CN107391876A - Helical gear pair time-variant mesh stiffness computational methods - Google Patents
Helical gear pair time-variant mesh stiffness computational methods Download PDFInfo
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Abstract
本发明属于机械动力学技术领域，具体涉及斜齿轮副时变啮合刚度计算方法。斜齿轮副时变啮合刚度计算方法包括如下步骤：步骤S1：获取斜齿轮副的基本参数；步骤S2：将斜齿轮副的轮齿模型沿齿宽方向分解为N个独立且均匀的薄片直齿轮副；步骤S3：确定N个薄片直齿轮在啮合位置j是否参与啮合；步骤S4：计算每一片薄片直齿轮的时变啮合刚度；步骤S5：对每一片薄片直齿轮的时变啮合刚度求和，得到斜齿轮副的时变啮合刚度。斜齿轮副时变啮合刚度计算方法考虑了非线性接触、修正基体刚度及延长啮合影响，改进了解析模型，提高了计算精度。
The invention belongs to the technical field of mechanical dynamics, and in particular relates to a calculation method for time-varying meshing stiffness of a helical gear pair. The calculation method of the time-varying meshing stiffness of the helical gear pair includes the following steps: Step S1: Obtain the basic parameters of the helical gear pair; Step S2: Decompose the tooth model of the helical gear pair into N independent and uniform spur gears along the tooth width direction pair; step S3: determine whether the N thin spur gears are engaged in meshing at the meshing position j; step S4: calculate the time-varying meshing stiffness of each thin spur gear; step S5: sum the time-varying meshing stiffness of each thin spur gear , to obtain the time-varying mesh stiffness of the helical gear pair. The calculation method of time-varying meshing stiffness of helical gear pairs takes into account the effects of nonlinear contact, modified matrix stiffness and extended meshing, improves the analytical model, and improves the calculation accuracy.
Description
Technical Field
The invention belongs to the technical field of mechanical dynamics, and particularly relates to a method for calculating time-varying meshing stiffness of a bevel gear pair.
Background
The gear transmission can be a transmission form which is most widely applied in the field of mechanical transmission, and the wide application not only has the advantages of accurate transmission, high efficiency, reliable work, long service life and the like, but also is beneficial to the fact that the gear transmission has a set of relatively complete international standards and the development of the gear transmission technology reaches a certain level.
Just because the gear transmission is indispensable in the mechanical field, the gear transmission is attracting more and more scholars at home and abroad to invest in the research of the gear transmission, and the development of the gear technology is dedicated to be promoted.
With the development and progress of gear research, the actual problems to be solved are more and more complex, and how to improve the precision, the efficiency and the bearing capacity of gear power transmission, how to reduce vibration and noise, how to select materials and the like are all problems to be solved urgently. The research direction is also gradually from straight gears to helical gears, and from healthy gear research to faulty gear research, some gear defects (such as tooth profile error, shaft misalignment and the like) need to be taken into consideration.
Although the existing research has a certain foundation, the requirements of engineering practice cannot be completely met, the influence of factors such as tooth profile errors on actual gear pair meshing is not deeply researched, and a deeper theory is needed in the actual engineering practice to guide design and reduce the loss rate.
The gear time-varying meshing stiffness is a basis for researching the meshing characteristics of the gear pair, and the development of a gear time-varying meshing stiffness modeling method is also an important index for measuring gear transmission research.
With the progressive improvement of the modeling method, a great number of scholars are also dedicated to generalizing the gear modeling method to be capable of considering more factors in actual meshing situations, such as matrix correction, prolonged meshing, load distribution, non-linear contact and the like.
The time-varying meshing stiffness of the gear has a great influence on the vibration response of the gear system, and how to calculate the time-varying meshing stiffness is important work for gear research.
At present, the time-varying meshing stiffness of the helical gear pair is calculated mainly in several ways: empirical formula method, analytic method, Finite Element (FE) method, and analytic-finite element method. Although the empirical formula method is simple and convenient to calculate, the calculation accuracy is not enough, and the empirical formula method can be used for rough calculation. The existing analytical method has limitations on the research of the shape-correcting gear pair, and the limitations are reflected in that the influences of nonlinear contact, correction of matrix rigidity and extension of meshing are not considered. The calculation results of the finite element method and the analytic-finite element method are closer to the real meshing process of the gear pair, but the finite element method and the analytic-finite element method have the defects of more complex modeling process, low calculation efficiency and the like.
Disclosure of Invention
Technical problem to be solved
In order to solve the above problems in the prior art, the invention provides a method for calculating time-varying meshing stiffness of a helical gear pair, which is used for solving the limitation existing in the conventional analytic method for calculating the time-varying meshing stiffness of the helical gear pair, and the limitation is that the nonlinear contact, the matrix stiffness correction and the meshing influence extension are not considered.
(II) technical scheme
In order to achieve the purpose, the invention adopts the main technical scheme that:
the invention provides a method for calculating time-varying meshing stiffness of a bevel gear pair, which comprises the following steps of:
step S1: acquiring basic parameters of a bevel gear pair;
step S2: decomposing a gear tooth model of the helical gear pair into N independent and uniform sheet straight gear pairs along the tooth width direction;
step S3: determining whether the N sheet spur gears participate in meshing at a meshing position j;
step S4: calculating the time-varying meshing stiffness of each sheet straight gear by a gear pair meshing characteristic analysis method considering nonlinear contact, matrix stiffness correction and meshing influence prolonging;
step S5: and summing the time-varying meshing stiffness of each sheet of straight gear to obtain the time-varying meshing stiffness of the bevel gear pair.
Further, step S3 includes the following steps:
step S31 determining maximum mesh angle α of the bevel gear pair at mesh position jmaxAnd minimum engagement angle αmin；
Step S32, determining a pressure angle of the nth-plate chip spur gear at the meshing position j (α)n)j；
Step S33: and judging whether the nth sheet straight gear participates in meshing.
Further, in step S31:
maximum meshing angle α of bevel gear pair at meshing position jmaxThe calculation formula of (a) is as follows:
helical gear pair at meshing position jMinimum engagement angle αminThe calculation formula of (a) is as follows:
in the formula, rb1The radius of the base circle of the driving wheel; r isb2α is the base radius of the driven wheel0Is a pressure angle; r isa1The radius of the addendum circle of the driving wheel αa2The pressure angle corresponding to the tooth crest meshing point of the driven wheel.
Further, in step S32:
pressure angle (α) of nth sheet spur gear at meshing position jn)jThe calculation formula of (a) is as follows:
wherein (α)n)jIs the pressure angle of the nth sheet spur gear at the meshing position j;the meshing pressure angle of the driving wheel of the ith pair of meshing teeth of the nth sheet straight gear at the meshing position j is shown; thetab1Is half of the base tooth angle of the driving wheel.
Further, in step S33:
the meshing stiffness judgment expression of the nth-plate spur gear at the meshing position j is as follows:
wherein (k)n)jShowing the time-varying meshing stiffness of the nth plate spur gear at the meshing position j.
Further, in step S4:
time-varying meshing stiffness (k) of the nth-plate thin-plate spur gear at meshing position jn)jThe specific calculation formula of (2) is as follows:
in the formula,the rigidity of the base body of the driving wheel is corrected at the meshing position j;the corrected driven wheel base stiffness at the meshing position j;the rigidity of the base body of the driving wheel of the nth sheet straight gear,the matrix rigidity of the nth sheet of straight gear driven wheel is obtained; lambda [ alpha ]1Correcting the coefficient for the base body of each sheet straight gear driving wheel; lambda [ alpha ]2Correcting the coefficient for the base body of each sheet straight gear driven wheel;the tooth stiffness of all pairs of teeth of the nth plate spur gear at meshing position j is shown.
Further, in step S4:
tooth stiffness of all tooth pairs of the nth sheet of spur gear at meshing position jThe calculation formula of (a) is as follows:
wherein,
wherein m represents the number of pairs of teeth simultaneously in mesh;representing the tooth stiffness of the ith pair of meshing teeth;hertzian contact stiffness at a meshing position j for an ith pair of meshing teeth of an nth sheet of thin straight gear pair;the gear tooth rigidity of the driving wheel of the nth sheet straight gear pair at the meshing position j is obtained;the gear tooth rigidity of the driven wheel of the nth sheet straight gear pair at the meshing position j is obtained; eeEffective modulus of elasticity; l is the tooth width;the meshing force of the ith pair of gears of the nth sheet straight gear at the meshing position j is obtained; f is the total meshing force;for the ith pair of thin sheet straight gears at the meshing position jThe load distribution coefficient of (2).
Further, in step S4:
load distribution coefficient of the ith pair of sheet spur gears at the meshing position j (lsr)i n)jThe calculation formula of (a) is as follows:
wherein,
in the formula, LsfiDistributing coefficient to the ith pair of gear teeth of the nth piece of straight gear; lsfjThe load distribution coefficient at the meshing position j of the nth straight gear is obtained;the total gear tooth rigidity of the nth sheet straight gear;the total tooth stiffness of the N pieces of gears.
Further, in step S5:
time-varying meshing stiffness K of bevel gear pairjThe calculation formula of (a) is as follows:
in the formula, KjIs the time-varying meshing stiffness of the helical gear pair at the meshing position j; n represents the total number of sheets.
(III) advantageous effects
The invention has the beneficial effects that:
the invention discloses a method for calculating time-varying meshing stiffness of a bevel gear pair, which comprises the steps of firstly decomposing a gear tooth model of the bevel gear pair into N independent and uniform sheet straight gear pairs along the tooth width direction, secondly determining whether N sheet straight gears participate in meshing at a meshing position j, then calculating the time-varying meshing stiffness of each sheet of sheet straight gear by considering nonlinear contact, matrix stiffness correction and prolonged meshing influence, and finally summing the time-varying meshing stiffness of each sheet of sheet straight gear to obtain the time-varying meshing stiffness of the bevel gear pair;
according to the method for calculating the time-varying meshing stiffness of the bevel gear pair, the time-varying meshing stiffness of each sheet of the straight gear is calculated by considering the nonlinear contact, the correction matrix stiffness and the prolonged meshing influence, so that the calculation accuracy of each sheet of the straight gear is improved, and the calculation accuracy of the time-varying meshing stiffness of the bevel gear pair is further improved.
In conclusion, the method for calculating the time-varying meshing stiffness of the bevel gear pair takes the nonlinear contact, the matrix stiffness is corrected and the meshing influence is prolonged into consideration, the analytic model is improved, and the calculation accuracy is improved.
Drawings
FIG. 1 is a flow chart of a method for calculating a time varying meshing stiffness of a helical gear pair according to an embodiment;
FIG. 2 is a schematic view of a bevel gear section according to an embodiment;
FIG. 3 is a schematic view of an engagement angle of a drive wheel according to an embodiment;
FIG. 4 is a graph of the time-varying meshing stiffness of the bevel gear pair obtained by different calculation methods when the helix angle is 12 ° in the first embodiment;
FIG. 5 is a graph of the time varying meshing stiffness of the bevel gear pair obtained by different calculation methods when the helix angle is 16 ° in the first embodiment;
FIG. 6 is a graph of the time varying meshing stiffness of the bevel gear pair obtained by different calculation methods when the helix angle is 20 ° in the first embodiment;
FIG. 7 is a graph of the time varying meshing stiffness of the bevel gear pair obtained by different calculation methods when the helix angle is 24 ° in the first embodiment;
FIG. 8 is a graph of a second helix angle versus contact ratio according to an embodiment;
FIG. 9 is a graph of pitch angle versus meshing stiffness variability for a second embodiment;
FIG. 10 is a graph showing the time-varying meshing stiffness of the helical gear pair in the third embodiment when the tooth widths are 10mm and 15 mm;
FIG. 11 is a graph showing the time-varying meshing stiffness of the bevel gear pair at 20mm and 25mm tooth widths in the third embodiment;
FIG. 12 is a graph of the relationship between the width of four teeth and the contact ratio of the embodiment;
FIG. 13 is a graph of embodiment four tooth widths vs. meshing stiffness fluctuations;
FIG. 14 is a graph of the time-varying meshing stiffness of the lower bevel gear pair according to five different shift coefficients of the embodiment;
FIG. 15 is a graph showing the relationship between the six-shift coefficient and the contact ratio according to the embodiment;
FIG. 16 is a graph of six deflection coefficients versus mesh stiffness for an embodiment;
FIG. 17 is a graph of the time-varying meshing stiffness for seven different coefficients of friction for the example embodiment.
Detailed Description
For the purpose of better explaining the present invention and to facilitate understanding, the present invention will be described in detail by way of specific embodiments with reference to the accompanying drawings.
The invention discloses a method for calculating time-varying meshing stiffness of a helical gear pair, which is characterized in that a helical gear model is dispersed into uniform sheet straight gears along the tooth width direction, a helical gear time-varying meshing stiffness solving model is established by calculating the time-varying meshing stiffness of the sheet straight gears and summing the time-varying meshing stiffness of each sheet of the sheet straight gears, and then the helical gear time-varying meshing stiffness is solved through the model.
Referring to fig. 1 to 3, a method for calculating time-varying meshing stiffness of a helical gear pair includes the following steps:
step S1: and acquiring basic parameters of the bevel gear pair.
The basic parameters of the helical gear pair in step S1 of the present invention include: tooth number, modulus of elasticity, Poisson's ratio, bore radius, modulus, tooth width, pressure angle, helix angle, crest coefficient of height, and crest coefficient.
Step S2: the tooth model of the helical gear pair is decomposed into N independent and uniform thin-plate spur gears along the tooth width direction, as shown in fig. 2.
Step S3: it is determined whether the N chip spur gears participate in the meshing at the meshing position j.
Step S31 referring to FIG. 3, the maximum mesh angle α of the helical gear pair at the mesh position j is determinedmaxAnd minimum engagement angle αminThe specific calculation formula is as follows:
wherein, NA is NP-AP rb1·tanα0-rb2·(tanαa2-tanα0)
r1＝mz1/cosβ
r2＝mz2/cosβ
rb1＝r1cosα0
rb2＝r2cosα0
ra1＝r1+ham
ra2＝r2+ham
αa2＝arccos(rb2/ra2)
In the formula, r1The reference circle radius of the driving wheel; r is2The radius of a reference circle of the driven wheel; m is a modulus; z is a radical of1The number of teeth of the driving wheel; z is a radical of2The number of teeth of the driven wheel, β the helical angle of the bevel gear, rb1The radius of the base circle of the driving wheel; r isb2α is the base radius of the driven wheel0Is a pressure angle; r isa1The radius of the addendum circle of the driving wheel; r isa2The radius of the addendum circle of the driven wheel; h isaα indicating the height of tooth topa2The pressure angle corresponding to the tooth crest meshing point of the driven wheel.
Step S32, determining a pressure angle of the nth-plate chip spur gear at the meshing position j (α)n)j；
Wherein (α)n)jIs the pressure angle of the nth sheet spur gear at the meshing position j;the meshing pressure angle of the driving wheel of the ith pair of meshing teeth of the nth sheet straight gear at the meshing position j is shown; thetab1Is half of the base tooth angle of the driving wheel.
Wherein, thetab1＝π/(2z1)+invαt
In the formula, αtAngle of pressure at end face, inv αtIs an involute function.
Wherein, the driving wheel of the ith pair of meshing teeth of the nth sheet straight gear is at the meshing pressure angle of the meshing position jThe following formula is used to obtain:
βb1＝arctan(rb1tanβ/r1)
in the formula,is the minimum roll angle; n is the nth sheet straight gear;the distance between the gear center and the central position o' of the ith pair of meshing teeth of the nth sheet straight gear (shown in figure 2) has the value range of L, βb1Is the base circle helix angle of the driving wheel; l is the tooth width.
Step S33: judging whether the straight gear of the nth slice participates in the meshing;
if αmin≤(αn)j≤αmaxThe nth sheet spur gear participates in the engagement if (α)n)j＜αminOr (α)n)j＞αmaxThen, the nth-plate spur gear does not participate in meshing, and specifically, the meshing stiffness determination expression of the nth-plate spur gear at the meshing position j is as follows:
step S4: and calculating the time-varying meshing stiffness of each sheet of the straight gear.
In the present invention, the time-varying meshing stiffness of the non-meshing chip spur gear is zero, and the time-varying meshing stiffness (k) of the meshing chip spur gear is zeron)jIs not zero.
Specifically, the time-varying meshing stiffness of the meshed sheet spur gear is obtained by a gear pair meshing characteristic analysis method considering nonlinear contact, finite element correction matrix stiffness and prolonged meshing influence.
More specifically, in the present invention, before calculating the time-varying meshing stiffness of each thin plate spur gear, it is necessary to determine the matrix correction coefficient λ of each thin plate spur gear in the double-tooth meshing and triple-tooth meshing processes based on the finite element methodiThe specific calculation formula is as follows:
λi＝1+rkfi
rkfi＝(kfAi-kfBi)×100％/kfBi
in which i takes the values 1 and 2, i.e. lambdaiIs λ1And λ2，λ1Correcting the coefficient for the base body of each sheet straight gear driving wheel; lambda [ alpha ]2Correcting the coefficient for the base body of each sheet straight gear driven wheel; r iskfiThe variation of the base body rigidity in the double-tooth meshing process relative to the base body rigidity in the single-tooth meshing process; k is a radical offAiMeshing rigidity of a double gear zone of the nth sheet straight gear at a meshing position j is obtained; k is a radical offBiThe meshing rigidity of the single-tooth area of the nth sheet straight gear at the meshing position j is obtained.
More specifically, the time-varying meshing stiffness (k) of the nth-plate chip spur gear at the meshing position jn)jThe specific calculation formula of (2) is as follows:
wherein (k)n)jRepresenting the time-varying meshing stiffness of the nth-plate spur gear at the meshing position j;the rigidity of the base body of the driving wheel is corrected at the meshing position j;the corrected driven wheel base stiffness at the meshing position j;the rigidity of the base body of the driving wheel of the nth sheet straight gear,the matrix rigidity of the nth sheet of straight gear driven wheel is obtained; lambda [ alpha ]1Correcting the coefficient for the base body of each sheet straight gear driving wheel; lambda [ alpha ]2Correcting the coefficient for the base body of each sheet straight gear driven wheel;the tooth stiffness of all pairs of teeth of the nth plate spur gear at meshing position j is shown.
Wherein,the calculation formula of (a) is as follows:
wherein m represents the number of pairs of teeth simultaneously in mesh;representing the tooth stiffness of the ith pair of meshing teeth;hertzian contact stiffness at a meshing position j for an ith pair of meshing teeth of an nth sheet of thin straight gear pair;the gear tooth rigidity of the driving wheel of the nth sheet straight gear pair at the meshing position j is obtained;the gear tooth rigidity of the driven wheel of the nth sheet straight gear pair at the meshing position j is obtained; eeEffective modulus of elasticity;the meshing force of the ith pair of gears of the nth sheet straight gear at the meshing position j is obtained;the bending rigidity of the gear teeth of the driving wheel at the meshing position j of the nth sheet straight gear is set;gear tooth scissors of driving wheel at meshing position j for nth sheet straight gearShear stiffness;the axial compression stiffness of the driving wheel of the nth sheet straight gear at the meshing position j is obtained;the gear tooth bending rigidity of the driven gear of the nth sheet straight gear at the meshing position j is set;the gear tooth shear stiffness of the driven gear of the nth sheet straight gear at the meshing position j is obtained;is the axial compressive stiffness of the driven wheel at the meshing position j for the nth plate spur gear.
More specifically, the effective modulus of elasticity E depends on whether the tooth is wide (plane strain) or narrow (plane stress)eThe calculation formula of (a) is as follows:
R＝L/HP
HP＝πm/2
wherein E is the modulus of elasticity; ν is the poisson ratio; the width-thickness ratio R is used for judging whether the gear teeth are wide teeth or narrow teeth; hPThe reference circle tooth thickness.
More specifically, the present invention is to provide a novel,the calculation formula of (a) is as follows:
wherein F is the total engaging force;and the load distribution coefficient of the ith pair of gears of the nth sheet straight gear at the meshing position j.
Wherein, the load distribution coefficient of the ith pair of thin-sheet straight gears at the meshing position jThe calculation formula of (a) is as follows:
in the formula, LsfiDistributing coefficient to the ith pair of gear teeth of the nth piece of straight gear; lsfjThe load distribution coefficient at the meshing position j of the nth straight gear is obtained;the total gear tooth rigidity of the nth sheet straight gear;the total tooth stiffness of the N pieces of gears.
The calculation formula of the total meshing force F is as follows:
in the formula, T is the torque applied to the driving wheel of the helical gear pair.
Step S5: and summing the time-varying meshing stiffness of each sheet of straight gear to obtain the time-varying meshing stiffness of the bevel gear pair.
The calculation formula of the time-varying meshing stiffness of the helical gear pair is as follows:
in the formula, KjIs the time-varying meshing stiffness of the helical gear pair at the meshing position j; n represents the total number of sheets.
The method for calculating the time-varying meshing stiffness of the bevel gear pair ensures certain calculation accuracy and certain calculation efficiency.
The method for calculating the time-varying meshing stiffness of the helical gear pair according to the present invention will be described below with reference to first to seventh embodiments. The effectiveness of the method for calculating the time-varying meshing stiffness of the helical gear pair of the embodiment is verified through comparison in the embodiment, and the second embodiment to the seventh embodiment illustrate specific influences of different gear pair basic parameters on the time-varying meshing stiffness.
Example one
In this embodiment, the basic parameters of the bevel gear pair are shown in table 1:
TABLE 1 basic parameters of bevel gear pairs
Specifically, the helix angles are 12 °, 16 °,20 °, 24 °, respectively, and when the entire tooth is engaged, the tooth width L is 60mm, and R is 12.738>5，Ee230.769 GPa.
In the present embodiment, fig. 4 is a graph showing the time-varying meshing stiffness of a helical gear obtained by an analytical method, a finite element method, ISO6336, or document 1 when the helix angle is 12 ° and the overlap ratio is 3.0614; fig. 5 is a graph showing the time-varying meshing stiffness of helical gears obtained by an analytical method, a finite element method, ISO6336, or document 1 when the helix angle is 16 ° and the overlap ratio is 3.4472; fig. 6 is a graph showing the time-varying meshing stiffness of helical gears obtained by an analytical method, a finite element method, ISO6336, or document 1 when the helix angle is 20 ° and the overlap ratio is 3.8124; fig. 7 is a graph showing the time-varying meshing stiffness of helical gears obtained by an analytical method, a finite element method, ISO6336, or document 1 when the helix angle is 24 ° and the overlap ratio is 4.1588.
Among them, reference 1 is Chang L H, Liu G, Wu L Y.A robust model for determining the mesh parameters of cylindral seeds [ J ]. Mechanism and Machine Theory 2015,87: 93-114.
As can be seen from fig. 4 to 7, the time-varying meshing stiffness of the helical gear pair obtained in the present embodiment is well matched with the time-varying meshing stiffness of the helical gear pair obtained by the finite element method, and the time-varying meshing stiffness calculation method of the helical gear pair of the present embodiment is effective.
Example two
In this embodiment, the basic parameters of the bevel gear pair are the same as those of the embodiment.
In this embodiment, the time-varying meshing stiffness fluctuation is measured by using a parameter η, and the expression is as follows:
wherein Δ K represents a difference between a maximum value and a minimum value of the time-varying engagement stiffness in one engagement period, KmeanRepresenting the average stiffness over an engagement period in the same units, parameter η is simply a dimensionless proportional value used to measure the fluctuation in engagement stiffness.
Specifically, the calculation accuracy of the "slicing idea" is affected after the spiral angle exceeds 30 °, so the spiral angle is selected to be 3 °, 5 °,8 °, 10 °, 13 °, 15 °, 18 °,20 °, 23 °, 25 °, 28 °, 30 °, and L is 60mm in this embodiment.
In the present embodiment, fig. 8 shows a relationship between a helix angle and a degree of overlap, and fig. 9 shows a relationship between a helix angle and a fluctuation in time-varying meshing stiffness.
As can be seen from fig. 8 and 9, 1, as the helix angle increases, the contact ratio increases, and the difference of the contact ratios has a large influence on the fluctuation of the meshing stiffness, and when the contact ratio is an integer, the fluctuation of the meshing stiffness is small; 2. the variation range of the axial contact ratio of the gear pair is larger than that of the end face contact ratio, and the fluctuation of the meshing rigidity can be reduced by controlling the axial contact ratio to be an integer.
EXAMPLE III
In this embodiment, the basic parameters of the bevel gear pair are different from those of the embodiment, and the other parameters are the same as those of the embodiment.
In this embodiment, the tooth widths L are 10mm, 15mm, 20mm, and 25mm, respectively.
In the present embodiment, fig. 10 is a graph showing the time-varying meshing stiffness of a helical gear pair when the tooth widths are 10mm and 15mm, and fig. 11 is a graph showing the time-varying meshing stiffness of a helical gear pair when the tooth widths are 20mm and 25 mm. For the operating mode of L ═ 10mm, there are single-tooth zones and double-tooth zones, the single-tooth zone (the proportion of the single-tooth zone is only 2.05% of a cycle) not being indicated in the figure.
As can be seen from fig. 10 and 11, the time-varying meshing stiffness increases with increasing tooth width, and when the tooth width increases from 10mm to 25mm, the time-varying meshing stiffness increases by 5% on average.
Example four
In this embodiment, the basic parameters of the bevel gear pair are different from those of the embodiment, and the other parameters are the same as those of the embodiment.
In this embodiment, the tooth width L is in the range of 30-85mm, and the pitch is 5 mm.
In the present embodiment, fig. 12 shows a relationship between the tooth width and the contact ratio, and fig. 13 shows a relationship between the tooth width and the meshing stiffness variation.
As can be seen from fig. 12 and 13, the variation in tooth width does not affect the face overlap ratio, and as the tooth width increases the total overlap ratio and the longitudinal overlap ratio increase, the meshing stiffness fluctuation is minimized as the longitudinal overlap ratio is closer to unity.
EXAMPLE five
In this embodiment, the basic parameters of the bevel gear pair are the same as those of the embodiment.
In the embodiment, according to the difference of the deflection coefficients, the gear deflection transmission is divided into positive transmission, zero transmission and negative transmission.
In the present embodiment, fig. 14 shows a time-varying meshing stiffness map with different displacement coefficients.
As can be seen from fig. 14, the deflection coefficient affects the tendency and the value of the time-varying mesh stiffness, and the deflection coefficient affects the mesh tooth zone distribution of the mesh stiffness.
EXAMPLE six
In this embodiment, the basic parameters of the bevel gear pair are the same as those of the embodiment.
In the embodiment, according to the difference of the deflection coefficients, the gear deflection transmission is divided into positive transmission, zero transmission and negative transmission.
In the present embodiment, fig. 15 shows a relationship between the displacement coefficient and the degree of overlap, and fig. 16 shows a relationship between the displacement coefficient and the average meshing stiffness.
As can be seen from fig. 15 and 16, the shift coefficient has no effect on the longitudinal contact ratio, and for the end-to-end contact ratio and the total contact ratio, the shift coefficient increases and then decreases as the drive wheel shift coefficient increases, and the effect on the average meshing stiffness also increases and then decreases.
EXAMPLE seven
In this embodiment, the basic parameters of the bevel gear pair are the same as those of the embodiment.
In the present example, the friction coefficient μ is 0, 0.01, 0.02, 0.03, 0.04, 0.05, 0.06.
In the present embodiment, fig. 17 is a graph of time-varying meshing stiffness at different coefficients of friction.
As can be seen from fig. 17, the average meshing stiffness of the helical gear pair increases with an increase in the friction coefficient, and furthermore, a turning point occurs in the double-tooth meshing zone of the time-varying meshing stiffness due to a change in the direction of the tooth flank friction.
The above description is only a preferred embodiment of the present invention, and for those skilled in the art, the present invention should not be limited by the description of the present invention, which should be interpreted as a limitation.
Claims (9)
1. A method for calculating time-varying meshing stiffness of a bevel gear pair is characterized by comprising the following steps:
step S1: acquiring basic parameters of a bevel gear pair;
step S2: decomposing a gear tooth model of the helical gear pair into N independent and uniform sheet straight gear pairs along the tooth width direction;
step S3: determining whether the N sheet spur gears participate in meshing at a meshing position j;
step S4: calculating the time-varying meshing stiffness of each sheet straight gear by a gear pair meshing characteristic analysis method considering nonlinear contact, matrix stiffness correction and meshing influence prolonging;
step S5: and summing the time-varying meshing stiffness of each sheet of straight gear to obtain the time-varying meshing stiffness of the bevel gear pair.
2. The method of calculating the time-varying meshing stiffness of a helical gear pair according to claim 1,
step S3 includes the following steps:
step S31 determining maximum mesh angle α of the bevel gear pair at mesh position jmaxAnd minimum engagement angle αmin；
Step S32, determining a pressure angle of the nth-plate chip spur gear at the meshing position j (α)n)j；
Step S33: and judging whether the nth sheet straight gear participates in meshing.
3. The method of calculating the time-varying meshing stiffness of a helical gear pair according to claim 2,
in step S31:
maximum meshing angle α of bevel gear pair at meshing position jmaxThe calculation formula of (a) is as follows:
<mrow> <msub> <mi>&alpha;</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mo>&angle;</mo> <mi>N</mi> <mi>O</mi> <mi>D</mi> <mo>=</mo> <mi>arctan</mi> <mfrac> <msub> <mi>r</mi> <mrow> <mi>a</mi> <mn>1</mn> </mrow> </msub> <msub> <mi>r</mi> <mrow> <mi>b</mi> <mn>1</mn> </mrow> </msub> </mfrac> </mrow>
minimum meshing angle α of bevel gear pair at meshing position jminThe calculation formula of (a) is as follows:
<mrow> <msub> <mi>&alpha;</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mo>&angle;</mo> <mi>N</mi> <mi>O</mi> <mi>A</mi> <mo>=</mo> <mi>arctan</mi> <mfrac> <mrow> <msub> <mi>r</mi> <mrow> <mi>b</mi> <mn>1</mn> </mrow> </msub> <mo>&CenterDot;</mo> <msub> <mi>tan&alpha;</mi> <mn>0</mn> </msub> <mo>-</mo> <msub> <mi>r</mi> <mrow> <mi>b</mi> <mn>2</mn> </mrow> </msub> <mo>&CenterDot;</mo> <mrow> <mo>(</mo> <msub> <mi>tan&alpha;</mi> <mrow> <mi>a</mi> <mn>2</mn> </mrow> </msub> <mo>-</mo> <msub> <mi>tan&alpha;</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> </mrow> <msub> <mi>r</mi> <mrow> <mi>b</mi> <mn>1</mn> </mrow> </msub> </mfrac> </mrow>
in the formula, rb1The radius of the base circle of the driving wheel; r isb2α is the base radius of the driven wheel0Is a pressure angle; r isa1The radius of the addendum circle of the driving wheel αa2The pressure angle corresponding to the tooth crest meshing point of the driven wheel.
4. The method of calculating the time-varying meshing stiffness of a helical gear pair according to claim 2,
in step S32:
pressure angle (α) of nth sheet spur gear at meshing position jn)jThe calculation formula of (a) is as follows:
<mrow> <msub> <mrow> <mo>(</mo> <msub> <mi>&alpha;</mi> <mi>n</mi> </msub> <mo>)</mo> </mrow> <mi>j</mi> </msub> <mo>=</mo> <mi>arctan</mi> <mrow> <mo>(</mo> <msub> <mrow> <mo>(</mo> <msubsup> <mi>&tau;</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi>i</mi> </mrow> <mi>n</mi> </msubsup> <mo>)</mo> </mrow> <mi>j</mi> </msub> <mo>+</mo> <msub> <mi>&theta;</mi> <mrow> <mi>b</mi> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> </mrow>
wherein (α)n)jIs the pressure angle of the nth sheet spur gear at the meshing position j;the meshing pressure angle of the driving wheel of the ith pair of meshing teeth of the nth sheet straight gear at the meshing position j is shown; thetab1Is half of the base tooth angle of the driving wheel.
5. The method of calculating the time-varying meshing stiffness of a helical gear pair according to claim 2,
in step S33:
the meshing stiffness judgment expression of the nth-plate spur gear at the meshing position j is as follows:
<mrow> <msub> <mrow> <mo>(</mo> <msup> <mi>k</mi> <mi>n</mi> </msup> <mo>)</mo> </mrow> <mi>j</mi> </msub> <mo>=</mo> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mrow> <mo>(</mo> <msup> <mi>k</mi> <mi>n</mi> </msup> <mo>)</mo> </mrow> <mi>j</mi> </msub> <mo>,</mo> </mrow> </mtd> <mtd> <mrow> <mo>(</mo> <msub> <mi>&alpha;</mi> <mi>min</mi> </msub> <mo>&le;</mo> <msub> <mrow> <mo>(</mo> <msub> <mi>&alpha;</mi> <mi>n</mi> </msub> <mo>)</mo> </mrow> <mi>j</mi> </msub> <mo>&le;</mo> <msub> <mi>&alpha;</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn>0</mn> <mo>,</mo> </mrow> </mtd> <mtd> <mrow> <mo>(</mo> <msub> <mrow> <mo>(</mo> <msub> <mi>&alpha;</mi> <mi>n</mi> </msub> <mo>)</mo> </mrow> <mi>j</mi> </msub> <mo><</mo> <msub> <mi>&alpha;</mi> <mi>min</mi> </msub> <mi>o</mi> <mi>r</mi> <msub> <mrow> <mo>(</mo> <msub> <mi>&alpha;</mi> <mi>n</mi> </msub> <mo>)</mo> </mrow> <mi>j</mi> </msub> <mo>></mo> <msub> <mi>&alpha;</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> </mrow>
wherein (k)n)jShowing the time-varying meshing stiffness of the nth plate spur gear at the meshing position j.
6. The method of calculating the time-varying meshing stiffness of a helical gear pair according to claim 5,
in step S4:
time-varying meshing stiffness (k) of the nth-plate thin-plate spur gear at meshing position jn)jThe specific calculation formula of (2) is as follows:
<mrow> <msub> <mrow> <mo>(</mo> <msup> <mi>k</mi> <mi>n</mi> </msup> <mo>)</mo> </mrow> <mi>j</mi> </msub> <mo>=</mo> <mn>1</mn> <mo>/</mo> <mrow> <mo>(</mo> <mfrac> <mn>1</mn> <msub> <mrow> <mo>(</mo> <mrow> <msub> <mi>&lambda;</mi> <mn>1</mn> </msub> <msubsup> <mi>k</mi> <mrow> <mi>f</mi> <mn>1</mn> </mrow> <mi>n</mi> </msubsup> </mrow> <mo>)</mo> </mrow> <mi>j</mi> </msub> </mfrac> <mo>+</mo> <mfrac> <mn>1</mn> <msub> <mrow> <mo>(</mo> <msubsup> <mi>k</mi> <mrow> <mi>t</mi> <mi>o</mi> <mi>o</mi> <mi>t</mi> <mi>h</mi> </mrow> <mi>n</mi> </msubsup> <mo>)</mo> </mrow> <mi>j</mi> </msub> </mfrac> <mo>+</mo> <mfrac> <mn>1</mn> <msub> <mrow> <mo>(</mo> <msub> <mi>&lambda;</mi> <mn>2</mn> </msub> <msubsup> <mi>k</mi> <mrow> <mi>f</mi> <mn>2</mn> </mrow> <mi>n</mi> </msubsup> <mo>)</mo> </mrow> <mi>j</mi> </msub> </mfrac> <mo>)</mo> </mrow> </mrow>
in the formula,the rigidity of the base body of the driving wheel is corrected at the meshing position j;the corrected driven wheel base stiffness at the meshing position j;the rigidity of the base body of the driving wheel of the nth sheet straight gear is set;the matrix rigidity of the nth sheet of straight gear driven wheel is obtained; lambda [ alpha ]1Correcting the coefficient for the base body of each sheet straight gear driving wheel; lambda [ alpha ]2Correcting the coefficient for the base body of each sheet straight gear driven wheel;the tooth stiffness of all pairs of teeth of the nth plate spur gear at meshing position j is shown.
7. The method of calculating the time-varying meshing stiffness of a bevel gear pair according to claim 6,
in step S4:
tooth stiffness of all tooth pairs of the nth sheet of spur gear at meshing position jThe calculation formula of (a) is as follows:
<mrow> <msub> <mrow> <mo>(</mo> <msubsup> <mi>k</mi> <mrow> <mi>t</mi> <mi>o</mi> <mi>o</mi> <mi>t</mi> <mi>h</mi> </mrow> <mi>n</mi> </msubsup> <mo>)</mo> </mrow> <mi>j</mi> </msub> <mo>=</mo> <munderover> <mo>&Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>m</mi> </munderover> <msubsup> <mi>k</mi> <mrow> <mi>t</mi> <mi>o</mi> <mi>o</mi> <mi>t</mi> <mi>h</mi> </mrow> <mi>i</mi> </msubsup> </mrow>
wherein,
<mrow> <msub> <mrow> <mo>(</mo> <msubsup> <mi>k</mi> <mrow> <mi>h</mi> <mi>i</mi> </mrow> <mi>n</mi> </msubsup> <mo>)</mo> </mrow> <mi>j</mi> </msub> <mo>=</mo> <mfrac> <mrow> <msup> <msub> <mi>E</mi> <mi>e</mi> </msub> <mn>0.9</mn> </msup> <msup> <mi>L</mi> <mn>0.8</mn> </msup> <msup> <msub> <mrow> <mo>(</mo> <msubsup> <mi>F</mi> <mi>i</mi> <mi>n</mi> </msubsup> <mo>)</mo> </mrow> <mi>j</mi> </msub> <mn>0.1</mn> </msup> </mrow> <mn>1.275</mn> </mfrac> </mrow>
<mrow> <msub> <mrow> <mo>(</mo> <msubsup> <mi>F</mi> <mi>i</mi> <mi>n</mi> </msubsup> <mo>)</mo> </mrow> <mi>j</mi> </msub> <mo>=</mo> <mi>F</mi> <mo>&CenterDot;</mo> <msub> <mrow> <mo>(</mo> <msubsup> <mi>lsr</mi> <mi>i</mi> <mi>n</mi> </msubsup> <mo>)</mo> </mrow> <mi>j</mi> </msub> </mrow>
wherein m represents the number of pairs of teeth simultaneously in mesh;representing the tooth stiffness of the ith pair of meshing teeth;hertzian contact stiffness at a meshing position j for an ith pair of meshing teeth of an nth sheet of thin straight gear pair;the gear tooth rigidity of the driving wheel of the nth sheet straight gear pair at the meshing position j is obtained;the n-th sheet of straight gear pair is at the meshing positionThe gear tooth rigidity of the driven wheel at the position j; eeEffective modulus of elasticity; l is the tooth width;the meshing force of the ith pair of gears of the nth sheet straight gear at the meshing position j is obtained; f is the total meshing force;and the load distribution coefficient of the ith pair of gears of the nth sheet straight gear at the meshing position j.
8. The method of calculating the time-varying meshing stiffness of a bevel gear pair according to claim 7,
in step S4:
load distribution coefficient of the ith pair of sheet spur gears at the meshing position j (lsr)i n)jThe calculation formula of (a) is as follows:
<mrow> <msub> <mrow> <mo>(</mo> <msubsup> <mi>lsr</mi> <mi>i</mi> <mi>n</mi> </msubsup> <mo>)</mo> </mrow> <mi>j</mi> </msub> <mo>=</mo> <msup> <mi>Lsf</mi> <mi>i</mi> </msup> <mo>&times;</mo> <msub> <mi>Lsf</mi> <mi>j</mi> </msub> </mrow>
wherein,
<mrow> <msub> <mi>Lsf</mi> <mi>j</mi> </msub> <mo>=</mo> <mfrac> <msubsup> <mi>K</mi> <mrow> <mi>t</mi> <mi>o</mi> <mi>o</mi> <mi>t</mi> <mi>h</mi> </mrow> <mi>n</mi> </msubsup> <mrow> <munderover> <mo>&Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>N</mi> </munderover> <msubsup> <mi>K</mi> <mrow> <mi>t</mi> <mi>o</mi> <mi>o</mi> <mi>t</mi> <mi>h</mi> </mrow> <mi>n</mi> </msubsup> </mrow> </mfrac> </mrow>
in the formula, LsfiDistributing coefficient to the ith pair of gear teeth of the nth piece of straight gear; lsfjThe load distribution coefficient at the meshing position j of the nth straight gear is obtained;the total gear tooth rigidity of the nth sheet straight gear;the total tooth stiffness of the N pieces of gears.
9. The method of calculating the time-varying meshing stiffness of a bevel gear pair according to claim 6,
in step S5:
time-varying meshing stiffness K of bevel gear pairjThe calculation formula of (a) is as follows:
<mrow> <msub> <mi>K</mi> <mi>j</mi> </msub> <mo>=</mo> <munderover> <mo>&Sigma;</mo> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>N</mi> </munderover> <msub> <mrow> <mo>(</mo> <msup> <mi>k</mi> <mi>n</mi> </msup> <mo>)</mo> </mrow> <mi>j</mi> </msub> </mrow>
in the formula, KjIs the time-varying meshing stiffness of the helical gear pair at the meshing position j; n represents the total number of sheets.
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| CN108416120B (en) * | 2018-02-12 | 2021-10-22 | 武汉理工大学 | Method for determining load distribution rate of double-tooth meshing area of straight-toothed spur gear |
| CN109871652A (en) * | 2019-03-14 | 2019-06-11 | 东北大学 | A kind of gear pair Abrasion prediction method based on dynamic engagement power |
| CN109871652B (en) * | 2019-03-14 | 2022-10-04 | 东北大学 | Gear pair wear loss prediction method based on dynamic meshing force |
| CN110059287A (en) * | 2019-04-16 | 2019-07-26 | 江苏省金象传动设备股份有限公司 | Consider to extend engagement and gear ring internal gear pair mesh stiffness calculation method flexible |
| CN110059287B (en) * | 2019-04-16 | 2023-01-24 | 江苏省金象传动设备股份有限公司 | Method for calculating meshing stiffness of internal gear pair by considering prolonged meshing and gear ring flexibility |
| CN111488682B (en) * | 2020-04-09 | 2022-11-08 | 北京理工大学 | Involute helical gear pair tooth width modification dynamic model establishing method |
| CN111488682A (en) * | 2020-04-09 | 2020-08-04 | 北京理工大学 | Involute helical gear pair tooth width modification dynamic model establishing method |
| CN112036049A (en) * | 2020-09-15 | 2020-12-04 | 株洲齿轮有限责任公司 | Rapid calculation method for time-varying meshing stiffness of bevel gear pair under actual working condition |
| CN112036049B (en) * | 2020-09-15 | 2024-04-23 | 株洲齿轮有限责任公司 | Rapid calculation method for time-varying meshing stiffness of helical gear pair under actual working condition |
| CN112507485A (en) * | 2020-11-27 | 2021-03-16 | 江苏省金象传动设备股份有限公司 | Bevel gear time-varying meshing stiffness analysis method based on slice coupling theory |
| CN113051682A (en) * | 2021-03-25 | 2021-06-29 | 天津职业技术师范大学(中国职业培训指导教师进修中心) | Method and device for calculating thermal elastic meshing stiffness of helical gear pair |
| CN113051682B (en) * | 2021-03-25 | 2022-08-12 | 天津职业技术师范大学(中国职业培训指导教师进修中心) | Calculation method and device for thermal elastic meshing stiffness of helical spur gear pair |
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