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Nonlinear analysis of a rub-impact rotor supported by turbulent couple stress fluid film journal bearings under quadratic damping

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Abstract

This work reports a numerical study undertaken to investigate the dynamic response of a rotor supported by two turbulent flow model journal bearings with nonlinear suspension and lubricated with couple stress fluid under quadratic damping. This may be the first time that analysis of rotor-bearing system considered the quadratic damping effect. The dynamic response of the rotor center and bearing center are studied. The analysis methods employed in this study are inclusive of the dynamic trajectories of the rotor center and bearing center, power spectra, Poincaré maps and bifurcation diagrams. The maximum Lyapunov exponent analysis is also used to identify the onset of chaotic motion. The modeling results provide some useful insights into the design and development of rotor-bearing system for rotating machinery that operates at highly rotational speed and highly nonlinear regimes.

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Abbreviations

b :

velocity-related parameter

c :

radial clearance, c=R−r

c p :

dimensionless parameter, \(c_{p}=\frac{k_{s}}{k_{1}}\)

C 1 :

damping coefficient of supported structure

C 2 :

viscous damping of rotor disk

e :

eccentricity, \(e=\sqrt{X^{2}+Y^{2}}\)

f :

friction coefficient between rotor and stator

f e ,f φ :

components of fluid film force in radial and tangential directions

f r ,f t :

resulting bearing forces about the journal center in the radial and tangential directions

f1,f2:

radial impact force and tangential rub force

F x ,F y :

components of fluid film force in X- and Y-directions

R x ,R y :

rub-impact forces in the horizontal and vertical directions

G θ ,G z :

\(\frac{1}{G_{\theta}}=12+0.0260(\mathit{Re}^{*})^{0.8265},\frac{1}{G_{z}}=12+0.0198(\mathit{Re}^{*})^{0.741}\)

g :

acceleration of gravity

K1,K2:

stiffness coefficients of springs supporting bearing housings

K s :

stiffness coefficient of shaft

k c :

radial stiffness of the stator

l :

characteristic length of additives, \(l=(\frac{\eta}{\mu})^{1/2}\)

l * :

dimensionless couple-stress parameter, l *=l/c

L:

bearing length

m,m0:

masses lumped at rotor mid-point and bearing housing mid-point

O m :

center of rotor gravity

O1,O2,O3:

geometric centers of bearing, rotor and journal

p :

pressure distribution in fluid film

R :

inner radius of bearing housing

Re * :

local Reynolds number, \(\mathit{Re}^{*}=\frac{\rho Uh}{\mu}\)

r :

radius of journal

s :

rotational speed ratio, \(s=(\frac{\omega ^{2}}{\omega _{n}^{2}})^{1/2}\)

s 1 :

dimensionless parameter, s 1=(c om c p s 2)1/2

v :

relative slip velocity between rotor and stator, \(v=\sqrt{\dot{x}^{2}+\dot{y}^{2}}\)

X,Y,Z:

horizontal, vertical and axial coordinates x 1,y 1,x 2,y 2 X 1/c,Y 1/c,X 2/c,Y 2/c

α :

dimensionless parameter, \(\alpha =\frac{k_{2}c^{2}}{k_{s}C_{om}}\)

ρ :

mass eccentricity of rotor

φ :

rotational angle (φ=ωt)

ω :

rotational speed of rotor

φ :

attitude angle

θ :

angular position

μ :

oil dynamic viscosity

η :

new material constant peculiar to couple-stress fluid

ε :

eccentricity ratio, ε=e/c

δ :

clearance between rotor and stator

ω n :

natural frequency, (k s /m)1/2

β :

dimensionless unbalance parameter, ρ/c

ξ 1 :

dimensionless parameter, \(\xi_{1}=\frac{c_{1}}{2\sqrt{k_{1}m_{0}}}\)

ξ 2 :

dimensionless parameter, \(\xi_{2}=\frac{c_{2}}{2\sqrt{K_{s}m}}\)

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Authors and Affiliations

  1. Department of Mechanical and Automation Engineering, I-Shou University, Kaohsiung County, 840, Taiwan, R.O.C.

    Cai-Wan Chang-Jian

  2. Department of Mechanical Engineering, National Cheng-Kung University, Taiwan, 70101, Taiwan, R.O.C.

    Cha’o-Kuang Chen

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  1. Cai-Wan Chang-Jian
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Correspondence to Cai-Wan Chang-Jian.

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Chang-Jian, CW., Chen, CK. Nonlinear analysis of a rub-impact rotor supported by turbulent couple stress fluid film journal bearings under quadratic damping. Nonlinear Dyn 56, 297–314 (2009). https://doi.org/10.1007/s11071-008-9400-0

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