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Analysis of Rotor Supported in Double-Layer Porous Journal Bearing with Gyroscopic Effects

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Part of the Lecture Notes in Mechanical Engineering book series (LNME)

Abstract

In this paper, a numerical analysis on the dynamics of a multi-degree of freedom shaft–rotor, supported on bearings, is presented. The system is a shaft with multiple rotor discs attached to it and supported on double-layer porous journal bearings. The system is modelled using finite element methods. Euler-Bernoulli beam element theory is used for modelling the shaft. The discs are considered as rigid. The support bearings are modelled based on linear spring elements for stiffness and linear damping elements for viscous damping coefficients. The rotor dynamic model of the system is analysed by incorporating the gyroscopic effects due to the precession of the offset discs and the bearing stiffness and damping anisotropy. The fluid flow in double-layer porous film is analysed using Brinkman equations to consider lubricant additives influences. The pressure gradients with respect to linearized perturbation of displacements and velocities under dynamic conditions are derived using Reynolds modified equation for Ocvirk (short) bearing. The dynamic linear and cross-coupled coefficients (stiffness and damping) dependent on speed are calculated using dynamic pressure gradients for the double-layer porous journal bearings. The system is represented in reduced order state-space form, and eigen value problem is solved to calculate its whirl frequencies. The rotor system critical speeds are obtained by plotting the Campbell diagram. This paper provides the basis for rotor system design with support bearings, representative of a multi-stage centrifugal pump. The design helps to identify and prevent rotor vibrations.

Keywords

  • Shaft–rotor-bearing system
  • Dynamic coefficients
  • Campbell diagram
  • Critical speeds

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Abbreviations

C :

Damping

cij, Cij:

Damping coefficients, Ns/m; \({{C_{ij} ,c_{ij} C^{ 3} } \mathord{\left/ {\vphantom {{C_{ij} ,c_{ij} C^{ 3} } {\eta R^{3} L}}} \right. \kern-0pt} {\eta R^{3} L}}\); for i = x, y

Cijl, Cijh:

Nondimensional damping coefficients of double-layer porous and homogeneous layer

C :

Radial clearance, m

E :

Young’s modulus of the shaft material

g :

Gyroscopic moment

h, H:

Film thickness, m; H = h/C

I a :

Shaft area moment of inertia

I d :

Mass moment of inertia of disc

I pd :

Polar mass moment of inertia of disc

I p :

Shaft polar mass moment of inertia

k :

Stiffness

k i :

Permeability of layer in porous regions, m2; Ki = ki/h2; for i = 1, 2

kij, Kij:

Stiffness coefficients evaluated at equilibrium position, N/m; \({{K_{ij} = k_{ij} C^{3} } \mathord{\left/ {\vphantom {{K_{ij} = k_{ij} C^{3} } {\eta \omega R^{3} L}}} \right. \kern-0pt} {\eta \omega R^{3} L}}\); for i = x, y

Kijl, Kijh:

Nondimensional stiffness coefficients of double-layer porous and homogeneous layer

l :

Length of the shaft element

lʹ:

Shaft length

M :

Moments

m d :

Mass of discs

r :

Shaft radius

R :

Journal radius, m

w :

Static load capacity, N; \(W = {{wC^{ 2} } \mathord{\left/ {\vphantom {{wC^{ 2} } {\eta \omega R^{3} L}}} \right. \kern-0pt} {\eta \omega R^{3} L}}\)

Wl, Wh:

Double porous and homogeneous layer nondimensional load capacity

Y, x :

Vertical and horizontal coordinates, m; Y = y/C, X = x/C

\({\dot{Y}},\dot{X}\) :

Journal centre velocity (nondimensional) in y and x direction

ρ s :

Density of shaft material

ε :

Journal bearing eccentricity ratio

γ i :

Porous layer thickness ratio; γi = i/h; for i = 1, 2

η:

Fluid dynamic viscosity, Ns/m2

μ :

Mass of the shaft per unit length

θ:

Coordinate from maximum film thickness

ϴ:

Coordinate from load line

ϕ :

Attitude angle

ω :

Shaft spin frequency

s:

Shaft

e:

Element

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Acknowledgements

The authors appreciate the support of SRM Institute of Science and Technology.

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Correspondence to C. Shravankumar .

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Shravankumar, C., Jegadeesan, K., Rao, T.V.V.L.N. (2021). Analysis of Rotor Supported in Double-Layer Porous Journal Bearing with Gyroscopic Effects. In: Rao, J.S., Arun Kumar, V., Jana, S. (eds) Proceedings of the 6th National Symposium on Rotor Dynamics. Lecture Notes in Mechanical Engineering. Springer, Singapore. https://doi.org/10.1007/978-981-15-5701-9_6

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  • DOI: https://doi.org/10.1007/978-981-15-5701-9_6

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