Abstract
A nonlinear time-varying dynamic model for right-angle gear pair systems, considering both backlash and asymmetric mesh effects, is formulated. The mesh parameters that are characteristically time-varying and asymmetric include mesh stiffness, directional rotation radius and mesh damping. The period-one dynamic motions are obtained by solving the dimensionless equation of gear motion using an enhanced multi-term harmonic balance method (HBM) with a modified discrete Fourier Transform process and the numerical continuation method. The accuracy of the enhanced HBM solution is verified by comparison of its results to the more computational intensive, direct numerical integration calculations. Also, the Floquet theory is applied to determine the stability of the steady-state harmonic balance solutions. Finally, a set of parametric studies are performed to determine quantitatively the effects of the variation and asymmetry in mesh stiffness and directional rotation radius on the gear dynamic responses.
Article PDF
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.Avoid common mistakes on your manuscript.
Abbreviations
- b :
-
gear backlash
- c :
-
mesh damping coefficient
- e :
-
static transmission error
- f :
-
nonlinear displacement
- F :
-
forcing function
- h :
-
arc-length
- I p ,I g :
-
Pinion and gear inertias
- :
-
unit vector along shaft axis
- J :
-
Jacobian matrix
- k :
-
mesh stiffness function
- m e :
-
equivalent mass function
- :
-
line of action unit vector
- :
-
mesh point
- r :
-
mean value ratio
- s l :
-
pinion/gear coordinate system
- t :
-
time
- T p ,T g :
-
load on pinion and gear
- x :
-
δ−e
- Δx :
-
perturbation of x
- y :
-
perturbation state vector
- δ :
-
dynamic transmission error
- λ :
-
directional rotation radius
- ω :
-
frequency
- ς :
-
damping ratio
- θ :
-
rotational displacement
- c :
-
coast side
- d :
-
drive side
- da :
-
damping
- k :
-
label for stiffness
- l :
-
pinion (l=p) and gear (l=g)
- m :
-
drive (m=d) and coast (m=c) sides
- n :
-
natural mode
- 1:
-
mean value
- 2,3,…:
-
alternative value
- •:
-
derivative w.r.t. time
- ′:
-
derivative w.r.t. dimensionless time
- →:
-
vector quantities
- ∼:
-
dimensionless quantities
References
Comparin, R.J., Singh, R.: Non-linear frequency response characteristics of an impact pair. J. Sound Vib. 134(2), 259–290 (1989)
Kahraman, A., Singh, R.: Non-linear dynamics of a spur gear pair. J. Sound Vib. 142(1), 49–75 (1990)
Kahraman, A., Singh, R.: Interactions between time-varying mesh stiffness and clearance non-linearity in a geared system. J. Sound Vib. 146(1), 136–156 (1991)
Blankenship, G.W., Kahraman, A.: Steady state forced response of a mechanical oscillator with combined parametric excitation and clearance type non-linearity. J. Sound Vib. 185(5), 743–765 (1995)
Kahraman, A., Blankenship, G.W.: Interaction between commensurate parametric and forcing excitations in a system with clearance. J. Sound Vib. 194(3), 317–336 (1996)
Theodossiades, S., Natsiavas, S.: Non-linear dynamics of gear-pair systems with periodic stiffness and backlash. J. Sound Vib. 229(2), 287–310 (2000)
Ma, Q., Kahraman, A.: Period-one motions of a mechanical oscillator with periodically time-varying, piecewise non-linear stiffness. J. Sound Vib. 284, 893–914 (2005)
Ma, Q., Kahraman, A.: Subharmonic resonances of a mechanical oscillator with periodically time-varying, piecewise non-linear stiffness. J. Sound Vib. 294, 624–636 (2006)
Al-shyyab, A., Kahraman, A.: Non-linear dynamic analysis of a multi-mesh gear train using multi-term harmonic balance method: period-one motions. J. Sound Vib. 284, 151–172 (2005)
Al-shyyab, A., Kahraman, A.: Non-linear dynamic analysis of a multi-mesh gear train using multi-term harmonic balance method: sub-harmonic motions. J. Sound Vib. 279, 417–451 (2005)
Kahraman, A., Singh, R.: Non-linear dynamics of a geared rotor-bearing system with multiple clearances. J. Sound Vib. 144(3), 469–506 (1991)
Al-shyyab, A., Kahraman, A.: A non-linear dynamic model for planetary gear sets. Proc. Inst. Mech. Eng., Part K: J. Multi-Body Dyn. 221(4), 567–576 (2007)
Maatar, M., Velex, P.: An analytical expression for time-varying contact length in perfect cylindrical gears: some possible applications in gear dynamics. J. Mech. Des. 118, 586–589 (1996)
Ajim, M., Velex, P.: A model for simulating the quasi-static and dynamic behavior of solid wide-faced spur and helical gears. Mech. Mach. Theory 40, 173–190 (2005)
Ozguven, H.N., Houser, D.R.: Mathematical models used in gear dynamics—a review. J. Sound Vib. 121(3), 383–411 (1988)
Kiyono, S., Fujii, Y., Suzuki, Y.: Analysis of vibration of bevel gears. Bull. JSME 14, 441–446 (1981)
Abe, E., Hagiwara, H.: Advanced method for reduction in axle gear noise, Gear Design, Manufacturing and Inspection Manual (Society of Automotive Engineerings, Warrendale, PA), 223–236 (1990)
Lim, T.C., Cheng, Y.: A theoretical study of the effect of pinion offset on the dynamics of hypoid geared rotor system. J. Mech. Des. 121, 594–601 (1999)
Cheng, Y., Lim, T.C.: Vibration analysis of hypoid transmissions applying an exact geometry-based gear mesh theory. J. Sound Vib. 240, 519–543 (2001)
Cheng, Y., Lim, T.C.: Dynamic analysis of high speed hypoid gears with emphasis on the automotive axle noise problem. In: Proceeding of the ASME Power Transmission and Gearing conference, DETC98/PTG-5784, Atlanta, GA (1998)
Cheng, Y., Lim, T.C.: Dynamics of hypoid gears transmission with time-varying mesh. In: Proceeding of the ASME Power Transmission and Gearing conference, DETC2000/PTG-14432, Baltimore, MD (2000)
Cheng, Y.: Dynamics of high-speed hypoid and bevel geared rotor systems. Ph.D. dissertation, Ohio State University (2000)
Jiang, X.: Non-linear torsional dynamic analysis of hypoid gear pairs. M.S. thesis, University of Alabama (2002)
Wang, H.: Gear mesh characteristics and dynamics of hypoid gear rotor system. Ph.D. dissertation, University of Alabama (2002)
Wang, J., Lim, T.C., Li, M.: Dynamics of a hypoid gear pair considering the effects of time-varying mesh parameters and backlash nonlinearity. J. Sound Vib. 308, 302–329 (2007)
Wang, J., Lim, T.C.: Effect of tooth mesh stiffness asymmetric nonlinearity for drive and coast sides on hypoid gear dynamics. J. Sound Vib. 319, 885–903 (2009)
Kim, T.C., Rook, T.E., Singh, R.: Super- and sub-harmonic response calculations for a torsional system with clearance nonlinearity using harmonic balance method. J. Sound Vib. 281, 965–993 (2005)
Allgower, E.L., Georg, K.: Numerical Continuation Methods: An Introduction. Springer, New York (1990)
Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley, New York (1979)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Yang, J., Peng, T. & Lim, T.C. An enhanced multi-term harmonic balance solution for nonlinear period-one dynamic motions in right-angle gear pairs. Nonlinear Dyn 67, 1053–1065 (2012). https://doi.org/10.1007/s11071-011-0048-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-011-0048-9