Stability and bifurcation phenomena of Laval/Jeffcott rotors in semi-floating ring bearings

Abstract

Within this contribution, a linear-elastic Laval/Jeffcott rotor is considered, which is symmetrically supported in two identical semi-floating ring bearings. Run-up simulations and bifurcation analyses are carried out to investigate the stability and bifurcation phenomena of the rotor-bearing system. In particular, the methods of numerical continuation are applied to identify the nonlinear phenomena (jump phenomena, coexistence of solutions, etc.) and the corresponding bifurcations. The occurrence of subsynchronous oscillations is examined, which is caused by an oil whirl/whip instability due to the inner oil films. In this case, the main damping is provided by the outer oil films so that the oscillation amplitudes usually remain moderate. Besides these well-known subsynchronous oscillations with moderate amplitudes (oil whirl/whip instability due to the inner oil films), it is shown that self-excited oscillations with very high amplitudes also exist. This effect resembles Total Instability known from rotors in full-floating ring bearings. A detailed bifurcation analysis proves the coexistence of a so-called critical limit cycle with high amplitudes in the case of the perfectly balanced rotor which represents Total Instability. Finally, a variation of rotor and bearing parameters shows the influence on both the subsynchronous oscillations of tolerable amplitudes and the critical limit cycle oscillations.

Appendix: Hydrodynamic bearing model

Within the scope of this appendix, the hydrodynamic bearing model is described which is applied to calculate the nonlinear forces of the semi-floating ring bearing in this paper. By using the established short bearing assumption, the Reynolds equation can be simplified to obtain an analytical solution for the nonlinear bearing forces [14]. In the following, the basic relationships are presented for the plain cylindrical journal bearing which can be rewritten to the case of the semi-floating ring bearing.

1.1 Plain journal bearing

1.1.1 Reynolds equation

A sketch of the plain cylindrical journal bearing is illustrated in Fig. 23 where the bearing shell (radius $R=D/2$) and the shaft (Radius $r=R-C\approx R$, radial bearing clearance $C$) are assumed to be rigid. Thereby, the shaft rotates with a constant angular velocity $\omega$ about its geometrical center $O_\mathrm{R}$ and moves translationally only in the plane perpendicular to the rotation axis. Furthermore, the axial direction of the bearing shell is assumed to remain aligned with the corresponding shaft axis so that within the calculation of the nonlinear bearing forces the inclination of both bodies toward each other is neglected. The displacement of the bearing shaft in the shell is given by the eccentricity $e$ and $\varepsilon =e/C$, respectively, which is normalized with respect to the bearing clearance $C$. The angle $\phi$ describes the line of the minimum film thickness $h_\mathrm{min}$ where the gap function $h$ and the corresponding dimensionless variable $H=h/C$ can be very well approximated with

$$\begin{aligned} h= C+e\cos \theta \quad \mathrm {and}\quad H= 1+\varepsilon \cos \theta \end{aligned}$$

(4)

in dependence of the circumferential angle $\theta$ for a plain circular bore geometry.

Fig. 23

View of a plain journal bearing

To determine the dimensionless pressure distribution $\varPi$ in the bearing, the non-dimensional Reynolds equation can be written in the following form:

$$\begin{aligned}&\frac{\partial }{\partial \theta } \left( H^{3}\frac{\partial \varPi }{\partial \theta } \right) +\frac{1}{4}\left( \frac{D}{B}\right) ^2 \frac{\partial }{\partial \bar{x}} \left( H^{3} \frac{\partial \varPi }{\partial \bar{x}}\right) \nonumber \\&\quad =2\varepsilon '\cos \theta +2\varepsilon \sin \theta \left( \phi '-\frac{1}{2}\varOmega _0\right) . \end{aligned}$$

(5)

Therefore, a dimensionless time $\tau =\omega _0 t$ is introduced by means of an arbitrary reference angular velocity $\omega _0$. The differentiations with respect to the dimensionless time $\tau$ are denoted by $( \ldots )' = \frac{\text {d}}{\text {d} \tau }(\ldots )$. $\bar{x}=x/B$ denotes the dimensionless $x$-coordinate in axial direction and $\varOmega _0=\omega /\omega _0$ the shaft speed ratio. Moreover, the relationship between the dimensionless and the physical pressure distribution $p$ is given by $\varPi =\left( \frac{C}{R}\right) ^2\frac{p}{6\eta \omega _0}$.

1.1.2 Impedance method

To calculate the nonlinear bearing forces, the impedance method [9] is applied which is based on the mobility method of Booker [4]. The main idea is to choose a transformation into a reference system in which only a pure squeeze velocity remains on the right-hand side of the Reynolds equation (5). By means of the introduction of a dimensionless squeeze velocity $\bar{v}_s$ the transformations $\bar{v}_\mathrm{s}\cos \alpha =\varepsilon '$ and $\bar{v}_\mathrm{s}\sin \alpha =\varepsilon \left( \phi '-\frac{1}{2}\varOmega _0\right)$ yield the Reynolds equation in this form:

$$\begin{aligned}&\frac{\partial }{\partial \theta } \left( H^{3} \frac{\partial \varPi }{\partial \theta } \right) + \frac{1}{4}\left( \frac{D}{B}\right) ^2 \frac{\partial }{\partial \bar{x}} \left( H^{3} \frac{\partial \varPi }{\partial \bar{x}}\right) \nonumber \\&\quad =2 \bar{v}_s\cos (\alpha +\theta ) \end{aligned}$$

(6)

Note that $\alpha$ denotes the angle between the minimum line of thickness and the direction of the pure squeeze velocity $\bar{v}_\mathrm{s}$. Then, the nonlinear bearing forces are determined by integration:

$$\begin{aligned} f_\mathrm{r}^*&=3\int \limits _{\theta _1}^{\theta _2} \int \limits _{-\frac{1}{2}}^{\frac{1}{2}} \varPi \cos \theta \,\mathrm{{d}} \bar{x}\,\mathrm{{d}}\theta , \end{aligned}$$

(7)

$$\begin{aligned} f_{\phi }^*&=3\int \limits _{\theta _1}^{\theta _2} \int \limits _{-\frac{1}{2}}^{\frac{1}{2}} \varPi \sin \theta \,\mathrm {d} \bar{x}\,\mathrm {d}\theta . \end{aligned}$$

(8)

The integration limits for the angle $\theta$ follow from the boundary conditions which are chosen in circumferential direction. The Sommerfeld number $So$ can be defined by

$$\begin{aligned} So=\left( \frac{C}{R}\right) ^2\frac{W}{\eta \omega _0 B D} \end{aligned}$$

(9)

to obtain a relationship between the dimensionless and physical bearing force components

$$\begin{aligned} f_i^*={So} \frac{F_i}{W}\quad \quad i=r,\phi \, \end{aligned}$$

(10)

for a given bearing load $W$. Since the bearing force is proportional to the pure squeeze velocity, for the dimensionless bearing force components $f_i^*$ it can also be written

$$\begin{aligned} f_i^*=-\bar{v}_s W_{I,i}^{^*} \quad \mathrm {}\quad i=r,\phi \end{aligned}$$

(11)

where $W_{I,i}^{^*}$ are the components of the so-called impedance vector. After determining the impedance vector in dependence of $\epsilon$ and $\alpha$ the obtained bearing force components $f_i^*$ can be transformed into the reference systems of the rotor-bearing system.

1.1.3 Short bearing theory

Here, the analytical solutions of the impedance vector components are presented for the short bearing theory (see e.g., [9, 29]) where the gradient of the pressure distribution $\partial \varPi /\partial \theta$ in circumferential direction is neglected in the Reynolds equation (6). To solve the simplified Reynolds equation, the pressure distribution in axial direction vanishes at both boundaries: $\varPi (\bar{x}=-\frac{1}{2})=\varPi (\bar{x}=\frac{1}{2})=0$. In circumferential direction the so-called Half-Sommerfeld or Gümbel boundary conditions are assumed where a positive pressure distribution exists between the two angles $\theta _1=\frac{1}{2}\varPi -\alpha$ and $\theta _2=\frac{3}{2}\varPi -\alpha$.

Hence, the components $W_{I,r}^{^*}$ and $W_{I,\phi }^{^*}$ of the impedance vector are given by

$$\begin{aligned} W_{I,r}^{^*}&=2\left( I_1\cos \alpha -I_2\sin \alpha \right) \left( \frac{B}{D}\right) ^2\,,\end{aligned}$$

(12)

$$\begin{aligned} W_{I,\phi }^{^*}&=2\left( I_2\cos \alpha -I_3\sin \alpha \right) \left( \frac{B}{D}\right) ^2. \end{aligned}$$

(13)

The evaluation of the integrals (cf. [5]) leads to the following expressions

$$\begin{aligned} I_1&= \frac{1}{2(1-\varepsilon ^2)^2}\Bigg [(1+2\varepsilon ^2)I_0\nonumber \\&+\frac{2\varepsilon \cos \alpha \left( 3+(2-5\varepsilon ^2)\sin ^2\alpha \right) }{(1-\varepsilon ^2\sin ^2\alpha )^2}\Bigg ]\,,\nonumber \\ I_2&= -\frac{2\varepsilon \sin ^3\alpha }{(1-\varepsilon ^2\sin ^2\alpha )^2}\,, \nonumber \\ I_3\!&= \!\frac{1}{2(1\!-\!\varepsilon ^2)}\!\left[ \!I_0+\frac{2\varepsilon \cos \alpha \left( 1-(2-\varepsilon ^2)\sin ^2\alpha \right) }{(1-\varepsilon ^2\sin ^2\alpha )^2}\!\right] ,\nonumber \\ \end{aligned}$$

(14)

where

$$\begin{aligned} I_0=\frac{\arccos (-\delta A)+\arccos (-\delta B)}{\sqrt{(1-\varepsilon ^2)}}\, \end{aligned}$$

(15)

and

$$\begin{aligned} \begin{aligned} A=\frac{\varepsilon +\sin \alpha }{1+\varepsilon \sin \alpha }\,,\\ B=\frac{\varepsilon -\sin \alpha }{1-\varepsilon \sin \alpha }\,, \end{aligned}\quad \delta = \bigg \{ \begin{array}{lll} &{}\displaystyle 1,&{}\displaystyle \quad \cos \alpha \ge 0 \\ &{}\displaystyle -1,&{}\displaystyle \quad \cos \alpha <0. \end{array} \end{aligned}$$

(16)

1.2 Semi-floating ring bearing

The analytical short bearing solution of the Reynolds equation is applied for the inner and outer oil film (cf. Fig. 24). Therefore, the dimensionless quantities as the eccentricity $\varepsilon$, the time derivative of the eccentricity $\varepsilon '$, the time derivative of the line of minimum film thickness $\phi '$ and additionally for the impedance method the angle $\alpha$ between the line of minimum film thickness and the direction of the pure squeeze velocity must be evaluated for both oil films. Note that the defined speed ratio for the outer oil film is $\varOmega _0=0$ since the ring does not rotate in the case of the semi-floating ring bearing. Then, the impedance vectors $W_{I,r}^{^*}$ and $W_{I,\phi }^{^*}$ for the short bearing assumption can be calculated analytically and the nonlinear bearing forces are obtained.

Fig. 24

View of a semi-floating ring bearing