C-L Method and Its Application to Engineering Nonlinear Dynamical Problems

CHEN Yu-shu; DING Qian

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 2001 > 22(2): 127-134

CHEN Yu-shu, DING Qian. C-L Method and Its Application to Engineering Nonlinear Dynamical Problems[J]. Applied Mathematics and Mechanics, 2001, 22(2): 127-134.

Citation:

CHEN Yu-shu, DING Qian. C-L Method and Its Application to Engineering Nonlinear Dynamical Problems[J]. Applied Mathematics and Mechanics, 2001, 22(2): 127-134.

Citation:

CHEN Yu-shu, DING Qian. C-L Method and Its Application to Engineering Nonlinear Dynamical Problems[J]. Applied Mathematics and Mechanics, 2001, 22(2): 127-134.

PDF( 349 KB)

C-L Method and Its Application to Engineering Nonlinear Dynamical Problems

Department of Mechanics, Tianjin University, Tianjin 300072, P. R. China

Received Date: 2000-04-25
Rev Recd Date: 2000-09-11
Publish Date: 2001-02-15

Abstract

Abstract

The C-L method was generalized from Liapunov-Schmidt reduction method, combined with theory of singularities, for study of non-autonomous dynamical systems to obtain the typical bifurcating response curves in the system parameter spaces. This method has been used, as an example, to analyze the engineering nonlinear dynamical problems by obtaining the bifurcation programs and response curves which are useful in developing techniques of control to subharmonic instability of large rotating machinery.
- C-L method,
- nonlinear dynamics,
- nonlinear oscillations,
- bifurcation and chaos

FullText(HTML)

References(22)

References

[1]	Krylov N,Bogoliubov N.Les methodes de la mecarique nonlineaire[J].Chaire de la Phys,and Math of Academic Science U K,1934,8,44-51.
[2]	Chen Y S,Langford W F.The subharmonic bifurcation solution of nonlinear Mathieu's equation and Euler dynamically buckling problem[J].Acta Mech Sinica,1988,4(4):350-362.
[3]	Noah S T,Sundararajan P.Significance of considering nonlinear effects in predicting the dynamic behavior of rotating machinery[J].J Vib Control,1995,1(1):431-458.
[4]	Nataraj C,Nelson H D,Arkere N.The effect of a Coulomb spine on rotor dynamics/analysis[A].In:Instability in Rotating Machinery NASA CP-2409[C].New York:Springer,1985,225-233.
[5]	Shaw J,Shaw S W.The effects of unbalance on oil whirl[J].Nonlinear Dynamics,1990,1(4):293-311.
[6]	Sundararajan P,Noah S T.Dynamics of forced nonlinear systems using shooting arc length continuation method[J].ASME J Vib Acous,1997,119(1):9-20.
[7]	Chen Y S,Ding Q.Stability and bifurcation of nonlinear rotor dynamics[J].J Nonlinear Dynamics in Sci Tech,1996,3(1):13-22.
[8]	Chen Y S,Meng Q.Bifurcations of a nonlinear rotor/bearing system[J].J Vib Engng,1996,9(3):266-275.
[9]	Chen Y S,Ding Q.Stability and Hopf bifurcation of nonlinear rotor/bearing system[J].J Vib Engng,1997,10(3):368-374.
[10]	Ding Q,Chen Y S.Study on mechanism subharmonic instability of nonlinear rotor/bearing system[J].J Vib Engng,1997,10(4):404-412.
[11]	Yu P,Huseyin K.Parametrically excited nonlinear systems:a comparison of certain methods[J].Int J Nonli Mech,1998,33(6):967-978.
[12]	Chow S Y,Hale J K.Methods of Bifurcation Theory[M].New York:Springer,1982.
[13]	Golubisky M,Schaeffer D G.Singularities and Groups in Bifurcation Theory[M].Vol.1.New York:Springer,1985.
[14]	Bogoliubov N,Mitropolsky Y A.Asymptotic Methods in the Theory of Nonlinear Oscillations[M].New York:Gordon & Breach,1961.
[15]	陈予恕,詹凯君.《非线性Mathieu方程亚谐共振分叉理论》的一些推广[J].应用数学和力学,1990,11(3):239-245.
[16]	Chen Y S,Zhan K J,Langford W F.New results on bifurcation theory of subharmonic resonance in nonlinear system with parametric excitation:degenerate bifurcation solution[J].J Vib Engng,1990,3(2):38-47.
[17]	Chen Y S,Xu J.Global bifurcations and chaos in Van der Pol-Duffing-Mathieu's system with three-well potential oscillator[J].Acta Mech Sinica,1995,11(4):357-372.
[18]	Chen Y S,Xu J.Bifurcations in nonlinear systems with parametric excitation[J].Doklady Mathematics,Russia,1997,56(3):880-883.
[19]	Muszynska A.Improvements in lightly loaded rotor/bearing and rotor/seal models[J].ASME J Vib Acous,1988,110(2):129-136.
[20]	Carr J.Application of Center Manifold Theory[M].New York:Springer,1981.
[21]	Sethna P R.On averaged and normal form equations[J].Nonlinear Dynamics,1995,7(1):1-10.
[22]	CHEN Yu-shu,Andrew Leung.Bifurcation and Chaos in Engineering[M].London:Springer-Verlag,1988.