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Investigations on the nonlinear dynamic characteristics of a rotor supported by porous tilting pad bearings

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Abstract

Porous tilting pad bearings (PTPBs) combine the advantages of tilting pad and porous gas bearings and can be used in rotary motion and precision machine tools. This research mathematically investigates the dynamic characteristics of a rotor supported by PTPBs. A nonlinear rotor dynamics model that considers the unsteady gas film as well as translational and angular gyroscopic motions is established in this paper. The predicted results agree well with the experimental data when the rotor accelerates to ~ 30 krpm. Subsynchronous responses with large vibration amplitude are observed at high rotational speeds. The subsynchronous vibrations are related to both the fluid circumferential average velocity and eccentricity ratio. Meanwhile, the nonlinear dynamic responses of the system are analyzed by using waterfall plots, fast Fourier transforms, rotor orbits, Poincaré maps, and bifurcation diagrams. The results suggest that the adaptive motions of pads can enhance the stability of the system and increase the onset speed of the instability. Given that the eccentricity ratio increases along with the rotor mass (within a certain scale) and the pressure ratio of unload pads, the system stability is enhanced at the same time. Both the preload of PTPBs and pad installation can influence the system stability. Those PTPBs without preload and those pads with loads in between can also benefit the stability of the system. The findings of this work can help designers avoid subsynchronous vibrations at high rotational speeds.

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Abbreviations

\( W \) :

Pad width (m)

\( L \) :

Pad axial length (m)

\( r_{0} ,\;r_{\text{d}} \) :

Pad radius and rotor diameter (m)

\( \varTheta_{\text{p}} \) :

Pad arc angle (rad)

\( \theta_{s} ,\;\theta_{e} \) :

Pad leading edge and pad trailing edge (rad)

\( \theta_{\text{p}} \) :

Angular position of pivot (rad)

\( r_{\text{p}} \) :

The preload of the pads

\( \xi \) :

Pad tilting angle (rad)

\( I_{\text{pad}} \) :

Moment of inertia of pad (kg m2)

\( p_{\text{a}} ,\;p_{\text{s}} \) :

Ambient and supply pressures (pa)

\( p \) :

Pressure (pa)

\( P \) :

Dimensionless pressure (\( p/p_{\text{a}} \))

\( r,\;\theta ,\;z \) :

Coordinates in the \( r \), \( \theta \), and \( z \) directions

\( R,\;Z \) :

Dimensionless coordinates in the \( r \) and \( z \) directions

\( C \) :

Nominal clearance (m)

\( h,\;H \) :

Film thickness (m) and dimensionless film thickness (\( h/C \))

\( e_{x} ,\;e_{y} \) :

Components of rotor eccentricity

\( \omega \) :

Angular velocity of shaft (rad/s)

\( m_{\text{r}} \) :

Mass flow rates in the \( r \) direction

\( x,\;y \) :

x and y axes

\( F_{x} ,\;F_{y} \) :

Forces acting on shaft along the x and y axes caused by gas pressure (N)

\( M_{\xi } \) :

Tilting moment (N m)

\( K_{\xi } \) :

Tilting stiffness of the pivot (N m/rad)

\( \eta \) :

Porosity of porous materials

\( k \) :

Permeability of porous materials (m2)

\( \mu ,\;\rho \) :

Gas viscosity (Pa s) and density (kg/m3)

\( \Re \) :

Air gas constant (J/kg K)

\( T \) :

Temperature of the supply gas (K)

\( m_{s} \) :

Rotor mass (kg)

\( F_{bx} ,\;F_{by} \) :

Dynamic forces of PTPBs in x- and y- directions (N)

\( F_{ux} ,\;F_{uy} \) :

Imbalance forces in x- and y- directions (N)

\( I_{\text{t}} \) :

Translational torque of the rotor inertia (kg m2)

\( I_{\text{p}} \) :

Polar moment of the rotor inertia (kg m2)

\( M_{{{\text{b}}\xi }} ,\;M_{{{\text{b}}\psi }} \) :

Rotor rotational moments caused by bearings

\( M_{u\xi } ,\;M_{u\psi } \) :

Rotor rotational moments caused by imbalance mass of the rotor (N m)

\( Z_{{u\_{\text{left}}}} ,\;Z_{{u\_{\text{right}}}} \) :

The axial distances between the imbalance locations and the rotor mass center (m)

\( Z_{{b\_{\text{lef}}t}} ,\;Z_{{b\_{\text{right}}}} \) :

The axial distances between the PTPB locations and the rotor mass center (m)

\( \phi_{\text{left}} ,\;\phi_{\text{right}} \) :

The imbalance phase angles at the different sides of the rotor (rad)

\( M_{\text{left}} \) :

The imbalance mass at the left side of the rotor (kg)

\( U_{\text{left}} \) :

Imbalance radius at the left side of the rotor (m)

\( \omega_{\text{ins}} \) :

Onset speed of the instability

\( \lambda \) :

Fluid circumferential average velocity ratio

\( \omega_{\text{sys}} \) :

System natural frequency (Hz)

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Acknowledgements

The authors acknowledge the financial support from 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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Authors and Affiliations

  1. State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha, 410082, China

    Yihua Wu, Mingwei Deng, Kai Feng, Hanqing Guan & Yuanlong Cao

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  1. Yihua Wu
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Correspondence to Kai Feng.

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Wu, Y., Deng, M., Feng, K. et al. Investigations on the nonlinear dynamic characteristics of a rotor supported by porous tilting pad bearings. Nonlinear Dyn 100, 2265–2286 (2020). https://doi.org/10.1007/s11071-020-05652-0

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