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Nonlinear Dynamics

, Volume 79, Issue 1, pp 749–763 | Cite as

Bifurcation and chaos of gear-rotor–bearing system lubricated with couple-stress fluid

  • Cai-Wan Chang-Jian
Original Paper

Abstract

This study performs a systematic analysis of the dynamic behaviors of a gear pair system mounted on rotor–bearing system with strongly nonlinear effects including nonlinear suspension effect, nonlinear couple-stress fluid film force, nonlinear rub-impact force and nonlinear gear mesh force. The dynamic orbits of the system are observed using bifurcation diagrams plotted using the dimensionless rotational speed ratio as a control parameter. The onset of chaotic motion is specified from the phase diagrams, Poincaré maps, Lyapunov exponents and fractal dimension of the system. There exist various forms of periodic, sub-harmonic and chaotic motions at different bifurcation parameters. The simulation results revealed that highly non-periodic motions do exist in gear-rotor–bearing systems under those nonlinear effects. The results also confirm that the couple-stress fluid can suppress non-periodic vibrations, especially at lower rotational speeds. The results of this study allow suitable system parameters to be defined such that undesirable behaviors of the system can be avoided and the machinery life extended as a result.

Keywords

Couple-stress fluid Rub-impact Gear-rotor–bearing system 

List of symbols

\(c\)

Radial clearance, \(c=R-r\)

\(C\)

Viscous damping of the gear or the pinion

\(C_{1}\)

Damping coefficient of the supported structure for bearing \(1\)

\(C_{2}\)

Damping coefficient of the supported structure for bearing \(2\)

\(C_\mathrm{m}\)

Damping coefficient of the gear mesh

\(C_\mathrm{r}\)

Viscous damping of the rotor disk

\(C_{1\mathrm{p}}\)

Dimensionless parameter, \(C_{1\mathrm{p}} =\frac{M_1}{M_\mathrm{p}}\)

\(C_{01}\)

Dimensionless parameter, \(C_{01}=\frac{K}{K_{11} }\)

\(C_{2\mathrm{p}}\)

Dimensionless parameter, \(C_{2\mathrm{p}} =\frac{M_2 }{M_\mathrm{p}}\)

\(C_{02}\)

Dimensionless parameter, \(C_{02}=\frac{K}{K_{21} }\)

\(C_{\mathrm{r}2}\)

Dimensionless parameter, \(C_{\mathrm{r}2}=\frac{K_\mathrm{r}}{K_{21}}\)

\(C_\mathrm{rp}\)

Dimensionless parameter, \(C_\mathrm{rp} =\frac{M_\mathrm{r}}{M_\mathrm{p}}\)

\(E_\mathrm{p}\)

Static transmission error

\(e\)

Static transmission error and varies as a function of time

\(e_\mathrm{i}\)

Offset of the journal center of the rotor relative to the X-coordinate direction

\(e_\mathrm{p}\)

Dimensionless parameter, \(e_\mathrm{p}=E_\mathrm{p} /c\)

\(F_\mathrm{x1}, F_\mathrm{y1}\)

Components of the fluid film force in the horizontal and vertical directions for bearing \(1\)

\(F_\mathrm{x2}, F_\mathrm{y2}\)

Components of the fluid film force in the horizontal and vertical directions for bearing \(2\)

\(f\)

Dimensionless parameter, \(f=\frac{M_\mathrm{p} g}{cK}\)

\(f_\mathrm{g}\)

Dimensionless parameter, \(f_\mathrm{g}=\frac{Kg}{cM_\mathrm{p}}\)

\(f_\mathrm{e1}, f_{\varphi 1}\)

Components of the fluid film force in radial and tangential directions for bearing \(1\)

\(f_\mathrm{e2}, f_{\varphi 2}\)

Components of the fluid film force in radial and tangential directions for bearing \(2\)

\(G_\mathrm{gy}\)

The inertia forces in the vertical gear mesh direction for gear, \(G_\mathrm{gy} =M_\mathrm{g} E_\mathrm{g} \ddot{\theta }_2 \cos \theta _2 \)

\(G_\mathrm{py}\)

The inertia forces in the vertical gear mesh direction for pinion, \(G_\mathrm{py} =M_\mathrm{p} E_\mathrm{p} \ddot{\theta }_1 \cos \theta _1 \)

\(g\)

Acceleration of gravity

\(K\)

Stiffness coefficient of the shafts

\(K_{11}, K_{12}\)

Stiffness coefficients of the springs supporting the two bearing housings for bearing \(1\)

\(K_{21}, K_{22}\)

Stiffness coefficients of the springs supporting the two bearing housings for bearing \(2\)

\(K_\mathrm{m} \)

Stiffness coefficient of the gear mesh

\(K_\mathrm{r}\)

Stiffness coefficient of the rotor disk

\(L\)

Bearing length

\(L_\mathrm{gy} \)

The centrifugal forces in the vertical gear mesh direction for gear, \(L_\mathrm{gy} =M_\mathrm{g} E_\mathrm{g} \omega _\mathrm{g}^2 \sin \theta _2\)

\(L_\mathrm{py} \)

The centrifugal forces in the vertical gear mesh direction for pinion, \(L_\mathrm{py} =M_\mathrm{p} E_\mathrm{p} \omega _\mathrm{p}^2 \sin \theta _1 \)

\(l\)

Characteristic length of additives, \(l=(\frac{\eta }{\mu })^{1/2}\)

\(l^{*}\)

Dimensionless couple-stress parameter, \(l^{*}=l/c\)

\(M_1 \)

Mass of the bearing housing for bearing \(1\)

\(M_2 \)

Mass of the bearing housing for bearing \(2\)

\(M_\mathrm{p}\)

Mass of the pinion

\(M_\mathrm{g}\)

Mass of the gear

\(M_\mathrm{r}\)

Mass of the rotor

\(O_{1}\)

Geometric centers of the bearing \(1\)

\(O_{2}\)

Geometric centers of the bearing \(2\)

\(O_\mathrm{j1}\)

Geometric centers of the journal \(1\)

\(O_\mathrm{j2}\)

Geometric centers of the journal \(2\)

\(O_\mathrm{g}\)

Center of gravity of the gear

\(O_\mathrm{p}\)

Center of gravity of the pinion

\(O_\mathrm{r}\)

Center of gravity of the rotor disk

\(p\)

Pressure distribution in the fluid film

\(R\)

Inner radius of the bearing housing

\(R_\mathrm{x}\)

Component of rub-impact force in the horizontal direction

\(R_\mathrm{y}\)

Component of rub-impact force in the vertical direction

\(r\)

Radius of the journal.

\(s\)

Rotational speed ratio, \(s\!=\!(\frac{\omega ^{2}}{\omega _\mathrm{n}^{2}})^{1/2}\)

\(s_1\)

Dimensionless parameter, \(s_1 ^{2}\!=\! C_\mathrm{o1} C_{1\mathrm{p}}s^{2}\)

\(s_2 \)

Dimensionless parameter, \(s_2 ^{2}\!=\! C_\mathrm{o2} C_{2\mathrm{p}}s^{2}\)

\(s_3 \)

Dimensionless parameter, \(s_3 ^{2}\!=\! C_\mathrm{r2} C_\mathrm{rp}s^{2}\)

\(W_\mathrm{cx}\)

The dynamic gear mesh force in the horizontal direction

\(W_\mathrm{cy}\)

The dynamic gear mesh force in the vertical direction

\(X, Y, Z\)

Horizontal, vertical and axial coordinates

\(x_\mathrm{j}, y_\mathrm{j}\)

\(X_\mathrm{j}/c, Y_\mathrm{j}/c ,\, j=1,\;2,\;\mathrm{j}1,\;\mathrm{j}2,\;p,\;g,\;r\)

\(\alpha _\mathrm{a}\)

Dimensionless parameter, \(\alpha _\mathrm{a} =\frac{K_{12} c^{2}K}{M_1 M_\mathrm{p} }\)

\(\alpha _\mathrm{b}\)

Dimensionless parameter, \(\alpha _\mathrm{b} =\frac{K_{22} c^{2}K}{M_2 M_\mathrm{p} }\)

\(\beta \)

Dimensionless unbalance parameter, \(\beta =E_\mathrm{p} /16\)

\(\beta _\mathrm{g} \)

Dimensionless unbalance parameter, \(\beta _\mathrm{g} =E_\mathrm{p} /16\)

\(\beta _\mathrm{r} \)

Dimensionless unbalance parameter, \(\beta _\mathrm{r} =\rho /c\)

\(\xi _1 \)

Dimensionless parameter, \(\xi _1 =\frac{C_1 }{2\sqrt{K_1 M_1 }}\)

\(\xi _2 \)

Dimensionless parameter, \(\xi _2 =\frac{C}{2\sqrt{KM_\mathrm{p} }}\)

\(\xi _3 \)

Dimensionless parameter, \(\xi _3 =\frac{C_\mathrm{m} }{2\sqrt{KM_\mathrm{p} }}\)

\(\xi _4 \)

Dimensionless parameter, \(\xi _4 =\frac{C}{2\sqrt{\frac{K}{M_\mathrm{p} }}M_\mathrm{g} }\)

\(\xi _5 \)

Dimensionless parameter, \(\xi _5 =\frac{C_\mathrm{m} \sqrt{M_\mathrm{p} }}{2M_\mathrm{g} \sqrt{K}}\)

\(\xi _6 \)

Dimensionless parameter, \(\xi _6 =\frac{C_2 }{2\sqrt{K_2 M_2 }}\)

\(\Lambda \)

Dimensionless parameter, \(\Lambda =\frac{K_\mathrm{m} }{K}\)

\(\Lambda _\mathrm{g}\)

Dimensionless parameter, \(\Lambda _\mathrm{g} =\frac{K_\mathrm{m} M_\mathrm{p}^2 }{M_\mathrm{g} K^{2}}\)

\(\rho \)

Mass eccentricity of the rotor

\(\phi \)

Rotational angle, \(\phi =\omega t\)

\(\omega \)

Rotational speed of the shaft

\(\theta \)

The angular position

\(\mu \)

Oil dynamic viscosity

\(\eta \)

A new material constant peculiar to fluids with couple stresses

\(\varepsilon _\mathrm{i}\)

Eccentricity ratio, \(\varepsilon =e_\mathrm{i}/c\)

\(\omega _\mathrm{n}\)

Natural frequency, \(\omega _\mathrm{n} =\sqrt{K/M_\mathrm{p} }\)

\(\omega _\mathrm{g}\)

Dimensionless parameter, \(\omega _\mathrm{g} =\omega _\mathrm{n} /8\)

\(\omega _\mathrm{p} \)

Dimensionless parameter, \(\omega _\mathrm{p} =\omega _\mathrm{n} /4\)

\(\varphi _\mathrm{i}\)

Attitude angle of the rotor relative to the X-coordinate direction

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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Department of Mechanical and Automation EngineeringI-Shou UniversityKaohsiungTaiwan, R.O.C

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