Nonlinear dynamic responses of a rigid rotor supported by active bump-type foil bearings

Abstract

Active bump-type foil bearings (ABFBs) enhance the rotordynamic characteristics of rotor–bearing systems with the advantage of controllable mechanical preloads. However, the coupling of controllable mechanical preloads, compressible gas film, and foil structures induce strong nonlinear characteristics and affect the dynamic responses of rotor. In this study, a nonlinear theoretical model that considers the gyroscopic effect of rotor, nonlinear Reynolds equation, complicated foil structures, and dynamic motions of active substructures is presented. This model is verified by a corresponding rotordynamic test. The nonlinear dynamic responses of a rotor–ABFB system are discussed on the basis of waterfall plots, orbit simulations, and Poincaré maps of rotor center, fast Fourier transform, minimum film thickness, and power loss during one cycle of journal orbit. The effects of voltage on piezoelectric actuators, nominal clearance, width of flexure hinge, and static load on the nonlinear rotordynamic responses of rotor are analyzed to provide guidelines on selecting the design and control parameters of rotor–ABFB systems.

Appendices

Appendix A: Modified assembly clearance for ABFB

As shown in Fig. 19, the radial displacements of the stations on one-lever amplifier are varied with their circumferential positions. Linear interpolation is used to simulate the corresponding assembly clearance within the circumferential span of lever amplifier. The modified assembly clearance is calculated by using the following equation

$$\begin{aligned} \begin{array}{l} \hbox {If} \quad c(\theta ) < C \\ c (\theta )=\left\{ \begin{array}{l} C-\Delta R_{a}^{n} \left( 1-\text {sin}\left( \frac{2\pi }{\theta _{p}} \frac{(\theta -\theta _{1} -(n-1)\theta _{p} )}{2}\right) ^{2} w\times 1.4\right) \\ \quad \theta \in (\theta _{1} -\theta _{p} /2+(n-1)\theta _{p}, \theta _{1} -\theta _{s} /2+(n-1)\theta _{p}) \\ C-\Delta R_{a}^{n} \left( \frac{\theta _{1} +\theta _{s} /2-\theta +(n-1)\theta _{p}}{\theta _{s}}\right) \left( 1\text {-sin} \left( \frac{2\pi }{\theta _{p}}\frac{(\theta -\theta _{1} -(n-1)\theta _{p})}{2}\right) ^{2}w \times 1.4\right) \\ -\Delta R_{b}^{n} \left( {\frac{\theta -\theta _{1} +\theta _{s} /2-(n-1)\theta _{p} }{\theta _{s} }} \right) \left( {\text {1-sin} \left( {\frac{2\pi }{\theta _{p} }\frac{(\theta -\theta _{1} -(n-1)\theta _{p} )}{2}} \right) ^{2}w\times 1.4} \right) \\ \quad \theta \in \left( {\theta _{1} -\theta _{s} /2+(n-1)\theta _{p}, \theta _{1} +\theta _{s} /2+(n-1)\theta _{p} } \right) \\ C-\Delta R_{b}^{n} \left( {1-\text {sin}\left( {\frac{2\pi }{\theta _{p}} \frac{(\theta -\theta _{1} -(n-1)\theta _{p} )}{2}} \right) ^{2}w\times 1.4} \right) \\ \quad \theta \in \left( {\theta _{1} +\theta _{s} /2+(n-1)\theta _{p}, \theta _{1} +\theta _{p} /2+(n-1)\theta _{p} }\right) \\ \end{array}\right. \\ n=1,2,3 \quad \theta \in [0,2\pi ]\\ If \quad c(\theta ) > C \\ c(\theta )=C; \\ \end{array}, \end{aligned}$$

(A.1)

where C is the nominal bearing clearance. $\Delta R_{a}^{n}$ and $\Delta R_{b}^{n}$ are the radial displacements of the two edges along the circumferential direction of the lever amplifier. Subscripts a and b stand for the two edges along each lever amplifier, and superscript n stands for the sequence number of the lever amplifier. Points a and b are the stations of the mesh, and $\Delta R_{a}^{n}$ and $\Delta R_{b}^{n}$ can be calculated with Eq. (20). Both of them are determined by the angle location and angle region of the lever amplifier. When the geometry of the lever amplifier is selected, their initial values are determined by U and the preload of the lock screw. Besides, $w=\theta _{s} /\theta _{p}$ is the circumferential span parameter of lever amplifier.

Fig. 19

Shape schematic of modified assembly clearance

Appendix B: Static load tests and parameters of tested ABFB and GFB

Fig. 20

Measurement tests on bearing clearance

Table 3 Parameters of the two test journal bearings

The results of static load tests of tested ABFB and GFB are shown in Fig. 20. The hysteresis curves of ABFB are in coincidence with that of GFB. Both tested bearings have identical clearance of $40\ \upmu \hbox {m}$. The parameters of the two bearings are shown in Table 3.