CN113010975A - Gear clearance optimization design method comprehensively considering machining cost and motion stability - Google Patents

Gear clearance optimization design method comprehensively considering machining cost and motion stability Download PDF

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CN113010975A
CN113010975A CN202110087695.1A CN202110087695A CN113010975A CN 113010975 A CN113010975 A CN 113010975A CN 202110087695 A CN202110087695 A CN 202110087695A CN 113010975 A CN113010975 A CN 113010975A
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高志慧
王昀
田刚
刘鹏
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Abstract

The invention relates to a method for optimally designing gear clearances of a gear transmission system needing long-term operation, such as wind power generation and the like, which can effectively solve the influence of environmental factors such as system load, rotating speed and the like and clearance coupling effect on clearance design. According to the invention, by counting in a gap coupling model, adopting a single-degree-of-freedom radial meshing force model and utilizing a meshing force model counting in energy loss, a solving result can better reflect the actual time domain and frequency domain characteristics of the system. The invention utilizes a clearance dynamics model to count and solve different radial clearances and tooth side clearances, and selects a more reasonable gear clearance by analyzing a spectrogram and comprehensively considering motion stability and processing difficulty and cost of different clearances of the gear. The invention abandons the traditional gear transmission system clearance design to select the tooth side clearance and the radial clearance according to the center distance, and considers the influence on the clearance under the working conditions of different rotating speeds, loads and the like, thereby realizing the clearance with lower cost and having good motion stability.

Description

Gear clearance optimization design method comprehensively considering machining cost and motion stability
Technical Field
The invention relates to a gear clearance optimization design method of a gear transmission system, which starts from the fact that multiple clearances exist in the actual gear transmission system, accurately describes the clearances and the coupling effect thereof, analyzes the motion stability by utilizing the frequency domain characteristics, considers the difficulty and the cost of gear clearance processing and adjustment, optimally designs the gear clearances, and particularly can be applied to the clearance optimization field of the gear transmission system which needs long-term operation and mass production, such as wind power generation and the like.
Background
The gear transmission system is widely applied in the fields of wind power and the like, and the transmission performance of the gear transmission system directly influences the integral running state of mechanical equipment. The mutual coupling of the clearances can seriously affect the dynamic performance and the motion stability of the gear transmission system and restrict the application of the gear transmission system.
Gear backlash is an important design parameter in gear drive system design, and mainly includes backlash and radial backlash. The reasonability of the clearance design profoundly influences the service life and the motion stability of the gear transmission system. Too large clearance can lead to the reduction of transmission performance of the gear transmission system, and too small clearance can lead to the increase of processing difficulty and cost and the deterioration of lubrication condition of the gear transmission system. The motion stability is an important characteristic for measuring the system reasonableness, the motion stability is mainly reflected in time domains and frequency domains of parameters such as meshing force and gear rotating speed, and the concentration degree of meshing frequency can be used as an important standard for judging the motion stability of the gear transmission system. At present, the value ranges of the tooth side clearance and the radial clearance are mainly determined according to the center distance in the traditional gear transmission system clearance design. The influence of the gap coupling effect on the transmission stability of the gear transmission system is mostly ignored, and the influence of factors such as the load and the rotating speed of the gear transmission system on the gap design is ignored, so that the selected gap range is large, and the effect optimization of the design of the gear transmission system cannot be realized. The backlash design of the gear system has been a design difficulty.
At present, most of collision force is greatly simplified in the dynamic research of a gear transmission system, and the research on the gap coupling effect is not common, so that the multi-gap coupling dynamic characteristic of the actual gear transmission system is difficult to accurately describe, and the dynamic research cannot provide effective guiding significance for the optimal design of the gear gap.
In view of this, the invention provides a gear clearance optimization design method of a gear transmission system based on a multi-clearance coupling dynamic model of a gear. The method not only can effectively account for the influence of system load, rotating speed and the like on the clearance design, but also better accounts for the influence of the clearance coupling effect on the clearance design, thereby optimizing the benefit of the clearance design on the gear transmission system.
Disclosure of Invention
The technical problem solved by the invention is as follows: the invention provides a gear clearance optimization design method of a gear transmission system, which is mainly used for clearance optimization design of single-stage or multi-stage spur gears and aims to solve the benefit problem of clearance design of the existing gear transmission system. Specific gear pair parameters are selected for a gear transmission system, motion parameters such as the rotating speed and the load of the gear transmission system are considered, a kinetic equation is established, the frequency domain characteristic of the kinetic equation is analyzed, the gear processing difficulty and the gear processing cost are comprehensively considered, and a gear transmission system optimal clearance value is selected.
A gear clearance optimization design method comprehensively considering machining cost and motion stability comprises the following specific steps:
the method comprises the following steps: a gap coupling model is established, and the gap coupling model,
the gear transmission system has radial clearance and tooth flank clearance, and the motion state of the radial clearance is judged by adopting a two-state method, which can be described as
Figure BDA0002909402280000021
Wherein e isabFor the gap vectors, describe the model transient state, crIs the bearing radial clearance.
When the gear transmission system operates, the radial clearance and the tooth side clearance are in a coupling relation, and the coupling relation can be described as
Figure BDA0002909402280000022
Wherein, btFor dynamic backlash, A0、α0Is the center distance and pressure angle in ideal gear transmission system (no clearance), A 'and alpha' are the center distance and pressure angle in actual state, b0Is the initial flank clearance;
step two: establishing a radial collision force model, and describing the radial force of the bearing by adopting a nonlinear spring-damping impact collision model with energy loss taken into account to obtain a required radial collision force model;
step three: establishing a dynamic meshing force model, and describing the dynamic meshing force of the gear according to the gear parameters and the tooth flank dynamic clearance model aiming at the gear pair needing to be optimized and processed
Figure BDA0002909402280000023
Wherein, KtIs the time-varying meshing stiffness of the gear pair, ftFor dynamic backlash of the gear, DtIn order to have a non-linear meshing damping coefficient,
Figure BDA0002909402280000024
the relative speed of the two gears when colliding;
step four: establishing a multi-gap coupled gear system kinetic equation, establishing a torsional freedom degree along the axial direction of each gear, performing kinetic modeling along two freedom degrees in an axial vertical plane, establishing a kinetic equation, and establishing a six-freedom-degree gap kinetic equation by a single pair of gears, which can be described as
Figure BDA0002909402280000031
Wherein, IpAnd IgRotational inertia, T, of the driving and driven wheels respectivelypIs the driving torque of the driving gear, RpAnd RgRespectively the radius of the driving wheel and the driven wheel,
Figure BDA0002909402280000032
and
Figure BDA0002909402280000033
is the angular acceleration of the driving and driven wheels, mpAnd mgThe masses of the driving and driven wheels, respectively, TgIs the driving torque of the driving gear and,
Figure BDA0002909402280000034
and
Figure BDA0002909402280000035
the components of the acceleration of the driving and driven wheels in the x and y directions, Frpx(t)、Frpy(t)、Frgx(t)、Frgy(t) components of the radial impact forces of the driving and driven wheels in the x and y directions, respectively, Ft(t) is dynamic engagement force, Ftpx(t)、Ftpy(t)、Ftgx(t)、Ftgy(t) are the components of the dynamic engaging force of the driving wheel and the driven wheel in the x direction and the y direction respectively;
step five: solving a multi-gap coupling nonlinear dynamics differential equation to obtain a time history graph of the gear meshing force, performing Fourier transform on the time history graph to obtain a frequency spectrum graph of the meshing force, finding a main peak value of the meshing frequency in the graph, and setting a half-power bandwidth as a frequency width under different radial gap and tooth side gap combinations to serve as a reference value of motion stability.
Step six: and changing the values of the radial clearance and the backlash, and repeating the steps from the first step to the fifth step to obtain the bandwidth values under different clearance combinations. The machining cost and the frequency width are comprehensively considered, and a proper gap combination is selected. For example, as shown in the following table, with reference to group 1, both the flank clearance and the radial clearance values for this group are the greatest. Comparing group 1 with group 2, C2/C1 is 2, D1/D2 is 4, i.e. compared with group 1, the cost is increased by 2 times, the bandwidth is reduced by 0.25 times, so group 2 gap combination is better than group 1. Comparing group 1 with group 3, C3/C1 is 8, D1/D3 is 5, i.e. compared with group 1, the cost is increased by 8 times, the bandwidth is reduced by 0.2 times, so group 1 gap combination is better than group 3. And comprehensively comparing, and identifying the clearance of the 2 nd group as the optimal combination under the load and the rotating speed.
Serial number Backlash of teeth Radial clearance Cost of Bandwidth of
1 A1 B1 C1 D1
2 A2 B2 C2 D2
3 A3 B3 C3 D3
Preferably, the involute straight gear is taken as a main optimization target, and under different use environments (including load, rotating speed and the like), even the same pair of gears can be designed by selecting different proper gaps.
Preferably, in step one, the description of the coupling effect of the radial clearance and the backlash takes into account the relative displacement g of the gear in the direction of the meshing line on the basis of the dynamic backlashtObtaining an accurate flank clearance bump function of
Figure BDA0002909402280000041
Preferably, the description of the radial impact force takes into account a single-degree-of-freedom model of the geometric characteristics between the shaft and the shaft sleeve, and the impact forces of two degrees of freedom in a plane are coupled through the geometric characteristics between the shaft and the shaft sleeve to establish the radial impact force model.
Preferably, in the description of the meshing force, the influence of the dynamic flank clearance is taken into account, and a clearance coupling term is added, so that the equation dynamic characteristic is more accurate.
Preferably, on the basis of specific gear parameters needing to be machined, combined solving analysis of different gaps is carried out, and a meshing force time history chart and a frequency spectrum under various combinations are obtained.
Preferably, in the fifth step, the half-power bandwidth is used as a reference basis for motion stationarity, and the smaller the bandwidth, the higher the motion stationarity.
Preferably, for the comprehensive consideration of the motion stability and the clearance, the comprehensive consideration is carried out on the reduction range of the half-power frequency width of the meshing frequency on the basis of the gear processing difficulty and the cost improvement caused by the clearance reduction, and a more proper gear is selected to design the clearance.
Compared with the prior art, the invention has the advantages that:
(1) the traditional gear transmission system mainly determines the value ranges of the tooth side clearance and the radial clearance according to the center distance, and even if the load and the rotating speed are different, the selected values of the clearances are the same, so that the motion stability of the system has great difference when the load and the rotating speed are different. The invention takes the parameters of load, rotating speed and the like as optimization parameters from the viewpoint of motion stability, comprehensively considers the processing cost and selects reasonable gaps, thereby leading the gap selection to be more targeted and leading the motion stability to be better under specific working conditions.
(2) The multi-gap coupling collision dynamics model provided by the invention gives the design steps of the specific parameters of the model, adopts the radial force model considering the geometric characteristics between the shaft and the shaft sleeve, adopts the gap coupling model to describe the dynamic change characteristics of the radial gap and the tooth side gap, improves the fit degree of the solution result and the actual situation, and ensures that the optimal design of the gap is more practical.
Drawings
FIG. 1 is a simplified model schematic of a gear system according to the present invention;
FIG. 2 is a flow chart of a method for optimally designing a clearance of a gear transmission system according to the present invention;
FIG. 3 is a time history chart of the gear meshing force under the condition that the radial clearance of the driving wheel is 0mm, the radial clearance of the driven wheel is 0.01mm, and the tooth flank clearance is 0.05 mm;
fig. 4 is a graph of the frequency spectrum corresponding to the gear mesh force of fig. 2.
The symbols in fig. 1 represent the following:
1 denotes a driven gear, 2 denotes a driven wheel inertial load, 3 denotes a driven wheel collision shaft, 4 denotes a driving wheel, 5 denotes a motor and an output shaft, and 6 denotes a system frame.
Detailed Description
The present invention will be described in further detail with reference to the following drawings and specific examples, but the present invention is not limited thereto.
The invention provides a gear clearance optimization design method suitable for a gear transmission system needing long-term operation, such as wind power generation and the like, wherein a multi-clearance coupling dynamic model is used for carrying out dynamic modeling on a gear to be machined, a practical application scene is simulated, the gear motion stability is analyzed, the corresponding clearance machining cost and difficulty are integrated, and clearance optimal solution design is realized.
In order to clearly and concisely illustrate the optimized gap machining method proposed by the present invention, a specific kinetic modeling analysis is performed herein. The method comprises the following specific steps:
the method comprises the following steps: and establishing a gap coupling model.
For the simplified model shown in fig. 1, the radial gap runout of the central shaft and the bearing of the gear transmission system is simplified into a collision model of the shaft and the shaft sleeve, the configuration block on the driven wheel is used for simulating the inertia load on the gear output shaft, the gear pair has a tooth side gap, the center distance is changed due to the existence of the radial gap, the tooth side gap is changed, the gear radial runout is influenced by the change of the meshing force, and the two have a mutual coupling effect. When the gear transmission system works, the shaft and the shaft sleeve are in two states of collision and separation, so a two-state model method is adopted for describing the radial clearance of the bearing.
Known bearing radial clearance crClearance vector e between shaft and sleeveabAnd when the determination is carried out, the current state of the model is judged. The gear radial clearance impact function can be described as
Figure BDA0002909402280000051
During the collision of the shaft with the shaft sleeve, the relative collision velocity is an important parameter in modeling the radial force, and can be described as
Figure BDA0002909402280000052
Wherein the content of the first and second substances,
Figure BDA0002909402280000053
is the critical velocity vector when the shaft collides with the sleeve.
And obtaining the geometric relation between the actual center distance and the tooth side clearance through the gear meshing relation. When gears are meshed, the pressure angle is changed due to the change of the center distance caused by radial runout, and the change relation of the pressure angle is described as
Figure BDA0002909402280000061
Obtaining dynamic tooth flank clearance b in a coupling state through the change relation between the geometrical relation of gear meshing and the center distancetCan be described as
Figure BDA0002909402280000062
Wherein A is0、α0The center distance and the pressure angle of an ideal gear transmission system (without clearance) state are obtained; a 'and alpha' are the center distance and the pressure angle in the actual state; b0Is the initial flank clearance.
When the gears rotate, the relative displacement along the meshing line is mainly generated by the deviation of the rotation angular displacement of the driving wheel and the driven wheel and the displacement along the meshing line direction caused by radial run-out, so that the relative displacement g of the driving gear and the driven gear in the meshing line directiontCan be described as
gt＝r1θ1-r2θ2+(x1-x2)sinα'+(y1-y2)cosα' (5)
In the formula: r is1、r2The pitch circle radius of the driving wheel and the driven wheel respectively; theta1、θ2Angular displacement of the driving wheel and the driven wheel respectively; x is the number of1Respectively y of1The displacement of the driving wheel along the x direction and the y direction when the radial runout occurs; x is the number of2、y2Respectively the displacement of the driven wheel in the x and y directions when radial runout occurs.
Taking into account the relative displacement g of the gear in the direction of the meshing line on the basis of the dynamic backlashtObtaining an accurate flank clearance bump function of
Figure BDA0002909402280000063
Step two: and establishing a radial collision force model.
Bearing radial contact force FrCan be tracedIs described as
Figure BDA0002909402280000064
Wherein, KrFor non-linear contact stiffness coefficient, DrFor nonlinear damping coefficients, n is the force index (typically taken to be 1.5).
In order to accurately depict the radial collision effect of the gear, a nonlinear spring-damping impact collision model with energy loss is adopted, the influence of the size of a gap on the relation between load and deformation is effectively taken into account, and the energy loss in the collision process is fully considered. By rigidity K for non-linear contactrAnd nonlinear damping coefficient DrThe correction of the model makes the correlation coefficient of the model better reflect the viscous phenomenon in the collision process and is not limited by gaps, loads and recovery coefficients.
Figure BDA0002909402280000065
Figure BDA0002909402280000071
Wherein L is the axial dimension of the bearing, namely the length of the shaft sleeve, Delta R is the radial difference between the shaft and the shaft sleeve, namely the radial clearance, E*Is an equivalent modulus of elasticity, CrTo restore the coefficient, KrIn order to have a non-linear contact stiffness,
Figure BDA0002909402280000072
when δ is 0, that is, the relative collision velocity of the shaft and the sleeve in the critical contact state.
The combined formulas (5), (6) and (7) can obtain a bearing radial force model of
Figure BDA0002909402280000073
On the basis of the radial force model of the formula (8), the geometric characteristics of a shaft/shaft sleeve are taken into account, a simple two-degree-of-freedom model with the radial impact force of the gear being mutually independent in the orthogonal direction in a two-dimensional coordinate system is converted into a single-degree-of-freedom model with the orthogonal direction having coupling property caused by the geometric characteristics of the shaft/shaft sleeve, and the radial force of the gear is obtained
Figure BDA0002909402280000074
Figure BDA0002909402280000075
Figure BDA0002909402280000076
Step three: and establishing a dynamic meshing force model.
For a single pair of gears requiring optimized machining, the dynamic gear mesh force can be described as the gear dynamic mesh force according to the gear parameters and the flank dynamic clearance model
Figure BDA0002909402280000077
Wherein, KtIs the time-varying meshing stiffness of the gear pair, ftFor dynamic backlash of the gear, DtIn order to have a non-linear meshing damping coefficient,
Figure BDA0002909402280000078
is the relative speed of the two gears at the time of impact.
During the transmission, the time-varying meshing stiffness K of the gearstThe gear tooth comprehensive time-varying meshing stiffness is approximately a periodic function of a discharge waveform, so that the time-varying stiffness is expressed in a Fourier series form
Figure BDA0002909402280000079
Figure BDA0002909402280000081
Figure BDA0002909402280000082
Figure BDA0002909402280000083
Wherein k ispFor single tooth pair mesh stiffness, kaIs the average meshing rigidity, and epsilon is the overlap ratio of the end faces of the gear pair, omegamFor the gear mesh frequency, m is the module of the gear pair and B is the tooth width.
Nonlinear mesh damping coefficient DtMainly used for describing energy loss in the gear tooth meshing process in a model, and the calculation formula is
Figure BDA0002909402280000084
Wherein xi is a damping ratio and ranges from 0.03 to 0.17, xi is 0.06 according to experience,
Figure BDA0002909402280000085
is equivalent mass.
For the calculation of gear mesh forces with backlash, the backlash collision function f is usedtJudging the current state between the driving gear and the driven gear, and determining the dynamic meshing force, wherein the expression of the dynamic meshing force of the gears obtained by combining the formulas (6), (13a), (13b), (13c), (13d) and (14) is shown as
Figure BDA0002909402280000086
Step four: and establishing a dynamic equation of the multi-gap coupled gear system.
A torsional freedom degree along the axial direction of each gear is established, dynamic modeling is carried out along two freedom degrees in the vertical plane of the shaft, a dynamic equation is established, a clearance dynamic equation with six freedom degrees can be established by a single pair of gears, and the clearance dynamic equation can be described as
Figure BDA0002909402280000087
Wherein, IpAnd IgRotational inertia, T, of the driving and driven wheels respectivelypIs the driving torque of the driving gear, RpAnd RgRespectively the radius of the driving wheel and the driven wheel,
Figure BDA0002909402280000088
and
Figure BDA0002909402280000089
is the angular acceleration of the driving and driven wheels, mpAnd mgThe masses of the driving and driven wheels, respectively, TgIs the driving torque of the driving gear and,
Figure BDA00029094022800000810
and
Figure BDA00029094022800000811
the components of the acceleration of the driving and driven wheels in the x and y directions, Frpx(t)、Frpy(t)、Frgx(t)、Frgy(t) components of the radial impact forces of the driving and driven wheels in the x and y directions, respectively, Ft(t) is dynamic engagement force, Ftpx(t)、Ftpy(t)、Ftgx(t)、FtgyAnd (t) are the components of the dynamic meshing force of the driving wheel and the driven wheel in the x direction and the y direction respectively.
The system dynamics equation can be described as the integration of the radial impact and meshing force models
Figure BDA0002909402280000091
Wherein, alpha'pAnd alpha'gThe actual pressure angles of the driving wheel and the driven wheel are respectively; and x and y are unit vectors in the directions of x and y in the inertial coordinate system respectively.
Step five: according to gear model parameters of a gap to be optimized, a specific dynamic model is established, a multi-gap coupling nonlinear dynamic differential equation is solved, a time history graph of gear meshing force is obtained, Fourier transformation is carried out on the time history graph, a frequency spectrum graph of the meshing force is obtained, a main peak value of meshing frequency in the graph is found, and a half-power bandwidth is set as a frequency width under different radial gap and tooth side gap combinations and serves as a reference value of motion stability.
Step six: and (4) for designing a gear model, changing the values of the radial clearance and the backlash, repeating the steps from the first step to the fifth step, and solving the frequency width values under different clearance combinations. The machining cost and the frequency width are comprehensively considered, and a proper gap combination is selected. For example, as shown in the following table, with reference to group 1, both the flank clearance and the radial clearance values for this group are the greatest. Comparing group 1 with group 2, C2/C1 is 2, D1/D2 is 4, i.e. compared with group 1, the cost is increased by 2 times, the bandwidth is reduced by 0.25 times, so group 2 gap combination is better than group 1. Comparing group 1 with group 3, C3/C1 is 8, D1/D3 is 5, i.e. compared with group 1, the cost is increased by 8 times, the bandwidth is reduced by 0.2 times, so group 1 gap combination is better than group 3. And comprehensively comparing, and identifying the clearance of the 2 nd group as the optimal combination under the load and the rotating speed.
Serial number Backlash of teeth Radial clearance Cost of Bandwidth of
1 A1 B1 C1 D1
2 A2 B2 C2 D2
3 A3 B3 C3 D3
To further illustrate the superiority of the present invention, the following examples are given. The calculation example starts from the angle of motion stability, and calculates the bandwidth by taking the radial clearance and the tooth side clearance as known, and counting a collision model and a clearance coupling model.
The parameters of the gear pair required to be machined are designed as follows: the number of teeth of the driving wheel and the driven wheel is 25, the modulus is 2, the pressure angle is 20 degrees, the gear tooth thickness is 15mm, the inertial load of the balancing weight is 0.41kg, and the rotational inertia is 200 kg.mm2The mass of the driving wheel and the driven wheel is 0.207kg, and the rotational inertia is 70.791kg mm2The driving wheel speed is 60 rpm. Establishing a gear pair dynamic model through the parameters, combining and substituting different clearance values, solving a time history graph of the gear pair meshing force by a Runge-Kutta method, carrying out Fourier transformation on the time history graph to obtain a meshing force spectrogram, carrying out comparative analysis on the gear meshing force spectrograms under different clearance combinations, and carrying out meshing frequency analysis on the gear meshing force spectrogram by using meshing frequencyThe nearby bandwidth value is the dominant consideration value. FIG. 2 is a time history of the gear meshing force when the radial clearance of the driving wheel is 0mm, the radial clearance of the driven wheel is 0.01mm, and the backlash is 0.05 mm. Performing spectrum analysis to obtain the spectrogram of FIG. 3, and solving to obtain the required meshing frequency frThe nearby bandwidth is 10 Hz. The required bandwidth is obtained through multiple clearance combination, the gear clearance machining and adjusting difficulty and cost are integrated, and a proper clearance value is selected to serve as a design value of the gear clearance in the operating environment, so that the operating life of the gear transmission system is further prolonged.
The above description is only a specific embodiment of the gear clearance optimization design method for a gear transmission system requiring long-term operation, such as wind power generation, etc., and is for facilitating those skilled in the art to understand and apply the present invention within the technical scope of the present disclosure, but the protection scope of the present invention is not limited thereto, and variations or various substitutions of equivalent principles and methods without creative efforts based on the technical solution are within the protection scope of the present invention.

Claims (8)

1. A gear optimization design method comprehensively considering processing cost and motion stability is characterized by comprising the following specific steps:
the method comprises the following steps: a gap coupling model is established, and the gap coupling model,
the gear transmission system has radial clearance and tooth flank clearance, and the motion state of the radial clearance is judged by adopting a two-state method, which can be described as
Figure FDA0002909402270000011
Wherein e isabFor the gap vectors, describe the model transient state, crIs the bearing radial clearance.
When the gear transmission system operates, the radial clearance and the tooth side clearance are in a coupling relation, and the coupling relation can be described as
Figure FDA0002909402270000012
Wherein, btFor dynamic backlash, A0、α0Is the center distance and pressure angle in ideal gear transmission system (no clearance), A 'and alpha' are the center distance and pressure angle in actual state, b0Is the initial flank clearance;
step two: establishing a radial collision force model, and describing the radial force of the bearing by adopting a nonlinear spring-damping impact collision model with energy loss taken into account to obtain a required radial collision force model;
step three: establishing a dynamic meshing force model, and describing the dynamic meshing force of the gear according to the gear parameters and the tooth flank dynamic clearance model aiming at the gear pair needing to be optimized and processed
Figure FDA0002909402270000013
Wherein, KtIs the time-varying meshing stiffness of the gear pair, ftFor dynamic backlash of the gear, DtIn order to have a non-linear meshing damping coefficient,
Figure FDA0002909402270000014
the relative speed of the two gears when colliding;
step four: establishing a multi-gap coupled gear system kinetic equation, establishing a torsional degree of freedom along the axial direction of each gear, performing kinetic modeling along two degrees of freedom in an axial vertical plane, establishing a kinetic equation, and establishing a six-degree-of-freedom gap kinetic equation by a single pair of gears, which can be described as
Figure FDA0002909402270000015
Wherein, IpAnd IgRotational inertia, T, of the driving and driven wheels respectivelypIs the driving torque of the driving gear, RpAnd RgRespectively the radius of the driving wheel and the driven wheel,
Figure FDA0002909402270000021
and
Figure FDA0002909402270000022
is the angular acceleration of the driving and driven wheels, mpAnd mgThe masses of the driving and driven wheels, respectively, TgIs the driving torque of the driving gear and,
Figure FDA0002909402270000023
and
Figure FDA0002909402270000024
the components of the acceleration of the driving and driven wheels in the x and y directions, Frpx(t)、Frpy(t)、Frgx(t)、Frgy(t) components of the radial impact forces of the driving and driven wheels in the x and y directions, respectively, Ft(t) is dynamic engagement force, Ftpx(t)、Ftpy(t)、Ftgx(t)、Ftgy(t) are the components of the dynamic engaging force of the driving wheel and the driven wheel in the x direction and the y direction respectively;
step five: solving a multi-gap coupling nonlinear dynamics differential equation to obtain a time history graph of the gear meshing force, performing Fourier transform on the time history graph to obtain a frequency spectrum graph of the meshing force, finding a main peak value of the meshing frequency in the graph, and setting a half-power bandwidth as a frequency width under different radial gap and tooth side gap combinations to serve as a reference value of motion stability.
Step six: and changing the values of the radial clearance and the backlash, and repeating the steps from the first step to the fifth step to obtain the bandwidth values under different clearance combinations. The machining cost and the frequency width are comprehensively considered, and a proper gap combination is selected. For example, as shown in the following table, with reference to group 1, both the flank clearance and the radial clearance values for this group are the greatest. Comparing group 1 with group 2, C2/C1 is 2, D1/D2 is 4, i.e. compared with group 1, the cost is increased by 2 times, the bandwidth is reduced by 0.25 times, so group 2 gap combination is better than group 1. Comparing group 1 with group 3, C3/C1 is 8, D1/D3 is 5, i.e. compared with group 1, the cost is increased by 8 times, the bandwidth is reduced by 0.2 times, so group 1 gap combination is better than group 3. And comprehensively comparing, and identifying the clearance of the 2 nd group as the optimal combination under the load and the rotating speed.
Serial number Backlash of teeth Radial clearance Cost of Bandwidth of 1 A1 B1 C1 D1 2 A2 B2 C2 D2 3 A3 B3 C3 D3
2. The optimal machining method for the gears comprehensively considering the machining cost and the motion stability as claimed in claim 1 is characterized in that an involute straight gear is taken as a main optimization target, and under different use environments (including loads, rotating speeds and the like), even the gears of the same pair can be designed by selecting different proper gaps.
3. The method for optimizing the machining of the gear according to the combination of the machining cost and the motion stability as claimed in claim 1, wherein in the step one, the description of the coupling effect of the radial clearance and the backlash takes into account the relative displacement g of the gear in the direction of the meshing line on the basis of the dynamic backlashtObtaining an accurate flank clearance bump function of
Figure FDA0002909402270000025
4. The optimal gear machining method comprehensively considering machining cost and motion stability as claimed in claim 1, wherein a single-degree-of-freedom model of geometric features between a shaft and a shaft sleeve is considered for describing the radial impact force, and the impact forces of two degrees of freedom in a plane are coupled through the geometric features between the shaft and the shaft sleeve to establish the radial impact force model.
5. The method for optimizing the machining of the gear with the comprehensive consideration of the machining cost and the motion stability as claimed in claim 1, wherein the influence of dynamic backlash is taken into account in the description of the meshing force, and a backlash coupling term is added, so that the equation dynamic characteristic is more accurate.
6. The optimal gear machining method comprehensively considering machining cost and motion stability as claimed in claim 1, wherein on the basis of specific gear parameters to be machined, combined solution analysis of different gaps is performed to obtain meshing force time history maps and frequency spectrograms under various combinations.
7. The method as claimed in claim 1, wherein in step five, the half power bandwidth is used as a reference for motion stability, and the smaller the bandwidth, the higher the motion stability.
8. The optimal gear machining method with comprehensive consideration of machining cost and motion stability as claimed in claim 7, wherein for the comprehensive consideration of motion stability and backlash, the comprehensive consideration is given to the reduction range of half-power bandwidth of meshing frequency on the basis of the gear machining difficulty and cost increase caused by backlash reduction, and a proper gear is selected to design the backlash.
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