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

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.

Appendices

Appendix 1

In this study, the non-Newtonian Reynolds-type equation for couple-stress fluid is as follows:

$$\begin{aligned}&\frac{\partial }{\partial x}\left( g(h,l)\frac{\partial p}{\partial x}\right) +\frac{\partial }{\partial z}\left( g(h,l)\frac{\partial p}{\partial z}\right) \nonumber \\&\quad =6\mu U\frac{\partial h}{\partial x}+12\mu \frac{\partial h}{\partial t}[20] \end{aligned}$$

(35)

where $g(h,l)=h^{3}-12l^{2}h+24l^{3}\tanh (\frac{h}{2l})\,,\quad \frac{\partial h}{\partial x}=-\frac{c\varepsilon }{R}\sin \theta \, , \frac{\partial h}{\partial t}=c\dot{\varepsilon }\cos \theta +c\varepsilon \dot{\phi }\sin \theta , x{=}R\theta ,U=R\omega \,,\quad \varepsilon =\frac{e}{c}$ and $h=c(1+\varepsilon \cos (\gamma -\varphi (t)))=c(1+\varepsilon \cos \theta )$. Thus, $g(h,l)$ can also be performed as $g(h,l)=c^{3}(1+\varepsilon \cos \theta )^{3}-12l^{2}c(1+\varepsilon \cos \theta )+24l^{3}\tanh (\frac{c(1+\varepsilon \cos \theta )}{2l})$, where $l=(\frac{\eta }{\mu })^{1/2}$, in which $\mu$ is the classical viscosity parameter and $\eta$ is a new material constant peculiar to fluids with couple stresses, and Reynolds equation can be rewritten as

$$\begin{aligned}&\!\!\!\frac{\partial }{R\partial \theta }\left( g(h,l)\frac{\partial p}{R\partial \theta }\right) +\frac{\partial }{\partial z}\left( g(h,l)\frac{\partial p}{\partial z}\right) \nonumber \\&\quad \!\!\!=-6\mu \omega c\varepsilon \sin \theta +12\mu (c\dot{\varepsilon }\cos \theta +c\varepsilon \dot{\varphi }\sin \theta )\nonumber \\ \end{aligned}$$

(36)

Using the “short bearing approximation” $(\frac{L}{D}<0.25, \frac{\partial p}{\partial \theta }<<\frac{\partial p}{\partial z})$, we can set $\frac{\partial p}{\partial \theta }=0$. The following equation can be introduced.

$$\begin{aligned} \frac{\partial ^{2}p}{\partial z^{2}}=\frac{-6\mu \omega c\varepsilon \sin \theta +12\mu (c\dot{\varepsilon }\cos \theta +c\varepsilon \dot{\varphi }\sin \theta )}{g(h,l)}\nonumber \\ \end{aligned}$$

(37)

with B.C. $\left\{ {{\begin{array}{l} {\frac{\partial p}{\partial z}=0,\quad z=0} \\ {p=0,\quad z=\pm \frac{L}{2}} \\ \end{array} }} \right.$ then

$$\begin{aligned} p\!=\!-\frac{3\mu c}{g(h,l)}[(\omega \!-\!2\dot{\varphi })\varepsilon \sin \theta \!-\!2\dot{\varepsilon }\cos \theta ]\left( z^{2}\!-\!\frac{L^{2}}{4}\right) \nonumber \\ \end{aligned}$$

(38)

The resulting damping forces about the journal center in the radial and tangential directions are determined by integrating Eq. (38) over the area of the journal sleeve.

$$\begin{aligned}&\!\!\!f_\mathrm{r} =\int _0^\pi {\int _{-\frac{L}{2}}^{\frac{L}{2}} p } R\cos \theta \hbox {d}z\hbox {d}\theta \end{aligned}$$

(39)

$$\begin{aligned}&\!\!\!f_\mathrm{t} =\int _0^\pi {\int _{-\frac{L}{2}}^{\frac{L}{2}} p } R\sin \theta \hbox {d}z\hbox {d}\theta \end{aligned}$$

(40)

$$\begin{aligned} f_e =-f_\mathrm{r} ,\quad f_\varphi =-f_\mathrm{t} \end{aligned}$$

Therefore,

$$\begin{aligned}&f_e =-\frac{\mu L^{3}R}{2c^{2}}\int _0^\pi {\left\{ {\frac{[(\omega -2\dot{\varphi })\varepsilon \sin \theta -2\dot{\varepsilon }\cos \theta ]\cos \theta }{[(1+\varepsilon \cos \theta )^{3}-12(l^{*})^{2}(1+\varepsilon \cos \theta )+24(l^{*})^{3}\tanh (\frac{1+\varepsilon \cos \theta }{2l^{*}})]}} \right\} } d\theta \end{aligned}$$

(41)

$$\begin{aligned}&f_\varphi =-\frac{\mu L^{3}R}{2c^{2}}\int _0^\pi {\left\{ {\frac{[(\omega -2\dot{\varphi })\varepsilon \sin \theta -2\dot{\varepsilon }\cos \theta ]\sin \theta }{[(1+\varepsilon \cos \theta )^{3}-12(l^{*})^{2}(1+\varepsilon \cos \theta )+24(l^{*})^{3}\tanh (\frac{1+\varepsilon \cos \theta }{2l^{*}})]}} \right\} } d\theta \end{aligned}$$

(42)

Substituting Eqs. (41) and (42) into Eqs. (11)–(14) enables the values of $F_\mathrm{x1}, F_\mathrm{y1}, F_\mathrm{x2 }$ and $F_\mathrm{y2}$ to be obtained. The radial impact force $f_\mathrm{n}$ and the tangential rub force $f_\mathrm{t}$ could be expressed as

$$\begin{aligned}&f_\mathrm{n} =(e-\delta )k_\mathrm{c}\end{aligned}$$

(43)

$$\begin{aligned}&f_\mathrm{t} =(f+bv)f_\mathrm{n} , \quad \hbox { if } \; e\ge \delta \end{aligned}$$

(44)

Then, we could get the rub-impact forces in the horizontal and vertical directions.

$$\begin{aligned} R_\mathrm{x} =\frac{(e-\delta )k_\mathrm{c} }{e}[X-(f+bv)Y]\end{aligned}$$

(45)

$$\begin{aligned} R_\mathrm{y} =\frac{(e-\delta )k_\mathrm{c} }{e}[(f+bv)X+Y] \end{aligned}$$

(46)

Appendix 2

$$\begin{aligned}&\alpha _1 =-\frac{\mu L^{3}R}{2c^{2}}\\&\beta _1 =\int _0^\pi {\frac{\sin \theta \cos \theta }{[(1+\varepsilon _1 \cos \theta )^{3}-12(l^{*})^{2}(1+\varepsilon _1 \cos \theta )+24(l^{*})^{3}\tanh (\frac{1+\varepsilon _1 \cos \theta }{2l^{*}})]}} d\theta \\&\gamma _1 =\int _0^\pi {\frac{\cos ^{2}\theta }{[(1+\varepsilon _1 \cos \theta )^{3}-12(l^{*})^{2}(1+\varepsilon _1 \cos \theta )+24(l^{*})^{3}\tanh (\frac{1+\varepsilon _1 \cos \theta }{2l^{*}})]}} d\theta \\&\delta _1 =\int _0^\pi {\frac{\sin ^{2}\theta }{[(1+\varepsilon _1 \cos \theta )^{3}-12(l^{*})^{2}(1+\varepsilon _1 \cos \theta )+24(l^{*})^{3}\tanh (\frac{1+\varepsilon _1 \cos \theta }{2l^{*}})]}} d\theta \\&\alpha _2 =-\frac{\mu L^{3}R}{2c^{2}}\\&\beta _2 =\int _0^\pi {\frac{\sin \theta \cos \theta }{[(1+\varepsilon _2 \cos \theta )^{3}-12(l^{*})^{2}(1+\varepsilon _2 \cos \theta )+24(l^{*})^{3}\tanh (\frac{1+\varepsilon _2 \cos \theta }{2l^{*}})]}} d\theta \\&\gamma _2 =\int _0^\pi {\frac{\cos ^{2}\theta }{[(1+\varepsilon _2 \cos \theta )^{3}-12(l^{*})^{2}(1+\varepsilon _2 \cos \theta )+24(l^{*})^{3}\tanh (\frac{1+\varepsilon _2 \cos \theta }{2l^{*}})]}} d\theta \\&\delta _2 =\int _0^\pi {\frac{\sin ^{2}\theta }{[(1+\varepsilon _2 \cos \theta )^{3}-12(l^{*})^{2}(1+\varepsilon _2 \cos \theta )+24(l^{*})^{3}\tanh (\frac{1+\varepsilon _2 \cos \theta }{2l^{*}})]}} d\theta \end{aligned}$$

where $l^{*}=l/c$ is the dimensionless parameter for $l$; c is the radial clearance; $l=(\frac{\eta }{\mu })^{1/2}$, in which $\mu$ is the classical viscosity parameter and $\eta$ is a new material constant peculiar to fluids with couple stresses.