CN103198226A - Method for analyzing vibration characteristics of cycloid bevel gear based on the consideration of friction - Google Patents

Abstract

The invention discloses a method for analyzing vibration characteristics of a cycloid bevel gear based on the consideration of friction and belongs to the field of analysis on gear nonlinear vibration. The method includes: firstly, simplifying a cycloid bevel gear system into a gear pair torsional vibration system model; secondly, introducing friction factor to the gear pair torsional vibration system model, and obtaining a torsional vibration balance equation for driving and driven gears according to the principle of Lagrange; thirdly, nondimensionalizing the gear pair torsional vibration balance equation to obtain a vibration model dimensionless equation; and fourthly, researching and analyzing the regular pattern of the friction factor and the regular pattern of cycloid bevel gear vibration characteristics according to the dimensionless equation of the cycloid bevel gear pair vibration model. The method theoretically supports vibration and noise reduction for the bevel gear drive system, and provides references for manufacturing high-precision high-bearing-capacity cycloid bevel gears and improving drive precision, lifetime and reliability of a cycloid bevel gear drive system.

Description

A kind ofly consider the cycloid bevel gears vibration characteristics analytical approach that rubs

Technical field

The invention belongs to gear nonlinear vibration analysis field, relate to a kind of cycloid bevel gears vibration characteristics analytical approach, more specifically relate to a kind of cycloid bevel gears vibration characteristics analytical approach of rubbing considered.

Background technology

Cycloid bevel gears is as one of two canine tooth systems of spiral bevel gear, have characteristics such as stable drive, load-bearing capacity height, hard surface skiving technology, thereby being specially adapted to high-power and high pulling torque heavy load transmission field, is the core transmission component in the key areas such as heavy high-grade, digitally controlled machine tools, car transmissions, Aero-Space equipment.Along with machine driven system develops towards directions such as high speed, precisions day by day, cycloid bevel gears is as the crucial drive disk assembly in the kinematic train, and its vibration characteristics will be more remarkable for the influence of transmission system performance.Therefore, research cycloid bevel gears vibration characteristics has important practical value and academic significance for designing and make efficient drive disk assembly such as high precision, high-durability, low noise.

In recent years, domestic and international many scholars are based on the theory of nonlinear oscillation, with in the gear engagement process the time to become non-linear factors such as rigidity and backlash be core, more extensive and deep research has been carried out in the nonlinear vibration of gear train.But Recent study shows: the engagement friction between teeth also is one of gear nonlinear vibration influence factor.But majority is studied all at straight spur gear at present, and less to the research of cycloid bevel gears.How in the secondary kinetic model of cycloid bevel gears, to introduce friction factor and the correct friction factor of analyzing to the rule that influences of cycloid bevel gears vibration characteristics, still has very big research potential, research and explore new kinetic model and analytical approach, be still one of the important content in this field.The research friction factor is to the influence of cycloid bevel gears vibration characteristics, not only the vibration and noise reducing for bevel gear tooth system provides theoretical support, and for making the cycloid bevel gears of high precision, high bearing capacity, the transmission accuracy, life-span and the reliability that promote the cycloid bevel gears kinematic train provide reference.

Summary of the invention

The purpose of this invention is to provide a kind of cycloid bevel gears vibration characteristics analytical approach of rubbing considered, explore friction factor to the rule that influences of cycloid bevel gears vibration characteristics, thereby for the vibration and noise reducing of bevel gear tooth system provides theoretical support, and for making the cycloid bevel gears of high precision, high bearing capacity, the transmission accuracy, life-span and the reliability that promote the cycloid bevel gears kinematic train provide reference.

The present invention adopts following technological means to realize:

1, the cycloid bevel gears system simplification is treated as the torsional vibration system model of gear pair;

2, in the torsional vibration system model of cycloid bevel gears pair, introduce friction factor, obtained the twisting vibration balance equation of driving and driven gear by the Lagrange principle respectively.Balance equation is as follows:

$I_p{\stackrel{\stackrel{\·\·}{~}}{\mathrm{\θ}}}_p+{\mathrm{\λ}}_p\tilde{C}\left(\tilde{t}\right)\[{\mathrm{\λ}}_p{\stackrel{\stackrel{\·}{~}}{\mathrm{\θ}}}_p-{\mathrm{\λ}}_g{\stackrel{\stackrel{\·}{~}}{\mathrm{\θ}}}_g-\stackrel{\stackrel{\·}{~}}{e}\left(\tilde{t}\right)\]+{\mathrm{\λ}}_p\tilde{K}\left(\tilde{t}\right)\tilde{f}({\mathrm{\λ}}_p{\widetilde{\mathrm{\θ}}}_p-{\mathrm{\λ}}_g{\widetilde{\mathrm{\θ}}}_g-\tilde{e}\left(\tilde{t}\right))={\tilde{T}}_p-{\tilde{T}}_{f,p}\left(\tilde{t}\right)$

$I_g{\stackrel{\stackrel{\·\·}{~}}{\mathrm{\θ}}}_g-{\mathrm{\λ}}_g\tilde{C}\left(\tilde{t}\right)\[{\mathrm{\λ}}_p{\stackrel{\stackrel{\·}{~}}{\mathrm{\θ}}}_p-{\mathrm{\λ}}_g{\stackrel{\stackrel{\·}{~}}{\mathrm{\θ}}}_g-\stackrel{\stackrel{\·}{~}}{e}\left(\tilde{t}\right)\]-{\mathrm{\λ}}_p\tilde{K}\left(\tilde{t}\right)\tilde{f}({\mathrm{\λ}}_p{\widetilde{\mathrm{\θ}}}_p-{\mathrm{\λ}}_g{\widetilde{\mathrm{\θ}}}_g-\tilde{e}\left(\tilde{t}\right))=-{\tilde{T}}_g+{\tilde{T}}_{f,g}\left(\tilde{t}\right)$

Wherein, ~ be the dimension symbol; I _i(i=p g) is moment of inertia main, driven gear; λ _i(i=p g) is gear direction radius of turn main, driven gear; θ _i(i=p g) is angular displacement main, driven gear; T _i(i=p g) is moment of torsion on main, the driven gear; T _{F, i}(i=p g) is moment of friction on main, the driven gear; C (t) is gear pair engagement damping; K (t) is the gear pair mesh stiffness; F () is gap function; E (t) is the static transmission error function of gear pair.

3, with the twisting vibration balance equation nondimensionalization of gear pair, obtain the nondimensionalization form of model of vibration;

3.1. introducing new variables ${\tilde{x}}_i={\mathrm{\λ}}_i{\widetilde{\mathrm{\θ}}}_i(i=p,g),$ $m_i=\frac{I_i}{{\mathrm{\λ}}_i^2}(i=p,g),$ $\tilde{F}=\frac{{\tilde{T}}_i}{{\mathrm{\λ}}_i}(i=p,g),$

In the balance equation in the substitution step 2:

$m_p{\stackrel{\stackrel{\·\·}{~}}{x}}_p+\tilde{C}\left(\tilde{t}\right)\[{\stackrel{\stackrel{\·}{~}}{x}}_p-{\stackrel{\stackrel{\·}{~}}{x}}_g-\stackrel{\stackrel{\·}{~}}{e}\left(\tilde{t}\right)\]+\tilde{\text{K}}\left(\tilde{t}\right)\tilde{f}({\tilde{x}}_p-{\tilde{x}}_g-\tilde{e}\left(\tilde{t}\right))=\tilde{F}-{\tilde{F}}_f\left(\tilde{t}\right)$

$m_g{\stackrel{\stackrel{\·\·}{~}}{x}}_g-\tilde{C}\left(\tilde{t}\right)\[{\stackrel{\stackrel{\·}{~}}{x}}_p-{\stackrel{\stackrel{\·}{~}}{x}}_g-\stackrel{\stackrel{\·}{~}}{e}\left(\tilde{t}\right)\]-\tilde{\text{K}}\left(\tilde{t}\right)\tilde{f}({\tilde{x}}_p-{\tilde{x}}_g-\tilde{e}\left(\tilde{t}\right))=\widetilde{-F}+{\tilde{F}}_f\left(\tilde{t}\right)$

Wherein, x _i(i=p g) is main, the dynamic transmission error of the driven gear gear teeth; m _i(i=p g) is quality main, driven gear; F is external applied load; F _fBe average friction power;

3.2. introducing new variables

Two equilibrium equations in the step 3.1 are subtracted each other and merge obtain:

$M\stackrel{\stackrel{\·\·}{~}}{x}+\tilde{C}\left(\tilde{t}\right)\stackrel{\stackrel{\·}{~}}{x}+\tilde{K}\left(\tilde{t}\right)\tilde{f}\left(\tilde{x}\right)=(\tilde{F}-{\tilde{F}}_f\left(\tilde{t}\right))-M\stackrel{\stackrel{\·\·}{~}}{e}\left(\tilde{t}\right)$

Wherein, ${\tilde{F}}_f\left(\tilde{t}\right)=\mathrm{\μ}(\tilde{K}\left(\tilde{t}\right)\tilde{f}\left(\tilde{x}\right)+\tilde{C}\left(\tilde{t}\right)\stackrel{\stackrel{\·}{~}}{x})$ Bring in the following formula and get:

$M\stackrel{\stackrel{\·\·}{~}}{x}+(1+\mathrm{\μ})\tilde{C}\left(\tilde{t}\right)\stackrel{\stackrel{\·}{~}}{x}+(1+\mathrm{\μ})\tilde{K}\left(\tilde{t}\right)\tilde{f}\left(\tilde{x}\right)=\tilde{F}-M\stackrel{\stackrel{\·\·}{~}}{e}\left(\tilde{t}\right)$

Wherein, x is the meshing point displacement; M is gear relative mass; μ is friction factor.

3.3. rigidity, damping and static transmission error are pressed Fourier expansion, and only consider that the main harmonic form has:

And order:

The nondimensionalization form that finally obtains model of vibration is:

Wherein, α is the harmonic wave ratio of damping; ρ is the harmonic wave stiffness coefficient; γ is the transmission error factor; ξ is damping factor;

Be the phasing degree; ω _nBe natural frequency; ω is excitation frequency, and b is gear clearance.

4, according to the nondimensionalization equation of cycloid bevel gears secondary undulation model, the rule of research and analysis friction factor μ and cycloid bevel gears vibration characteristics.

The objective of the invention is at friction the influence of cycloid bevel gears vibration characteristics, proposed a kind of cycloid bevel gears vibration characteristics analytical approach of rubbing considered.Characteristics are the twisting vibration model from the cycloid bevel gears pair, introduce the nondimensionalization equation that friction factor obtains containing friction factor in its kinetic balance equation, the rule of last research and analysis friction factor μ and cycloid bevel gears vibration characteristics.Summary of the invention comprises three parts.In first, it mainly is the twisting vibration model of setting up the cycloid bevel gears pair; In second portion, it mainly is the nondimensionalization equation of the cycloid bevel gears secondary undulation model obtain containing friction factor of deriving; In third part, mainly be the nondimensionalization equation according to cycloid bevel gears secondary undulation model, the rule of research and analysis friction factor μ and cycloid bevel gears vibration characteristics.

Description of drawings

Fig. 1 considers the cycloid bevel gears vibration characteristics analytical approach process flow diagram that rubs

The secondary kinetic model figure of Fig. 2 embodiment of the invention cycloid bevel gears

Fig. 3 embodiment of the invention gap function illustraton of model

Fig. 4 embodiment of the invention friction factor influences meshing point oscillating curve figure

Embodiment

The embodiment of the invention a kind of considers that the cycloid bevel gears vibration characteristics analytical approach process flow diagram that rubs as shown in Figure 1, elaborates to step of the present invention below in conjunction with process flow diagram.Concrete implementation step is as follows:

The first step: the torsional vibration system model that the cycloid bevel gears system simplification is treated as gear pair;

Present embodiment is research object with aviation with the cycloid bevel gears pair, and its concrete parameter sees Table 1.Consider the secondary kinetic model of the cycloid bevel gears that rubs between the flank of tooth as shown in Figure 2.In this model, suppose that the support stiffness of two gears is bigger, and do not consider the elastic deformation of transmission shaft, block bearing and casing etc. to the influence of cycloid bevel gears system that the cycloid bevel gears system simplification is treated as the torsional vibration system model of gear pair the most at last.

Table 1 cycloid bevel gears systematic parameter

Second step: in the torsional vibration system model of cycloid bevel gears pair, introduce friction factor, obtained the twisting vibration balance equation of driving and driven gear by the Lagrange principle respectively.Balance equation is as follows:

$I_p{\stackrel{\stackrel{\·\·}{~}}{\mathrm{\θ}}}_p+{\mathrm{\λ}}_p\tilde{C}\left(\tilde{t}\right)\[{\mathrm{\λ}}_p{\stackrel{\stackrel{\·}{~}}{\mathrm{\θ}}}_p-{\mathrm{\λ}}_g{\stackrel{\stackrel{\·}{~}}{\mathrm{\θ}}}_g-\stackrel{\stackrel{\·}{~}}{e}\left(\tilde{t}\right)\]+{\mathrm{\λ}}_p\tilde{K}\left(\tilde{t}\right)\tilde{f}({\mathrm{\λ}}_p{\widetilde{\mathrm{\θ}}}_p-{\mathrm{\λ}}_g{\widetilde{\mathrm{\θ}}}_g-\tilde{e}\left(\tilde{t}\right))={\tilde{T}}_p-{\tilde{T}}_{f,p}\left(\tilde{t}\right)$

$I_g{\stackrel{\stackrel{\·\·}{~}}{\mathrm{\θ}}}_g-{\mathrm{\λ}}_g\tilde{C}\left(\tilde{t}\right)\[{\mathrm{\λ}}_p{\stackrel{\stackrel{\·}{~}}{\mathrm{\θ}}}_p-{\mathrm{\λ}}_g{\stackrel{\stackrel{\·}{~}}{\mathrm{\θ}}}_g-\stackrel{\stackrel{\·}{~}}{e}\left(\tilde{t}\right)\]-{\mathrm{\λ}}_g\tilde{K}\left(\tilde{t}\right)\tilde{f}({\mathrm{\λ}}_p{\widetilde{\mathrm{\θ}}}_p-{\mathrm{\λ}}_g{\widetilde{\mathrm{\θ}}}_g-\tilde{e}\left(\tilde{t}\right))={-\tilde{T}}_g+{\tilde{T}}_{f,g}\left(\tilde{t}\right)$

The 3rd step: with the twisting vibration balance equation nondimensionalization of gear pair, obtain the nondimensionalization form of model of vibration;

1), order ${\tilde{x}}_i={\mathrm{\λ}}_i{\widetilde{\mathrm{\θ}}}_i(i=p,g),$ $m_i=\frac{I_i}{{\mathrm{\λ}}_i^2}(i=p,g),$ $\tilde{F}=\frac{{\tilde{T}}_i}{{\mathrm{\λ}}_i}(i=p,g),$ ${\tilde{F}}_f\left(\tilde{t}\right)=\frac{{\tilde{T}}_{f,i}\left(\tilde{t}\right)}{{\mathrm{\λ}}_i}(i=p,g)$ Obtain in the balance equation in second step of substitution respectively:

$m_p{\stackrel{\stackrel{\·\·}{~}}{x}}_p+\tilde{C}\left(\tilde{t}\right)\[{\stackrel{\stackrel{\·}{~}}{x}}_p-{\stackrel{\stackrel{\·}{~}}{x}}_g-\stackrel{\stackrel{\·}{~}}{e}\left(\tilde{t}\right)\]+\tilde{\text{K}}\left(\tilde{t}\right)\tilde{f}({\tilde{x}}_p-{\tilde{x}}_g-\tilde{e}\left(\tilde{t}\right))=\tilde{F}-{\tilde{F}}_f\left(\tilde{t}\right)$

$m_g{\stackrel{\stackrel{\·\·}{~}}{x}}_g-\tilde{C}\left(\tilde{t}\right)\[{\stackrel{\stackrel{\·}{~}}{x}}_p-{\stackrel{\stackrel{\·}{~}}{x}}_g-\stackrel{\stackrel{\·}{~}}{e}\left(\tilde{t}\right)\]-\tilde{\text{K}}\left(\tilde{t}\right)\tilde{f}({\tilde{x}}_p-{\tilde{x}}_g-\tilde{e}\left(\tilde{t}\right))=\widetilde{-F}+{\tilde{F}}_f\left(\tilde{t}\right)$

2), with 1) in two formulas respectively except m _p, m _gAnd subtract each other and obtain:

${\stackrel{\stackrel{\·\·}{~}}{x}}_p-{\stackrel{\stackrel{\·\·}{~}}{x}}_g+\frac{m_p+m_g}{m_p{m}_g}\tilde{C}\left(\tilde{t}\right)\[{\stackrel{\stackrel{\·}{~}}{x}}_p-{\stackrel{\stackrel{\·}{~}}{x}}_g-\stackrel{\·}{\tilde{e}}\left(\tilde{t}\right)\]+\frac{m_p+m_g}{m_p{m}_g}\tilde{K}\left(\tilde{t}\right)\tilde{f}({\tilde{x}}_p-{\tilde{x}}_g-\tilde{e}\left(\tilde{t}\right))=\frac{m_p+m_g}{m_p{m}_g}(\tilde{F}-{\tilde{F}}_f\left(\tilde{t}\right))$

Order $\tilde{x}={\tilde{x}}_p-{\tilde{x}}_g-\tilde{e}\left(\tilde{t}\right),$ $M=\frac{m_p{m}_g}{m_p+m_g}$ Obtain in the substitution following formula:

$M\stackrel{\stackrel{\·\·}{~}}{x}+\tilde{C}\left(\tilde{t}\right)\stackrel{\stackrel{\·}{~}}{x}+\tilde{K}\left(\tilde{t}\right)\tilde{f}\left(\tilde{x}\right)=(\tilde{F}-{\tilde{F}}_f\left(\tilde{t}\right))-M\stackrel{\stackrel{\·\·}{~}}{e}\left(\tilde{t}\right)$

Again, ${\tilde{F}}_f\left(\tilde{t}\right)=\mathrm{\μ}(\tilde{K}\left(\tilde{t}\right)\tilde{f}\left(\tilde{x}\right)+\tilde{C}\left(\tilde{t}\right)\stackrel{\·}{\tilde{x}})$ Obtain in the substitution following formula:

$M\stackrel{\stackrel{\·\·}{~}}{x}+(1+\mathrm{\μ})\tilde{C}\left(\tilde{t}\right)\stackrel{\stackrel{\·}{~}}{x}+(1+\mathrm{\μ})\tilde{K}\left(\tilde{t}\right)\tilde{f}\left(\tilde{x}\right)=\tilde{F}-M\stackrel{\stackrel{\·\·}{~}}{e}\left(\tilde{t}\right)$

3), with rigidity, damping and static transmission error by Fourier expansion, and only consider the main harmonic form, namely

The substitution following formula obtains:

Order $x=\frac{\tilde{x}}{b},$ $t={\mathrm{\&omega;}}_n\tilde{t},$ ${\mathrm{\&omega;}}_n=\sqrt{\frac{k_m}{M}},$ $\mathrm{\&gamma;}=\frac{e_1}{b},$ $\mathrm{\&xi;}=\frac{c_m}{M{\mathrm{\&omega;}}_n},$ $\mathrm{\&alpha;}=\frac{c_1}{M{\mathrm{\&omega;}}_n},$ $\mathrm{\&rho;}=\frac{k_1}{M{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="HDA00003067617500022.png"\; state="\{\{state\}\}">n</figure-callout>}}^2},$ $F=\frac{\tilde{F}}{\mathrm{bM}{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="HDA00003067617500022.png"\; state="\{\{state\}\}">n</figure-callout>}}^2},$ $\mathrm{\&omega;}=\frac{\widetilde{\mathrm{\&omega;}}}{{\mathrm{\&omega;}}_n},$ $f\left(x\right)=\frac{\tilde{f}\left(\tilde{x}\right)}{b},$

The nondimensionalization form that the substitution following formula finally obtains model of vibration is:

Wherein, $f\left(x\right)=\left\{\begin{array}{cc}x-1& x\&GreaterEqual;1\\ 0& -1<x<1,\\ x+1& x\&le;-1\end{array}\right.$ Its model as shown in Figure 3.

The 4th step: according to the nondimensionalization equation of cycloid bevel gears secondary undulation model, the rule of research and analysis friction factor and cycloid bevel gears vibration characteristics.

Selected parameter ξ=0.1, α=0.01, ρ=0.1, γ=0.2, F=2,

,

With

, probe into friction factor to the influence of cycloid bevel gears vibration characteristics.Wherein when μ equaled 0,0.1,0.2,0.3 respectively, friction factor influenced point of mesh oscillating curve figure as shown in Figure 4.

As can be seen from Figure 4, do not consider friction, i.e. μ=0 o'clock, displacement response curve amplitude is about 3.33, and crest frequency is between ω=0.9～1.0; When μ=0.1, displacement response curve amplitude is about 3.18, and crest frequency is at ω=1.0 places; Be increased to 0.2 and 0.3 with μ, its response amplitude is respectively 3.02 and 2.84, and crest frequency is respectively ω=1.0 and ω=1.1.Hence one can see that, and along with the increase of friction factor, meshing point displacement vibration amplitude decreases, and drift appears in crest frequency, and the trend that thereupon increases is arranged, and other Frequency point response all increases with friction factor and reduces.Friction energy changes the motion state of gear train, increases the complicacy of system motion.

Sum up by above instance analysis: the inventive method can apply in the analysis of cycloid bevel gears vibration characteristics, and can draw friction factor to the rule that influences of cycloid bevel gears vibration characteristics.The inventive method not only provides theoretical support for the vibration and noise reducing of bevel gear tooth system, and for making the cycloid bevel gears of high precision, high bearing capacity, the transmission accuracy, life-span and the reliability that promote the cycloid bevel gears kinematic train provide reference.

Claims (2)

1. consider the cycloid bevel gears vibration characteristics analytical approach that rubs to it is characterized in that this method may further comprise the steps for one kind:

1) the cycloid bevel gears system simplification is treated as the torsional vibration system model of gear pair;

2) introduce friction factor in the torsional vibration system model of cycloid bevel gears pair, obtain the twisting vibration balance equation of driving and driven gear respectively by the Lagrange principle, balance equation is as follows:

$I_p{\stackrel{\stackrel{\&CenterDot;\&CenterDot;}{~}}{\mathrm{\&theta;}}}_p+{\mathrm{\&lambda;}}_p\tilde{C}\left(\tilde{t}\right)\&lsqb;{\mathrm{\&lambda;}}_p{\stackrel{\stackrel{\&CenterDot;}{~}}{\mathrm{\&theta;}}}_p-{\mathrm{\&lambda;}}_g{\stackrel{\stackrel{\&CenterDot;}{~}}{\mathrm{\&theta;}}}_g-\stackrel{\stackrel{\&CenterDot;}{~}}{e}\left(\tilde{t}\right)\&rsqb;+{\mathrm{\&lambda;}}_p\tilde{K}\left(\tilde{t}\right)\tilde{f}({\mathrm{\&lambda;}}_p{\widetilde{\mathrm{\&theta;}}}_p-{\mathrm{\&lambda;}}_g{\widetilde{\mathrm{\&theta;}}}_g-\tilde{e}\left(\tilde{t}\right))={\tilde{T}}_p-{\tilde{T}}_{f,p}\left(\tilde{t}\right)$

$I_g{\stackrel{\stackrel{\&CenterDot;\&CenterDot;}{~}}{\mathrm{\&theta;}}}_g-{\mathrm{\&lambda;}}_g\tilde{C}\left(\tilde{t}\right)\&lsqb;{\mathrm{\&lambda;}}_p{\stackrel{\stackrel{\&CenterDot;}{~}}{\mathrm{\&theta;}}}_p-{\mathrm{\&lambda;}}_g{\stackrel{\stackrel{\&CenterDot;}{~}}{\mathrm{\&theta;}}}_g-\stackrel{\stackrel{\&CenterDot;}{~}}{e}\left(\tilde{t}\right)\&rsqb;-{\mathrm{\&lambda;}}_g\tilde{K}\left(\tilde{t}\right)\tilde{f}({\mathrm{\&lambda;}}_p{\widetilde{\mathrm{\&theta;}}}_p-{\mathrm{\&lambda;}}_g{\widetilde{\mathrm{\&theta;}}}_g-\tilde{e}\left(\tilde{t}\right))={\widetilde{-T}}_g+{\tilde{T}}_{f,g}\left(\tilde{t}\right)$

Wherein, ~ be the dimension symbol; I _i(i=p g) is moment of inertia main, driven gear; λ _i(i=p g) is gear direction radius of turn main, driven gear; θ _i(i=p g) is angular displacement main, driven gear; T _i(i=p g) is moment of torsion on main, the driven gear; T _{F, i}(i=p g) is moment of friction on main, the driven gear; C (t) is gear pair engagement damping; K (t) is the gear pair mesh stiffness; F () is gap function; E (t) is the static transmission error function of gear pair;

3) with the twisting vibration balance equation nondimensionalization of gear pair, obtain the nondimensionalization form of torsional vibration system model;

4) according to the nondimensionalization equation of the secondary torsional vibration system model of cycloid bevel gears, the rule of research and analysis friction factor μ and cycloid bevel gears vibration characteristics.

2. cycloid bevel gears vibration characteristics analytical approach according to claim 1 is characterized in that, wherein, the concrete steps that obtain the nondimensionalization form of torsional vibration system model in the step 3) are:

3.1) the introducing new variables ${\tilde{x}}_i={\mathrm{\&lambda;}}_i{\widetilde{\mathrm{\&theta;}}}_i(i=p,g),$ $m_i=\frac{I_i}{{\mathrm{\&lambda;}}_i^2}(i=p,g),$ $\tilde{F}=\frac{{\tilde{T}}_i}{{\mathrm{\&lambda;}}_i}(i=p,g),$

In the balance equation in the substitution step 2:

$m_p{\stackrel{\stackrel{\&CenterDot;\&CenterDot;}{~}}{x}}_p+\tilde{C}\left(\tilde{t}\right)\&lsqb;{\stackrel{\stackrel{\&CenterDot;}{~}}{x}}_p-{\stackrel{\stackrel{\&CenterDot;}{~}}{x}}_g-\stackrel{\stackrel{\&CenterDot;}{~}}{e}\left(\tilde{t}\right)\&rsqb;+\tilde{\text{K}}\left(\tilde{t}\right)\tilde{f}({\tilde{x}}_p-{\tilde{x}}_g-\tilde{e}\left(\tilde{t}\right))=\tilde{F}-{\tilde{F}}_f\left(\tilde{t}\right)$

$m_g{\stackrel{\stackrel{\&CenterDot;\&CenterDot;}{~}}{x}}_g-\tilde{C}\left(\tilde{t}\right)\&lsqb;{\stackrel{\stackrel{\&CenterDot;}{~}}{x}}_p-{\stackrel{\stackrel{\&CenterDot;}{~}}{x}}_g-\stackrel{\stackrel{\&CenterDot;}{~}}{e}\left(\tilde{t}\right)\&rsqb;-\tilde{\text{K}}\left(\tilde{t}\right)\tilde{f}({\tilde{x}}_p-{\tilde{x}}_g-\tilde{e}\left(\tilde{t}\right))=\widetilde{-F}+{\tilde{F}}_f\left(\tilde{t}\right)$

Wherein, x _i(i=p g) is main, the dynamic transmission error of the driven gear gear teeth; m _i(i=p g) is quality main, driven gear; F is external applied load; F _fBe average friction power;

3.2) the introducing new variables

Two equilibrium equations in the step 3.1 are subtracted each other and merge obtain:

$M\stackrel{\stackrel{\&CenterDot;\&CenterDot;}{~}}{x}+\tilde{C}\left(\tilde{t}\right)\stackrel{\stackrel{\&CenterDot;}{~}}{x}+\tilde{K}\left(\tilde{t}\right)\tilde{f}\left(\tilde{x}\right)=(\tilde{F}-{\tilde{F}}_f\left(\tilde{t}\right))-M\stackrel{\stackrel{\&CenterDot;\&CenterDot;}{~}}{e}\left(\tilde{t}\right)$

Wherein, ${\tilde{F}}_f\left(\tilde{t}\right)=\mathrm{\&mu;}(\tilde{K}\left(\tilde{t}\right)\tilde{f}\left(\tilde{x}\right)+\tilde{C}\left(\tilde{t}\right)\stackrel{\&CenterDot;}{\tilde{x}})$ Bring in the following formula and get:

$M\stackrel{\stackrel{\&CenterDot;\&CenterDot;}{~}}{x}+(1+\mathrm{\&mu;})\tilde{C}\left(\tilde{t}\right)\stackrel{\stackrel{\&CenterDot;}{~}}{x}+(1+\mathrm{\&mu;})\tilde{K}\left(\tilde{t}\right)\tilde{f}\left(\tilde{x}\right)=\tilde{F}-M\stackrel{\stackrel{\&CenterDot;\&CenterDot;}{~}}{e}\left(\tilde{t}\right)$

Wherein, x is the meshing point displacement; M is gear relative mass; μ is friction factor;

3.3) rigidity, damping and static transmission error are pressed Fourier expansion, and only consider that the main harmonic form has:

And order:

The nondimensionalization form that finally obtains model of vibration is:

Wherein, α is the harmonic wave ratio of damping; ρ is the harmonic wave stiffness coefficient; γ is the transmission error factor; ξ is damping factor;

Be the phasing degree; ω _nBe natural frequency; ω is excitation frequency, and b is gear clearance.

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