CN103198226B - A kind of cycloid bevel gears Analysis of Vibration Characteristic method considering to rub - Google Patents

Abstract

The invention discloses a kind of cycloid bevel gears Analysis of Vibration Characteristic method considering to rub, belong to nonlinear gear system vibration analysis field, the method comprises: cycloid bevel gears system simplify processes is become the torsional vibration system model of gear pair by (1); (2) in the torsional vibration system of cycloid bevel gears pair, introduce friction factor, obtained the twisting vibration balance equation of driving and driven gear by Lagrange principle respectively; (3) by the twisting vibration balance equation nondimensionalization of gear pair, the dimensionless equation formula of model of vibration is obtained; (4) according to the dimensionless equation formula of cycloid bevel gears secondary undulation model, the rule of research and analysis friction factor and cycloid bevel gears vibration characteristics.The vibration and noise reducing that the inventive method is not only bevel gear tooth system provides theories integration, and for manufacturing the cycloid bevel gears of high precision, high bearing capacity, promoting the transmission accuracy of cycloid bevel gears kinematic train, life-span and reliability and providing reference.

Description

A kind of cycloid bevel gears Analysis of Vibration Characteristic method considering to rub

Technical field

The invention belongs to nonlinear gear system vibration analysis field, relate to a kind of cycloid bevel gears Analysis of Vibration Characteristic method, more specifically relate to a kind of cycloid bevel gears Analysis of Vibration Characteristic method considering to rub.

Background technology

Cycloid bevel gears is as one of two canine tooths of spiral bevel gear, there is the features such as stable drive, load-bearing capacity are high, hard surface skiving technology, thus be specially adapted to high-power and high pulling torque heavy load transmission field, be 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 is day by day towards future developments such as high speed, precisions, cycloid bevel gears is as the crucial drive disk assembly in kinematic train, and its vibration characteristics will be more remarkable for the impact of transmission system performance.Therefore, study cycloid bevel gears vibration characteristics and have important practical value and academic significance for the efficient drive disk assembly such as Design and manufacture high precision, high-durability, low noise.

In recent years, domestic and international many scholars, based on the theory of nonlinear oscillation, with non-linear factors such as the time-varying rigidity in Meshing Process of Spur Gear and backlash for core, have carried out more extensive and deep research to the nonlinear vibration of gear train.But Recent study shows: engagement friction between teeth is also one of nonlinear gear system vibration effect factor.But most research is all for straight spur gear at present, and less to the research of cycloid bevel gears.How in the secondary kinetic model of cycloid bevel gears, introduce friction factor and Correct Analysis friction factor to the affecting laws of cycloid bevel gears vibration characteristics, still there is very large research potential, study and explore new kinetic model and analytical approach, one of important content being still this field.Research friction factor is on the impact of cycloid bevel gears vibration characteristics, the vibration and noise reducing not being only bevel gear tooth system provides theories integration, and for manufacturing the cycloid bevel gears of high precision, high bearing capacity, promoting the transmission accuracy of cycloid bevel gears kinematic train, life-span and reliability and reference is provided.

Summary of the invention

The object of this invention is to provide a kind of cycloid bevel gears Analysis of Vibration Characteristic method considering to rub, explore friction factor to the affecting laws of cycloid bevel gears vibration characteristics, thus provide theories integration for the vibration and noise reducing of bevel gear tooth system, and for manufacturing the cycloid bevel gears of high precision, high bearing capacity, promoting the transmission accuracy of cycloid bevel gears kinematic train, life-span and reliability and reference is provided.

The present invention adopts following technological means to realize:

1, cycloid bevel gears system simplify processes is become 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 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 dimension symbol; I _i(i=p, g) is the moment of inertia of active and passive gear; λ _ithe gear direction radius of turn that (i=p, g) is active and passive gear; θ _i(i=p, g) is the angular displacement of active and passive gear; T _i(i=p, g) is the moment of torsion on active and passive gear; T _f,i(i=p, g) is the moment of friction on active and passive gear; C (t) is gear pair engagement damping; K (t) is gear pair mesh stiffness; F () is gap function; E (t) is gear pair Static transmissions error function.

3, by the twisting vibration balance equation nondimensionalization of gear pair, the nondimensionalization form of model of vibration is obtained;

3.1. new variables is introduced ${\tilde{x}}_i={\mathrm{\λ}}_i{\widetilde{\mathrm{\θ}}}_i(i=p,g),$ $m_i=\frac{I_i}{{\mathrm{\λ}}_i^2}(i=p,g),$ $\tilde{F}=\frac{{\tilde{T}}_i}{{\mathrm{\λ}}_i}(i=p,g),$ substitute in the balance equation in step 2 and obtain:

$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 the dynamic transmission error of active and passive gear tooth; m _i(i=p, g) is the quality of active and passive gear; F is external applied load; F _ffor average friction force;

3.2. new variables is introduced two equilibrium equations in step 3.1 are subtracted each other and merge and obtains:

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

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 above formula and obtain:

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

Wherein, x is engaging point; M is gear relative mass; μ is friction factor.

3.3. rigidity, damping and Static transmissions error are pressed Fourier expansion, and only consider that main harmonic form has:

And order: the nondimensionalization form finally obtaining model of vibration is:

Wherein, α is harmonic wave ratio of damping; ρ is harmonic wave stiffness coefficient; γ is the transmission error factor; ξ is damping factor; for phasing degree; ω _nfor natural frequency; ω is excitation frequency, and b is gear clearance.

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

The object of the invention is, for the impact of friction on cycloid bevel gears vibration characteristics, to propose a kind of cycloid bevel gears Analysis of Vibration Characteristic method considering to rub.Feature is the twisting vibration model from cycloid bevel gears pair, introduces the dimensionless equation that friction factor obtains containing friction factor, the rule of last research and analysis friction factor μ and cycloid bevel gears vibration characteristics in its kinetic balance equation.Summary of the invention comprises three parts.In a first portion, the twisting vibration model of cycloid bevel gears pair is mainly set up; In the second portion, mainly derivation obtains the dimensionless equation formula of the cycloid bevel gears secondary undulation model containing friction factor; In Part III, mainly according to the dimensionless equation formula of cycloid bevel gears secondary undulation model, the rule of research and analysis friction factor μ and cycloid bevel gears vibration characteristics.

Accompanying drawing explanation

Fig. 1 considers the cycloid bevel gears Analysis of Vibration Characteristic method flow diagram rubbed

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

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

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

Embodiment

A kind of cycloid bevel gears Analysis of Vibration Characteristic method flow diagram considering to rub of the embodiment of the present invention 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: torsional vibration system model cycloid bevel gears system simplify processes being become gear pair;

The present embodiment is secondary for research object with aviation cycloid bevel gears, and its design parameter is in table 1.The secondary kinetic model of the cycloid bevel gears rubbed between the consideration flank of tooth as shown in Figure 2.In the model, suppose that the support stiffness of two gears is comparatively large, and do not consider that the elastic deformation of transmission shaft, block bearing and casing etc. is on the impact of cycloid bevel gears system, cycloid bevel gears system simplify processes becomes the torsional vibration system model of gear pair the most at last.

Table 1 cycloid bevel gears systematic parameter

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

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

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

$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), by 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 $\tilde{x}={\tilde{x}}_p-{\tilde{x}}_g-\tilde{e}\left(\tilde{t}\right),$ $M=\frac{m_p{m}_g}{m_p+m_g}$ Substitute in above formula and obtain:

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

Again, ${\tilde{F}}_f\left(\tilde{t}\right)=\mathrm{\μ}(\tilde{K}\left(\tilde{t}\right)\tilde{f}\left(\tilde{x}\right)+\tilde{C}\left(\tilde{t}\right)\stackrel{\·}{\tilde{x}})$ Substitute in above formula and obtain:

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

3), by rigidity, damping and Static transmissions error by Fourier expansion, and only main harmonic form is considered, namely substitution above formula obtains:

Order $x=\frac{\tilde{x}}{b},$ $t={\mathrm{\ω}}_n\tilde{t},$ ${\mathrm{\ω}}_n=\sqrt{\frac{k_m}{M}},$ $\mathrm{\γ}=\frac{e_1}{b},$ $\mathrm{\ξ}=\frac{c_m}{M{\mathrm{\ω}}_n},$ $\mathrm{\α}=\frac{c_1}{M{\mathrm{\ω}}_n},$ $\mathrm{\ρ}=\frac{k_1}{M{\mathrm{\ω}}_n^2},$ $F=\frac{\tilde{F}}{\mathrm{bM}{\mathrm{\ω}}_n^2},$ $\mathrm{\ω}=\frac{\widetilde{\mathrm{\ω}}}{{\mathrm{\ω}}_n},$ $f\left(x\right)=\frac{\tilde{f}\left(\tilde{x}\right)}{b},$ the nondimensionalization form that substitution above 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.

4th step: according to the dimensionless equation formula 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 the impact of friction factor on cycloid bevel gears vibration characteristics.Wherein when μ equals 0,0.1,0.2,0.3 respectively, friction factor affects point of mesh oscillating curve figure as shown in Figure 4.

As can be seen from Figure 4, do not consider friction, namely during μ=0, dynamic respond amplitude is about 3.33, and crest frequency is between ω=0.9 ~ 1.0; When μ=0.1, dynamic respond amplitude is about 3.18, and crest frequency is at ω=1.0 place; 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.It can thus be appreciated that along with the increase of friction factor, engaging point vibration amplitude decreases, there is drift in crest frequency, has the trend thereupon increased, 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.

Summed up by above instance analysis: the inventive method can apply in cycloid bevel gears Analysis of Vibration Characteristic, and the affecting laws of friction factor to cycloid bevel gears vibration characteristics can be drawn.The vibration and noise reducing that the inventive method is not only bevel gear tooth system provides theories integration, and for manufacturing the cycloid bevel gears of high precision, high bearing capacity, promoting the transmission accuracy of cycloid bevel gears kinematic train, life-span and reliability and providing reference.

Claims (1)

1. consider the cycloid bevel gears Analysis of Vibration Characteristic method rubbed, it is characterized in that, the method comprises the following steps:

1) cycloid bevel gears system simplify processes is become 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 active and passive gear respectively by Lagrange principle, balance equation is as follows:

Wherein, ~ be dimension symbol; I _i(i=p, g) is the moment of inertia of active and passive gear; λ _ithe gear direction radius of turn that (i=p, g) is active and passive gear; θ _i(i=p, g) is the angular displacement of active and passive gear; T _i(i=p, g) is the moment of torsion on active and passive gear; T _f,i(i=p, g) is the moment of friction on active and passive gear; C (t) is gear pair engagement damping; K (t) is gear pair mesh stiffness; F () is gap function; E (t) is gear pair Static transmissions error function;

3) by the twisting vibration balance equation nondimensionalization of gear pair, the nondimensionalization form of torsional vibration system model is obtained;

4) according to the dimensionless equation formula 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;

Wherein, step 3) in obtain the nondimensionalization form of torsional vibration system model concrete steps be:

3.1) new variables is introduced substitute in the balance equation in step 2 and obtain:

Wherein, x _i(i=p, g) is the dynamic transmission error of active and passive gear tooth; m _i(i=p, g) is the quality of active and passive gear; F is external applied load; F _ffor average friction force;

3.2) new variables is introduced two equilibrium equations in step 3.1 are subtracted each other and merge and obtains:

Wherein, bring in above formula and obtain:

Wherein, x is engaging point; M is gear relative mass; μ is friction factor;

3.3) rigidity, damping and Static transmissions error are pressed Fourier expansion, and only consider that main harmonic form has:

And order: the nondimensionalization form finally obtaining model of vibration is:

Wherein, α is harmonic wave ratio of damping; ρ is harmonic wave stiffness coefficient; γ is the transmission error factor; ξ is damping factor; for phasing degree; ω _nfor natural frequency; ω is excitation frequency, and b is gear clearance.