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
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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