CN103577687A - Time-varying characteristic quantitative calculation method for meshing stiffness of gear with minor defect - Google Patents

Abstract

The invention relates to a quantitative calculation method for the meshing stiffness of a gear with a minor defect. In order to describe the influence of a typical gear fault on the time-varying stiffness characteristic, a meshing stiffness energy method calculation model is firstly introduced, wherein five kinds of elastic strain energy, which refers to bending, shearing, radial compression, contact and base deformation, are respectively considered, and five corresponding stiffnesses are further formed. The quantitative calculation method is based on the energy method, the influences of case crush, tooth root crack and tooth breakage on the stiffness distribution are discussed one after another. Aiming at spalling defects, the influences of spalling length (in the meshing direction) and spalling width (in the tooth width direction) on a stiffness distribution curve is researched, and the quantitative relationship between the spalling size and the stiffnesses degradation is obtained. In the aspect of flexural fatigue crack, the change rule of the stiffness curve along with the crack depth, and the quantitative relationship between the stiffness curve and the crack depth are discussed. In the aspect of broken gear tooth, the influence of missing of a single tooth on the stiffness distribution is discussed. By adopting the quantitative calculation method, the actual meshing situation can be really reflected, the complexity and computation in the process of solving can be lowered.

Description

A kind of time-varying characteristics quantitative calculation method of local defect Gear Meshing Stiffness

Technical field

The present invention relates to a kind of gear rigidity time-varying characteristics quantitative calculation method, belong to Gear Fault Diagnosis field, relate in particular to a kind of time-varying characteristics quantitative calculation method of local defect Gear Meshing Stiffness.

Background technology

Time to become mesh stiffness be one of main driving source of gear train assembly vibratory response, while therefore calculating effectively and accurately local defect gear, becoming mesh stiffness has important meaning to research fault gear vibration Response Mechanism.

At present, both at home and abroad the method for mesh stiffness research is mainly contained to following 5 kinds of (1) international standard methods: National Standard Method can be calculated the average mesh stiffness of gear effectively and accurately, but become mesh stiffness can not calculate time; (2) experimental method: the solving result of experimental method is more accurate, but therefore complicated operation and high to experimental facilities requirement is difficult to widespread use; (3) Ishikawa method: utilizing Ishikawa formula to calculate mesh stiffness is that gear is reduced to by the trapezoidal semi-girder forming with rectangle, and does not consider the rigidity that wheel body deformability causes; (4) linear programming technique: when linear programming technique solves for normal Gear Meshing Stiffness, result is also more accurate, but the method is studied seldom to fault Gear Meshing Stiffness, the reliability of its research and the degree of accuracy of calculating need textual criticism; (5) finite element method: finite element method is conventionally by setting up the solid model of gear train assembly, apply again finite element method and calculate this gear-driven deflection, finally obtain this and become mesh stiffness when gear-driven, finite element can reflect actual engagement situation more really, but it is relatively large to solve process computation amount.Generally speaking, the mean rigidity that Ishikawa method is calculated is larger compared with National Standard Method gap, becomes the actual conditions of mesh stiffness in the time of can not effectively reflecting gear; Experimental method, finite element method calculation of complex; Linear programming technique is still immature.

Abroad to time become solving of mesh stiffness and mainly contain based on energy method, suppose that elastic strain energy is converted into several or whole in a hertz energy, shear energy, flexional and radial compression energy entirely, but seldom have the wheel body of consideration elastic deformation.A kind of time-varying characteristics quantitative calculation method of local defect Gear Meshing Stiffness is proposed based on above analysis the present invention, the method be take energy method as support, on peeling off gear, has considered to peel off hole along the facewidth and along two kinds of change in size of flank engagement direction, has become the impact of mesh stiffness when comprehensive; To Gear with Crack, considered the different crack depths impact on shear energy and flexional respectively; Finally, to containing local defect gear time become mesh stiffness and carried out solving quantitatively.

Summary of the invention

The object of the present invention is to provide a kind of time-varying characteristics quantitative calculation method of local defect Gear Meshing Stiffness, become the quantitative calculating of mesh stiffness when systematically inquiring into fault gear, a kind of energy that the elastic strain energy that is stored in meshing gear centering is transformed to quinquepartite has been proposed, then obtain five kinds of rigidity of answering in contrast, when the result of gained is final gear mesh comprehensive after the series connection of every kind of rigidity, become the method for mesh stiffness.

To achieve these goals, the technical solution used in the present invention is a kind of time-varying characteristics quantitative calculation method of local defect Gear Meshing Stiffness, and the performing step of the method is: set up Gear Meshing Stiffness computation model; Gear Meshing Stiffness computation model is peeled off in foundation; Set up Gear with Crack mesh stiffness computation model; Set up broken teeth Gear Meshing Stiffness computation model; The calculating of mesh stiffness in swing circle of fault gear.

Compare with existing computing method, the present invention has the following advantages:

1, the method be take energy method as support, on peeling off gear, has considered to peel off hole along the facewidth and along two kinds of change in size of flank engagement direction, has become the impact of mesh stiffness when comprehensive.

2, to Gear with Crack, considered the different crack depths impact on shear energy and flexional respectively, while effectively reflecting gear, become mesh stiffness actual conditions, more approach actual conditions.

3, to containing local defect gear time become mesh stiffness and carried out solving quantitatively, reflect really actual engagement situation, reduced complexity and the calculated amount in solution procedure.

Accompanying drawing explanation

Fig. 1 is tooth force figure;

Fig. 2 is for peeling off gear graph;

Fig. 3 is Gear with Crack figure;

Fig. 4 is broken teeth gear graph;

Fig. 5 is diagram of gears, Basic parameters of gear: pinion wheel number of teeth z ₁=22, gear wheel number of teeth z ₂=30, addendum coefficient h _a*=1, tip clearance coefficient c*=0.25, facewidth L ₁=L ₂=20mm, pressure angle of graduated circle a ₀=20 degree, elastic modulus E=209MP, Poisson ratio u=0.269, in figure, pinion wheel is driving wheel, number of teeth z ₁=22, arrow indication place is engagement starting point;

Fig. 6 is workflow diagram;

Fig. 7 is the geometric parameter of wheel body deformability;

Fig. 8 is the mesh stiffness of normal gear in two mesh cycles;

Fig. 9 is for peeling off the contrast of gear and normal gear obtains overall meshing stiffness curve along facewidth direction different faults size;

Figure 10 is for peeling off gear and the normally contrast of Gear Meshing Stiffness curve along flank engagement direction different faults size;

Figure 11 is Gear with Crack correlation parameter concrete meaning;

Figure 12 is the fault gear of three kinds of different crackles and the contrast of normal gear bending stiffness;

Figure 13 is the fault gear of three kinds of different crackles and the contrast of normal gear shearing rigidity;

Figure 14 is the size that normal gear subtracts Gear with Crack mesh stiffness, and normally gear and Gear with Crack mesh stiffness is poor;

Figure 15 is three kinds of different crack depths fault gears and the normally contrast of gear obtains overall meshing stiffness;

Figure 16 is broken teeth diagram of gears;

Figure 17 is broken teeth gear and the normally contrast of Gear Meshing Stiffness.

embodiment

Below with reference to compound case-study, the invention will be further described.

The performing step of the method is as follows:

1, set up Gear Meshing Stiffness computation model

1.1 determine the basic parameter of gear mesh

1.2 determine that gear mesh list bi-tooth gearing is interval

1.3 calculate respectively five kinds of rigidity of meshing gear

Be illustrated in figure 1 tooth force figure, when gear pair is meshed F, do the used time, F is decomposed into two the power Fs parallel with vertical with gear teeth center line _aand F _b.Because gear pair exists line contact, therefore there is a hertz contact stiffness; The power F that is parallel to gear teeth center line _aeffect, there is radial compression in gear, therefore there is radial compression rigidity; Be subject to perpendicular to gear teeth center line F _beffect, because of F _bbe equal to shearing and make wheel body bear moment of flexure with respect to gear centre, therefore there is shearing rigidity and bending stiffness; Finally, because of tooth force, there is elastic deformation in basis, therefore there is wheel body deformation rigidity.The strain energy that is stored in meshing gear centering is converted into a hertz energy U _h, flexional U _b, radial compression energy U _a, shear energy U _swith wheel body strain energy of distortion U _f, by energy conservation, just can be calculated the hertz rigidity k answering in contrast _h, bending stiffness k _b, radial compression rigidity k _a, shearing rigidity k _swith wheel body deformation rigidity k _f.

The calculating of 1.4 global stiffnesses

By just obtaining normal spur gear after five kinds of rigidity series connection of gained in 1.3, become mesh stiffness when comprehensive:

$k_t=\frac{1}{\frac{1}{k_h}+\frac{1}{{\mathrm{<figure-callout\; id="1"\; label="k\; b"\; filenames="DEST\_PATH\_HDA0000423208030000062.png,DEST\_PATH\_HDA0000423208030000063.png"\; state="\{\{state\}\}">k</figure-callout>}}_b}+\frac{1}{k_{a1}}+\frac{1}{{\mathrm{<figure-callout\; id="1"\; label="k\; s"\; filenames="DEST\_PATH\_HDA0000423208030000062.png,DEST\_PATH\_HDA0000423208030000063.png"\; state="\{\{state\}\}">k</figure-callout>}}_s}+\frac{1}{{\mathrm{<figure-callout\; id="1"\; label="k\; f"\; filenames="DEST\_PATH\_HDA0000423208030000062.png,DEST\_PATH\_HDA0000423208030000063.png"\; state="\{\{state\}\}">k</figure-callout>}}_f}+\frac{1}{{\mathrm{<figure-callout\; id="2"\; label="k\; b"\; filenames="DEST\_PATH\_HDA0000423208030000051.png,DEST\_PATH\_HDA0000423208030000052.png"\; state="\{\{state\}\}">k</figure-callout>}}_b}+\frac{1}{k_{a2}}+\frac{1}{{\mathrm{<figure-callout\; id="2"\; label="k\; s"\; filenames="DEST\_PATH\_HDA0000423208030000051.png,DEST\_PATH\_HDA0000423208030000052.png"\; state="\{\{state\}\}">k</figure-callout>}}_s}+\frac{1}{k_{f2}}}---\left(1\right)$

Wherein, subscript 1,2 represents respectively driving and driven gear.

2, peel off the foundation of Gear Meshing Stiffness computation model

Be illustrated in figure 2 and peel off gear graph, when gear pair is meshed F, do the used time, F is decomposed into two the power Fs parallel with vertical with gear teeth center line _aand F _b.The power F that is parallel to gear teeth center line _aeffect, there is radial compression in gear, because of F _aconstant, therefore radial compression rigidity is constant; Be subject to perpendicular to gear teeth center line F _beffect, because of F _bbe equal to shearing and make wheel body bear moment of flexure with respect to gear centre, F _bconstant, therefore shearing rigidity and bending stiffness are also constant; Finally, because of tooth force, there is elastic deformation in basis, and because of integral body, stressed F is constant, therefore exist wheel body deformation rigidity also constant.Finally the contact line length because of gear engagement changes, and hertz rigidity is main relevant with contact line length, therefore has the variation of mainly considering hertz rigidity while peeling off, now normal hertz rigidity k during gear _hbecome the hertz rigidity k while peeling off gear _hchip.Wherein, consider to peel off and cheat when tooth is very little to be changed and change along flank engagement direction along facewidth direction, k _hchiprespectively about peeling off hole size w _sand α _sfunction.

By peeling off gear hertz rigidity, substitute to obtain after normal gear hertz rigidity and become mesh stiffness while peeling off gear comprehensive:

$k_{\mathrm{tchip}}=\frac{1}{\frac{1}{k_{\mathrm{hchip}}}+\frac{1}{{\mathrm{<figure-callout\; id="1"\; label="k\; b"\; filenames="DEST\_PATH\_HDA0000423208030000062.png,DEST\_PATH\_HDA0000423208030000063.png"\; state="\{\{state\}\}">k</figure-callout>}}_b}+\frac{1}{{\mathrm{<figure-callout\; id="1"\; label="k\; s"\; filenames="DEST\_PATH\_HDA0000423208030000062.png,DEST\_PATH\_HDA0000423208030000063.png"\; state="\{\{state\}\}">k</figure-callout>}}_s}+\frac{1}{k_{a1}}+\frac{1}{{\mathrm{<figure-callout\; id="1"\; label="k\; f"\; filenames="DEST\_PATH\_HDA0000423208030000062.png,DEST\_PATH\_HDA0000423208030000063.png"\; state="\{\{state\}\}">k</figure-callout>}}_f}+\frac{1}{{\mathrm{<figure-callout\; id="2"\; label="k\; b"\; filenames="DEST\_PATH\_HDA0000423208030000051.png,DEST\_PATH\_HDA0000423208030000052.png"\; state="\{\{state\}\}">k</figure-callout>}}_b}+\frac{1}{{\mathrm{<figure-callout\; id="2"\; label="k\; s"\; filenames="DEST\_PATH\_HDA0000423208030000051.png,DEST\_PATH\_HDA0000423208030000052.png"\; state="\{\{state\}\}">k</figure-callout>}}_s}+\frac{1}{k_{a2}}+\frac{1}{k_2}}---\left(2\right)$

3, the foundation of Gear with Crack mesh stiffness computation model

Be illustrated in figure 3 Gear with Crack figure, because of Surface Contact area and component F _ado not become, therefore hertz rigidity and radial compression rigidity are constant; Whole stressed F is also constant again, therefore the rigidity of wheel body deformability is also constant; While having crackle, mainly consider the variation of bending stiffness and shearing rigidity.With step 2: the bending stiffness of normal gear is by k _bwith shearing rigidity k _sbecome respectively the crooked k of Gear with Crack _bcrackwith shearing rigidity k _scrack.K now _bcrackand k _scrackall the function about crack depth q and crackle and gear teeth center line angle v, finally Gear with Crack become when comprehensive mesh stiffness into:

$k_{\mathrm{tcrack}}=\frac{1}{\frac{1}{k_h}+\frac{1}{k_{\mathrm{bcrack}}}+\frac{1}{k_{\mathrm{scrack}}}+\frac{1}{k_{a1}}+\frac{1}{{\mathrm{<figure-callout\; id="1"\; label="k\; f"\; filenames="DEST\_PATH\_HDA0000423208030000062.png,DEST\_PATH\_HDA0000423208030000063.png"\; state="\{\{state\}\}">k</figure-callout>}}_f}+\frac{1}{{\mathrm{<figure-callout\; id="2"\; label="k\; b"\; filenames="DEST\_PATH\_HDA0000423208030000051.png,DEST\_PATH\_HDA0000423208030000052.png"\; state="\{\{state\}\}">k</figure-callout>}}_b}+\frac{1}{{\mathrm{<figure-callout\; id="2"\; label="k\; s"\; filenames="DEST\_PATH\_HDA0000423208030000051.png,DEST\_PATH\_HDA0000423208030000052.png"\; state="\{\{state\}\}">k</figure-callout>}}_s}+\frac{1}{k_{a2}}+\frac{1}{k_2}}---\left(3\right)$

4, the foundation of broken teeth Gear Meshing Stiffness computation model

Be illustrated in figure 4 broken teeth gear graph, in the position of gear tooth breakage, lose contact, original bi-tooth gearing district becomes monodentate engagement.Therefore, obtains overall meshing stiffness only by monodentate to forming, broken teeth Gear Meshing Stiffness computing formula is:

${\mathrm{kt}}_{\mathrm{broken}}=\frac{1}{\frac{1}{k_h}+\frac{1}{{\mathrm{<figure-callout\; id="1"\; label="k\; b"\; filenames="DEST\_PATH\_HDA0000423208030000062.png,DEST\_PATH\_HDA0000423208030000063.png"\; state="\{\{state\}\}">\; <figure-callout\; id="1"\; label="k\; b"\; filenames="DEST\_PATH\_HDA0000423208030000062.png,DEST\_PATH\_HDA0000423208030000063.png"\; state="\{\{state\}\}">k</figure-callout>\; </figure-callout>}}_b}+\frac{1}{{\mathrm{<figure-callout\; id="2"\; label="k\; b"\; filenames="DEST\_PATH\_HDA0000423208030000051.png,DEST\_PATH\_HDA0000423208030000052.png"\; state="\{\{state\}\}">\; <figure-callout\; id="1"\; label="k\; b"\; filenames="DEST\_PATH\_HDA0000423208030000062.png,DEST\_PATH\_HDA0000423208030000063.png"\; state="\{\{state\}\}">k</figure-callout>\; </figure-callout>}}_b}+\frac{1}{{\mathrm{<figure-callout\; id="1"\; label="k\; s"\; filenames="DEST\_PATH\_HDA0000423208030000062.png,DEST\_PATH\_HDA0000423208030000063.png"\; state="\{\{state\}\}">\; <figure-callout\; id="1"\; label="k\; s"\; filenames="DEST\_PATH\_HDA0000423208030000062.png,DEST\_PATH\_HDA0000423208030000063.png"\; state="\{\{state\}\}">k</figure-callout>\; </figure-callout>}}_s}+\frac{1}{{\mathrm{<figure-callout\; id="2"\; label="k\; s"\; filenames="DEST\_PATH\_HDA0000423208030000051.png,DEST\_PATH\_HDA0000423208030000052.png"\; state="\{\{state\}\}">\; <figure-callout\; id="1"\; label="k\; s"\; filenames="DEST\_PATH\_HDA0000423208030000062.png,DEST\_PATH\_HDA0000423208030000063.png"\; state="\{\{state\}\}">k</figure-callout>\; </figure-callout>}}_s}+\frac{1}{k_{a\mathrm{1,1}}}+\frac{1}{k_{a\mathrm{2,1}}}+\frac{1}{{\mathrm{<figure-callout\; id="1"\; label="k\; f"\; filenames="DEST\_PATH\_HDA0000423208030000062.png,DEST\_PATH\_HDA0000423208030000063.png"\; state="\{\{state\}\}">\; <figure-callout\; id="1"\; label="k\; f"\; filenames="DEST\_PATH\_HDA0000423208030000062.png,DEST\_PATH\_HDA0000423208030000063.png"\; state="\{\{state\}\}">k</figure-callout>\; </figure-callout>}}_f}+\frac{1}{{\mathrm{<figure-callout\; id="2"\; label="k\; f"\; filenames="DEST\_PATH\_HDA0000423208030000051.png,DEST\_PATH\_HDA0000423208030000052.png"\; state="\{\{state\}\}">k</figure-callout>}}_f}}---\left(4\right)$

Wherein comma presubscript 1,2 represents driving and driven gear; The gear pair on left side when subscript 1 represents bi-tooth gearing after comma.

5, the calculating of mesh stiffness in swing circle of fault gear

Be illustrated in figure 5 diagram of gears, establishing meshing gear centering pinion wheel is driving wheel, and when the initial two pairs of gears of take mesh simultaneously, the gear mesh on the left side is benchmark.If the fault gear teeth are first teeth that are rotated counterclockwise on pinion wheel, when pinion wheel rotates a circle, gear pair has z ₁(z ₁for the pinion wheel number of teeth) individual mesh cycle.At [3, z ₁] mesh stiffness of individual mesh cycle is identical with normal gear.Within first and second mesh cycle, by step 2～4, can be obtained the size of fault gear tooth rigidity value.So far, can obtain the mesh stiffness in swing circle of fault gear.

Be illustrated in figure 6 the workflow diagram of the present invention to local defect Gear Meshing Stiffness time-varying characteristics quantitative calculation method.Specific implementation process is as follows:

1, the calculating of healthy Meshing Stiffness of Spur Gears

1.1 determine the basic parameter of gear mesh

Parameter and the material behavior of selected standard involute spur, take Fig. 5 as example.The number of teeth of driving and driven gear is respectively z ₁=22, z ₂=30; Addendum coefficient h _a*=1; Tip clearance coefficient c*=0.25; Facewidth L ₁=L ₂=20mm; Pressure angle of graduated circle α ₀=20.; Elastic modulus E=2.09*10 ¹¹; Poisson ratio u=0.269.

1.2 determine a mesh cycle

By the fundamental formular of gear:

$\begin{array}{c}{\mathrm{\&theta;}}_d=\tan \left(\arccos \frac{z_1\cos {\mathrm{\&alpha;}}_0}{z_1+2}\right)-\frac{2\mathrm{\&pi;}}{z_1}-\\ \tan \left[\arccos \frac{z_1\cos {\mathrm{\&alpha;}}_0}{\sqrt{{(z_2+2)}^2+{(z_1+z_2)}^2-2(z_2+2)(z_1+z_2)\cos (\arccos \frac{z_2\cos {\mathrm{\&alpha;}}_0}{z2+2}-{\mathrm{\&alpha;}}_0)}}\right]\end{array}$

θ ₁∈ [0, θ _d] Shi Wei bi-tooth gearing district,

time be monodentate region of engagement.As calculated: θ _d=10.2 °, this belongs to bi-tooth gearing district to gear when [0,10.2 °], belongs to monodentate region of engagement when [10.2 °, 16.4 °].

1.3 calculate respectively the rigidity of hertz rigidity, bending stiffness, shearing rigidity, radial compression rigidity and the wheel body deformability of normal gear engagement in the mesh cycle.

By mechanics ABC, can be obtained

Hertz rigidity: $k_{h,i}=\frac{\mathrm{\&pi;EW}}{4(1-{\mathrm{\&mu;}}^2)}---\left(6\right)$

Bending stiffness: $\frac{1}{{\mathrm{<figure-callout\; id="1"\; label="k\; b"\; filenames="DEST\_PATH\_HDA0000423208030000062.png,DEST\_PATH\_HDA0000423208030000063.png"\; state="\{\{state\}\}">k</figure-callout>}}_b}={\&Integral;}_{{-\mathrm{\&alpha;}}_{1,i}}^{{\mathrm{\&alpha;}}_2}\frac{3{\{1+\cos {\mathrm{\&alpha;}}_{1,i}[({\mathrm{\&alpha;}}_2-\mathrm{\&alpha;})\sin \mathrm{\&alpha;}\left]\right\}}^2({\mathrm{\&alpha;}}_2-\mathrm{\&alpha;})\cos \mathrm{\&alpha;}}{2\mathrm{EL}{[\sin \mathrm{\&alpha;}+({\mathrm{\&alpha;}}_2-\mathrm{\&alpha;})\cos \mathrm{\&alpha;}]}^3}\mathrm{d\&alpha;}---\left(7\right)$

$\frac{1}{{\mathrm{<figure-callout\; id="2"\; label="k\; b"\; filenames="DEST\_PATH\_HDA0000423208030000051.png,DEST\_PATH\_HDA0000423208030000052.png"\; state="\{\{state\}\}">k</figure-callout>}}_b}={\&Integral;}_{{-\mathrm{\&alpha;}}_{1,i}^{\&prime;}}^{{\mathrm{\&alpha;}}_2^{\&prime;}}\frac{3{\{1+\cos {\mathrm{\&alpha;}}_{1,i}^{\&prime;}[({\mathrm{\&alpha;}}_2^{\&prime;}-\mathrm{\&alpha;})\sin \mathrm{\&alpha;}\left]\right\}}^2({\mathrm{\&alpha;}}_2^{\&prime;}-\mathrm{\&alpha;})\cos \mathrm{\&alpha;}}{2\mathrm{EL}{[\sin \mathrm{\&alpha;}+({\mathrm{\&alpha;}}_2^{\&prime;}-\mathrm{\&alpha;})\cos \mathrm{\&alpha;}]}^3}\mathrm{d\&alpha;}---\left(8\right)$

Shearing rigidity: $\frac{1}{{\mathrm{<figure-callout\; id="1"\; label="k\; s"\; filenames="DEST\_PATH\_HDA0000423208030000062.png,DEST\_PATH\_HDA0000423208030000063.png"\; state="\{\{state\}\}">k</figure-callout>}}_s}={\&Integral;}_{{-\mathrm{\&alpha;}}_{1,i}}^{{\mathrm{\&alpha;}}_2}\frac{1.2(1+\mathrm{\&mu;})({\mathrm{\&alpha;}}_2-\mathrm{\&alpha;})\cos \mathrm{\&alpha;}{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="DEST\_PATH\_HDA0000423208030000051.png,DEST\_PATH\_HDA0000423208030000052.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2{\mathrm{\&alpha;}}_{1,i}}{\mathrm{EL}[\sin \mathrm{\&alpha;}+({\mathrm{\&alpha;}}_2-\mathrm{\&alpha;})\cos \mathrm{\&alpha;}]}\mathrm{d\&alpha;}---\left(9\right)$

$\frac{1}{{\mathrm{<figure-callout\; id="2"\; label="k\; s"\; filenames="DEST\_PATH\_HDA0000423208030000051.png,DEST\_PATH\_HDA0000423208030000052.png"\; state="\{\{state\}\}">k</figure-callout>}}_s}={\&Integral;}_{{-\mathrm{\&alpha;}}_{1,i}^{\&prime;}}^{{\mathrm{\&alpha;}}_2^{\&prime;}}\frac{1.2(1+\mathrm{\&mu;})({\mathrm{\&alpha;}}_2^{\&prime;}-\mathrm{\&alpha;})\cos \mathrm{\&alpha;}{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="DEST\_PATH\_HDA0000423208030000051.png,DEST\_PATH\_HDA0000423208030000052.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2{\mathrm{\&alpha;}}_{1,i}^{\&prime;}}{\mathrm{EL}[\sin \mathrm{\&alpha;}+({\mathrm{\&alpha;}}_2^{\&prime;}-\mathrm{\&alpha;})\cos \mathrm{\&alpha;}]}\mathrm{d\&alpha;}---\left(10\right)$

Radial compression rigidity: $\frac{1}{k_{a1,i}}={\&Integral;}_{{-\mathrm{\&alpha;}}_{1,i}}^{{\mathrm{\&alpha;}}_2}\frac{({\mathrm{\&alpha;}}_2-\mathrm{\&alpha;})\cos \mathrm{\&alpha;}{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="DEST\_PATH\_HDA0000423208030000051.png,DEST\_PATH\_HDA0000423208030000052.png"\; state="\{\{state\}\}">sin</figure-callout>}}^2{\mathrm{\&alpha;}}_{1,\mathrm{<figure-callout\; id="2"\; label="i"\; filenames="DEST\_PATH\_HDA0000423208030000051.png,DEST\_PATH\_HDA0000423208030000052.png"\; state="\{\{state\}\}">i</figure-callout>}}}{2\mathrm{EL}[\sin \mathrm{\&alpha;}+({\mathrm{\&alpha;}}_2-\mathrm{\&alpha;})\cos \mathrm{\&alpha;}]}\mathrm{d\&alpha;}---\left(11\right)$

$\frac{1}{k_{\mathrm{\&alpha;}2,i}}={\&Integral;}_{{-\mathrm{\&alpha;}}_{1,i}^{\&prime;}}^{{\mathrm{\&alpha;}}_2^{\&prime;}}\frac{({\mathrm{\&alpha;}}_2^{\&prime;}-\mathrm{\&alpha;})\cos \mathrm{\&alpha;}{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="DEST\_PATH\_HDA0000423208030000051.png,DEST\_PATH\_HDA0000423208030000052.png"\; state="\{\{state\}\}">sin</figure-callout>}}^2{\mathrm{\&alpha;}}_{1,i}^{\&prime;}}{2\mathrm{EL}[\sin \mathrm{\&alpha;}+({\mathrm{\&alpha;}}_2^{\&prime;}-\mathrm{\&alpha;})\cos \mathrm{\&alpha;}]}\mathrm{d\&alpha;}---\left(12\right)$

In each formula, i=1,2 two pairs of gear pairs while representing bi-tooth gearing respectively, and

${\mathrm{\&alpha;}}_2=\frac{\mathrm{\&pi;}}{{2\mathrm{<figure-callout\; id="1"\; label="z"\; filenames="DEST\_PATH\_HDA0000423208030000062.png,DEST\_PATH\_HDA0000423208030000063.png"\; state="\{\{state\}\}">z</figure-callout>}}_1}+{\mathrm{inv\&alpha;}}_0=\frac{\mathrm{\&pi;}}{{2\mathrm{<figure-callout\; id="1"\; label="z"\; filenames="DEST\_PATH\_HDA0000423208030000062.png,DEST\_PATH\_HDA0000423208030000063.png"\; state="\{\{state\}\}">z</figure-callout>}}_1}+{\tan \mathrm{\&alpha;}}_0-{\mathrm{\&alpha;}}_0$

${\mathrm{\&alpha;}}_2^{\&prime;}=\frac{\mathrm{\&pi;}}{{2\mathrm{<figure-callout\; id="2"\; label="z"\; filenames="DEST\_PATH\_HDA0000423208030000051.png,DEST\_PATH\_HDA0000423208030000052.png"\; state="\{\{state\}\}">z</figure-callout>}}_2}+{\mathrm{inv\&alpha;}}_0=\frac{\mathrm{\&pi;}}{{2\mathrm{<figure-callout\; id="2"\; label="z"\; filenames="DEST\_PATH\_HDA0000423208030000051.png,DEST\_PATH\_HDA0000423208030000052.png"\; state="\{\{state\}\}">z</figure-callout>}}_2}+{\tan \mathrm{\&alpha;}}_0-{\mathrm{\&alpha;}}_0---\left(13\right)$

$\begin{array}{c}{\mathrm{\&alpha;}}_{\mathrm{1,1}}={\mathrm{\&theta;}}_1-\frac{\mathrm{\&pi;}}{{2z}_1}-(\tan {\mathrm{\&alpha;}}_0-{\mathrm{\&alpha;}}_0)-\\ \tan [\mathrm{<figure-callout\; id="1"\; label="arccos\; z"\; filenames="DEST\_PATH\_HDA0000423208030000062.png,DEST\_PATH\_HDA0000423208030000063.png"\; state="\{\{state\}\}">arccos</figure-callout>}\frac{z_{}}{}\frac{{}_1\cos {\mathrm{\&alpha;}}_0}{\sqrt{{({\mathrm{<figure-callout\; id="2"\; label="z"\; filenames="DEST\_PATH\_HDA0000423208030000051.png,DEST\_PATH\_HDA0000423208030000052.png"\; state="\{\{state\}\}">z</figure-callout>}}_2+2)}^2+{({\mathrm{<figure-callout\; id="1"\; label="z"\; filenames="DEST\_PATH\_HDA0000423208030000062.png,DEST\_PATH\_HDA0000423208030000063.png"\; state="\{\{state\}\}">z</figure-callout>}}_1+z_2)}^2-2({\mathrm{<figure-callout\; id="2"\; label="z"\; filenames="DEST\_PATH\_HDA0000423208030000051.png,DEST\_PATH\_HDA0000423208030000052.png"\; state="\{\{state\}\}">z</figure-callout>}}_2+2)({\mathrm{<figure-callout\; id="1"\; label="z"\; filenames="DEST\_PATH\_HDA0000423208030000062.png,DEST\_PATH\_HDA0000423208030000063.png"\; state="\{\{state\}\}">z</figure-callout>}}_1+z_2)\cos (\mathrm{<figure-callout\; id="2"\; label="arccos\; z"\; filenames="DEST\_PATH\_HDA0000423208030000051.png,DEST\_PATH\_HDA0000423208030000052.png"\; state="\{\{state\}\}">arccos</figure-callout>}\frac{z_{}}{}\frac{{}_2\cos {\mathrm{\&alpha;}}_0}{{\mathrm{<figure-callout\; id="2"\; label="z"\; filenames="DEST\_PATH\_HDA0000423208030000051.png,DEST\_PATH\_HDA0000423208030000052.png"\; state="\{\{state\}\}">z</figure-callout>}}_2+2}-{\mathrm{\&alpha;}}_0)}}---\left(14\right)\end{array}$

${\mathrm{\&alpha;}}_{\mathrm{1,1}}^{\&prime;}=\tan \left(\mathrm{<figure-callout\; id="2"\; label="arccos\; z"\; filenames="DEST\_PATH\_HDA0000423208030000051.png,DEST\_PATH\_HDA0000423208030000052.png"\; state="\{\{state\}\}">arccos</figure-callout>}\frac{z_{}}{}\frac{{}_2\cos {\mathrm{\&alpha;}}_0}{{\mathrm{<figure-callout\; id="2"\; label="z"\; filenames="DEST\_PATH\_HDA0000423208030000051.png,DEST\_PATH\_HDA0000423208030000052.png"\; state="\{\{state\}\}">z</figure-callout>}}_2+2}\right)-\frac{\mathrm{\&pi;}}{{2z}_2}-(\tan {\mathrm{\&alpha;}}_0-{\mathrm{\&alpha;}}_0)-\frac{z_1}{{\mathrm{<figure-callout\; id="2"\; label="z"\; filenames="DEST\_PATH\_HDA0000423208030000051.png,DEST\_PATH\_HDA0000423208030000052.png"\; state="\{\{state\}\}">z</figure-callout>}}_2}{\mathrm{\&theta;}}_1---\left(15\right)$

$\begin{array}{c}{\mathrm{\&alpha;}}_{1,2}={\mathrm{\&theta;}}_1-\frac{3\mathrm{\&pi;}}{{2z}_1}-(\tan {\mathrm{\&alpha;}}_0-{\mathrm{\&alpha;}}_0)+\\ \tan [\mathrm{<figure-callout\; id="1"\; label="arccos\; z"\; filenames="DEST\_PATH\_HDA0000423208030000062.png,DEST\_PATH\_HDA0000423208030000063.png"\; state="\{\{state\}\}">arccos</figure-callout>}\frac{z_{}}{}\frac{{}_1\cos {\mathrm{\&alpha;}}_0}{\sqrt{{({\mathrm{<figure-callout\; id="2"\; label="z"\; filenames="DEST\_PATH\_HDA0000423208030000051.png,DEST\_PATH\_HDA0000423208030000052.png"\; state="\{\{state\}\}">z</figure-callout>}}_2+2)}^2+{({\mathrm{<figure-callout\; id="1"\; label="z"\; filenames="DEST\_PATH\_HDA0000423208030000062.png,DEST\_PATH\_HDA0000423208030000063.png"\; state="\{\{state\}\}">z</figure-callout>}}_1+z_2)}^2-2({\mathrm{<figure-callout\; id="2"\; label="z"\; filenames="DEST\_PATH\_HDA0000423208030000051.png,DEST\_PATH\_HDA0000423208030000052.png"\; state="\{\{state\}\}">z</figure-callout>}}_2+2)({\mathrm{<figure-callout\; id="1"\; label="z"\; filenames="DEST\_PATH\_HDA0000423208030000062.png,DEST\_PATH\_HDA0000423208030000063.png"\; state="\{\{state\}\}">z</figure-callout>}}_1+z_2)\cos (\mathrm{<figure-callout\; id="2"\; label="arccos\; z"\; filenames="DEST\_PATH\_HDA0000423208030000051.png,DEST\_PATH\_HDA0000423208030000052.png"\; state="\{\{state\}\}">arccos</figure-callout>}\frac{z_{}}{}\frac{{}_2\cos {\mathrm{\&alpha;}}_0}{{\mathrm{<figure-callout\; id="2"\; label="z"\; filenames="DEST\_PATH\_HDA0000423208030000051.png,DEST\_PATH\_HDA0000423208030000052.png"\; state="\{\{state\}\}">z</figure-callout>}}_2+2}-{\mathrm{\&alpha;}}_0)}}---\left(16\right)\end{array}$

${\mathrm{\&alpha;}}_{1,2}^{\&prime;}=\tan \left(\mathrm{<figure-callout\; id="2"\; label="arccos\; z"\; filenames="DEST\_PATH\_HDA0000423208030000051.png,DEST\_PATH\_HDA0000423208030000052.png"\; state="\{\{state\}\}">arccos</figure-callout>}\frac{z_{}}{}\frac{{}_2\cos {\mathrm{\&alpha;}}_0}{{\mathrm{<figure-callout\; id="2"\; label="z"\; filenames="DEST\_PATH\_HDA0000423208030000051.png,DEST\_PATH\_HDA0000423208030000052.png"\; state="\{\{state\}\}">z</figure-callout>}}_2+2}\right)-\frac{5\mathrm{\&pi;}}{{2z}_2}-(\tan {\mathrm{\&alpha;}}_0-{\mathrm{\&alpha;}}_0)-\frac{z_1}{{\mathrm{<figure-callout\; id="2"\; label="z"\; filenames="DEST\_PATH\_HDA0000423208030000051.png,DEST\_PATH\_HDA0000423208030000052.png"\; state="\{\{state\}\}">z</figure-callout>}}_2}{\mathrm{\&theta;}}_1---\left(17\right)$

Wheel body deformation rigidity:

$\frac{1}{k_f}=\frac{{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="DEST\_PATH\_HDA0000423208030000051.png,DEST\_PATH\_HDA0000423208030000052.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2\mathrm{\&alpha;}}{\mathrm{EL}}\{L^{*}{\left(\frac{u_f}{s_f}\right)}^2+M^{*}\left(\frac{u_f}{s_f}\right)+P^{*}(1+Q^{*}{\tan }^2\mathrm{\&alpha;})\}---\left(18\right)$

Coefficient L* wherein, M*, P* and Q* can be expressed as by polynomial function:

$X^{*}=\frac{A}{{\mathrm{\&theta;}}_{\mathrm{<figure-callout\; id="2"\; label="f"\; filenames="DEST\_PATH\_HDA0000423208030000051.png,DEST\_PATH\_HDA0000423208030000052.png"\; state="\{\{state\}\}">f</figure-callout>}}^2}+{\mathrm{<figure-callout\; id="2"\; label="Bh\; f"\; filenames="DEST\_PATH\_HDA0000423208030000051.png,DEST\_PATH\_HDA0000423208030000052.png"\; state="\{\{state\}\}">Bh</figure-callout>}}_f^{}+\frac{{\mathrm{<figure-callout\; id="2"\; label="Ch\; f"\; filenames="DEST\_PATH\_HDA0000423208030000051.png,DEST\_PATH\_HDA0000423208030000052.png"\; state="\{\{state\}\}">Ch</figure-callout>}}_f^{}}{{\mathrm{\&theta;}}_{\mathrm{<figure-callout\; id="2"\; label="f"\; filenames="DEST\_PATH\_HDA0000423208030000051.png,DEST\_PATH\_HDA0000423208030000052.png"\; state="\{\{state\}\}">f</figure-callout>}}^2}+\frac{D}{{\mathrm{\&theta;}}_f}+{\mathrm{Eh}}_{\mathrm{<figure-callout\; id="2"\; label="f"\; filenames="DEST\_PATH\_HDA0000423208030000051.png,DEST\_PATH\_HDA0000423208030000052.png"\; state="\{\{state\}\}">f</figure-callout>}}^2+F---\left(19\right)$

X ^*represent coefficient L ^*, M ^*, P ^*and Q ^*.H _f=r _f/ r, r _ffor root radius, u _f, θ _fand s _fmeaning as shown in Figure 7.A, B, C, D, the value of E and F is as shown in following table one.

The value of parameter A, B, C, D, E, F in table one formula (19)

?	L *	M *	P *	Q *
					A	-5.574×10 -5	60.111×10 -5	-50.952×10 -5	-6.2042×10 -5
B	-1.9986×10 -3	28.100×10 -3	185.50×10 -3	9.0889×10 -3
					C	-2.3015×10 -4	-83.431×10 -4	0.0538×10 -4	-4.0964×10 -4
D	4.7702×10 -3	-9.9256×10 -3	53.300×10 -3	7.8297×10 -3
					E	0.0271	0.1624	0.2895	-0.1472
F	6.8045	0.9086	0.9236	0.6904

1.4 obtains overall meshing stiffness calculate

$k_{t,1}=\frac{1}{\frac{1}{k_h}+\frac{1}{{\mathrm{<figure-callout\; id="1"\; label="k\; b"\; filenames="DEST\_PATH\_HDA0000423208030000062.png,DEST\_PATH\_HDA0000423208030000063.png"\; state="\{\{state\}\}">\; <figure-callout\; id="1"\; label="k\; b"\; filenames="DEST\_PATH\_HDA0000423208030000062.png,DEST\_PATH\_HDA0000423208030000063.png"\; state="\{\{state\}\}">k</figure-callout>\; </figure-callout>}}_b}+\frac{1}{{\mathrm{<figure-callout\; id="2"\; label="k\; b"\; filenames="DEST\_PATH\_HDA0000423208030000051.png,DEST\_PATH\_HDA0000423208030000052.png"\; state="\{\{state\}\}">\; <figure-callout\; id="1"\; label="k\; b"\; filenames="DEST\_PATH\_HDA0000423208030000062.png,DEST\_PATH\_HDA0000423208030000063.png"\; state="\{\{state\}\}">k</figure-callout>\; </figure-callout>}}_b}+\frac{1}{{\mathrm{<figure-callout\; id="1"\; label="k\; s"\; filenames="DEST\_PATH\_HDA0000423208030000062.png,DEST\_PATH\_HDA0000423208030000063.png"\; state="\{\{state\}\}">\; <figure-callout\; id="1"\; label="k\; s"\; filenames="DEST\_PATH\_HDA0000423208030000062.png,DEST\_PATH\_HDA0000423208030000063.png"\; state="\{\{state\}\}">k</figure-callout>\; </figure-callout>}}_s}+\frac{1}{{\mathrm{<figure-callout\; id="2"\; label="k\; s"\; filenames="DEST\_PATH\_HDA0000423208030000051.png,DEST\_PATH\_HDA0000423208030000052.png"\; state="\{\{state\}\}">\; <figure-callout\; id="1"\; label="k\; s"\; filenames="DEST\_PATH\_HDA0000423208030000062.png,DEST\_PATH\_HDA0000423208030000063.png"\; state="\{\{state\}\}">k</figure-callout>\; </figure-callout>}}_s}+\frac{1}{k_{a\mathrm{1,1}}}+\frac{1}{k_{a\mathrm{2,1}}}+\frac{1}{k_{f1}}}---\left(20\right)$

$k_{t,2}=\frac{1}{\frac{1}{k_h}+\frac{1}{{\mathrm{<figure-callout\; id="1"\; label="k\; b"\; filenames="DEST\_PATH\_HDA0000423208030000062.png,DEST\_PATH\_HDA0000423208030000063.png"\; state="\{\{state\}\}">\; <figure-callout\; id="2"\; label="k\; b"\; filenames="DEST\_PATH\_HDA0000423208030000051.png,DEST\_PATH\_HDA0000423208030000052.png"\; state="\{\{state\}\}">k</figure-callout>\; </figure-callout>}}_b}+\frac{1}{{\mathrm{<figure-callout\; id="2"\; label="k\; b"\; filenames="DEST\_PATH\_HDA0000423208030000051.png,DEST\_PATH\_HDA0000423208030000052.png"\; state="\{\{state\}\}">\; <figure-callout\; id="2"\; label="k\; b"\; filenames="DEST\_PATH\_HDA0000423208030000051.png,DEST\_PATH\_HDA0000423208030000052.png"\; state="\{\{state\}\}">k</figure-callout>\; </figure-callout>}}_b}+\frac{1}{{\mathrm{<figure-callout\; id="1"\; label="k\; s"\; filenames="DEST\_PATH\_HDA0000423208030000062.png,DEST\_PATH\_HDA0000423208030000063.png"\; state="\{\{state\}\}">\; <figure-callout\; id="2"\; label="k\; s"\; filenames="DEST\_PATH\_HDA0000423208030000051.png,DEST\_PATH\_HDA0000423208030000052.png"\; state="\{\{state\}\}">k</figure-callout>\; </figure-callout>}}_s}+\frac{1}{{\mathrm{<figure-callout\; id="2"\; label="k\; s"\; filenames="DEST\_PATH\_HDA0000423208030000051.png,DEST\_PATH\_HDA0000423208030000052.png"\; state="\{\{state\}\}">\; <figure-callout\; id="2"\; label="k\; s"\; filenames="DEST\_PATH\_HDA0000423208030000051.png,DEST\_PATH\_HDA0000423208030000052.png"\; state="\{\{state\}\}">k</figure-callout>\; </figure-callout>}}_s}+\frac{1}{k_{a\mathrm{1,2}}}+\frac{1}{k_{a\mathrm{2,2}}}+\frac{1}{k_{f2}}}---\left(21\right)$

K _{t, 1}, k _{t, 2}the mesh stiffness of the right and left while representing bi-tooth gearing respectively, obtains overall meshing stiffness k _t=k _{t, 1}+ k _{t, 2}.Stiffness curve as shown in Figure 8.In Meshing Process of Spur Gear, along with the alternately variation of single biconjugate tooth engagement, mesh stiffness also periodically changes thereupon.

2, peel off the calculating of Gear Meshing Stiffness

In Meshing Process of Spur Gear, suppose that peeling off position, hole is present near reference circle.If peel off the rectangle that is shaped as in hole, the flank of tooth exists while peeling off, because being mainly that variation has occurred gear pair contact length, therefore the stiffness variation of bending stiffness, shearing rigidity, radial compression rigidity and wheel body deformability is less, and the now main variation of considering hertz rigidity.By

: exist while peeling off, facewidth W will become actual engagement facewidth W _c, W now _c=W-w _s, w wherein _sfor peeling off hole in the size of facewidth direction,

global stiffness becomes:

$k_{\mathrm{tchip}}=\frac{1}{\frac{1}{k_{\mathrm{hchip}}}+\frac{1}{{\mathrm{<figure-callout\; id="1"\; label="k\; b"\; filenames="DEST\_PATH\_HDA0000423208030000062.png,DEST\_PATH\_HDA0000423208030000063.png"\; state="\{\{state\}\}">k</figure-callout>}}_b}+\frac{1}{{\mathrm{<figure-callout\; id="1"\; label="k\; s"\; filenames="DEST\_PATH\_HDA0000423208030000062.png,DEST\_PATH\_HDA0000423208030000063.png"\; state="\{\{state\}\}">k</figure-callout>}}_s}+\frac{1}{k_{a1}}+\frac{1}{{\mathrm{<figure-callout\; id="1"\; label="k\; f"\; filenames="DEST\_PATH\_HDA0000423208030000062.png,DEST\_PATH\_HDA0000423208030000063.png"\; state="\{\{state\}\}">k</figure-callout>}}_f}+\frac{1}{{\mathrm{<figure-callout\; id="2"\; label="k\; b"\; filenames="DEST\_PATH\_HDA0000423208030000051.png,DEST\_PATH\_HDA0000423208030000052.png"\; state="\{\{state\}\}">k</figure-callout>}}_b}+\frac{1}{{\mathrm{<figure-callout\; id="2"\; label="k\; s"\; filenames="DEST\_PATH\_HDA0000423208030000051.png,DEST\_PATH\_HDA0000423208030000052.png"\; state="\{\{state\}\}">k</figure-callout>}}_s}+\frac{1}{k_{a2}}+\frac{1}{k_{f2}}}---\left(22\right)$

Now, becoming solving of mesh stiffness during this fault gear minute peels off hole size and only along facewidth direction, changes and only along flank engagement direction, change two kinds of situations:

2.1 peel off hole size only considers along the variation of facewidth direction

Take Fig. 2 as example, normal gear tooth width W=20mm, whole depth h=11.25mm.If the size α along flank engagement direction _s=4mm is certain value, facewidth direction w _sget respectively 3mm, 6mm, 9mm, tetra-groups of different values of 12mm, w _sthe ratio of four shared facewidth of value be respectively 15%, 30%, 45%, 60%.Finally obtain peeling off gear obtains overall meshing stiffness as shown in Figure 9, by Fig. 9, obviously drawn: facewidth direction is peeled off size w _slarger, rigidity is less.

By the width W c substitution hertz rigidity formula of actual participation engagement:

if to a pair of gear of determining, E, L, μ are constant, so λ, and A is constant.If w _sfor m, k _hchipfor n, finally obtaining peeling off gear hertz rigidity n is the linear function that peels off size m about changing along facewidth direction, and its expression formula is: λ m+n-A=0.To a pair of gear of determining, bending stiffness, shearing rigidity, radial compression rigidity and wheel body deformation rigidity at a time rigidity value do not change along with the change of x again, therefore establish $B=\frac{1}{k_{b1}}+\frac{1}{k_{s1}}+\frac{1}{k_{a1}}+\frac{1}{k_{f1}}+\frac{1}{k_{b2}}+\frac{1}{k_{s2}}+\frac{1}{k_{a2}}+\frac{1}{k_{f2}},$ B will not change along with the variation of u yet.Now can directly obtain peeling off gear obtains overall meshing stiffness k _tchipwith the pass of peeling off size u along the facewidth be:

formula calculates the value that can directly obtain as u and is respectively 3mm thus, and 6mm, when 9mm and 12mm, peels off the mean rigidity of gear within a mesh cycle and be respectively 99.60%, 99.06%, 98.26% and 96.98% of the average mesh stiffness of normal gear.

2.2 peel off hole size only considers to change along flank engagement direction

If along facewidth direction w _s=10mm is certain value, and the size of flank engagement direction is got respectively α _s1=1.5mm, α _s2=3mm, α _s3=4.5mm, α _s4tetra-groups of different values of=6mm, the correlation curve that obtains peeling off gear obtains overall meshing stiffness and normal Gear Meshing Stiffness along flank engagement direction different faults size, as Figure 10, can be obtained by Figure 10: facewidth direction size w _swhen constant, fault Gear Meshing Stiffness is identical in the minimum value of synchronization; α _swhile changing along flank engagement direction, the angle range that mesh stiffness value reduces is respective change also.Work as α _svalue be respectively 1.5mm, 3mm, when 4.5mm and 6mm, determine after the angle range that rigidity value reduces, by formula (22), try to achieve the mean rigidity of fault gear within a mesh cycle and be respectively 99.26%, 98.44%, 97.63% and 96.70% of normal gear mean rigidity.

3, become the calculating of mesh stiffness during Gear with Crack

When gear has crackle, because of Surface Contact area and component F _aall do not become, therefore hertz rigidity and radial compression rigidity are constant; Whole stressed F is also constant again, therefore the rigidity of wheel body deformability is also constant; Now mainly consider the variation of bending stiffness and shearing rigidity.There is the fault gear correlation parameter concrete meaning of crackle as shown in figure 11.Wherein, h _c, h _rbe respectively tooth root crackle to the distance of gear teeth center line and half of point circle place chordal thickness; α ₁, α _gbe respectively the complementary angle of the complementary angle of action line and gear teeth center line angle and meshing point action line and gear teeth center line angle when point circle place.

Work as h _c<h _ror h _c>=h _rand α ₁≤ α _gtime,

$\frac{1}{k_{\mathrm{bcrack}}}={\&Integral;}_{{-\mathrm{\&alpha;}}_1}^{{\mathrm{\&alpha;}}_2}\frac{12{\{1+\cos {\mathrm{\&alpha;}}_1[({\mathrm{\&alpha;}}_2-\mathrm{\&alpha;})\sin \mathrm{\&alpha;}\left]\right\}}^2({\mathrm{\&alpha;}}_2-\mathrm{\&alpha;})\cos \mathrm{\&alpha;}}{\mathrm{EL}{[\sin {\mathrm{\&alpha;}}_2-\frac{\mathrm{<figure-callout\; id="1"\; label="q\; R\; b"\; filenames="DEST\_PATH\_HDA0000423208030000062.png,DEST\_PATH\_HDA0000423208030000063.png"\; state="\{\{state\}\}">q</figure-callout>}}{R_b}\sin v+\sin \mathrm{\&alpha;}+({\mathrm{\&alpha;}}_2-\mathrm{\&alpha;})\cos \mathrm{\&alpha;}]}^3}---\left(23\right)$

$\frac{1}{k_{\mathrm{scrack}}}={\&Integral;}_{{-\mathrm{\&alpha;}}_1}^{{\mathrm{\&alpha;}}_2}\frac{2.4({\mathrm{\&alpha;}}_2-\mathrm{\&alpha;})\cos \mathrm{\&alpha;}{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="DEST\_PATH\_HDA0000423208030000051.png,DEST\_PATH\_HDA0000423208030000052.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2{\mathrm{\&alpha;}}_1}{\mathrm{EL}[\sin {\mathrm{\&alpha;}}_2-\frac{\mathrm{<figure-callout\; id="1"\; label="q\; R\; b"\; filenames="DEST\_PATH\_HDA0000423208030000062.png,DEST\_PATH\_HDA0000423208030000063.png"\; state="\{\{state\}\}">q</figure-callout>}}{R_b}\sin v+\sin \mathrm{\&alpha;}+({\mathrm{\&alpha;}}_2-\mathrm{\&alpha;})\cos \mathrm{\&alpha;}]}\mathrm{d\&alpha;}---\left(24\right)$

Work as h _c>=h _rand α ₁> α _gtime,

$\begin{array}{c}\frac{1}{k_{\mathrm{bcrack}}}={\&Integral;}_{-\mathrm{\&alpha;g}}^{{\mathrm{\&alpha;}}_2}\frac{12{\{1+\cos {\mathrm{\&alpha;}}_1[({\mathrm{\&alpha;}}_2-\mathrm{\&alpha;})\sin \mathrm{\&alpha;}-\cos \mathrm{\&alpha;}\left]\right\}}^2({\mathrm{\&alpha;}}_2-\mathrm{\&alpha;})\cos \mathrm{\&alpha;}}{\mathrm{EL}{[\sin {\mathrm{\&alpha;}}_2-\frac{\mathrm{<figure-callout\; id="1"\; label="q\; R\; b"\; filenames="DEST\_PATH\_HDA0000423208030000062.png,DEST\_PATH\_HDA0000423208030000063.png"\; state="\{\{state\}\}">q</figure-callout>}}{R_b}\sin v+\sin \mathrm{\&alpha;}+({\mathrm{\&alpha;}}_2-\mathrm{\&alpha;})\cos \mathrm{\&alpha;}]}^3}\mathrm{d\&alpha;}+\\ {\&Integral;}_{{-\mathrm{\&alpha;}}_1}^{-\mathrm{\&alpha;g}}\frac{3{\{1+\cos {\mathrm{\&alpha;}}_1[({\mathrm{\&alpha;}}_2-\mathrm{\&alpha;})\sin \mathrm{\&alpha;}-\cos {\mathrm{\&alpha;}}_2\left]\right\}}^2({\mathrm{\&alpha;}}_2-\mathrm{\&alpha;})\cos \mathrm{\&alpha;}}{2\mathrm{EL}{[\sin {\mathrm{\&alpha;}}_2+({\mathrm{\&alpha;}}_2-\mathrm{\&alpha;})\cos \mathrm{\&alpha;}]}^3}\mathrm{d\&alpha;}\end{array}---\left(25\right)$

$\frac{1}{k_{\mathrm{scrack}}}={\&Integral;}_{-\mathrm{\&alpha;g}}^{{\mathrm{\&alpha;}}_2}\frac{2.4(1+v)({\mathrm{\&alpha;}}_2-\mathrm{\&alpha;})\cos \mathrm{\&alpha;}{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="DEST\_PATH\_HDA0000423208030000051.png,DEST\_PATH\_HDA0000423208030000052.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2{\mathrm{\&alpha;}}_1}{\mathrm{EL}[\sin {\mathrm{\&alpha;}}_2-\frac{\mathrm{<figure-callout\; id="1"\; label="q\; R\; b"\; filenames="DEST\_PATH\_HDA0000423208030000062.png,DEST\_PATH\_HDA0000423208030000063.png"\; state="\{\{state\}\}">q</figure-callout>}}{R_b}\sin v+\sin \mathrm{\&alpha;}+({\mathrm{\&alpha;}}_2-\mathrm{\&alpha;})\cos \mathrm{\&alpha;}]}\mathrm{d\&alpha;}+$

${\&Integral;}_{{-\mathrm{\&alpha;}}_1}^{{-\mathrm{\&alpha;}}_g}\frac{1.2(1+v)({\mathrm{\&alpha;}}_2-\mathrm{\&alpha;})\cos \mathrm{\&alpha;}{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="DEST\_PATH\_HDA0000423208030000051.png,DEST\_PATH\_HDA0000423208030000052.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2{\mathrm{\&alpha;}}_1}{\mathrm{EL}[\sin \mathrm{\&alpha;}+({\mathrm{\&alpha;}}_2-\mathrm{\&alpha;})\cos \mathrm{\&alpha;}]}---\left(26\right)$

For the ease of calculating, suppose that crackle appears at tooth root and locates with the angle v=45 ° of gear teeth center line, as shown in Figure 3.The transverse tooth thickness of the gear teeth at dedendum circle place is 13.92mm, now considers that different crack depths q is respectively 2mm, 4mm, 6mm.Now can obtain respectively the fault gear and the normal bending stiffness of gear and the correlation curve of shearing rigidity of three kinds of different crack depths, as shown in figure 12 with shown in Figure 13.

In order systematically to analyze bending stiffness and shearing rigidity, become respectively the influence degree of mesh stiffness when comprehensive, while considering crack depth q=6mm, calculate respectively normal gear in the mesh cycle, only consider crooked, only consider to shear and consider to shear simultaneously and crooked these four kinds of situations under comprehensive time become after mesh stiffness, during by normal gear, become mesh stiffness and deduct respectively only consideration bending, only consider to shear, consider simultaneously crooked and shear in these three kinds of situations time become mesh stiffness, by these three differences, (be called crooked difference, shear difference and total difference) stool and urine can obtain because of bending, shear or consider crooked simultaneously and shear the impact that becomes mesh stiffness when comprehensive, corresponding curve as shown in figure 14.With the impact of shearing, increasing when crooked difference, shearing difference and total difference all move from tooth root to tooth top along with the position of engagement by bending; Crooked difference is mainly subject to gear pair to the impact of tooth root displacement, and when driving wheel drives engaged wheel to rotate, gear pair increases more and more sooner to tooth root displacement, therefore crooked difference is along with the rotation of gear also increases also rapidly; Shear difference main with by gear pair and relevant perpendicular to the cross-sectional sizes of gear teeth center line, therefore along with the variation of the position of engagement, shear difference increase comparatively steady, be approximated to linear relationship; But after 26.5 ° (adding a rear bi-tooth gearing mesh cycle), have the gear teeth of crackle will no longer participate in engagement, the large young pathbreaker of mesh stiffness is no longer subject to shear and crooked impact.In order more specifically to analyze crooked and to shear which kind of factor to become the impact of mesh stiffness when comprehensive larger, crooked difference from Figure 14 and shearing difference can be found out, when pinion wheel corner is [0,21.7 °] when (from just starting to be engaged to 1.533 times of bi-tooth gearing cycles), the impact that mesh stiffness value is sheared is larger; When corner is at [21.7 °, 26.5 °] when (add and add a bi-tooth gearing cycle 0.533 times of bi-tooth gearing cycle to first mesh cycle from a mesh cycle), the impact by bending of mesh stiffness value is larger.

As seen from Figure 12, tooth root place is clearly owing to there is crackle on the impact of tooth bending rigidity.As seen from Figure 13, tooth root crackle also has a certain impact to gear shearing rigidity.To three kinds of different crack depths, can obviously find out that crackle is to increase along with the increase of the degree of depth on the impact of rigidity.Within a mesh cycle, when crack depth, q is respectively 2mm, 4mm, during 6mm, the mean value of shearing rigidity be respectively healthy gear shearing rigidity mean value 86.22%, 69.94%, 52.38%.Last consideration is simultaneously crooked and shear the impact on mesh stiffness, when obtaining the Gear with Crack of three kinds of different crack depths and normally gear is comprehensive, becomes mesh stiffness correlation curve as shown in figure 15.

4, become the calculating of mesh stiffness during broken teeth gear

During gear tooth breakage, in the position of broken teeth, lose contact, as shown in figure 14.Original bi-tooth gearing district becomes monodentate engagement.Therefore, total mesh stiffness only by monodentate to forming, its computing formula becomes:

${\mathrm{kt}}_{\mathrm{broken}}=\frac{1}{\frac{1}{k_h}+\frac{1}{{\mathrm{<figure-callout\; id="1"\; label="k\; b"\; filenames="DEST\_PATH\_HDA0000423208030000062.png,DEST\_PATH\_HDA0000423208030000063.png"\; state="\{\{state\}\}">\; <figure-callout\; id="1"\; label="k\; b"\; filenames="DEST\_PATH\_HDA0000423208030000062.png,DEST\_PATH\_HDA0000423208030000063.png"\; state="\{\{state\}\}">k</figure-callout>\; </figure-callout>}}_b}+\frac{1}{{\mathrm{<figure-callout\; id="2"\; label="k\; b"\; filenames="DEST\_PATH\_HDA0000423208030000051.png,DEST\_PATH\_HDA0000423208030000052.png"\; state="\{\{state\}\}">\; <figure-callout\; id="1"\; label="k\; b"\; filenames="DEST\_PATH\_HDA0000423208030000062.png,DEST\_PATH\_HDA0000423208030000063.png"\; state="\{\{state\}\}">k</figure-callout>\; </figure-callout>}}_b}+\frac{1}{{\mathrm{<figure-callout\; id="1"\; label="k\; s"\; filenames="DEST\_PATH\_HDA0000423208030000062.png,DEST\_PATH\_HDA0000423208030000063.png"\; state="\{\{state\}\}">\; <figure-callout\; id="1"\; label="k\; s"\; filenames="DEST\_PATH\_HDA0000423208030000062.png,DEST\_PATH\_HDA0000423208030000063.png"\; state="\{\{state\}\}">k</figure-callout>\; </figure-callout>}}_s}+\frac{1}{{\mathrm{<figure-callout\; id="2"\; label="k\; s"\; filenames="DEST\_PATH\_HDA0000423208030000051.png,DEST\_PATH\_HDA0000423208030000052.png"\; state="\{\{state\}\}">\; <figure-callout\; id="1"\; label="k\; s"\; filenames="DEST\_PATH\_HDA0000423208030000062.png,DEST\_PATH\_HDA0000423208030000063.png"\; state="\{\{state\}\}">k</figure-callout>\; </figure-callout>}}_s}+\frac{1}{k_{a\mathrm{1,1}}}+\frac{1}{k_{a\mathrm{2,1}}}+\frac{1}{{\mathrm{<figure-callout\; id="1"\; label="k\; f"\; filenames="DEST\_PATH\_HDA0000423208030000062.png,DEST\_PATH\_HDA0000423208030000063.png"\; state="\{\{state\}\}">\; <figure-callout\; id="1"\; label="k\; f"\; filenames="DEST\_PATH\_HDA0000423208030000062.png,DEST\_PATH\_HDA0000423208030000063.png"\; state="\{\{state\}\}">k</figure-callout>\; </figure-callout>}}_f}+\frac{1}{{\mathrm{<figure-callout\; id="2"\; label="k\; f"\; filenames="DEST\_PATH\_HDA0000423208030000051.png,DEST\_PATH\_HDA0000423208030000052.png"\; state="\{\{state\}\}">k</figure-callout>}}_f}}---\left(27\right)$

The correlation curve of final broken teeth gear and normal Gear Meshing Stiffness as shown in figure 12.Broken teeth gear is only monodentate to engagement, and in two mesh cycles that reduce at mesh stiffness, broken teeth gear mean rigidity value is half left and right of normal gear mean rigidity value.

Comprehensive above full-fledged research content, the present invention be take energy method as basic, take standard straight spur geer as research object, systematically discussed single gear have peel off, the quantitative calculating of mesh stiffness when crackle, three kinds of different local defects of broken teeth.But when multipair gear meshes jointly, time to become mesh stiffness be not single simple addition to Gear Meshing Stiffness, in reality, also will be taken turns the distribution of Transverse Load, and then be also subject to the impact of tooth error.

Claims (5)

1. a time-varying characteristics quantitative calculation method for local defect Gear Meshing Stiffness, is characterized in that: the performing step of the method is for setting up Gear Meshing Stiffness computation model; Gear Meshing Stiffness computation model is peeled off in foundation; Set up Gear with Crack mesh stiffness computation model; Set up broken teeth Gear Meshing Stiffness computation model; The calculating of mesh stiffness in swing circle of fault gear; When gear pair is meshed F, do the used time, F can be decomposed into two the power Fs parallel with vertical with gear teeth center line _aand F _b; Because gear pair exists contact, therefore there is a hertz contact stiffness; The power F that is parallel to gear teeth center line _aeffect, there is radial compression in gear, therefore there is radial compression rigidity; Be subject to perpendicular to gear teeth center line F _beffect, because of F _bbe equal to shearing and make wheel body bear moment of flexure with respect to gear centre, therefore there is shearing rigidity and bending stiffness; Finally, because of tooth force, there is elastic deformation in basis, therefore there is wheel body deformation rigidity; The strain energy that is stored in meshing gear centering is converted into a hertz energy U _h, flexional U _b, radial compression energy U _a, shear energy U _swith wheel body strain energy of distortion U _f, by energy conservation, just can be calculated the hertz rigidity k answering in contrast _h, bending stiffness k _b, radial compression rigidity k _a, shearing rigidity k _swith wheel body deformation rigidity k _f.

2. according to right 1, require the time-varying characteristics quantitative calculation method of described a kind of local defect Gear Meshing Stiffness, it is characterized in that: when gear pair is meshed F, do the used time, F is decomposed into two the power Fs parallel with vertical with gear teeth center line _aand F _b.The power F that is parallel to gear teeth center line _aeffect, there is radial compression in gear, because of F _aconstant, therefore radial compression rigidity is constant; Be subject to perpendicular to gear teeth center line F _beffect, because of F _bbe equal to shearing and make wheel body bear moment of flexure with respect to gear centre, F _bconstant, therefore shearing rigidity and bending stiffness are also constant; Finally, because of tooth force, there is elastic deformation in basis, and because of integral body, stressed F is constant, therefore exist wheel body deformation rigidity also constant; Finally the contact line length because of gear engagement changes, and hertz rigidity is main relevant with contact line length, therefore has the variation of mainly considering hertz rigidity while peeling off, now normal hertz rigidity k during gear _hbecome the hertz rigidity k while peeling off gear _hchip.Wherein, consider to peel off and cheat when tooth is very little to be changed and change along flank engagement direction along facewidth direction, k _hchiprespectively about peeling off hole size w _sand α _sfunction.

3. according to right 1, require the time-varying characteristics quantitative calculation method of described a kind of local defect Gear Meshing Stiffness, it is characterized in that: Surface Contact area and component F _ado not become, therefore hertz rigidity and radial compression rigidity are constant; Whole stressed F is also constant again, therefore the rigidity of wheel body deformability is also constant; While having crackle, mainly consider the variation of bending stiffness and shearing rigidity; The bending stiffness of normal gear is by k _bwith shearing rigidity k _sbecome respectively the crooked k of Gear with Crack _bcrackwith shearing rigidity k _scrack.K now _bcrackand k _scrackall the function about crack depth q and crackle and gear teeth center line angle v, finally Gear with Crack becomes mesh stiffness when comprehensive.

4. according to right 1, require the time-varying characteristics quantitative calculation method of described a kind of local defect Gear Meshing Stiffness, it is characterized in that: the position at gear tooth breakage loses contact, original bi-tooth gearing district becomes monodentate engagement; Obtains overall meshing stiffness only by monodentate to forming, broken teeth Gear Meshing Stiffness can be calculated.

5. according to the time-varying characteristics quantitative calculation method of a kind of local defect Gear Meshing Stiffness described in claim 1 or 2 or 3 or 4, it is characterized in that: if the fault gear teeth are first teeth that are rotated counterclockwise on gear, when gear rotates a circle, gear pair has z ₁(z ₁for the pinion wheel number of teeth) individual mesh cycle; At [3, z ₁] mesh stiffness of individual mesh cycle is identical with normal gear; Within first and second mesh cycle, can obtain the size of fault gear tooth rigidity value, can obtain thus the mesh stiffness in swing circle of fault gear.

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