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Non-linear dynamic analysis of a HSFD mounted gear-bearing system

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Abstract

An investigation is carried out on the systematic analysis of the dynamic behavior of the hybrid squeeze-film damper (HSFD) mounted a gear-bearing system with strongly non-linear oil-film force and gear meshing force in the present study. The dynamic orbits of the system are observed using bifurcation diagrams plotted using the dimensionless unbalance coefficient, damping coefficient and the dimensionless rotating speed ratio as control parameters. The non-dimensional equations of the gear-bearing system are solved using the fourth order Runge-Kutta method. The onset of chaotic motion is identified from the phase diagrams, power spectra, Poincaré maps, bifurcation diagrams, maximum Lyapunov exponents and fractal dimension of the gear-bearing system. The results presented in this study provide some useful insights into the design and development of a gear-bearing system for rotating machinery that operates in highly rotating speed and highly non-linear regimes.

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Abbreviations

B :

bearing parameter = \(\frac{6\mu R^{2}L^{2}}{m\delta ^{3}\omega _{n}}\)

c :

radial clearance, c=R−r

C :

damping coefficient of the gear mesh

d :

viscous damping of disk

D :

\(\frac{d}{m\omega _{n}}\)

e :

static transmission error and varies as a function of time

f x ,f y :

components of the fluid film force in horizontal and vertical coordinates

F r ,F t :

components of the fluid film force in radial and tangential directions

h :

oil-film thickness, h=δ(1+εcos θ)

k :

stiffness of the retaining springs

K :

stiffness coefficient of the gear mesh

k d :

proportional gain of PD controller

k p :

derivative gain of PD controller

L :

bearing length

m :

masses lumped at the mid-point

m p :

mass of the pinion

m g :

mass of the gear

O g :

center of gravity of the gear

O p :

center of gravity of the pinion

O b ,O j :

geometric center of the bearing and journal

p(θ):

pressure distribution in the fluid film

p s :

pressure of supplying oil

p c,i :

pressure in the static pressure chamber

Q in,i :

the volumetric flow rate into oil chamber i (i=1–4) from the controllable orifice

R :

inner radius of the bearing housing

r :

radius of the journal

r,t:

radial and tangent coordinates

s :

speed parameter = \(\frac{\omega}{\omega _{n}}\); \(\omega_{n}=\sqrt{\frac{k}{m}}\)

U :

\(\frac{\rho}{\delta}\)

x,y,z:

horizontal, vertical and axial coordinates

x0,y0:

damper static displacements

x j ,y j :

X j /c, Y j /c, j=p,g

ρ :

mass eccentricity

φ :

rotational angle (φ=ω t)

ω :

rotating speed of the shaft

φ b :

angle displacement of line O b O j from the x-coordinate (see Fig. 1)

Ω :

\(\dot{\varphi}_{b}\)

δ :

radial clearance = R−r,

θ :

the angular position along the oil film from line O b O j (see Fig. 1)

θ 1 :

the rotating angle of gear

θ 2 :

the rotating angle of pinion

μ :

oil dynamic viscosity

ε 0 :

\(\sqrt{X_{0}^{2}+Y_{0}^{2}}\)

β :

distribution angle of static pressure region

(⋅), (′):

derivatives with respect to t and φ

References

  1. Holmes, A.G., Ettles, C.M., Mayes, I.W.: Aperiodic behavior of a rigid shaft in short journal bearings. Int. J. Numer. Method Eng. 12, 695–702 (1978)

    Article  MATH  Google Scholar 

  2. Nikolajsent, J.I., Holmes, R.: Investigation of squeeze-film isolators for the vibration control of a flexible rotor. ASME J. Mech. Sci. 21(4), 247–252 (1979)

    Google Scholar 

  3. Sykes, J.E.H., Holmes, R.: The effect of bearing misalignment on the non-linear vibration of aero-engine rotor-damper assemblies. Proc. Inst. Mech. Eng. 204, 83–99 (1990)

    Google Scholar 

  4. Kim, Y.B., Noah, S.T.: Bifurcation analysis of a modified jeffcot rotor with bearing clearances. Nonlinear Dyn. 1, 221–241 (1990)

    Article  Google Scholar 

  5. Ehrich, F.F.: Some observations of chaotic vibration phenomena in high-speed rotordynamics. ASME J. Vib. Acoust. 113, 50–57 (1991)

    Article  Google Scholar 

  6. Zhao, J.Y., Linnett, I.W., Mclean, L.J.: Subharmonic and quasi-periodic motion of an eccentric squeeze film damper-mounted rigid rotor. ASME J. Vib. Acoust. 116, 357–363 (1994)

    Article  Google Scholar 

  7. Brown, R.D., Addison, P., Chan, A.H.C.: Chaos in the unbalance response of journal bearings. Nonlinear Dyn. 5, 421–432 (1994)

    Article  Google Scholar 

  8. Adiletta, G., Guido, A.R., Rossi, C.: Chaotic motions of a rigid rotor in short journal bearings. Nonlinear Dyn. 10, 251–269 (1996)

    Article  Google Scholar 

  9. Adiletta, G., Guido, A.R., Rossi, C.: Nonlinear dynamics of a rigid unbalanced rotor in short bearings. Part I: Theoretical analysis. Nonlinear Dyn. 14, 57–87 (1997)

    Article  MATH  Google Scholar 

  10. Adiletta, G., Guido, A.R., Rossi, C.: Nonlinear dynamics of a rigid unbalanced rotor in short bearings. Part II: Experimental analysis. Nonlinear Dyn. 14, 157–189 (1997)

    Article  Google Scholar 

  11. Sundararajan, P., Noah, S.T.: Dynamics of forced nonlinear systems using shooting/arc-length continuation method—Application to rotor systems. ASME J. Vib. Acoust. 119, 9–20 (1997)

    Article  Google Scholar 

  12. Chang-Jian, C.W., Chen, C.K.: Chaos and bifurcation of a flexible rub-impact rotor supported by oil film bearings with non-linear suspension. Mech. Mach. Theory 42(3), 312–333 (2007)

    Article  MATH  Google Scholar 

  13. Chang-Jian, C.W., Chen, C.K.: Bifurcation and chaos of a flexible rotor supported by turbulent journal bearings with non-linear suspension. Trans. IMechE Part J.: J. Eng. Tribol. 220, 549–561 (2006)

    Article  Google Scholar 

  14. Chang-Jian, C.W., Chen, C.K.: Nonlinear dynamic analysis of a flexible rotor supported by micropolar fluid film journal bearings. Int. J. Eng. Sci. 44, 1050–1070 (2006)

    Article  Google Scholar 

  15. Chang-Jian, C.W., Chen, C.K.: Bifurcation and chaos analysis of a flexible rotor supported by turbulent long journal bearings. Chaos Solitons Fractals 34, 1160–1179 (2007)

    Article  Google Scholar 

  16. Chang-Jian, C.W., Chen, C.K.: Nonlinear numerical analysis of a flexible rotor equipped with squeeze couple stress fluid film journal bearings. Acta Mech. Solida Sin. 20(4), 283–290 (2007)

    Google Scholar 

  17. Chen, C.L., Yau, H.T.: Subharmonic and chaotic motion of a hybrid squeeze-film damper-mounted rigid rotor with active control. ASME J. Vib. Acoust. 124, 198–208 (2002)

    Article  Google Scholar 

  18. Li, Z.P., Luo, Y.G., Yao, H.L., Wen, B.C.: Nonlinear dynamic study of the elastic rotor-bearing systems with rub-impact fault. J. Northeast. Univ. (Nat. Sci.) 23, 980–983 (2002)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mechanical and Automation Engineering, I-Shou University, 1, Section 1, Hsueh-Cheng Rd., Ta-Hsu Hsiang, Kaohsiung County, Taiwan, 840, ROC

    Cai-Wan Chang-Jian

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  1. Cai-Wan Chang-Jian
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Chang-Jian, CW. Non-linear dynamic analysis of a HSFD mounted gear-bearing system. Nonlinear Dyn 62, 333–347 (2010). https://doi.org/10.1007/s11071-010-9720-8

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  • DOI: https://doi.org/10.1007/s11071-010-9720-8

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