# Use of nonlinear journal-bearing impedance descriptions to evaluate linear analysis of the steady-state imbalance response for a rigid symmetric rotor supported by two identical finite-length hydrodynamic journal bearings at high eccentricities

## Abstract

This paper concerns the investigation of validity limits of linear models in predicting rotor trajectory inside the bearing clearance for a rigid symmetric rotor supported by two identical journal bearings operating at high eccentricities.

The inherent nonlinearity of hydrodynamic journal bearings becomes strong for eccentricities grater than 60% of the bearing clearance where most existing linear models are not able to accurately predict the rotor trajectory.

The usefulness of nonlinear journal-bearing impedance description method in this investigation is due to the analytical formulations of the linearised bearing coefficients, and the analytical nonlinear bearing models. These analytically derived bearing coefficients do not require any numerical differentiation (or integration) and are therefore more accurate for large eccentricities. The analytically derived nonlinear bearing models markedly decrease the simulation time while valid for all *L*/*D* (length to diameter ratios) and all eccentricities.

The results contained in this paper show that linear models derived from the nonlinear impedance descriptions of the Moes-cavitated (*π*-film) finite-length bearing can predict the steady-state imbalance response of a symmetric rigid rotor supported by two identical journal bearings at high eccentricities. This is, however, only the case when operating conditions are below the threshold speed of instability and when the system has period-one solutions. The error will become larger closer to the resonance speed.

## Keywords

Nonlinear model Linear model Journal bearing Impedance descriptions Dynamics Imbalance response Rotor## Abbreviations

*C*=*C*_{r}Bearing radial clearance [m]

*C*_{ij},*i*,*j*=*X*,*Y*Dimensional bearing damping coefficients [Ns/m]

*c*_{ij},*i*,*j*=*X*,*Y*Non-dimensional bearing damping coefficients [–]

*D*Bearing inner diameter [m]

*E*Rotor mass eccentricity [m]

*e*Journal (rotor) dimensional eccentricity [m]

*F*_{0}Half-static load (for a symmetrically loaded rotor) [N]

*F*_{x},*F*_{y}Bearing reaction force \(\hat{F}\) [N] components in the

*x*and*y*directions*F*_{X},*F*_{Y}Bearing reaction force \(\hat{F}\) [N] components in the

*X*and*Y*directions*g*Gravity acceleration [m/s

^{2}]*K*_{ij},*i*,*j*=*X*,*Y*Dimensional bearing stiffness coefficients [N/m]

*k*_{ij},*i*,*j*=*X*,*Y*Non-dimensional bearing stiffness coefficients [–]

- \(\hat{k}\)
Unit vector in

*z*and*Z*directions [–]*L*Bearing length [m]

*M*Rotor mass [kg]

*O*_{1}Bearing centre

*O*_{2}Rotor (journal) centre

*R*Bearing inner radius [m]

- rpm
Rounds per minute

*S*Sommerfeld number [–]

*t*Time [s]

- \(\hat{V}_{j}\)
Journal (rotor) velocity vector [m/s]

- \(\hat{V}_{S}\)
Journal’s pure-squeeze-velocity vector [m/s]

*V*_{S}Journal’s pure-squeeze-velocity magnitude [m/s]

*W*_{0}Bearing impedance \(\hat{W}\) magnitude at the static equilibrium position

*W*_{x},*W*_{y}Bearing impedance \(\hat{W}\) magnitude components in

*x, y*directions*x*,*y*,*z*Coordinate system with abscissa (

*x*-*axis*) fixed to the vector \(\hat{V}_{S}\), rotating with \(\bar{\omega}\) relative to rotor inertial coordinates*X*,*Y*,*Z**X*,*Y*Rotor displacement components in

*X*and*Y*directions [m]- \(\dot{X},\dot{Y}\)
Rotor velocity components in

*X*and*Y*directions- \(\ddot{X},\ddot{Y}\)
Rotor acceleration components in

*X*and*Y*directions- \(\bar{X},\bar{Y}\)
Rotor non-dimensional displacement components in

*X*and*Y*directions [–]- \(\bar{X}',\bar{Y}'\)
Rotor non-dimensional velocity components in

*X*and*Y*directions- \(\bar{X}'',\bar{Y}''\)
Rotor non-dimensional acceleration components in

*X*and*Y*directions*γ*Journal (rotor) attitude angle [radians]

*γ*_{0}Journal (rotor) attitude angle at static equilibrium position

*ε*Journal (rotor) non-dimensional eccentricity [–]

- \(\hat{\varepsilon}\)
Journal (rotor) non-dimensional eccentricity vector

*ε*_{0}Journal (rotor) non-dimensional eccentricity at static equilibrium position

*ς*Angle between

*x*and*X*[radians]*μ*Oil dynamic viscosity [Pa×s]

*ω*Rotor angular speed [rad/s]

- \(\bar{\omega}\)
\(\bar{\omega}=\frac{\omega}{2}\), the angular speed of

*x*,*y*,*z*coordinates relative to*X*,*Y*,*Z*[radians/s]

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## References

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