Theoretical analysis of the natural frequency of a geared system under the influence of a variable mesh stiffness

Abstract

Owing to the change in the number of contact tooth pairs, the mesh stiffness is periodically time varying, which leads to additional time-variant coefficients in the corresponding governing equations of a geared dynamic system. The natural frequency of such a system differs distinctly from that of a time-invariant system. However, it has not gained sufficient attention in geared dynamic analysis, and the natural frequency of a geared system is usually assumed to be the average frequency obtained using the time-average mesh stiffness. Whether it is reasonable to obtain the natural frequency of a geared system in this way should be analysed seriously. Thus, the natural frequency of parametric vibration for a spur-gear-pair system will be researched using the Floquet theory in this paper. The influences of the periodically time-varying mesh stiffness parameters (including the mesh frequency and the contact ratio) on the natural frequency will be discussed in detail. Contrasts between the natural frequency and the average frequency for the system in the stable and unstable regions will be processed and the scope of application for taking the average frequency as a substitution for the natural frequency of a geared system will then be presented.

Keywords

geared system periodically time-varying mesh stiffness natural frequency Floquet theory

Get full access to this article

View all access options for this article.

References

Wang

J. J.

R. F.

Peng

X. H.

‘Survey of nonlinear vibration of gear transmission systems.’

Appl. Mechanics Rev. 56 (2003): 309–329

R. F.

Wang

J. J.

Geared system dynamic: Vibration, shock and noise (in Chinese) 1997 (Science Press, Beijing).)

Benton

Seireg

‘Simulation of resonances and instability conditions in pinion-gear systems.’

Trans. ASME, J. Mech. Des. 100 (1978): 26–30

Tordion

G. V.

Gauvin

‘Dynamic stability of a two-stage gear train under the influence of variable meshing stiffness.’

Trans. ASME, J. Engng Industry 99 (1977): 785–791

Lin

Parker

R. G.

‘Mesh stiffness variation instabilities in two-stage gear systems.’

Trans, ASME, J. Vibr. Acous. 124 (2002): 68–76

Hbaieb

Chaari

Fakhfakh

Haddar

‘Dynamic stability of a planetary gear train under the influence of variable meshing stiffness.’

Proc. IMechE, Part D: J. Automobile Engineering 220 (2006): 1711–1725

Johnson

Helicopter theory (New York Princeton University Press 1980)

Pipes

L. A.

‘Matrix solution of equations of the Mathieu—Hill type.’

J. Appl. Physics 24 (1953): 902–910

Sinha

S. C.

Pandiyan

Bibb

J. S.

‘Liapunov—Floquet transformation: Computation and applications to periodic systems.’

Trans. ASME, J. Vibr. Acoust. 118 (1996): 209–219

10.

Peters

D. A.

Hohenemser

K. H.

‘Application of the Floquet transition matrix to problems of lifting rotor stability.’

J. Am. Helicopter Soc. 16 (1971): 25–33

11.

Nagabhushanam

Gaonkar

G. H.

‘Automatic identification of modal damping from Floquet analysis.’

J. Am. Helicopter Soc. 40 (1995): 39–42

12.

Stol

Balas

Bir

‘Floquet modal analysis of a teetered-rotor wind turbine.’

Trans. ASME, J. Solar Energy Engng. 124 (2002): 364–371

13.

Saracho

C. M.

Santos

I. F.

‘Modal analysis in periodic, time-varying systems with emphasis to the coupling between flexible rotating beams and non-rotating flexible structures.’ In Proceedings of the Tenth International Symposium on

Dynamic problems in mechanics (X-DINAME), Sao Paulo, Brazil, 10–14 March 2003, pp. 399–404 (Brazilian Society of Engineering and Mechanical Sciences, Rio de Janeiro).

14.

Richards

J. A.

Analysis of periodically time-varying systems (New York Springer-Verlag 1983)

15.

Hsu

C. S.

‘On approximating a general linear periodic system.’

J. Math. Analysis Applic. 45 (1974): 234–251

16.

Friedmann

Hammond

Woo

T. H.

‘Efficient numerical treatment of periodic systems with application to stability problems.’

Int. J. Numer. Meth. Engng. 11 (1977): 1117–1136

17.

Sinha

S. C.

D.-H.

‘An efficient computational scheme for the analysis of periodic systems.’

J. Sound Vibr. 151 (1991): 91–117

18.

Bauchau

O. A.

Nikishkov

Y. G.

‘An implicit Floquet analysis for rotorcraft stability evaluation.’

J. Am. Helicopter Soci. 46 (2001): 200–209

19.

Kahraman

Blankenship

G. W.

‘Effect of involute contact ratio on spur gear dynamics.’

Trans. ASME, J. Mech. Des. 121 (1999): 112–118

20.

Ozguven

H. N.

Houser

D. R.

‘Mathematical models used in gear dynamics — a review.’

J. Sound Vibr. 121 (1988): 383–411

21.

Benton

Seireg

‘Factors influencing instability and resonances in geared systems.’

Trans. ASME, J. Mech. Des. 103 (1981): 372–378

22.

Kahraman

Blankenship

G. W.

‘Experiments on nonlinear dynamic behavior of an oscillator with clearance and periodically time-varying parameters.’

Trans. ASME, J. Appl. Mechanics 64 (1997): 217–226