# Order reduction and bifurcation analysis of a flexible rotor system supported by a full circular journal bearing

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## Abstract

Journal bearing has been widely used in the wide range of rotating machineries to support large loads and to add significant damping to the system. Conventionally, its fluid film force is represented by the linear model of spring and damper around its equilibrium position in the vibration analysis of the rotor system. However, the fluid film force of the journal bearing is essentially nonlinear, and it is necessary to consider its nonlinearity to expect the characteristics of the limit cycle at around the instability point. This paper investigated the order reduction and bifurcation analysis of a flexible rotor system supported by a full circular journal bearing. The fluid film force is derived under the conditions of both infinitesimal length approximation and half-Zommerfeld boundary condition, and its polynomial function approximation expression is used. Order reduction in the FEM rotor model retaining the nonlinearity of the journal bearing was performed by utilizing the substructure synthesis method. Then, its bifurcation phenomena at around the instability point are investigated by applying the center manifold theory and using the normal form theory. The influences of various parameters, such as kinematic viscosity, bearing length, and the disk position, on the bifurcation phenomena at around the instability point were investigated and explained. Furthermore, the validity of the derived analytical observation was confirmed by numerical simulation and experiment. By invoking these analytical techniques and obtained results, the bifurcation characteristics can be expected theoretically at the design stage of the journal bearing and rotor system.

## Keywords

Flexible rotor Journal bearing Nonlinear fluid film force Bifurcation analysis Order reduction Experimental vilification## List of symbols

- \(\wedge \)
Symbol designating dimensional value

- \(\hat{{d}}\)
Shaft cross-sectional diameter (m)

- \(\hat{{\omega }}\)
Angular rotational speed (rad/s)

- \(\hat{{X}},\,\hat{{Y}}\)
Horizontal and vertical coordinates denoted in Fig. 1

- \(\hat{{x}},\hat{{y}}\)
Horizontal and vertical displacements of the journal from the bearing center in vertical cross section (m)

- \(\dot{\hat{{x}}},\dot{\hat{{y}}}\)
Horizontal and vertical components of the velocity of the journal from the bearing center in vertical cross section (m/s)

- \(\hat{{F}}_{\mathrm{JB}x} \)
Fluid film force component in \(\hat{{X}}\) direction (N)

- \(\hat{{F}}_{\mathrm{JB}y} \)
Fluid film force component in \(\hat{{Y}}\) direction (N)

- \(\hat{{e}}\)
Eccentricity of the journal from the bearing center in vertical cross section (m)

- \(\varepsilon \)
Eccentricity ratio of the journal from the bearing center in vertical cross section (–)

- \(\beta \)
Angle of the eccentricity of the journal from the bearing center in vertical cross section in relation to horizontal direction (rad)

- \(\hat{{F}}_\varepsilon \)
Fluid film force component in \(\hat{{e}}\) direction (N)

- \(\hat{{F}}_\beta \)
Fluid film force component in \(\beta \) direction (N)

- \(\hat{{R}}\)
Bearing radius (m)

- \(\hat{{D}}\)
Bearing diameter \((=2\hat{{R}})\) (m)

- \(\hat{{r}}\)
Journal radius (m)

- \(\hat{{L}}\)
Bearing length (m)

- \(\hat{{\mu }}\)
Dynamic viscosity of the fluid (Pas)

- \(\hat{{\nu }}\)
Kinematic viscosity of the fluid (\(\hbox {mm}^{2}\)/s)

- \(\hat{{h}}\)
Radial gap between the journal and the bearing (m)

- \(\hat{{C}}_\mathrm{r} \)
Radial clearance of the journal bearing (m)

- \(\hat{{F}}_{\mathrm{JB}x0} \)
Fluid film force component in \(\hat{{X}}\) direction at the equilibrium position (N)

- \(\hat{{F}}_{\mathrm{JB}y0} \)
Fluid film force component in \(\hat{{Y}}\) direction at the equilibrium position (N)

- \(\hat{{k}}_{**} \)
Linear spring coefficients of fluid film force (N/m)

- \(\hat{{c}}_{**} \)
Linear damping coefficients of fluid film force (Ns/m)

- \(\hat{{\alpha }}_{x^{****}},\hat{{\alpha }}_{y^{****}} \)
Coefficient of the nonlinear term of fluid film force

- \(\hat{{l}}_{\mathrm{full}} \)
Total length of the rotor system (m)

- \(\hat{{l}}_*\)
Part length of the rotor system (m)

- \({\hat{\mathbf{q}}}\)
Displacement vector

- \({\hat{\mathbf{M}}}\)
Total mass matrix

- \({\hat{\mathbf{C}}}\)
Total damping matrix

- \({\hat{\mathbf{G}}}\)
Total gyro matrix

- \({\hat{\mathbf{K}}}\)
Total stiffness matrix

- \({\hat{\mathbf{F}}}\)
Total external force vector

- \(\mathbf{K}_\mathrm{s} \)
Stiffness matrix representing the shaft’s deflection

- \(\mathbf{K}_\mathrm{b} \)
Stiffness matrix representing the linear stiffness of the ball bearing

- \({\hat{\mathbf{F}}}_{\mathrm{mg}} \)
Total gravitational force vector

- \({\hat{\mathbf{F}}}_{\mathrm{un}} \)
Total unbalance force vector

- \({\hat{\mathbf{F}}}_{\mathrm{jb}} \)
Total journal bearing force vector

- \({\hat{\mathbf{q}}}_{\mathrm{eq}} (\hat{{\omega }})\)
Position vector of the equilibrium position

- \({\hat{\mathbf{Q}}}\)
Displacement vector from the equilibrium position

- \({\hat{\mathbf{F}}}_{\mathrm{jb}0} \)
Constant force component vector of the fluid film force

- \(\mathbf{F}_{\mathrm{jb}n} \)
Nonconstant component vector of total journal bearing reaction force

- \(\hat{{p}} *\)
Natural frequency of *th mode (rad/s)

- \(\hat{{\omega }}_\mathrm{b} \)
Angular velocity of bifurcation point (rad/s)

- \(\hat{{m}}_{\mathrm{jb}} \)
Component of the mass matrix \({\hat{\mathbf{M}}}\) corresponding to the element of the journal part

- \({\hat{\mathbf{q}}}_{\mathrm{st}} \)
Standard displacement vector for nondimensionalization

- \(\mathbf{Q}_\mathrm{B} \)
Displacement vector for boundary degree of freedom at the journal bearing in displacement vector \(\mathbf{Q}\)

- \(\mathbf{Q}_\mathrm{I} \)
Displacement vector for internal degree of freedom of the other shaft parts in displacement vector \(\mathbf{Q}\)

- \({\hat{\mathbf{M}}}_\mathrm{B} \)
Matrix for boundary degree of freedom at the journal bearing in total mass matrix \({\hat{\mathbf{M}}}\)

- \({\hat{\mathbf{C}}}_\mathrm{B} \)
Matrix for boundary degree of freedom at the journal bearing in total damping matrix \({\hat{\mathbf{C}}}\)

- \({\hat{\mathbf{G}}}_\mathrm{B} \)
Matrix for boundary degree of freedom at the journal bearing in total gyro matrix \({\hat{\mathbf{G}}}\)

- \({\hat{\mathbf{K}}}_\mathrm{B} \)
Matrix for boundary degree of freedom at the journal bearing in total stiffness matrix \({\hat{\mathbf{K}}}\)

- \(\mathbf{F}_{\mathrm{jb}n\mathrm{B}} \)
Vector for boundary degree of freedom at the journal bearing in nonconstant component vector of total journal bearing reaction force \(\mathbf{F}_{\mathrm{jb}n} \)

- \(\delta _\omega \)
Variation in the rotational speed \(\omega \) from the instability point \(\omega _\mathrm{b} (\omega =\omega _\mathrm{b} +\delta _\omega )\)

- \(\mathbf{K}_{\mathrm{linb}} \)
Linearized stiffness matrix of journal bearing reaction force \(\mathbf{F}_{\mathrm{jb}n\mathrm{B}} \)

- \(\mathbf{C}_{\mathrm{linb}} \)
Linearized damping matrix of journal bearing reaction force \(\mathbf{F}_{\mathrm{jb}n\mathrm{B}} \vspace{-1.8pt}\)

- \(\mathbf{N}^{\prime }\)
Nonlinear component vector at the instability point \(\omega _\mathrm{b} \)

- \(\mathbf{P}\)
Modal matrix consists of eigenvectors

- \(n_1 , n_2 \)
Nonlinear terms in the dynamical equation for the mode which contributes to the bifurcation at instability point

- \(\kappa _{****}\)
Coefficients of nonlinear terms of \(n_1 \) and \(n_2 \)

- \(r_\mathrm{f} \)
Analytical expression of the amplitude of the limit cycle due to Hopf bifurcation at instability point

## Notes

### Acknowledgements

The author would like to express their gratitude to Dr. Hitoshi Sakakida for his invaluable comments in the discussion regarding experiment.

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

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