CN102314534A - Gear profile modification method based on vibration reliability and genetic algorithm - Google Patents

Abstract

The invention discloses a gear profile modification method based on vibration reliability and a genetic algorithm, belonging to the technical field of reliability designs. The gear profile modification method can not only reduce meshing impact due to elastic deformation and manufacturing errors, but also reduce the meshing excitation of a gear so that a gear system has smooth transmission, the vibration and the noises are reduced, and the reliability and service life of the gear system can be improved. The gear profile modification method comprises the following steps of: (1), establishing an elastic modification virtual prototype for meshing between a gear and a modifying gear; (2) carrying out dynamic simulation on gear transmission errors in the process of an involute profile and the profile modification; (3) analyzing the reliability sensitivity of the transmission errors at the time of gear pair random parameter meshing; (4) determining optimal parameters of the gear profile modification by utilizing the genetic algorithm; and (5) verifying the correctness of the parameters.

Description

A kind of gear-profile correction method based on vibration reliability and genetic algorithm

Technical field

The invention belongs to the reliability design technology field, particularly relate to a kind of gear-profile correction method based on vibration reliability and genetic algorithm.

Background technology

The gear modification technology is classified as I class tackling key problem problem in 16 gordian techniquies of gear.The gear modification technology is the gordian technique of high class gear transmission design and manufacturing, and it is most important means that reduce vibration and improve high-speed overload gear drive reliability.Profile modification is a kind of mode of gear modification technology; Profile modification is at tooth top or near repairing the variation that a part relaxes mesh stiffness in the position of root fillet original involute profile; Reduce because base pitch error and stand under load are out of shape caused engaging-in and nibble out impact; Improve flank of tooth lubricating status, prevent that gummed from taking place.Gear-profile correction of the flank shape technology has been done a large amount of research both at home and abroad; Tang Zengbao etc. with gear vibration acceleration root-mean-square value minimum as optimization aim; Propose gear dynamic performance optimization method for designing, and utilized this method to try to achieve dynamic property best profile modification amount and correction of the flank shape length.Tavakoli etc. as the effective means that reduces the tooth mesh excitation, are used to profile modification to eliminate the engaging-in of the gear teeth and nibble out the fluctuation that impact also reduces gear drive error to greatest extent.Though these research methods have been confirmed the size of profiling quantity and the selection at correction of the flank shape position etc.; But because the computation model of tooth mesh rigidity and elastic deformation is accurate not enough; And the correction of the flank shape parameter generally can both be accurate to micron, also can't the correction of the flank shape of accurate guidance gear-profile design so confirm the method for correction of the flank shape parameter; And owing to consider that it is very difficult that the actual dynamic perfromance of gear is come design gear correction of the flank shape parameter, so most of at present method that reduces gear vibration is based on static calculation.The most researchs of this class methods all concentrate on the problem such as the geometrical interference, load sudden change of the gear teeth, yet confirm that accurately the method for profile modification parameter is still immature, be difficult to satisfy large complicated mechanical drive heavy duty, at a high speed, high-precision requirement.

The continuous development of Along with computer technology, the computing velocity of large-scale finite element software can be accepted.Therefore; Reliability theory, optimized Algorithm and three-dimensional gear finite element model are combined the optimized parameter of confirming the gear-profile correction of the flank shape; This has important actual directive significance to research gear-profile correction of the flank shape design, for new approaches and new method have been opened up in the theoretical research work of gear-profile correction of the flank shape.

Summary of the invention

Above-mentioned deficiency to the prior art existence; The present invention provides a kind of gear-profile correction method based on vibration reliability and genetic algorithm; This method not only can reduce the mesh impact that caused by elastic deformation and foozle, but also can reduce the gearing mesh excitation, makes the gear train stable drive; Reduce vibration and noise, improved the reliability and the serviceable life of gear train.

To achieve these goals, the present invention adopts following technical scheme, and a kind of gear-profile correction method based on vibration reliability and genetic algorithm comprises following step:

Step 1: set up gear and profile modifying gear engagement elastic deformation virtual prototype;

Step 2: the dynamic simulation of gear transmission error when carrying out involute profile and profile modification;

Step 3: the reliability susceptibility of analyzing gear pair stray parameter engagement transmission error;

Step 4: adopt genetic algorithm to confirm the optimized parameter of gear-profile correction of the flank shape;

Step 5: the correctness of inspection parameter.

Described gear and the profile modifying gear engagement elastic deformation virtual prototype set up of step 1, it specifically comprises following step:

Steps A: parametric equation and the full-depth tooth model of setting up tooth curve;

Step B: divide the gear finite element grid;

Step C: handle gear teeth boundary condition and load, set up the gear finite element model;

Step D: select the profile modification mode, if linear dressing is then set up the linear dressing finite element model; If para-curve correction of the flank shape finite element model is then set up in the para-curve correction of the flank shape.

The dynamic simulation of gear transmission error that step 2 is described when carrying out involute profile and profile modification, it specifically comprises following step:

Steps A: the dynamic simulation that carries out involute profile gear transmission error;

Step B: the dynamic simulation that carries out profile modification gear transmission error;

Step C: the transmission error curve is compared analysis.

The reliability susceptibility of the described analysis gear pair of step 3 stray parameter engagement transmission error, it specifically comprises following step:

Steps A: confirm the number and the distribution pattern thereof of input variable at random, and judge the number N of required sample point;

Step B: randomly draw input variable, specify output variable, the finite element model that obtains confirming;

Step C: utilize the determinacy Finite Element Method to construct input variable and the corresponding relation between the output variable at random at random, and the output result;

Step D: the output of calculating all sample points;

Step e: utilize the response surface method with quadratic polynomial match sample point, response surface function expression and limit state function expression formula between definite output variable at random and the input variable;

Step F: the response surface function is carried out reliability sensitivity analysis, the output result.

The described employing genetic algorithm of step 4 is confirmed the optimized parameter of gear-profile correction of the flank shape, and it specifically comprises following step:

Steps A: the correction of the flank shape parameter is carried out gene code, and random initializtion colony;

Step B: estimate colony;

Step C: judge whether to finish according to stopping criterion,, then change and remove execution in step E if finish; Otherwise, execution in step D;

Step D: colony is used chromosome select operator, crossover operator and mutation operator; And then estimate colony, and change and remove execution in step C;

Step e: optimize the correction of the flank shape parameter, the parameter of linear dressing and the parameter of para-curve correction of the flank shape are optimized.

The correctness of the described inspection parameter of step 5, it specifically comprises following step:

Steps A: carry out the modeling of gear nonlinear kinetics;

Step B: gear nonlinear kinetics equation is found the solution;

Step C: the vibration damping demonstration of carrying out optimal case.

Beneficial effect of the present invention:

1, the present invention has considered this key factor of vibration reliability:

The present invention carries out profile modification to gear and eliminates the engaging-in of theoretical involute gear and nibble out impact; Reduce vibration and noise; And the influence of the stochastic variable in the consideration various engineering practical structures; This is a kind of dynamic computing method, and result of calculation is more accurate, more meets the actual requirement of Modern High-Speed, heavy-duty gear.Drawn the reliability susceptibility of each correction of the flank shape parameter at random through analytical calculation, obtained susceptibility figure and scatter diagram transmission error.Susceptibility figure shows which design variable will change reliability of structure should revise, and scatter diagram then further shows how to change the approximate range of these design variables and change.Reduce the discrete range and the raising of each stochastic variable or reduce relevant parameters, can effectively improve the transmission error reliability of gear, the reliability susceptibility is significant for instructing the actual reliability design of engineering.

2, the present invention adopts genetic algorithm to carry out gear-profile correction of the flank shape Parameter Optimization:

The correction of the flank shape parameter of optimal design can significantly reduce the vibration of gear, the generation of avoiding meshing moment impact.The gear modification parameter of the present invention through respectively to linear dressing and para-curve correction of the flank shape the time is optimized design, confirmed this two kinds of best vibration damping effect that modification curve can reach, and obtains optimum modification curve, and correction of the flank shape is designed with important meaning to gear-profile.

3, the present invention tests to the correctness of parameter:

The present invention is on gear pair gap twisting vibration non-linear dynamic model basis; Found the solution and drawn before the correction of the flank shape and the amplitude frequency curve after the correction of the flank shape to equation; Through contrast to amplitude before and after it, verified the correctness of optimizing backgear correction of the flank shape parameter by the angle of vibration, have theoretical foundation.

In sum, the present invention has not only reduced the mesh impact that caused by elastic deformation and foozle, but also has reduced vibration and noise; Effectively raise the transmission error reliability of gear, make the gear train stable drive, the reliability and the serviceable life of having improved gear train, the actual reliability Optimum Design of engineering is significant for instructing.

Description of drawings

Fig. 1 is the program flow diagram of correction method of the present invention;

Fig. 2 is a program flow diagram of setting up gear and profile modifying gear engagement elastic deformation virtual prototype;

Fig. 3 is the program flow diagram of the dynamic simulation of gear transmission error when carrying out involute profile and profile modification;

Fig. 4 is for analyzing the program flow diagram that the gear pair stray parameter meshes the reliability susceptibility of transmission error;

Fig. 5 confirms the program flow diagram of the optimized parameter of gear-profile correction of the flank shape for adopting genetic algorithm;

Fig. 6 is the program flow diagram of the correctness of inspection parameter;

Fig. 7 is the generating principle figure of flank profil involute urve;

Fig. 8 is rack type cutter structure figure;

Fig. 9 is the procedure chart of rack type cutter Machining of Gear;

Figure 10 is the solid model figure of gearing mesh elastic deformation virtual prototype;

Figure 11 is the finite element model figure of gearing mesh virtual test model machine;

Figure 12 is the gear modification Parameter Map;

Figure 13 is the illustraton of model of flank profil linear dressing;

Figure 14 is the tangent slope figure that the flank profil involute urve is ordered at b;

Figure 15 is the illustraton of model of flank profil para-curve correction of the flank shape;

Figure 16 is the gear solid model figure of linear dressing;

Figure 17 is the gear solid model figure of para-curve correction of the flank shape;

Figure 18 is linear dressing gear meshing virtual prototype finite element model figure;

Figure 19 is the engagement virtual prototype finite element model figure of para-curve profile modifying gear;

Figure 20 is the transmission error dynamic curve diagram of gear;

Figure 21 is the gear transmission error dynamics curve map of linear dressing;

Figure 22 is the gear transmission error dynamics curve map of para-curve correction of the flank shape;

Figure 23 is sampling sample figure;

Figure 24 is a histogram frequency distribution diagram;

Figure 25 is the probability graph of cumulative distribution function reflection gear failure;

Figure 26 is the histogram and the pie chart of stray parameter;

Figure 27 is limit state function value g (x) and each scatter diagram between input variable at random;

Figure 28 is gene code figure;

Figure 29 is the gear transmission error dynamics curve map of linear dressing;

Figure 30 is the gear transmission error dynamics curve map of para-curve correction of the flank shape;

Figure 31 is the kinetic model figure of the secondary pure twisting vibration of straight spur gear;

Figure 32 is amplitude frequency curve comparison diagram before and after the correction of the flank shape.

Embodiment

Be example with the spur gear wheel below, carry out the gear-profile correction of the flank shape based on vibration reliability and genetic algorithm, the material of gear is 16Cr3NiWMoVNbE, driving wheel number of teeth z ₁Be 26, engaged wheel number of teeth z ₂Be 40, modulus m is 3mm, pressure angle α ₀Be 20 °, facewidth B is 20mm, tip clearance coefficient c ^*Be 0.25, driving and driven form of gear tooth deviation is respectively Ef _p=8 μ m, Ef _g=8 μ m, comprehensive tooth pitch deviation ES=10 μ m, the maximum profiling quantity of driven wheel is S _pAnd S _g, the correction of the flank shape angle is α _pAnd α _g, S _p=0.06mm, α _p=63 °, S _g=0.06mm, α _g=63 °, program flow diagram of the present invention is as shown in Figure 1.

Step 1: set up gear and profile modifying gear engagement elastic deformation virtual prototype;

The transmission error of gear is relevant with the comprehensive deformation amount, and Accurate Analysis need be through setting up three-dimensional entity model and carrying out FEM calculation.In order to simulate the randomness of the original mismachining tolerance of gear, the present invention selects for use the ANSYS finite element software to carry out the parametric modeling of gear.

Steps A: parametric equation and the full-depth tooth model of setting up tooth curve;

The generating principle of flank profil involute urve, as shown in Figure 7, being limit, be in the polar coordinate system of pole axis with the x axle with the gear centre of gyration, the flank profil involute equation can be expressed as

$\left\{\begin{array}{c}{r}_k=r_b/\cos {\mathrm{\α}}_k\\ {\mathrm{\θ}}_k=\tan {\mathrm{\α}}_k-{\mathrm{\α}}_k\end{array}\right.---\left(1\right)$

In the formula, r _kBe any radius vector of any on the involute urve, r _bBe base radius, α _kBe the pressure angle of this point, θ _kExhibition angle for this point.

Through coordinate transform, obtain the parametric equation of involute urve under the rectangular coordinate system, promptly the parametric equation of tooth curve is:

$\left\{\begin{array}{c}{x}_k=\frac{r_b}{\cos {\mathrm{\α}}_k}\sin (\frac{\mathrm{\φ}}{2}+\mathrm{inv}{\mathrm{\α}}_0-\mathrm{inv}{\mathrm{\α}}_k)\\ y_k=\frac{r_b}{\cos {\mathrm{\α}}_k}\cos (\frac{\mathrm{\φ}}{2}+\mathrm{inv}{\mathrm{\α}}_0-\mathrm{inv}{\mathrm{\α}}_k)\end{array}\right.---\left(2\right)$

In the formula, x _k, y _kBe respectively any some horizontal stroke, ordinate under rectangular coordinate system on the involute urve; α ₀Be the pressure angle on the gear compound graduation circle; φ is the pairing central angle of reference circle transverse tooth thickness, is the standard straight spur gear of z for the number of teeth, φ=π/z.

According to the difference of processing mode and process tool flank profil, fillet curve has various ways.When adopting tooth bar slotting tool or hobboing cutter Machining of Gear, if the top of cutter tooth-profile only has a fillet, then transient curve is the equidistant curve of one whole section prolate involute.The rack type cutter structure, as shown in Figure 8, among the figure, α ₀Be the pressure angle on the gear compound graduation circle, a is cutter fillet center of circle C _ρDistance apart from center line; B is cutter fillet center of circle C _ρDistance apart from cutter teeth groove center line; r _ρBe the cutter radius of corner;

Be the tooth depth coefficient; c ^*Be the radial play coefficient; M is a modulus.The process of rack type cutter Machining of Gear, as shown in Figure 9, among the figure, C is a node, and nn is the common normal of cutter fillet and transient curve contact point, and a is cutter fillet center of circle C _ρApart from the distance of center line, b is cutter fillet center of circle C _ρApart from the distance of cutter teeth groove center line, r _ρBe cutter radius of corner, C _ρBe the cutter fillet center of circle, have following relation between the parameter of fillet curve:

$\left\{\begin{array}{c}a=h_a^{*}m+c^{*}m-r_{\mathrm{\ρ}}\\ b=\frac{\mathrm{\πm}}{2}\\ r_{\mathrm{\ρ}}=\frac{\mathrm{\πm}-4h_a^{*}m\tan {\mathrm{\α}}_0}{4\cos {\mathrm{\α}}_0}\\ c^{*}m=r_{\mathrm{\ρ}}(1-\sin {\mathrm{\α}}_0)\end{array}\right.---\left(3\right)$

In the formula, α ₀Be the pressure angle on the gear compound graduation circle, m is a modulus,

Be tooth depth coefficient, c ^*Be the radial play coefficient.For the standard straight spur gear, draw the parametric equation that coordinate system shown in Figure 9 extends below the involute urve equidistant curve and be:

In the formula,

Xk, yk are respectively any some horizontal stroke, ordinate under rectangular coordinate system on the involute urve, α ' _kBe pressure angle, m is a modulus, and z is the number of teeth,

Be central angle, a is cutter fillet center of circle C _ρApart from the distance of center line, b is cutter fillet center of circle C _ρApart from the distance of cutter teeth groove center line, r _ρBe the cutter radius of corner.

In ANSYS, set up relatively difficulty of accurate gear-profile,, utilize APDL to circulate to lay foundations order to set up key point based on formula (2) and formula (4).Involute urve built together found 1000 points, fillet curve is set up 300 points, then each key point line can be obtained the flank profil that is made up of standard involute urve and transient curve.With the monodentate model rotating mirror-image and the adult that stretches, can obtain the three-dimensional model of a gear.

Driving and driven gear respectively along its axis rotation, because transverse tooth thickness on the reference circle and space width are equal, so rotating to the node position of engagement, driving gear need be rotated 360/ (4z ₁), follower gear rotates to the node position of engagement need rotate 360/ (4z ₂).Set up the solid model of gearing mesh elastic deformation virtual prototype, promptly the full-depth tooth model is shown in figure 10.

Step B: divide the gear finite element grid;

The key of finite element automatic modeling is that the grid of structure generates automatically.The finite element grid generation technique mainly can be divided into the structured grid generation and unstructured grid generates two big types.The structured grid method of formation mainly adopts reflection method, and this method must become subregion with segmentation of structures in advance, in subregion, carries out grid and generates automatically, can be used in the generation of surface mesh.The unstructured grid method of formation can be realized robotization in various degree, can be divided into trigonometric ratio method, topological decomposition method, geometry decomposition method and free grid partitioning etc. by operating feature.The size of grid and the efficient of finite element analysis and precision are closely related, and grid is close more, and computational accuracy is high more, but counting yield is low more.Therefore, can adopt closeer grid at the gear contact area, adopt large-sized grid in other zones, the present invention selects the smart grid dividing mode to generate finite element model.According to material definition elastic modulus is 2.06 * 10 ⁵MPa, Poisson ratio is 0.3, the gear tooth friction coefficient is 0.05.Select three-dimensional hexahedral element SOLID45 that the solid model of model machine is divided grid.

Step C: handle gear teeth boundary condition and load, set up the gear finite element model;

In the gear drive process, external applied load is the working resistance of follower gear and the driving moment of driving gear.Engagement behavior for real simulated practical work process middle gear; Gear is done following processing: suppose moment in the contact of two gear tooths; Engaged wheel is fixed, and driving wheel rotates in fixed axis, and its radial displacement and axial displacement are zero restriction; Promptly, all nodes on the engaged wheel inner cylinder face are applied U being under the partial cylindrical coordinate system of Z axle with the engaged wheel axis _X, U _Y, U _ZConstraint is fixed; Be under the partial cylindrical coordinate system of Z axle with the driving wheel axis again, the node coordinate system of all nodes is cylindrical-coordinate system on the conversion driving wheel inner cylinder face, and all nodes on the driving wheel inner cylinder face are applied U _X, U _ZConstraint, and keep its rotary freedom U _YIf the radius of driving wheel inner cylinder face is r _z, the driving wheel input torque is T ₁, then in driving wheel, on average applying along the total size of sense of rotation on all nodes of axial cylindrical face is T ₁/ r _zPower.Create contact to, imposed load and boundary condition after, set up the finite element model of virtual test model machine, shown in figure 11.

Step D: select the profile modification mode, if linear dressing is then set up the linear dressing finite element model; If para-curve correction of the flank shape finite element model is then set up in the para-curve correction of the flank shape.

The gear modification parameter, shown in figure 12, among the figure, S _pBe the maximum profiling quantity of driving wheel, S _gBe the maximum profiling quantity of engaged wheel, α _pBe driving wheel correction of the flank shape angle, α _gBe engaged wheel correction of the flank shape angle.The profile modification mode comprises linear dressing and para-curve correction of the flank shape.

Linear dressing: can directly confirm the particular location of correction of the flank shape straight line according to the value of gear modification amount and correction of the flank shape angle, set up the wheel tooth model and press linear dressing that shown in figure 13, among the figure, a, 2 of b are respectively the intersection point of straight line and tooth top, involute profile.

The para-curve correction of the flank shape: the profiling quantity of linear dressing and para-curve correction of the flank shape is identical with the correction of the flank shape angle, is S _pAnd S _g, α _pAnd α _g, promptly the contour curve of two kinds of corrections of the flank shape is all crossed a, b 2 points among Figure 13.Be respectively (x if establish the coordinate of 2 of a, b _a, y _a) and (x _b, y _b), can get:

$\left\{\begin{array}{c}{y}_a=c_1{x}_a^2+c_2{x}_a+c_3\\ y_b=c_1{\mathrm{<figure-callout\; id="2"\; label="x\; b"\; filenames="HDA0000066144690000102.png,HDA0000066144690000111.png"\; state="\{\{state\}\}">x</figure-callout>}}_b^{}+c_2{x}_b+c_3\end{array}\right.---\left(5\right)$

On the tooth curve after the correction of the flank shape, involute urve and para-curve junction should be smooth, and gear-driven stable to guarantee, promptly para-curve equates with the tangent slope that involute urve is ordered at b.The tangent slope that the flank profil involute urve is ordered at b, shown in figure 14, among the figure, α ' is an engaging angle, θ is exhibition angle, r _bBe base radius, (x _b, y _b) coordinate of ordering for b.

Can know that by the involute urve characteristic tangent line cd is vertical with generating line of involute be, get final product to such an extent that cd is parallel to basic circle center of circle O and the line Oe that line point of contact e takes place.Can get the tangent slope K of involute urve thus at b point place _b

$K_b=K_{\mathrm{oe}}=\tan ({\mathrm{\&alpha;}}^{\&prime;}-\mathrm{\&theta;}+\frac{\mathrm{\&pi;}}{2})=-\cot ({\mathrm{\&alpha;}}^{\&prime;}-\mathrm{\&theta;})---\left(6\right)$

Can get by geometric relationship among the figure:

$\mathrm{\&theta;}=\arctan \left(\frac{x_b}{y_b}\right)---\left(7\right)$

${\mathrm{\&alpha;}}^{\&prime;}=\arccos \left(\frac{L_{\mathrm{oe}}}{L_{\mathrm{ob}}}\right)---\left(8\right)$

In the formula, $L_{\mathrm{Ob}}=\sqrt{{\mathrm{<figure-callout\; id="2"\; label="x\; b"\; filenames="HDA0000066144690000102.png,HDA0000066144690000111.png"\; state="\{\{state\}\}">x</figure-callout>}}_b^{}+{\mathrm{<figure-callout\; id="2"\; label="y\; b"\; filenames="HDA0000066144690000102.png,HDA0000066144690000111.png"\; state="\{\{state\}\}">y</figure-callout>}}_b^{}},$ L _Oe=r _b

Para-curve in the tangent slope expression formula at b point place is:

$y_b^{\&prime;}=\frac{\mathrm{dy}}{\mathrm{dx}}=2c_1{x}_b+c_2---\left(9\right)$

Equate at b point tangent slope according to para-curve and involute urve, can get by formula (5) again

$\left\{\begin{array}{c}{y}_a=c_1{x}_a^2+c_2{x}_a+c_3\\ y_b=c_1{\mathrm{<figure-callout\; id="2"\; label="x\; b"\; filenames="HDA0000066144690000102.png,HDA0000066144690000111.png"\; state="\{\{state\}\}">x</figure-callout>}}_b^{}+c_2{x}_b+c_3\\ K_b=2c_1{x}_b+c_2\end{array}\right.---\left(10\right)$

In the formula, K _bBe the tangent slope of involute urve at b point place.

Can solve coefficient c ₁, c ₂, c ₃, substitution para-curve expression formula get final product profile modification para-curve.Work out parametrization tooth curve generator program, obtain the model of flank profil para-curve correction of the flank shape, shown in figure 15.

After respectively tooth curve being carried out linear dressing and para-curve correction of the flank shape, the solid model of foundation such as Figure 16 and shown in Figure 17.

According to gear finite element grid partitioning and boundary condition and load Processing method, adopt closeer grid at the gear contact area, adopt large-sized grid in other zones.Create contact to, imposed load and boundary condition after, the engagement virtual prototype finite element model of the linear dressing of foundation and para-curve profile modifying gear is like Figure 18 and shown in Figure 19.

Step 2: the dynamic simulation of gear transmission error when involute profile and profile modification;

The finite element model that step 1 is obtained carries out the emulation of transmission error.

Steps A: the dynamic simulation that carries out involute profile gear transmission error;

Carry out simulation calculation according to the known conditions in the step 1, obtain the transmission error curve of gear and the undulating quantity of transmission error, shown in figure 20, visible by figure, gear is less at the transmission error of bidentate region of engagement, and transmission error is bigger in the monodentate region of engagement.So, produced very big fluctuation at single bidentate graded area, cause curve to be stairstepping, explain that the gear teeth can produce impact engaging-in when nibbling out, the maximum fluctuation of transmission error is 10 μ m.

Step B: the dynamic simulation that carries out profile modification gear transmission error;

Adopt the finite element modeling type to carry out simulation calculation to gear.The correction of the flank shape parameter is chosen in one set by step; For of the influence of more different modification curves to transmission error; Respectively different modification curves are carried out simulation calculation, Figure 21, Figure 22 be respectively with finite element analogy press straight line and para-curve correction of the flank shape the time gear the transmission error curve map.

Step C: the transmission error curve is compared analysis;

Visible by Figure 21, when carrying out linear dressing, the fluctuation Δ TE=7.98 μ m of the transmission error of gear train.Transmission error curve Figure 20 before the contrast correction of the flank shape; At this moment curve has not had tangible ladder property; Explain and engaging-inly nibble out impact and do not exist; And the crest instruction book bidentate of bidentate region of engagement alternately is to produce the main cause that fluctuates, and change violent explanation at the crest place generation of impact is arranged, and amplitude has reduced about 20%.The correction of the flank shape effect is bad, need further optimize the correction of the flank shape parameter and choose.

Visible by Figure 22, when carrying out the para-curve correction of the flank shape, the fluctuation Δ TE=7.13 μ m of the transmission error of gear train.Transmission error curve Figure 20 before the contrast correction of the flank shape; Curve does not have tangible ladder property, explains engaging-inly to nibble out impact and do not exist, and has milder crest in the bidentate region of engagement; The instruction book bidentate alternately is the main cause that produces fluctuation; But curve is mild, and it is about 28% that amplitude has reduced, and effect is better than linear dressing.But also need carry out the correction of the flank shape Parameter Optimization calculates.

Though during correction of the flank shape, the fluctuation of the transmission error curve of gear train is little parabolically, and the fluctuation when pressing linear dressing is big.But because corresponding dissimilar modification curves, the profiling quantity that should get also is different.These two kinds of modification curves optimal effectiveness that can reach does not embody in the present embodiment in other words.So the correction of the flank shape of illustrative para-curve is necessarily not good than linear dressing.And the emulation of various finite element models of this step and transmission error thereof lays the foundation for follow-up analytical calculation just.

Step 3: the reliability susceptibility of analyzing gear pair stray parameter engagement transmission error:

Can know that by step 2 transmission error receives the influence of each correction of the flank shape parameter at random, therefore is necessary to adopt the method for vibration reliability to come the randomness of each parameter is analyzed.

Steps A: confirm the number and the distribution pattern thereof of input variable at random, and judge the number N of required sample point;

In order to investigate of the influence of correction of the flank shape parameter, establish gearing mesh transmission error fluctuation Δ TE and receive the maximum profiling quantity S of driving wheel tooth top gearing mesh transmission error reliability _p, driving wheel tooth top correction of the flank shape angle α _p, the maximum profiling quantity S of engaged wheel tooth top _g, engaged wheel tooth top correction of the flank shape angle α _gInfluence, and S _p, α _p, S _g, α _gAll be stochastic variable, maximum allowable transmission error fluctuation Δ TE _MaxIt also is stochastic variable.Suppose the equal Normal Distribution of above stochastic variable, its average and standard deviation are as shown in table 1, and the number of sampling that needs according to Central Composite design (k=4) is NS=2 ^k+ 2k+1=25.

Table 1

Step B: randomly draw input variable, specify output variable, the finite element model that obtains confirming;

Randomly draw input variable S _p, α _p, S _g, α _g, specifying output variable is Δ TE _Max, can set up definite finite element model.

Step C: utilize the determinacy Finite Element Method to construct input variable and the corresponding relation between the output variable at random at random, and the output result;

Finite element model according to confirming meshes simulation calculation, obtains the fluctuation Δ TE of transmission error _Max

Step D: the output of calculating all sample points;

Probability level is respectively p ₁=0.01, p ₂=0.50, p ₃=0.99 o'clock, calculate sample point numerical value and list in the table 2.

Through 25 finite element analogys, obtain 25 responses of gear pair transmission error fluctuation according to the sample point in the table 2, list last row of table 2 in.

Table 2

Step e: utilize the response surface method with quadratic polynomial match sample point, response surface function expression and limit state function expression formula between definite output variable at random and the input variable;

The response surface function is all chosen the quadratic function that contains cross term usually and is described, and can be expressed as following expression formula:

$\hat{Y}={\mathrm{<figure-callout\; id="0"\; label="C"\; filenames="HDA0000066144690000101.png,HDA0000066144690000102.png"\; state="\{\{state\}\}">C</figure-callout>}}_0+\underset{i=1}{\overset{\mathrm{NR}}{\mathrm{\&Sigma;}}}{C}_i{X}_i+\underset{i=1}{\overset{\mathrm{NR}}{\mathrm{\&Sigma;}}}\underset{j=i}{\overset{\mathrm{NR}}{\mathrm{\&Sigma;}}}{C}_{\mathrm{ij}}{X}_i{X}_j---\left(11\right)$

In the formula, C ₀, C _i, C _Ij(i=1 ... NR; J=i ... NR) be undetermined coefficient.

NS sample point through to the stray parameter vector carries out numerical evaluation, obtains NS output point (y ₁, y ₂..., y _NS), with least square method stray parameter vector sum structural response is carried out regretional analysis

$s=\underset{i=1}{\overset{\mathrm{NS}}{\mathrm{\&Sigma;}}}{\mathrm{\&epsiv;}}^2=\underset{i=1}{\overset{\mathrm{NS}}{\mathrm{\&Sigma;}}}{[y_i-({\mathrm{<figure-callout\; id="0"\; label="C"\; filenames="HDA0000066144690000101.png,HDA0000066144690000102.png"\; state="\{\{state\}\}">C</figure-callout>}}_0+\underset{i=1}{\overset{\mathrm{NR}}{\mathrm{\&Sigma;}}}{C}_i{x}_i+\underset{i=1}{\overset{\mathrm{NR}}{\mathrm{\&Sigma;}}}\underset{j=i}{\overset{\mathrm{NR}}{\mathrm{\&Sigma;}}}{C}_{\mathrm{ij}}{x}_i{x}_j)]}^2---\left(12\right)$

In the formula, ε is an error term, and is minimum for making error term, then has

$\left\{\begin{array}{c}\frac{\&PartialD;s}{{\&PartialD;\mathrm{<figure-callout\; id="0"\; label="C"\; filenames="HDA0000066144690000101.png,HDA0000066144690000102.png"\; state="\{\{state\}\}">C</figure-callout>}}_0}=0\\ \frac{\&PartialD;s}{{\&PartialD;C}_i}=0,i=\mathrm{1,2},L,\mathrm{NR}\\ \frac{\&PartialD;s}{{\&PartialD;C}_{\mathrm{ij}}}=0,i=\mathrm{1,2},L,\mathrm{NR};j=i,L,\mathrm{NR}\end{array}\right.---\left(13\right)$

According to the data in the table 2 and formula (11), (12), (13), the response surface function that obtains the fluctuation of gear pair transmission error is:

$\hat{Y}=173.086+5.11S_p-14.122{\mathrm{\&alpha;}}_p-1.086S_g+5.325{\mathrm{\&alpha;}}_g-0.017{\mathrm{<figure-callout\; id="2"\; label="S\; p"\; filenames="HDA0000066144690000102.png,HDA0000066144690000111.png"\; state="\{\{state\}\}">S</figure-callout>}}_p^{}$

$-0.195S_p{\mathrm{\&alpha;}}_p+0.009S_p{S}_g+0.139S_p{\mathrm{\&alpha;}}_g-1.587{\mathrm{\&alpha;}}_{\mathrm{<figure-callout\; id="2"\; label="p"\; filenames="HDA0000066144690000102.png,HDA0000066144690000111.png"\; state="\{\{state\}\}">p</figure-callout>}}^2+0.164{\mathrm{\&alpha;}}_p{S}_g---\left(14\right)$

$+3.399{\mathrm{\&alpha;}}_p{\mathrm{\&alpha;}}_g-0.006S_g^2-0.143S_g{\mathrm{\&alpha;}}_g-1.728{\mathrm{\&alpha;}}_{\mathrm{<figure-callout\; id="2"\; label="g"\; filenames="HDA0000066144690000102.png,HDA0000066144690000111.png"\; state="\{\{state\}\}">g</figure-callout>}}^2$

Obtaining limit state function is:

$g\left(X\right)={\mathrm{\&Delta;TE}}_{\max }-\hat{Y}={\mathrm{\&Delta;TE}}_{\max }-(173.086+5.11S_p-14.122{\mathrm{\&alpha;}}_p-1.086S_g$

$+5.325{\mathrm{\&alpha;}}_g-0.017S_p^2-0.195S_p{\mathrm{\&alpha;}}_p+0.009S_p{S}_g+0.139S_p{\mathrm{\&alpha;}}_g---\left(15\right)$

$-1.587{\mathrm{\&alpha;}}_{\mathrm{<figure-callout\; id="2"\; label="p"\; filenames="HDA0000066144690000102.png,HDA0000066144690000111.png"\; state="\{\{state\}\}">p</figure-callout>}}^2+0.164{\mathrm{\&alpha;}}_p{S}_g+3.399{\mathrm{\&alpha;}}_p{\mathrm{\&alpha;}}_g-0.006S_g^2-0.143S_g{\mathrm{\&alpha;}}_g-1.728{\mathrm{\&alpha;}}_g^2)$

Step F: the response surface function is carried out reliability sensitivity analysis, the output result.

Specify the gear modification parameter S _p, α _p, S _g, α _gWith maximum allowable transmission error fluctuation Δ TE _MaxBe input variable at random, limit state function g (x) is as output variable at random, and selects Monte-Carlo probabilistic design method, carries out 5000 Monte-Carlo simulations, generated 5000 samples of the functional value of g (x), and the sampling sample is shown in figure 23.Limit state function g (x) is carried out statistical study, obtain histogram frequency distribution diagram, shown in figure 24.

Cumulative distribution function shown in Figure 25 reflection gear failure probability equals data at the numerical value of any point and the probability under this point occurs.Gear failure when g (x)＜0 checks that it is 0.32 that the result gets unreliable degree, explains that the gear fiduciary level is 0.68.Reduce vibration so must change the correction of the flank shape parameter of gear.

If do not consider stochastic variable Δ TE _Max, the correction of the flank shape coefficient relevant with the gear transmission error has 4, and changing which parameter and this parameter is to increase or reduce to the influence of gear transmission error reliability, and this all need judge through calculating the stray parameter susceptibility.Probability susceptibility can adopt statistical significance check expression, given level of significance α, and input variable is divided at random: g (x) is had significantly sex and do not make significant difference.Significance test supposes that the susceptibility of input variable is 0 at random, and calculates its probability, when this probable value surpasses level of significance α, this at random the influence of input variable will be left in the basket; Otherwise, just think this at random input variable g (x) is had conspicuousness influence.Get level of significance α=2.5%, with histogram and pie chart represent each at random input variable to the influence of g (x), shown in figure 26.

By the ratio contrast that can find out each stray parameter susceptibility in the pie chart.In histogram, can find out the influence degree of each parameter, the maximum input variable at random of susceptibility appears at Far Left, and accordings to order from big to small, arranges to the right successively; And; Susceptibility has positive and negative branch; The input variable positive axis explanation susceptibility that appears at the histogram longitudinal axis is being for just at random, expression response g (x) with this at random the increase of input variable increase, susceptibility be negative indication respond g (x) with this at random the increase of input variable reduce.

Visible by Figure 26, input variable S at random _p, α _p, S _g, α _gAll g (x) there is the conspicuousness influence, arranges according to influence degree order from big to small and be followed successively by α _p, α _g, S _p, S _g, and g (x) is with α _pIncrease and increase, with S _p, α _g, S _gIncrease and reduce.Table 3 has been listed the value of correction of the flank shape parametric reliability susceptibility.

Table 3

Susceptibility figure has pointed out and will change reliability of structure and should revise which design variable, and how scatter diagram change the approximate range of these design variables and change if then further having been pointed out.Draw limit state function value g (x) and each scatter diagram between input variable at random respectively, shown in figure 27.

Thus it is clear that, reduce the discrete range of each stochastic variable and pass through to improve α _pAnd reduction S _p, α _g, S _gValue, can effectively improve the transmission error reliability of gear.

Step 4: adopt genetic algorithm to confirm the optimized parameter of gear-profile correction of the flank shape:

Getting cicada by step 3 is variation tendency and the susceptibility that improves each stray parameter of reliability of gear train, and this has dwindled the interval of optimization searching parameter, and in the corresponding region of search, lays a good foundation for choosing of step-length.

Steps A: the correction of the flank shape parameter is carried out gene code, and random initializtion colony;

1) the correction of the flank shape parameter is carried out gene code

To repair altitude conversion and become the correction of the flank shape angle, the maximal value that obtains correction of the flank shape angle and profiling quantity is respectively 64.331 ° and 0.06mm.In order more to meet the interval at correction of the flank shape parameter place, with the suitable translation in hunting zone, in conjunction with the coding method of genetic algorithm, the span of definition correction of the flank shape parameter optimization, as shown in table 4.

Use the genetic algorithm for solving problem, the coding method of necessary first problem identificatioin for the convenience of representing, is generally all adopted the binary coding mode when encoding, and binary coding is the chromosome bit string of 0 and 1 formation with the parametric representation of problem space.Because 16 binary number can be represented the value between 0 to 65535.According to the span of table 4, if the correction of the flank shape parameter alpha _p, α _g, S _p, S _gRespectively account for 16, then their search span all is 65535.It is thus clear that precision is enough, gene code is shown in figure 28.

2) random initializtion colony

In the genetic algorithm, the initial work of population has very big influence to the operational effect of algorithm.I can be expressed as for colony:

P(i)＝(s ₁，s ₂...s _n)(16)

In the formula, i is for colony in P (i) expression, and S representes chromosome, and n is a population size.

In the genetic algorithm, the initial work of population has very big influence to the operational effect of algorithm.Because correction of the flank shape parameter rule of thumb formula is confirmed span roughly; So with correction of the flank shape values of parameters interval as boundary condition; Select the scale of n as colony; Produce n coincidence boundary condition at random, figure place be the chromosome of 64 2 scale codings as initial population, get n=30 here.

Step B: estimate colony;

According to coded system with the chromosome bit string decode profiling quantity S _pAnd S _g, the correction of the flank shape angle [alpha] _pAnd α _g

Genetic algorithm is estimated with fitness function a quality of separating, and fitness is big more, and the quality of separating is good more.Before the realization of using genetic algorithm, should define fitness function earlier.Generally speaking, there is dual mode available: a kind of from the suffered impact of the gear teeth minimum aspect consideration; Another kind is to consider from the transmission error fluctuation minimum of gear.Because gear friction when the gear transmission error is constant, so it is more reasonable that the transmission error fluctuation inverse of gear is estimated colony as adaptive value.Definition adaptive value function is following:

$f=\frac{1}{|\max \left(\mathrm{TE}\right)-\min \left(\mathrm{TE}\right)|}---\left(17\right)$

Step C: adaptive value is calculated, and judges whether to finish according to stopping criterion, if finish, then changes and removes execution in step E; Otherwise, execution in step D;

With the decoded result substitution finite element model of the top first step, calculate the minimum and maximum value of transmission error, bring adaptive value functional expression (17) into and just can calculate this chromosomal adaptive value.To each chromosome its adaptive value of calculating and according to the big young pathbreaker's chromosome ordering of adaptive value.When calculate K for the time, if colony convergence then stops, changeing and remove execution in step E, otherwise, execution in step D.

Step D: colony is used chromosome select operator, crossover operator and mutation operator; And then estimate colony, and change and remove execution in step C;

1) colony is used chromosome and selects operator:

Selection operation arrives follow-on individuality based on the fitness decision heredity of individuality, and at first the ineligible individuality of deletion in colony is filled the qualified individual new colony that forms again.At this moment just can adopt the adaptive value ratio to select to carry out individual selection; This is a kind of basic system of selection; Each individual selecteed expectation numerical value is relevant with the ratio of the average adaptive value of its adaptive value and colony in the wherein new colony, can adopt the mode of roulette dish to realize.For given scale is the colony of n, P=(s ₁, s ₂... s _n), individual fitness is f (s _j), the selecteed probability of this individuality is:

$p\left(s_j\right)=\frac{f\left(s_j\right)}{\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}f\left(s_i\right)},$ j＝1，2，L，n (18)

Promptly calculate each individual fitness shared ratio in colony's adaptive value summation, represent this individuality selected probability in selection course.

2) colony is used the chiasma operator:

Interlace operation is arranged in pairs or groups intragroup individuality at random with certain their chromosome dyad of probability exchange, obtains new individuality.With individual replicate in the former generation colony and pairing in twos at random, carry out interlace operation.Adopt the multiple spot cross method, select a pair of individuality of wanting mating, three crossover locations of picked at random.Continuous mutual exchange between the variable between the point of crossing, but do not do exchange before in first point of crossing.Be to exchange between first point of crossing and second point of crossing, exchange behind the 4th point of crossing that remaining does not exchange.Produce two new offsprings after intersecting, produce the n individuals at last and constitute offspring colony.

3) colony is used the chromosomal variation operator:

Certain jumping phenomenon that gene takes place above the chromosome during mutation operation simulating nature circle biosome carries out, thus chromosomal structure and physical behavior changed.Mutation operation is exactly for intragroup each individuality, and the genic value that changes on the locus with certain probability is other allele, thereby produces new individuality.The variation Probability p _mAffect the diversity of colony, its span is set at:

p _m＝(e ^x-1)/(e ^0.7-1)，x∈{0.0，L，0.7}(19)

Here get p _m=0.01.In genetic algorithm, mutation operator is through pressing the variation Probability p _mCertain the allelic binary-coded character that reverses is at random realized.Generally comprise two steps:

With original variation Probability p _mBe the basis, can calculate the probability P that individual in population morphs _N

P _N＝1-(1-p _m) ^w (20)

In the formula, w is chromosomal figure place, w=64 here.Calculate P _N=0.4744, getting stochastic variable is random, if random≤p _N, then this individuality is made a variation, otherwise expression is not morphed.

If chosen individuality, then each all presses the variation Probability p with selected individuality _mVariation is inserted into and goes among the next generation to get final product.

Through the above generating step P of colony (i+1) of a new generation, the purpose of these operators is to expand the coverage rate of limited individuality, embodies the thought of global search.

Step e: optimize the correction of the flank shape parameter, the parameter of linear dressing and the parameter of para-curve correction of the flank shape are optimized.

Under different correction of the flank shape parameter situation, the fluctuation of the transmission error of gear also can change accordingly.Make the minimum correction of the flank shape parameter value of transmission error fluctuation for searching, can on the basis of finite element model, utilize the solution space of genetic algorithm search correction of the flank shape parameter, find optimum correction of the flank shape parameter.If the parameter of a pair of gear is as shown in table 5.The fluctuation of input torque is ignored in the gap of ignoring bearing in this example.

Table 5

1) correction of the flank shape Parameter Optimization during linear dressing:

Employing flank profil linear dressing finite element model carries out simulation calculation and searches with genetic algorithm, and with the decoding of optimum dyeing body, it is as shown in table 6 to obtain the gear modification parameter.

Table 6

Carry out simulation calculation according to table 6 correction of the flank shape data, obtain the gear transmission error dynamics curve of linear dressing, shown in figure 29.Among the figure, dotted line is the transmission error curve before the correction of the flank shape, and visible gear transmission fluctuating error when single bidentate replaces is bigger; The engaging-in main cause that impact is transmission error fluctuation of nibbling out is described, its maximum fluctuation is 10 μ m, and solid line is to decide the transmission error curve after the parameter correction of the flank shape according to table 6; At this moment curve has not had tangible ladder property; The engaging-in main cause that impact no longer has been the transmission error fluctuation of nibbling out is described, its maximum fluctuation is reduced to 3.897 μ m, has reduced 61%.

2) correction of the flank shape Parameter Optimization during the para-curve correction of the flank shape:

Employing flank profil para-curve correction of the flank shape finite element model carries out simulation calculation and searches with genetic algorithm, and with the decoding of optimum dyeing body, it is as shown in table 7 to obtain the gear modification parameter.

Table 7

Carry out simulation calculation according to table 7 correction of the flank shape data, obtain the gear transmission error dynamics curve of para-curve correction of the flank shape, shown in figure 30.Dotted line is the transmission error curve before the correction of the flank shape among the figure; It is thus clear that the gear transmission fluctuating error when single bidentate replaces is bigger, the engaging-in main cause that impact is the transmission error fluctuation of nibbling out is described, its maximum fluctuation is 10 μ m; Solid line is to decide the transmission error curve after the correction of the flank shape of parameter para-curve according to table 7; At this moment curve has not had tangible ladder property, explain engaging-inly to nibble out impact and do not exist, and the crest instruction book bidentate of bidentate region of engagement alternately is the main cause that produces fluctuation.Its maximum fluctuation is reduced to 8.017 μ m, has reduced 20%, and the amplitude that transmission error reduces is little, explains that linear dressing is better.

Though it is pointed out that linear dressing is superior to the para-curve correction of the flank shape in this example, this also is not suitable for all scopes, and para-curve correction of the flank shape meeting is better than straight line in some cases.When carrying out the modification curve design, must confirm modification curve type and parameter of curve based on actual condition.

Step 5: the correctness of inspection parameter:

Can know that by step 4 flank profil linear dressing in this example is better than the correction of the flank shape of flank profil para-curve to the abated effect of gear transmission fluctuating error.But this linear dressing scheme just represented to transmission error fluctuation to weaken effect better.This correction of the flank shape scheme is to the abated effect of gear train vibration, the checking that also need come it is carried out vibration damping through gear nonlinear vibration movable model.

Steps A: carry out the modeling of gear nonlinear kinetics;

Adopt concentrated quality method to set up the kinetic model of the secondary pure twisting vibration of straight spur gear, shown in figure 31.Wherein, become mesh stiffness k the time _mWith engagement damping c _mAlong the action line direction; θ _pAnd θ _gBe the angular displacement of gear,

With

Be the moment that acts on the gear; I _pAnd I _gBe moment of inertia, R _pAnd R _gBe base radius; Be the transmission error of gear, backlash is 2b; Then the analysis on Torsional Vibration model of gear pair is:

$\left\{\begin{array}{c}{\mathrm{<figure-callout\; id="2"\; label="I\; p\; d"\; filenames="HDA0000066144690000102.png,HDA0000066144690000111.png"\; state="\{\{state\}\}">I</figure-callout>}}_p\frac{d^{}}{}\frac{{}^2{\mathrm{\&theta;}}_p}{d{\stackrel{\&OverBar;}{t}}^2}+R_p{c}_m(R_p\frac{{\mathrm{d\&theta;}}_p}{d\stackrel{\&OverBar;}{t}}-R_g\frac{{\mathrm{d\&theta;}}_g}{d\stackrel{\&OverBar;}{t}}-\frac{d\stackrel{\&OverBar;}{e}}{d\stackrel{\&OverBar;}{t}})+k_m{R}_p\stackrel{\&OverBar;}{f}(R_p{\mathrm{\&theta;}}_p-R_g{\mathrm{\&theta;}}_g-\stackrel{\&OverBar;}{e})={\stackrel{\&OverBar;}{T}}_p\left(\stackrel{\&OverBar;}{t}\right)\\ {\mathrm{<figure-callout\; id="2"\; label="I\; g\; d"\; filenames="HDA0000066144690000102.png,HDA0000066144690000111.png"\; state="\{\{state\}\}">I</figure-callout>}}_g\frac{d^{}}{}\frac{{}^2{\mathrm{\&theta;}}_g}{d{\stackrel{\&OverBar;}{t}}^2}+R_g{c}_m(R_p\frac{{\mathrm{d\&theta;}}_p}{d\stackrel{\&OverBar;}{t}}-R_g\frac{{\mathrm{d\&theta;}}_g}{d\stackrel{\&OverBar;}{t}}-\frac{d\stackrel{\&OverBar;}{e}}{d\stackrel{\&OverBar;}{t}})+k_m{R}_g\stackrel{\&OverBar;}{f}(R_p{\mathrm{\&theta;}}_p-R_g{\mathrm{\&theta;}}_g-\stackrel{\&OverBar;}{e})=-{\stackrel{\&OverBar;}{T}}_g\left(\stackrel{\&OverBar;}{t}\right)\end{array}\right.---\left(21\right)$

In the formula, I _p, I _gThe moment of inertia of-master, driven gear;

The transmission error of -meshing gear;

R _p, R _g-base radius;

c _m-engagement damping;

θ _p, θ _gThe twisting vibration displacement of-master, driven gear;

k _mIn-time, become mesh stiffness;

- acting on the active and passive external torque on the gear;

-nonlinear function of gear teeth engagement force when having backlash.

is the backlash nonlinearity described function of nondimensionalization, should be expressed as:

$f\left(\stackrel{\&OverBar;}{x}\left(\stackrel{\&OverBar;}{t}\right)\right)=\frac{\stackrel{\&OverBar;}{f}\left(\stackrel{\&OverBar;}{x}\left(\stackrel{\&OverBar;}{t}\right)\right)}{k_m}=\left\{\begin{array}{cc}\stackrel{\&OverBar;}{x}\left(\stackrel{\&OverBar;}{t}\right)-b& \stackrel{\&OverBar;}{x}\left(\stackrel{\&OverBar;}{t}\right)>b\\ 0& -b<\stackrel{\&OverBar;}{x}\left(\stackrel{\&OverBar;}{t}\right)<b\\ \stackrel{\&OverBar;}{x}\left(\stackrel{\&OverBar;}{t}\right)+b& \stackrel{\&OverBar;}{x}\left(\stackrel{\&OverBar;}{t}\right)<-b\end{array}\right.---\left(22\right)$

In the formula,

-transmission error.

For the convenience of finding the solution, can formula (21) be reduced to the single-degree-of-freedom kinetic model.Under the situation that does not have processing, alignment error, gear is in desirable engagement, and the relative deformation that the elastic deformation of gear in transmission process produces is called dynamic transmission error.Because main, driven gear is respectively R along the vibration displacement of action line direction _pθ _pAnd R _gθ _g, the dynamic transmission error of system then For:

${\stackrel{\&OverBar;}{x}}_d\left(\stackrel{\&OverBar;}{t}\right)=R_p{\mathrm{\&theta;}}_p\left(\stackrel{\&OverBar;}{t}\right)-R_g{\mathrm{\&theta;}}_g\left(\stackrel{\&OverBar;}{t}\right)---\left(23\right)$

In fact; Gear pair actual transfer error is the poor of dynamic transmission error and transmission error, is defined as

then:

$\stackrel{\&OverBar;}{x}\left(\stackrel{\&OverBar;}{t}\right)={\stackrel{\&OverBar;}{x}}_d\left(\stackrel{\&OverBar;}{t}\right)-\stackrel{\&OverBar;}{e}\left(\stackrel{\&OverBar;}{t}\right)---\left(24\right)$

If

m _pAnd m _gThe equivalent quality that is called gear, formula (23), (24) substitution (21) put in order:

$\frac{m_p{m}_g}{m_p+m_g}\frac{d^2\stackrel{\&OverBar;}{x}}{d{\stackrel{\&OverBar;}{t}}^2}+c_m\frac{d\stackrel{\&OverBar;}{x}}{d\stackrel{\&OverBar;}{t}}+k_m\stackrel{\&OverBar;}{f}\left(\stackrel{\&OverBar;}{x}\right)=\frac{m_p{m}_g}{m_p+m_g}(\frac{{\stackrel{\&OverBar;}{T}}_p\left(\stackrel{\&OverBar;}{t}\right)}{R_p{m}_p}+\frac{{\stackrel{\&OverBar;}{T}}_g\left(\stackrel{\&OverBar;}{t}\right)}{R_g{m}_g}-\frac{d^2\stackrel{\&OverBar;}{e}}{d{\stackrel{\&OverBar;}{t}}^2})---\left(25\right)$

If the equivalent mass of gear $m_e=\frac{m_p{m}_g}{m_p+m_g},$ The engagement force that acts on the driving gear ${\stackrel{\&OverBar;}{F}}_p\left(\stackrel{\&OverBar;}{t}\right)=\frac{{\stackrel{\&OverBar;}{T}}_p\left(\stackrel{\&OverBar;}{t}\right)}{R_p},$ The engagement force that acts on the follower gear ${\stackrel{\&OverBar;}{F}}_g\left(\stackrel{\&OverBar;}{t}\right)=\frac{{\stackrel{\&OverBar;}{T}}_g\left(\stackrel{\&OverBar;}{t}\right)}{R_g},$ Then following formula becomes:

$m_e\frac{d^2\stackrel{\&OverBar;}{x}}{d{\stackrel{\&OverBar;}{t}}^2}+c_m\frac{d\stackrel{\&OverBar;}{x}}{d\stackrel{\&OverBar;}{t}}+k_m\stackrel{\&OverBar;}{f}\left(\stackrel{\&OverBar;}{x}\right)=\frac{m_e}{m_p}{\stackrel{\&OverBar;}{F}}_p\left(\stackrel{\&OverBar;}{t}\right)+\frac{m_e}{m_g}{\stackrel{\&OverBar;}{F}}_g\left(\stackrel{\&OverBar;}{t}\right)-m_e\frac{d^2\stackrel{\&OverBar;}{e}}{d{\stackrel{\&OverBar;}{t}}^2}---\left(26\right)$

The excitation of gear train assembly is divided into two types; For the lower excitation of frequency that causes by the imbalance of gyrating mass, geometric eccentricity, prime mover and loading moment fluctuation; Generally be processed into external drive; And will become internal motivation by the higher energized process of frequency that mismachining tolerance, gear teeth elastic deformation etc. causes, order

$\stackrel{\&OverBar;}{F}\left(\stackrel{\&OverBar;}{t}\right)=\frac{m_e}{m_p}{\stackrel{\&OverBar;}{F}}_p\left(\stackrel{\&OverBar;}{t}\right)+\frac{m_e}{m_g}{\stackrel{\&OverBar;}{F}}_g\left(\stackrel{\&OverBar;}{t}\right)-m_e\frac{d^2\stackrel{\&OverBar;}{e}\left(\stackrel{\&OverBar;}{t}\right)}{d{\stackrel{\&OverBar;}{t}}^2}{\stackrel{\&OverBar;}{F}}_a\left(\stackrel{\&OverBar;}{t}\right)+{\stackrel{\&OverBar;}{F}}_h\left(\stackrel{\&OverBar;}{t}\right)---\left(27\right)$

In the formula,

is total excitation;

Be external drive power, ${\stackrel{\&OverBar;}{F}}_a\left(\stackrel{\&OverBar;}{t}\right)=\frac{m_e}{m_p}{\stackrel{\&OverBar;}{F}}_p\left(\stackrel{\&OverBar;}{t}\right)+\frac{m_e}{m_g}{\stackrel{\&OverBar;}{F}}_g\left(\stackrel{\&OverBar;}{t}\right);$

Be internal motivation power, ${\stackrel{\&OverBar;}{F}}_h\left(\stackrel{\&OverBar;}{t}\right)=-m_e\frac{d^2\stackrel{\&OverBar;}{e}\left(\stackrel{\&OverBar;}{t}\right)}{d{\stackrel{\&OverBar;}{t}}^2}.$

Then formula (26) is expressed as:

$m_e\frac{d^2\stackrel{\&OverBar;}{x}}{d{\stackrel{\&OverBar;}{t}}^2}+c_m\frac{d\stackrel{\&OverBar;}{x}}{d\stackrel{\&OverBar;}{t}}+k_m\stackrel{\&OverBar;}{f}\left(\stackrel{\&OverBar;}{x}\right)=\stackrel{\&OverBar;}{F}\left(\stackrel{\&OverBar;}{t}\right)---\left(28\right)$

Formula (28) be gear pair contain backlash and the time become the Nonlinear Vibration Differential Equations of mesh stiffness, it is the strong nonlinearity second order differential equation of single-degree-of-freedom, variable element.Under International System of Units, the order of magnitude in the differential equation between each physical quantity differs greatly, like the order of magnitude of stiffness coefficient 10 ⁷～10 ⁹Between, the order of magnitude of resistance coefficient is 10 ²～10 ⁴Between, the order of magnitude of vibratory response displacement is then in μ m level, and this makes that calculating brings very big difficulty to equation solution; Therefore, generally all will carry out the normalization dimensionless to picture formula (28) this type differential equation handles.Moreover, because strongly non-linear differential equation generally all will be found the solution with numerical method, when carrying out numerical solution,, then make error control value and step value be difficult to select if the order of magnitude of each amount differs greatly in the same equation.Can know that by formula (28) natural frequency of gear pair kinetic model does

In the formula, k ₀For the time become the mean value of mesh stiffness, m _eBe equivalent mass, the definition dimensionless time does

Then:

$\frac{d\stackrel{\&OverBar;}{x}\left(\stackrel{\&OverBar;}{t}\right)}{d\stackrel{\&OverBar;}{t}}=\frac{d\stackrel{\&OverBar;}{x}\left(\stackrel{\&OverBar;}{t}\right)}{\mathrm{dt}}\&CenterDot;\frac{\mathrm{dt}}{d\stackrel{\&OverBar;}{t}}=\frac{d\stackrel{\&OverBar;}{x}\left(\stackrel{\&OverBar;}{t}\right)}{\mathrm{dt}}{\mathrm{\&omega;}}_m---\left(29\right)$

$\frac{d^2\stackrel{\&OverBar;}{x}\left(\stackrel{\&OverBar;}{t}\right)}{d{\stackrel{\&OverBar;}{t}}^2}=\frac{d}{d\stackrel{\&OverBar;}{t}}\left[\frac{d\stackrel{\&OverBar;}{x}\left(\stackrel{\&OverBar;}{t}\right)}{d\stackrel{\&OverBar;}{t}}\right]=\frac{d}{\mathrm{dt}}{\mathrm{\&omega;}}_n\left[\frac{d\stackrel{\&OverBar;}{x}\left(\stackrel{\&OverBar;}{t}\right)}{\mathrm{dt}}{\mathrm{\&omega;}}_n\right]=\frac{d^2\stackrel{\&OverBar;}{x}\left(\stackrel{\&OverBar;}{t}\right)}{d{\mathrm{<figure-callout\; id="2"\; label="t"\; filenames="HDA0000066144690000102.png,HDA0000066144690000111.png"\; state="\{\{state\}\}">t</figure-callout>}}^2}{\mathrm{\&omega;}}_n^2---\left(30\right)$

Bring formula (29), (30) into formula (28), then have:

$m_e{\mathrm{\&omega;}}_n^2\frac{d^2\stackrel{\&OverBar;}{x}}{\mathrm{<figure-callout\; id="2"\; label="d\; t"\; filenames="HDA0000066144690000102.png,HDA0000066144690000111.png"\; state="\{\{state\}\}">d</figure-callout>}{t}^{}{}^2}+c_m{\mathrm{\&omega;}}_n\frac{d\stackrel{\&OverBar;}{x}}{\mathrm{dt}}+k_m\left(\stackrel{\&OverBar;}{t}\right)\stackrel{\&OverBar;}{f}\left(\stackrel{\&OverBar;}{x}\right)=\stackrel{\&OverBar;}{F}\left(\stackrel{\&OverBar;}{t}\right)---\left(31\right)$

If half b of backlash is the displacement nominal dimension; With

substitution formula (31):

$\frac{d^2\mathrm{<figure-callout\; id="2"\; label="x\; d\; t"\; filenames="HDA0000066144690000102.png,HDA0000066144690000111.png"\; state="\{\{state\}\}">x</figure-callout>}}{d{t}^{}}+\frac{c_m}{m_e{\mathrm{\&omega;}}_n}\frac{\mathrm{dx}}{\mathrm{dt}}+\frac{k_m\left(\stackrel{\&OverBar;}{t}\right)\stackrel{\&OverBar;}{f}\left(\stackrel{\&OverBar;}{x}\right)}{m_e{\mathrm{\&omega;}}_n^{2}b}=\frac{\stackrel{\&OverBar;}{F}\left(\stackrel{\&OverBar;}{t}\right)}{m_e{\mathrm{\&omega;}}_n^{2}b}---\left(32\right)$

If each variable corresponding dimensionless amount is respectively:

$\mathrm{\&zeta;}=\frac{c_m}{2m_e{\mathrm{\&omega;}}_n},k\left(t\right)=\frac{k_m\left(\stackrel{\&OverBar;}{t}\right)}{m_e{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="HDA0000066144690000102.png,HDA0000066144690000111.png"\; state="\{\{state\}\}">n</figure-callout>}}^2},f\left(x\right)=\frac{\stackrel{\&OverBar;}{f}\left(\stackrel{\&OverBar;}{x}\right)}{b},F\left(t\right)=\frac{\stackrel{\&OverBar;}{F}\left(\stackrel{\&OverBar;}{t}\right)}{m_{e}b{\mathrm{\&omega;}}_n^2}---\left(33\right)$

The gear pair analytical model that then can obtain the dimensionless form is:

$\stackrel{\&CenterDot;\&CenterDot;}{x}\left(t\right)+2\mathrm{\&zeta;}\stackrel{\&CenterDot;}{x}\left(t\right)+k\left(t\right)f\left(x\left(t\right)\right)=F\left(t\right)---\left(34\right)$

In the formula, f (x (t)) is the backlash nonlinear function of nondimensionalization.

$f\left(x\left(t\right)\right)=\frac{\stackrel{\&OverBar;}{f}\left(\stackrel{\&OverBar;}{x}\left(\stackrel{\&OverBar;}{t}\right)\right)}{b}=\left\{\begin{array}{cc}x\left(t\right)-1& x\left(t\right)>1\\ 0& -1\&le;x\left(t\right)\&le;1\\ x\left(t\right)+1& x\left(t\right)<-1\end{array}\right.---\left(35\right)$

Step B: gear nonlinear kinetics equation is found the solution;

Gear pair contain the gap and the time become the nonlinear kinetics equation of mesh stiffness method for solving be divided into solving method and the numerical simulation method of resolving.Analytical method mainly is harmonic wave equilibrium method, multiple dimensioned method, nibbling method etc.; Owing to receive the excitation supposed and the restriction of response forms; The computational accuracy of analytical method is not ideal, though and can under the small nonlinearity situation, find the solution nonlinear differential equation with analytical method, when system is in strong nonlinearity; Then be difficult to find the solution, so adopt numerical methods of solving.

Step C: the vibration damping demonstration of carrying out optimal case.

Dimensionless frequency

Explain that dimensionless frequency is exactly a frequency ratio, calculating the preceding parameter of correction of the flank shape is ω _e=0.6929, ξ=0.02, ε=0.283, φ _k=0, φ _e=π, F _m=0.1, e ₁/ b=0.05; Parameter after the correction of the flank shape is ω _e=0.7, ξ=0.02, ε=0.0962, φ _k=0, φ _e=π, F _m=0.1, e ₁/ b=0.015.Lay a good foundation for drawing amplitude frequency curve.

If starting condition is x (0)=0, With frequency ratio ω _eAs horizontal ordinate, calculate respectively before the correction of the flank shape with correction of the flank shape after the amplitude frequency curve of system, shown in figure 32, wherein, dotted line is the amplitude frequency curve before the correction of the flank shape, solid line is according to the amplitude frequency curve after the correction of the flank shape of data shown in the table 6.Visible by figure, system is at ω before the correction of the flank shape _eThe amplitude at=1 place has reached 2.8 * 10 ^-3After the correction of the flank shape, amplitude is reduced to 0.3 * 10 ^-3About, the visible correction of the flank shape parameter that obtains has effectively reduced the gear torsional vibration amplitude, and visible, the correctness of correction of the flank shape parameter has obtained strong proof.

Claims (6)

1. gear-profile correction method based on vibration reliability and genetic algorithm is characterized in that comprising following step:

Step 1: set up gear and profile modifying gear engagement elastic deformation virtual prototype;

Step 2: the dynamic simulation of gear transmission error when carrying out involute profile and profile modification;

Step 3: the reliability susceptibility of analyzing gear pair stray parameter engagement transmission error;

Step 4: adopt genetic algorithm to confirm the optimized parameter of gear-profile correction of the flank shape;

Step 5: the correctness of inspection parameter.

2. a kind of gear-profile correction method based on vibration reliability and genetic algorithm according to claim 1 is characterized in that described gear and the profile modifying gear engagement elastic deformation virtual prototype set up of step 1, and it specifically comprises following step:

Steps A: parametric equation and the full-depth tooth model of setting up tooth curve;

Step B: divide the gear finite element grid;

Step C: handle gear teeth boundary condition and load, set up the gear finite element model;

Step D: select the profile modification mode, if linear dressing is then set up the linear dressing finite element model; If para-curve correction of the flank shape finite element model is then set up in the para-curve correction of the flank shape.

3. a kind of gear-profile correction method according to claim 1 based on vibration reliability and genetic algorithm; The dynamic simulation of gear transmission error when it is characterized in that step 2 is described and carrying out involute profile and profile modification, it specifically comprises following step:

Steps A: the dynamic simulation that carries out involute profile gear transmission error;

Step B: the dynamic simulation that carries out profile modification gear transmission error;

Step C: the transmission error curve is compared analysis.

4. a kind of gear-profile correction method based on vibration reliability and genetic algorithm according to claim 1 is characterized in that the described analysis gear pair of step 3 stray parameter meshes the reliability susceptibility of transmission error, and it specifically comprises following step:

Steps A: confirm the number and the distribution pattern thereof of input variable at random, and judge the number N of required sample point;

Step B: randomly draw input variable, specify output variable, the finite element model that obtains confirming;

Step C: utilize the determinacy Finite Element Method to construct input variable and the corresponding relation between the output variable at random at random, and the output result;

Step D: the output of calculating all sample points;

Step e: utilize the response surface method with quadratic polynomial match sample point, response surface function expression and limit state function expression formula between definite output variable at random and the input variable;

Step F: the response surface function is carried out reliability sensitivity analysis, the output result.

5. a kind of gear-profile correction method based on vibration reliability and genetic algorithm according to claim 1 is characterized in that the described employing genetic algorithm of step 4 confirms the optimized parameter of gear-profile correction of the flank shape, and it specifically comprises following step:

Steps A: the correction of the flank shape parameter is carried out gene code, and random initializtion colony;

Step B: estimate colony;

Step C: judge whether to finish according to stopping criterion,, then change and remove execution in step E if finish; Otherwise, execution in step D;

Step D: colony is used chromosome select operator, crossover operator and mutation operator; And then estimate colony, and change and remove execution in step C;

Step e: optimize the correction of the flank shape parameter, the parameter of linear dressing and the parameter of para-curve correction of the flank shape are optimized.

6. a kind of gear-profile correction method based on vibration reliability and genetic algorithm according to claim 1 is characterized in that the correctness of the described inspection parameter of step 5, and it specifically comprises following step:

Steps A: carry out the modeling of gear nonlinear kinetics;

Step B: gear nonlinear kinetics equation is found the solution;

Step C: the vibration damping demonstration of carrying out optimal case.

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