,i+er,isin(2πωt)，e0，i＝0，
Figure BDA0002444145210000042
fpd,iFor gear base pitch deviation, ff,iIs a tooth profile tolerance;
said ep,iThe calculation formula of (2) is as follows: e.g. of the typep,i＝Ap,isin(αi+γ)；
Said em,iThe calculation formula of (2) is as follows:
Figure BDA0002444145210000043
said eβ,iThe calculation formula of (2) is as follows:
Figure BDA0002444145210000044
in another specific embodiment, the calculation formula of the comprehensive swinging moment of the bevel gear pair is as follows:
Figure BDA0002444145210000045
Tm,i＝Fm,i·di；
Tm,ithe deflection moment borne by the ith pair of sheet gear pairs.
In another specific embodiment, the helical gear set comprises a first gear and a second gear;
the dynamic equation of the first gear is as follows:
Figure BDA0002444145210000051
Figure BDA0002444145210000052
Figure BDA0002444145210000053
Figure BDA0002444145210000054
Figure BDA0002444145210000055
Figure BDA0002444145210000056
the dynamic equation of the second gear is as follows:
Figure BDA0002444145210000057
T1、T2input and load torques, k, respectively, of the systemix、kiy、kizAnd cix、ciy、cizRespectively the stiffness and damping of the central bearing of each gear Ixi,Iyi,IziThe rotational inertia of the gear around the x, y and z axes, i ═ 1 and 2, respectively.
In another particular embodiment of the process of the present invention,
Figure BDA0002444145210000058
Figure BDA0002444145210000059
mu is a tooth surface friction coefficient.
The various embodiments according to the invention can be combined as desired, and the embodiments obtained after these combinations are also within the scope of the invention and are part of the specific embodiments of the invention.
According to the technical scheme, the involute helical gear pair tooth width modification dynamic model building method provided by the invention has the advantages that the polynomial function modification method is adopted to modify the tooth width of the gear of the helical gear pair, so that the tooth surface deformation caused by uneven loads at different positions in the tooth width direction can be more accurately reflected; in addition, the invention disperses the helical gear pair into N sheet gear pairs with equal width along the tooth width direction, superposes the dynamic engaging force of the sheet gear pair in the engaging state to obtain the comprehensive dynamic engaging force of the helical gear pair, superposes the deflection moment of the sheet gear pair in the engaging state to obtain the comprehensive swinging moment of the helical gear pair, and substitutes the comprehensive dynamic engaging force of the helical gear pair and the comprehensive swinging moment of the helical gear pair into a kinetic equation to obtain an involute helical gear pair tooth width modification kinetic model, thereby improving the accuracy of the dynamic characteristics of the gear pair with uneven tooth width direction.
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In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings used in the description of the embodiments or the prior art will be briefly described below, it is obvious that the drawings in the following description are only embodiments of the present invention, and for those skilled in the art, other drawings can be obtained according to the provided drawings without creative efforts.
FIG. 1 is a flow chart of a method for establishing an involute helical gear pair tooth width modification dynamic model provided by the invention;
FIG. 2 is a schematic view of a tooth width modification parameter of the helical gear according to the present invention;
FIG. 3 is a schematic view of a gear slice cut provided by the present invention;
FIG. 4 is a schematic view of the gear angles provided by the present invention;
FIG. 5 is a three-dimensional dynamic model of helical gear meshing according to the present invention;
FIG. 6 shows the load distribution factor K under different modification amounts provided by the present inventionhA graph;
FIG. 7 shows the load distribution factor K for different bending indexes provided by the present inventionhA graph;
FIG. 8 shows the tooth dynamic load coefficient K according to the present inventionβA graph of variation with modification;
FIG. 9 shows the tooth direction dynamic load coefficient K provided by the present inventionβA graph of the relationship of the change with the rotating speed;
FIG. 10 shows the tooth dynamic load coefficient K according to the present inventionβA graph of variation with load;
FIG. 11 is a three-dimensional tooth surface contact stress map provided by the present invention;
FIG. 12 is a projection of tooth surface stress provided by the present invention;
FIG. 13 shows the tooth width modification amount Δ provided by the present inventiondWhen the stress distribution is 0, a tooth surface stress distribution three-dimensional graph is obtained;
FIG. 14 shows the tooth width modification amount Δ provided by the present inventiondWhen the stress distribution of the tooth surface is equal to 0, a two-dimensional contour map of the stress distribution of the tooth surface is obtained;
FIG. 15 shows the tooth width modification amount Δ provided by the present inventiondWhen the thickness is 8 mu m, a tooth surface stress distribution three-dimensional graph is obtained;
FIG. 16 shows the tooth width modification amount Δ provided by the present inventiondWhen the thickness is 8 mu m, a two-dimensional contour diagram of tooth surface stress distribution;
FIG. 17 shows the tooth width modification amount Δ provided by the present inventiondWhen the stress distribution is 16 mu m, a tooth surface stress distribution three-dimensional graph is obtained;
FIG. 18 shows the tooth width modification amount Δ provided by the present inventiondWhen the tooth surface stress distribution is 16 mu m, a two-dimensional contour map of the tooth surface stress distribution is obtained;
FIG. 19 shows the tooth width modification amount Δ provided by the present inventiondWhen the stress distribution is 24 mu m, a tooth surface stress distribution three-dimensional graph is obtained;
FIG. 20 shows the tooth width modification amount Δ provided by the present inventiondWhen the tooth surface stress distribution is 24 mu m, a two-dimensional contour map of the tooth surface stress distribution is obtained;
FIG. 21 is a three-dimensional graph of tooth surface stress distribution when the bending index s is 1.5 according to the present invention;
FIG. 22 is a two-dimensional contour plot of tooth surface stress distribution for a bending index s of 1.5 provided by the present invention;
fig. 23 is a three-dimensional graph of tooth surface stress distribution when the bending index s is 2 according to the present invention;
FIG. 24 is a two-dimensional contour plot of tooth flank stress distribution for a bending index s of 2 provided by the present invention;
FIG. 25 is a three-dimensional graph of tooth surface stress distribution when the bending index s is 2.5 according to the present invention;
FIG. 26 is a two-dimensional contour plot of tooth flank stress distribution for a 2.5 bend index s provided by the present invention;
FIG. 27 is a three-dimensional graph of tooth surface stress distribution when the bending index s is 3 according to the present invention;
fig. 28 is a two-dimensional contour diagram of tooth surface stress distribution when the bending index s is 3 according to the present invention.
Detailed Description
In order to make the technical solutions of the present invention better understood, the present invention will be further described in detail with reference to the accompanying drawings and specific embodiments.
As shown in FIG. 1, the invention discloses a method for establishing an involute helical gear pair tooth width modification dynamic model, which comprises the following steps:
step S1: and (5) modifying the gear tooth width of the bevel gear pair by adopting a polynomial function modification method.
It should be noted that the tooth width is subjected to the comprehensive modification, and the same modification parameters, including the tooth direction modification amount Δ, are applied to the left and right sides of the tooth width direction of the gear teeth of the helical gear pairdAxial modification length b and axial modification curve
Figure BDA0002444145210000082
As shown in fig. 2.
Specifically, the invention discloses a modification quantity formula of a polynomial function, which is as follows:
Figure BDA0002444145210000081
in the formula,. DELTA.d,iRepresentsModification amount at any position in tooth width direction, diIs the length of the tooth surface of the helical gear from the center position, s is the bending index of a polynomial function, b is the tooth direction modification length, deltadThe tooth direction modification amount is d, the tooth width is d, and s is more than or equal to 1 and less than or equal to 5.
Thus, a single gear tooth width modification, p ∈ { Δ } can be represented by a parameter set p containing 3 parametersd,b,s}。
Step S2: the helical gear pair is dispersed into N sheet gear pairs with equal width along the tooth width direction.
For a gear pair with uneven load distribution and tooth direction meshing deviation, the meshing center of the gear teeth is not in the middle of the gear in the tooth surface and tooth width direction, and even in the gear movement process, the meshing center varies left and right in the tooth width direction. The load response distribution of the gear tooth meshing force in the tooth width direction cannot be accurately analyzed by adopting the conventional method. The invention analyzes the dynamic characteristic law of the gear system caused by the uneven tooth width direction by establishing a more accurate meshing model.
The only fundamental difference between a helical gear pair and a spur gear pair is that the helical gear system design includes a helix angle parameter β. from another perspective, the spur gear pair is a special helical gear, namely the helix angle value β is 0 degrees, therefore, the helical gear is dispersed into N slices with equal width along the tooth width direction, as shown in fig. 3, each slice is a pair of helical gears with smaller tooth width, the tooth width is delta l, the helix angle is β, and the meshing stiffness of each slice gear tooth is β
Figure BDA0002444145210000091
(i＝1,…,N)。
Step S3: the comprehensive dynamic meshing force of the helical gear pair is obtained by calculating the dynamic meshing force superposition of the sheet gear pair in the meshing state.
For a single pair of gear pairs, the meshing stiffness of the gears can be obtained as long as the meshing position is determined. However, in the case of the helical gear, a series of helical gear pairs for the sheet after the cutting of the sheet are determined not only the meshing position of each sheet but also the respective positionsWhether or not the individual lamellae are in mesh, it is important to determine whether or not the teeth of the individual lamellae are in mesh. As shown in fig. 4, at the first gear input angular velocity ω1Driving the second gear is exemplified.
Figure BDA0002444145210000092
Is constant value, and
Figure BDA0002444145210000093
Figure BDA0002444145210000094
can be expressed as:
Figure BDA0002444145210000095
wherein, t is a time,
Figure BDA0002444145210000096
is composed of
Figure BDA0002444145210000097
The initial minimum angular phase of.
In each slicing gear
Figure BDA0002444145210000098
And
Figure BDA0002444145210000099
the sizes of (i ═ 1,2 for the first gear and the second gear, respectively) can be expressed as;
Figure BDA00024441452100000910
wherein i is the number of the gears in mesh, i is 1, …, N is α'0Representing the face pressure angle of the gear. diIndicating that the axial coordinate of each lamina gear is located in the middle of the lamina gear teeth, L is the center distance of the gear pair, β indicates the helix angle of the helical gear pair.
The instantaneous pressure angle of each thin sheet gear tooth is：
Figure BDA0002444145210000101
The instantaneous pressure angle of the gears of the meshed sheet gear pair is αL＜αt,i＜αU。αLMinimum pressure angle of helical gear, αt,iThe instantaneous pressure angle of the gear wheel of the laminated gear wheel pair, i ═ 1,2, represents two bevel gears in the bevel gear wheel pair, α respectivelyUIs the maximum pressure angle of the helical gear.
Further, the invention discloses a calculation formula of the comprehensive dynamic meshing force of the bevel gear pair, which is as follows:
Figure BDA0002444145210000102
in the formula (I), the compound is shown in the specification,
Figure BDA0002444145210000103
is the combined dynamic engagement force of a helical gear pair, Fm,iIs the dynamic meshing force of the ith pair of thin-plate gear pairs in the direction of the meshing line.
Specifically, the invention discloses a calculation formula of the dynamic meshing force of each pair of sheet gear pairs in the meshing line direction, which is as follows:
Figure BDA0002444145210000104
Figure BDA0002444145210000105
where i is 1, …, N represents the ith pair of sheet gears, keFor the time varying mesh stiffness of the ith pair of foil gears,
Figure BDA0002444145210000106
damping of the i-th pair of thin-plate gear pairs mesh, m1And m2Respectively the modulus, x, of the bevel gear pair1、x2、y1、y2、z1、z2The coordinate positions r of two bevel gears of the bevel gear pair on a coordinate systemb1And rb2The radiuses of two bevel gears of the bevel gear pair are respectively.i(b,Δit) is the deformation of the meshing line of the ith pair of the thin plate gears,
Figure BDA0002444145210000107
is the first derivative, Delta, of the distortion of the meshing line of the ith pair of thin-plate gear pairsi(t) is the dynamic transmission error of the ith pair of slice gear pairs,
Figure BDA0002444145210000108
is the first derivative of the dynamic transmission error of the ith pair of laminated gear pairs, is a sign function, with 1 representing the tooth flank mesh, with-1 representing the tooth flank mesh, αiIs the dynamic meshing angle, gamma, of the ith pair of laminar gear pairsiIs the relative position angle of the ith pair of sheet gears at any time. e.g. of the typeiThe tooth profile deviation caused by the combined error of the ith pair of sheet gear pairs and gear modification, es,iTooth profile error of ith pair of thin-plate gear pairs, ep,iFor the projected equivalent value of the assembly error of the ith pair of thin-plate gear pairs on the meshing line, em,iDeformation of bearing and housing parts to cause non-parallelism of drive shaft and errors in tooth width direction for manufacturing, mounting and wear, eβ,iThe tooth shape deviation caused by the tooth direction modification of the ith pair of the thin-plate gear pairs.
The invention fully considers various errors existing in the meshing of the bevel gear pair and further improves the modification precision.
In particular, the invention discloses
Figure BDA0002444145210000111
Figure BDA0002444145210000112
In the formula,. DELTA.i(t)≥bcosβbWhile being in tooth flank engagement, Δi(t)≤-bcosβbThe tooth back is engaged while the rest is disengaged.
Further, the present invention specifically discloses es,iThe calculation formula of (2) is as follows: e.g. of the types,i＝e0,i+er,isin(2πωt)，e0，i＝0，
Figure BDA0002444145210000116
fpd，iFor gear base pitch deviation, ff，iIs a tooth profile tolerance;
ep,ithe calculation formula of (2) is as follows: e.g. of the typep,i＝Ap,isin(αi+γ)；
em,iThe calculation formula of (2) is as follows:
Figure BDA0002444145210000114
eβ,ithe calculation formula of (2) is as follows:
Figure BDA0002444145210000115
step S4: and calculating the deflection moment superposition of the sheet gear pair in the meshing state and obtaining the comprehensive swing direction moment of the bevel gear pair.
The calculation formula of the comprehensive swinging moment of the bevel gear pair is as follows:
Figure BDA0002444145210000121
Tm,i＝Fm,i·di；Tm,ithe deflection moment borne by the ith pair of sheet gear pairs.
Step S5: and establishing a dynamic equation of the bevel gear pair, and substituting the comprehensive dynamic meshing force of the bevel gear pair and the comprehensive swinging moment of the bevel gear pair into the dynamic equation.
As shown in fig. 5, in particular, the present invention discloses that the helical gear pair comprises a first gear and a second gear, and the kinetic equation of the first gear is:
Figure BDA0002444145210000122
Figure BDA0002444145210000123
Figure BDA0002444145210000124
Figure BDA0002444145210000125
Figure BDA0002444145210000126
Figure BDA0002444145210000127
the dynamic equation of the second gear is as follows:
Figure BDA0002444145210000128
T1、T2input and load torques, k, respectively, of the systemix、kiy、kizAnd cix、ciy、cizRespectively the stiffness and damping of the central bearing of each gear Ixi,Iyi,IziThe rotational inertia of the gear around the x, y and z axes, i ═ 1 and 2, respectively.
Further, the invention discloses
Figure BDA0002444145210000129
Figure BDA00024441452100001210
Mu is a tooth surface friction coefficient.
In order to prove the accuracy of the method disclosed by the invention, the dynamic load distribution state of the helical gear pair in the tooth width direction is analyzed to verify, the dynamic load distribution factor of each thin-plate gear pair and the tooth-direction dynamic load coefficient of a gear system are defined, and the expressions are respectively as follows:
Figure BDA0002444145210000131
Figure BDA0002444145210000132
wherein (F)m,i)RMSFor the root mean square value of the dynamic meshing force of the ith pair of thin-plate gear pairs, [ (F)m,i)RMS]maxIs the maximum value of the root mean square value of the dynamic meshing force of the thin-plate gear pair, FnThe actual load on the tooth surface under the quasi-static condition, and N is the total number of the thin-plate gear pairs in the meshing state after slicing. The dynamic load distribution factor is a parameter related to the dynamic meshing force borne by the tooth surface of each thin-plate gear pair and can represent the distribution state of the dynamic load of the gear pair in the tooth width direction; the tooth direction dynamic load coefficient is a parameter related to the average value of dynamic loads borne by the tooth surface of each thin-plate gear pair and the maximum load, and can be used for expressing the uneven degree of the dynamic loads of the bevel gear pair distributed in the tooth width direction.
Example one
Under the conditions that the input rotation speed is 2000r/min and the load torque is 100Nm, the modification length b is d/2 (namely, full-tooth-width modification) and the bending index s of the s-polynomial function is 2, the distribution state of the dynamic load distribution factor in the tooth width direction under different modification amounts is studied, and the result is shown in fig. 6. Wherein the maximum modification amount delta of the tooth widthdThe values are respectively [0 μm,8 μm,16 μm and 24 μm]. It can be seen from the figure that in the non-deformed state, the load in the tooth width direction is not uniform in actual operation due to errors such as gear installation and bearing deformation. Maximum value of load (K)h,i1.41) is concentrated at one end in the tooth width direction, and the minimum value (K) ish,i0.62) is centered on the other end of the tooth width. As the modification amount increases, the load maximum value in the tooth width direction gradually approaches the center position of the tooth surface, and the maximum load distribution factor gradually decreases. The modification amount at the tooth width is deltadIn the 24 μm state, the maximum load distribution factor is located at 4.2mm from the center of the face width. At this time, the maximum load distribution factor is Kh,i1.19, the load fluctuation amplitude in the face-width direction is significantly reduced compared with the gear pair in the standard state. From the results, it can be seen that the tooth face load overconfluence can be improved by the tooth width modification method under the appropriate modification amountThe state of (1); meanwhile, the tooth load distribution center can be changed, so that the tooth load center is gradually close to the center of the tooth surface.
Under the working condition, the modification length b is d/2, and the maximum modification amount delta of the tooth widthdThe distribution state of the dynamic load distribution factor in the tooth width direction at the bending index of different polynomial functions under the condition of 24 μm is shown in fig. 7. Wherein the bending index s is [1.5,2,2.5,3 ]]. As can be seen from the figure, the change in the bending index does not significantly change the maximum and minimum values of the dynamic load in the tooth width direction, nor has a large influence on the center of the distribution of the dynamic load in the tooth width direction, but affects the distribution shape of the dynamic load in the tooth width direction. Along with the gradual increase of the bending index, the load change from the tooth to the position near the dynamic load distribution center is gradually smooth, and the load distribution is more uniform. The maximum dynamic load in the tooth direction is slightly reduced compared to the profile curve with the smaller bending index. From the results, it can be seen that with the tooth width modification method, under different bending indexes of polynomial functions, the tooth direction dynamic load center position does not change obviously, but the load change curvature near the load center position is influenced. That is, under this condition, the larger bending index of the polynomial function is used, so that the load distribution near the load center position is more uniform.
Under the same rotating speed and load working condition, the influence rule of the tooth width modification quantity on the tooth direction dynamic load coefficient under the bending index conditions of different polynomial functions is shown in fig. 8. Wherein the bending index s is [1.5,2,2.5,3 ]]And d/2 is the trimming length b. As can be seen from the figure, for each bending index, an optimal modification amount exists, so that the dynamic load coefficient reaches an optimal value, and the optimal modification amount is about 23 μm. Secondly, it can be seen that the dynamic load coefficient when the bending index s is 3 can optimally reach about 1.64 under the optimal modification amount; the dynamic load coefficient when the bending index s is 1.5 can be optimally about 1.81. This indicates that, with the optimum modification amount, a large bending index can reduce the uneven distribution of the dynamic load of the helical gear pair in the tooth width direction to some extent. It can also be derived from the figures when takingAt a smaller tooth width modification amount (modification amount Δ)d< 9 μm), the use of a larger bending index may result in a less effective profile modification than a profile modification gear corresponding to a smaller bending index. When a smaller modification amount is needed, the modification effect is better by adopting a modification curve with a lower bending index.
Under the condition that the bending index s is 3 and the modification length b is d/2 under the working condition that the load torque is 100Nm, the change rule of the tooth direction dynamic load coefficient of the gear system with different tooth width modification amounts along with the rotating speed is shown in FIG. 9. Wherein the maximum modification amount delta of the tooth widthdThe values are respectively [0 μm,8 μm,16 μm and 24 μm]The variation range of the rotating speed is [100r/min, 2500r/min]. The result shows that under different modification quantities, the tooth direction dynamic load coefficients are in a gradually increasing and finally stable variation trend along with the increase of the rotating speed. Compared with a standard gear system without modification, the gear pair with modified tooth direction can reduce dynamic load in the whole rotating speed working range. And the tooth direction dynamic load of the modified gear is reduced more obviously along with the increase of the rotating speed. When the rotating speed is 2500r/min, the tooth direction modification amount is deltadThe gear pair profile modification effect was most pronounced at 24 μm, with a tooth dynamic load factor of 1.96, compared to the results (K) in the non-modified stateβ2.21), the modification reduces the tooth direction dynamic load coefficient by 11.31%.
Under the working condition that the input rotating speed is 2000r/min, the tooth width modification quantity is deltadThe result of the law of the change of the tooth direction dynamic load factor of the gear system with the load torque under the condition that the modification length b is d/2 and the bending index of the polynomial function is as shown in fig. 10. Wherein the bending index s of the polynomial function takes on values of [1.5,2,2.5 and 3 respectively]The load variation range is [0Nm, 800Nm]. From the results, it can be seen that the tooth dynamic load coefficient can be better reduced by using a smaller bending index under a higher load condition. When the load is 800Nm, the optimal tooth direction dynamic load coefficient corresponding to the bending index s of 1.5 is about 1.71, and the tooth direction dynamic load coefficient corresponding to the bending index s of 3 is KβCompared with a modified gear with a bending index s of 1.5, the modified gear with the bending index s of 1.80 is optimized by 5.3 percent compared with a modified gear with the bending index s of 3. In addition, the method can be used for producing a composite materialNote that under the condition of lower load, the corresponding dynamic load coefficient is also larger. It can be seen that the tooth dynamic load factor can be better reduced by using a larger bending index. It can be analyzed by combining the results shown in fig. 7 that, under the condition of lower load, because the elastic deformation of the loaded tooth surface is smaller, the load is too concentrated at the position of the load distribution center of the tooth surface by adopting a smaller bending index, and the reduction of the dynamic load coefficient in the tooth direction is not as obvious as that of a larger bending index.
Therefore, under the bending indexes of different polynomial functions, the optimal tooth width modification amount can be found, so that the tooth direction dynamic load coefficient of the helical gear pair is minimum. By adopting the method provided by the invention, the uneven distribution degree of the dynamic load in the tooth width direction can be relieved, the maximum value of the dynamic load in the whole tooth width direction can be reduced, and the load distribution center in the tooth width direction can be improved, so that the load center position is gradually close to the center of the tooth surface, and the side deflection moment caused by the misalignment of the load center position can be reduced. Under the optimal modification amount, when the load is fixed, the corresponding modification gears under different bending indexes can reduce the dynamic load in the tooth width direction in the whole working rotating speed range. When the tooth width modification amount is large, the dynamic load change near the load center position is more smooth by adopting a large bending index and is superior to the modification gear corresponding to a small bending index; when the amount of the modification of the tooth width is small, the modification effect is better by using a smaller bending index. When the load is larger, because the tooth surface is deformed greatly under load, the shape modification effect with smaller bending index is due to the shape modification gear corresponding to the larger bending index; however, when the load is small, the use of the modification curve with a small bending index can lead to too concentrated load distribution, and the modification effect is not as good as that of the modification gear corresponding to the large bending index.
Example two
Considering the sliced sheet gear teeth as the curvature radius rho at the contact point1,ρ2Is the contact of a pair of cylinders with radius, when two cylinders with parallel axes are contacted and pressed with each other under the action of load (as shown in figures 11 and 12), the contact line is due to local elastic deformationTo become an elongated contact strip with a width H, the contact width H, which can be found by elasto-mechanical theory, can be expressed as:
Figure BDA0002444145210000161
in the formula, FnNormal pressure to the cylinder, i.e. engagement force F to the thin-plate gear pairm,i(ii) a l is the length of the contact line, namely the tooth width d/N of the sheet in the meshing state; mu.s1And mu1Is the poisson ratio of two cylinder materials; e1And E2Is the elastic modulus of two cylindrical materials; rho1And ρ2The radius of the two cylinders, namely the instantaneous radius of the teeth of the sheet gear pair.
When the two gears are in meshing extrusion, the pressure applied to the tooth surface contact area is equal in magnitude and opposite in direction, so that the stress and the strain applied to the tooth surface are equal in magnitude and opposite in direction, as shown in fig. 12. The surface-induced local stress in the contact region is called contact stress, and the maximum contact stressHOccurs on the theoretical contact line where the deformation is the largest and can be expressed as shown in formula:
Figure BDA0002444145210000171
the basic formula of the maximum contact stress of the ith pair of thin-plate gear pairs is as follows:
Figure BDA0002444145210000172
according to the meshing dynamics model calculation process and the tooth flank contact touch stress analysis method provided by the invention, firstly, under the steady-state working condition that the input rotating speed is 2000r/min and the load torque is 500Nm, the modification length b is d/2 (namely, full tooth width modification) and the bending index s of a polynomial function is 2.5, the dynamic contact stress distribution states of the tooth flank of the first gear under different tooth width modification amounts are respectively researched. The results of the three-dimensional graph and the two-dimensional contour graph of the stress distribution of the tooth surface at different tooth width relief quantities are shown in fig. 13-20. Wherein the maximum modification amount delta of the tooth widthdThe values are respectively [0 μm,8 μm,16 μm and 24 μm]. As can be seen from fig. 13 and 14, the relief amount Δ at the tooth widthdWhen the gear is in a standard gear state without modification, the tooth surface contact is uneven due to errors such as manufacturing installation and bearing deformation, and the tooth surface unbalance loading phenomenon occurs. Wherein the maximum flank stress occurs at one axial end of the tooth with a lower contact ratio and a maximum of about 419 MPa. As can be seen from the two-dimensional contour lines, the dynamic contact stress of the gear pair tooth surface in the non-modification state is mainly concentrated on the tooth surface width of [ -22.5mm, -10mm at the tooth surface contact position of more than 400MPa]Within the range of (1). As can be seen from FIGS. 15-16, the amount of modification ΔdUnder the condition of 8 mu m, the end face value with larger tooth width stress is reduced after the modification, and the maximum stress on the tooth surface is about 408 MPa. As can be seen from the two-dimensional contour lines, the dynamic tooth surface contact stress of the gear pair in the modification state is mainly concentrated on the tooth surface width of [ -22.5mm, -3mm at the tooth surface contact position exceeding 380MPa]Within the range of (1). As can be seen from FIGS. 17-18, the amount of modification ΔdUnder the condition of 16 mu m, the maximum stress on the tooth surface is about 379MPa after the modification. As can be seen from the two-dimensional contour lines, the dynamic tooth surface contact stress of the gear pair in the modification state is mainly concentrated on the tooth surface width of [ -19mm, 3mm at the tooth surface contact position exceeding 360MPa]Within the range of (1). As can be seen from FIGS. 19-20, the amount of modification ΔdUnder the condition of 24 mu m, the maximum stress on the tooth surface is about 368MPa after the modification. As can be seen from the two-dimensional contour lines, the dynamic tooth surface contact stress of the gear pair in the modification state is mainly concentrated on the tooth surface width of [ -15mm, 5mm ] at the tooth surface contact position exceeding 350MPa]Within the range of (1). The result shows that the tooth width modification can effectively reduce the stress concentration phenomenon at the tooth end and reduce the maximum contact stress of the tooth surface. In addition, the tooth width modification can also reduce the distance between the load distribution center and the tooth surface center position, and relieve the side deflection phenomenon of the gear.
Under the steady-state working condition that the input rotating speed is 2000r/min and the load torque is 500Nm, the modification length b is d/2, and the tooth width modification amount is deltadUnder the condition of 24 mu m, the tooth surface dynamic contact stress distribution state of the first gear under different bending indexes. The results of the three-dimensional and two-dimensional contour plots of the stress distribution of the tooth surface at different bending indices are shown in FIGS. 21-28. Wherein the bending index s of the polynomial function takes on values of [1.5,2,2.5 and 3 respectively]. FromIt can be seen from the three-dimensional graph that no matter which bending index modification is adopted, the tooth width end face contact stress concentration phenomenon can be remarkably reduced under the modification amount, and the tooth face contact stress center is enabled to be close to the tooth face center. In addition, the maximum value of the contact stress in all of the four bending indexes is about 412 MPa. Further comparing the two-dimensional contour diagrams of the contact stress under the four bending indexes, it can be seen that when the bending indexes are sequentially valued as [1.5,2,2.5,3 ]]The tooth width ranges of the tooth surface contact stress exceeding 400MPa are [ -4.5mm and 3.5mm respectively]，[-7mm，4mm]，[-12.5mm，5mm]，[-15mm，6mm]The tooth width span ranges of the contact stress peak are respectively 8mm, 11mm, 17.5mm and 21 mm. It can be seen that the peak distribution of the tooth surface contact stress under the working condition is relatively uniform along with the increase of the modification index. From the analysis result, under the condition of certain input working condition and modification quantity, the change of the bending index of the polynomial function does not obviously change the peak value of the tooth surface contact dynamic stress and the stress center position, but influences the distribution uniformity state near the peak value of the contact dynamic stress.
It is noted that, herein, relational terms such as first and second, and the like may be used solely to distinguish one entity or action from another entity or action without necessarily requiring or implying any actual such relationship or order between such entities or actions.
The previous description of the disclosed embodiments is provided to enable any person skilled in the art to make or use the present invention. Various modifications to the embodiments will be readily apparent to those skilled in the art, and the generic principles defined herein may be applied to other embodiments without departing from the spirit or scope of the invention. Thus, the present invention is not intended to be limited to the embodiments shown herein but is to be accorded the widest scope consistent with the principles and inventive features disclosed herein.

Claims (10)

1. A method for establishing an involute helical gear pair tooth width modification dynamic model is characterized by comprising the following steps:
modifying the gear tooth width of the bevel gear pair by adopting a polynomial function modification method;
dispersing the helical gear pair into N sheet gear pairs with equal width along the tooth width direction;
calculating the dynamic meshing force superposition of the sheet gear pair in a meshing state and obtaining the comprehensive dynamic meshing force of the bevel gear pair;
calculating the deflection moment superposition of the sheet gear pair in a meshing state and obtaining the comprehensive swinging moment of the bevel gear pair;
and establishing a dynamic equation of the bevel gear pair, and substituting the comprehensive dynamic meshing force of the bevel gear pair and the comprehensive swinging moment of the bevel gear pair into the dynamic equation.
2. The method for establishing the involute helical gear pair tooth width modification dynamic model according to claim 1, wherein a modification quantity formula of the polynomial function is as follows:
Figure FDA0002444145200000011
in the formula,. DELTA.d,iRepresents the modification amount at any position in the tooth width direction, diIs the length of the tooth surface of the bevel gear from the center position, s is the bending index of a polynomial function, b is the tooth direction modification length, deltadThe tooth direction modification amount is d, the tooth width is d, and s is more than or equal to 1 and less than or equal to 5.
3. The involute helical gear pair tooth width modification dynamic model building method according to claim 1, wherein the gear instantaneous pressure angle of the meshed thin plate gear pair is:
αL＜αt,i＜αU；
αLat the minimum pressure angle of the bevel gear, αt,iFor the instantaneous pressure angle of the gear of the thin-plate gear pair, i ═ 1,2 represents two bevel gears in the bevel gear pair, α, respectivelyUIs the maximum pressure angle of the helical gear.
4. The method for establishing the involute helical gear pair tooth width modification dynamic model according to claim 1, wherein a comprehensive dynamic meshing force calculation formula of the helical gear pair is as follows:
Figure FDA0002444145200000021
in the formula (I), the compound is shown in the specification,
Figure FDA0002444145200000022
is the combined dynamic engagement force of said bevel gear pair, Fm,iIs the dynamic meshing force of the ith pair of thin-plate gear pairs in the direction of the meshing line.
5. The method for establishing the involute helical gear pair tooth width modification dynamic model according to claim 4, wherein a dynamic meshing force calculation formula of each pair of thin plate gear pairs in a meshing line direction is as follows:
Figure FDA0002444145200000023
Figure FDA0002444145200000024
Figure FDA0002444145200000025
ei＝es,i+ep,i+em,i+eβ,i；
where i is 1, …, N represents the ith pair of sheet gears, keFor the time varying mesh stiffness of the ith pair of foil gears,
Figure FDA0002444145200000026
damping of the i-th pair of thin-plate gear pairs mesh, m1And m2Respectively the modulus, x, of the bevel gear pair1、x2、y1、y2、z1、z2The coordinate positions r of two bevel gears of the bevel gear pair on a coordinate systemb1And rb2The radiuses of two bevel gears of the bevel gear pair are respectively;
i(b,Δit) is the deformation of the meshing line of the ith pair of the thin plate gears,
Figure FDA0002444145200000027
is the first derivative, Delta, of the distortion of the meshing line of the ith pair of thin-plate gear pairsi(t) is the dynamic transmission error of the ith pair of slice gear pairs,
Figure FDA0002444145200000028
is the first derivative of the dynamic transmission error of the ith pair of the thin-plate gear pairs;
for the sign function, 1 stands for flank engagement, -1 stands for flank engagement, αiIs the dynamic meshing angle, gamma, of the ith pair of laminar gear pairsiIs the relative position angle of the ith pair of sheet gears at any time;
eithe tooth profile deviation caused by the combined error of the ith pair of sheet gear pairs and gear modification, es,iTooth profile error of ith pair of thin-plate gear pairs, ep,iFor the projected equivalent value of the assembly error of the ith pair of thin-plate gear pairs on the meshing line, em,iDeformation of bearing and housing parts to cause non-parallelism of drive shaft and errors in tooth width direction for manufacturing, mounting and wear, eβ,iThe tooth shape deviation caused by the tooth direction modification of the ith pair of the thin-plate gear pairs.
6. The involute helical gear pair tooth width modification dynamic model building method of claim 5, wherein,
Figure FDA0002444145200000031
Figure FDA0002444145200000032
in the formula,. DELTA.i(t)≥bcosβbWhile being in tooth flank engagement, Δi(t)≤-bcosβbThe tooth back is engaged while the rest is disengaged.
7. The method for establishing the involute helical gear pair tooth width modification kinetic model of claim 6, wherein e iss,iThe calculation formula of (2) is as follows: e.g. of the types,i＝e0,i+er,isin(2πωt)，e0,i＝0，
Figure FDA0002444145200000033
fpd,iFor gear base pitch deviation, ff,iIs a tooth profile tolerance;
said ep,iThe calculation formula of (2) is as follows: e.g. of the typep,i＝Ap,isin(αi+γ)；
Said em,iThe calculation formula of (2) is as follows:
Figure FDA0002444145200000034
said eβ,iThe calculation formula of (2) is as follows:
Figure FDA0002444145200000035
8. the method for establishing the involute helical gear pair tooth width modification dynamic model according to claim 7, wherein a calculation formula of a comprehensive swinging moment of the helical gear pair is as follows:
Figure FDA0002444145200000041
Tm,i＝Fm,i·di；
Tm,ithe deflection moment borne by the ith pair of sheet gear pairs.
9. The involute bevel gear pair tooth width modification kinetic model creation method of claim 8, wherein the bevel gear pair comprises a first gear and a second gear;
the dynamic equation of the first gear is as follows:
Figure FDA0002444145200000042
Figure FDA0002444145200000043
Figure FDA0002444145200000044
Figure FDA0002444145200000045
Figure FDA0002444145200000046
Figure FDA0002444145200000047
the dynamic equation of the second gear is as follows:
Figure FDA0002444145200000048
T1、T2input and load torques, k, respectively, of the systemix、kiy、kizAnd cix、ciy、cizRespectively the stiffness and damping of the central bearing of each gear Ixi,Iyi,IziThe rotational inertia of the gear around the x, y and z axes, i ═ 1 and 2, respectively.
10. The involute bevel gear pair tooth width modification dynamic die of claim 9A pattern creation method, characterized in that,
Figure FDA0002444145200000049
Figure FDA00024441452000000410
mu is a tooth surface friction coefficient.
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Cited By (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN112059082A (en) * 2020-09-11 2020-12-11 汉德车桥(株洲)齿轮有限公司 Involute spline key tooth modification design method and machining method and gear product thereof
CN112836319A (en) * 2021-03-11 2021-05-25 西南交通大学 Simulation method considering non-uniformly distributed tooth root crack faults
CN113158479A (en) * 2021-04-29 2021-07-23 清华大学 Method for calculating transmission efficiency of cylindrical gear, computer device and readable storage medium

Citations (4)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
JP2006090465A (en) * 2004-09-24 2006-04-06 Fuji Heavy Ind Ltd Conical shape involute gear pair
JP2014098438A (en) * 2012-11-14 2014-05-29 O-Oka Corp Double helical gear
CN107153736A (en) * 2017-05-11 2017-09-12 东北大学 The gear pair meshing characteristic analysis method to correction of the flank shape is roused in a kind of consideration of amendment
CN107391876A (en) * 2017-08-11 2017-11-24 东北大学 Helical gear pair time-variant mesh stiffness computational methods

Patent Citations (4)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
JP2006090465A (en) * 2004-09-24 2006-04-06 Fuji Heavy Ind Ltd Conical shape involute gear pair
JP2014098438A (en) * 2012-11-14 2014-05-29 O-Oka Corp Double helical gear
CN107153736A (en) * 2017-05-11 2017-09-12 东北大学 The gear pair meshing characteristic analysis method to correction of the flank shape is roused in a kind of consideration of amendment
CN107391876A (en) * 2017-08-11 2017-11-24 东北大学 Helical gear pair time-variant mesh stiffness computational methods

Non-Patent Citations (2)

* Cited by examiner, † Cited by third party
Title
王奇斌等: "齿向修形直齿轮系统动力学特性分析", 《振动工程学报》 *
王成: "基于齿轮传动系统横－扭－摆耦合", 《振动与冲击》 *

Cited By (5)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN112059082A (en) * 2020-09-11 2020-12-11 汉德车桥(株洲)齿轮有限公司 Involute spline key tooth modification design method and machining method and gear product thereof
CN112059082B (en) * 2020-09-11 2022-08-09 汉德车桥(株洲)齿轮有限公司 Involute spline key tooth modification design method and machining method and gear product thereof
CN112836319A (en) * 2021-03-11 2021-05-25 西南交通大学 Simulation method considering non-uniformly distributed tooth root crack faults
CN113158479A (en) * 2021-04-29 2021-07-23 清华大学 Method for calculating transmission efficiency of cylindrical gear, computer device and readable storage medium
CN113158479B (en) * 2021-04-29 2024-03-01 清华大学 Cylindrical gear transmission efficiency calculation method, computer device and readable storage medium

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