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.
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- \(A_\mathrm{b}\) :
-
Area of one bump
- B :
-
Center point on the fixed end of the flexure hinge B
- \({B}'\) :
-
Center point on the free end of the flexure hinge B
- \(\overline{c_\mathrm{b}}\) :
-
Dimensionless damping of the bump foil
- \(c_{\varphi }\) :
-
Rotational compliance of the flexure hinge B (rad/(N m))
- C :
-
Nominal clearance (m)
- \(C_{\varphi }\) :
-
Structure damping of the lever amplifier (N s/m)
- \(d_{33}\) :
-
Piezoelectric coefficient of the PZT (m/V)
- E :
-
Young’s modulus (Pa)
- \(\{F\}\) :
-
Force matrix of foil structure (N)
- \(F_{0}\) :
-
Output force of the PZTs (N)
- \(F_\mathrm{G}\) :
-
Static load of rotor (N)
- \(F_{N}\) :
-
Preload of the lock screw (N)
- \(F_{p}\) :
-
Force from the output side of the lever amplifier (N)
- \(\overline{h}\) :
-
Dimensionless film thickness (\(=h/C\))
- \(K_{\varphi \mathrm{equal}}\) :
-
Equivalent rotational stiffness of the lever amplifier (N/rad)
- \(K_\mathrm{b}\) :
-
Stiffness of one bump derived from the simplified link-spring model (N/m)
- \(k_\mathrm{b}\) :
-
Average stiffness of bump foil derived from the link-spring model (\(\hbox {N}/\hbox {m}^{3}\))
- \(\overline{k_\mathrm{b}}\) :
-
Dimensionless stiffness of the bump foil
- \(k_\mathrm{p}\) :
-
Equivalent stiffness of the two PZTs in parallel (N/m)
- \(I_\mathrm{T}\) :
-
Translational inertia moment of rotor (\(\hbox {kg m}^{2}\))
- \(I_\mathrm{P}\) :
-
Polar inertia moment of rotor (kg m\(^2\))
- \(I_{\varphi }\) :
-
Moment of inertia of the lever amplifier along the Z-axis
- L :
-
Bearing cartridge axial length (m)
- \(M_{z\mathrm{equal}}\) :
-
Equivalent moment of the lever amplifier (N m)
- \(M_{\xi \_U}\) :
-
Moments in \(\xi \) directions caused by unbalanced (N m)
- \(M_{\psi \_U}\) :
-
Moments in \(\psi \) directions caused by unbalanced (N m)
- \(M_{\xi \_\mathrm{JB}}\) :
-
Moments in \(\xi \) directions caused by the journal ABFB and the journal GFB(N m)
- \(M_{\psi \_\mathrm{JB}}\) :
-
Moments in \(\psi \) directions caused by the journal ABFB and the journal GFB(N m)
- \(O_\mathrm{B}\) :
-
Bearing center
- \(O_\mathrm{J}\) :
-
Journal center
- \(\overline{p}\) :
-
Dimensionless pressure (\(=p/p_{a}\))
- R :
-
Journal radius (m)
- \({R}'\) :
-
Radius of bearing inner surface (m)
- t :
-
Time (s)
- \(\overline{t}\) :
-
Dimensionless time (\(=\upsilon t\))
- \(t_{2}\) :
-
Width of flexure hinge B (m)
- U :
-
Voltage on PZTs (V)
- x, y, z :
-
Horizontal, vertical, and axial coordinate
- \(\overline{z}\) :
-
Dimensionless axial coordinate (\(=z/R\))
- \(Z^{A}\) :
-
Distance between center of bearing A to end surface of rotor
- \(Z^{B}\) :
-
Distance between center of bearing B to end surface of rotor
- \(Z_\mathrm{G}\) :
-
Distance between rotor CG to end surface of rotor
- \(\upsilon \) :
-
Excitation frequency (rad/s)
- \(\overline{\delta }\) :
-
Dimensionless deformation of the top foil
- \(\varepsilon \) :
-
Eccentricity ratio
- \(\xi \) :
-
Rotational conical motion of rotor around the X-axis
- \(\psi \) :
-
Rotational conical motion of rotor around the Y-axis
- \(\varphi \) :
-
Rotational deformation angle of flexure hinge B (rad)
- \(\Delta u_{p}\) :
-
Actual deformation of the electrified PZT(m)
- \(\Delta u_{p0}\) :
-
Initial compression of the equivalent spring for the electrified PZT (m)
- \(\theta \) :
-
Circumferential coordinate of calculation domain (rad)
- \(\theta _{\min }\) :
-
Attitude angle of rotor center (rad)
- \(\theta _{n}\) :
-
The center angle position of the nth lever amplifiers (rad)
- \(\theta _{p}\) :
-
Pitch angle between two lever amplifiers (rad)
- \(\theta _{s}\) :
-
Circumferential span of a lever amplifier (rad)
- \(\Lambda \) :
-
Bearing number (\(=\frac{6\mu \omega }{p_{a}}(\frac{R}{C})^{2}\))
- \(\tau \) :
-
Time increment (s)
- \(\tau _{\mathrm{s}}\) :
-
Shear stress of the gas film (Pa)
- \(\mu \) :
-
Air viscosity (Pa s)
- \(\omega \) :
-
Rotating angular speed (rad/s)
- \(\omega _\mathrm{e}\) :
-
Excitation frequency (rad/s)
- \(\eta \) :
-
Structural loss factor of the bump foil
- i:
-
Station sequence number along the circumferential direction
- G:
-
Represents the gravity center of rotor
- n :
-
Sequence number of the lever amplifier
- A :
-
Represents the bearing A
- B :
-
Represents the bearing B
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Acknowledgements
This work was supported by the National Key R&D Program of China (2018YFB2000100), the National Natural Science Foundation of China (51875185), and the Foundation of Hunan Province (2018JJ1006).
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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
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.
Appendix B: Static load tests and parameters of tested ABFB and GFB
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.
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Guan, HQ., Feng, K., Yu, K. et al. Nonlinear dynamic responses of a rigid rotor supported by active bump-type foil bearings. Nonlinear Dyn 100, 2241–2264 (2020). https://doi.org/10.1007/s11071-020-05608-4
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DOI: https://doi.org/10.1007/s11071-020-05608-4