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Stability and nonlinear dynamic analysis of gas-lubricated system with elastomer suspension

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Abstract

This paper presents the dynamic behavior of a rigid rotor supported by a pair of self-acting gas-lubricated bearings which is used in a turbo-expander, the key equipment in a large-scale cryogenic system. In order to restrict the vibration of the rotor relative to bearings, elastomers are mounted between bearings and shell. A finite difference method has been employed to solve the Reynolds equation in time-dependent states. The center orbits of the rotor are got by solving the motion equations. The system state trajectory, Poincare maps, logarithmic spectra maps and bifurcation diagrams are used to analyze the dynamic behavior of the system. The effect of the elastomer suspension to absorb the vibration is verified by comparison with the system with rigid suspension. A set of mass values, damping exponent values and stiffness of elastomer are calculated and compared in this paper. The results show that the rotor center loses its regular behavior gradually with the increase in the rotor mass. The square damping exponent model of the elastomer shows more stability than linear damping suspension model, and the quadratic damping exponent model has a similar motion behavior compared with a linear model. A suitable stiffness of the elastomer is important to the stability of the system. The elastomer with a low stiffness may cause the large amplitude of the vibration, and the system may lose its regular behavior when the stiffness is large enough.

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Abbreviations

\(A_{\mathrm{r}x},A_{\mathrm{r}y} \) :

Dimensionless horizontal and vertical acceleration of rotor

\(A_{\mathrm{b}x},A_{\mathrm{b}y} \) :

Dimensionless horizontal and vertical acceleration of bearing

Cr:

Average radial clearance between shaft and bearing (m)

D :

Diameter of the rotor (m)

\(e_{ub} \) :

Mass eccentricity of the rotor (m)

\(f_{\mathrm{g}x},f_{\mathrm{g}y} \) :

Supporting force components in horizontal and vertical directions (N)

\(f_{\mathrm{e}x},f_{\mathrm{e}y}\) :

External force components in horizontal and vertical directions (N)

\(f_\mathrm{d} \) :

Damping force of elastomer (N)

\(F_{\mathrm{g}x},F_{\mathrm{g}y} \) :

Dimensionless supporting force components in horizontal and vertical directions

F :

Defined as \(\sqrt{F_{\mathrm{g}x}^2 + F_{\mathrm{g}y}^2} \)

\(\varphi \) :

Defined as \(\tan ^{-1}(F_{\mathrm{e}x} /F_{\mathrm{e}y} )\)

\(F_{\mathrm{e}x}, F_{\mathrm{e}y} \) :

Dimensionless external force components in horizontal and vertical directions

H :

Dimensionless film thickness between shaft and bearing

h :

Film thickness (m)

k, c :

Stiffness (N/m) and damping (N s/m) of elastic damper

\(\gamma \) :

Damping exponent

K, C :

Dimensionless stiffness and damping of the damper

\(\xi \) :

Dimensionless damping coefficient of elastomer \(\xi =c/{2\sqrt{km_\mathrm{b} }}\)

L :

Length of the journal bearing (m)

\(m_\mathrm{r} \) :

One-half mass of rotor (kg)

\(m_\mathrm{b} \) :

Mass of bearing (kg)

\(M_\mathrm{r},M_\mathrm{b} \) :

Dimensionless mass of rotor and bearing

p :

Pressure \((\hbox {N}{/}\hbox {m}^{2})\)

P :

Dimensionless pressure

\(P_\mathrm{a} \) :

Atmosphere pressure \((\hbox {N}{/}\hbox {m}^{2})\)

R :

Radius of the shaft (m)

t :

Time (s)

\(\tau \) :

Dimensionless time, \(\varpi t\)

U :

Peripheral speed of the rotor (m/s)

\(\mu \) :

Fluid kinematic viscosity (Pa s)

u :

Deflection of the elastomer

\(\dot{u}\) :

Defined as \({\mathrm{d}u}{/}{\mathrm{d}t}\)

\(V_{\mathrm{r}x}, V_{\mathrm{r}y} \) :

Dimensionless horizontal and vertical velocity of the rotor

\(V_{\mathrm{b}x}, V_{\mathrm{b}y} \) :

Dimensionless horizontal and vertical velocity of bearing

\(\varpi \) :

Angular speed of the rotor (rad/s)

x, y :

Horizontal and vertical coordinates (m)

X, Y, Z :

Dimensionless displacement in Cartesian coordinate system

\(X_\mathrm{r},Y_\mathrm{r} \) :

Dimensionless horizontal and vertical displacement of the rotor

\(X_\mathrm{b} ,Y_\mathrm{b} \) :

Dimensionless horizontal and vertical displacement of bearing

\(X_\mathrm{rel} ,Y_\mathrm{rel} \) :

Dimensionless horizontal and vertical displacement of the rotor relative to the beating. (\(X_\mathrm{rel} =X_\mathrm{r} -X_\mathrm{b} ;Y_\mathrm{rel} =Y_\mathrm{r} -Y_\mathrm{b} )\)

\(\Lambda \) :

Bearing number

\(\varepsilon \) :

Eccentricity ratio of the system

\(\theta , \eta \) :

Dimensionless coordinate in circumferential and axial directions, respectively

n :

Time level

i, j :

Grid location in circumferential and axial directions

b :

Parameters of bearing

r :

Parameters of rotor

0:

Initial parameter

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Acknowledgements

This work is supported by the fund of National Research and Development Project for Key Scientific Instruments, ZDYZ2014-1; and the fund of the State Key Laboratory of Technologies in Space Cryogenic Propellants, SKLTSCP1604.

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Authors and Affiliations

  1. School of Energy Power and Engineering, Huazhong University of Science and Technology, Wuhan, 430074, Hubei, People’s Republic of China

    J. Li & Q. Li

  2. State Key Laboratory of Technologies in Space Cryogenic Propellants, Beijing, 100190, People’s Republic of China

    J. Li, Q. Li, S. Q. Yang, F. Yu & X. M. Li

  3. University of Chinese Academy of Sciences, Beijing, 100049, People’s Republic of China

    Q. Li, S. Q. Yang, F. Yu & X. M. Li

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  1. J. Li
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Correspondence to Q. Li.

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Li, J., Li, Q., Yang, S.Q. et al. Stability and nonlinear dynamic analysis of gas-lubricated system with elastomer suspension. Nonlinear Dyn 94, 2161–2176 (2018). https://doi.org/10.1007/s11071-018-4481-x

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