Principal parametric resonance of axially accelerating viscoelastic strings with an integral constitutive law

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

Volumn 461, Issue 2061, 2005, Pages 2701-2720

  Chen, Li Qun ^a  

^a SHANGHAI UNIVERSITY   (China)

Author keywords

Axially moving string; Method of multiple scales; Nonlinear parametric vibration; Stability; Viscoelasticity

Indexed keywords

ELASTICITY; PARAMETRIC DEVICES; PIPE FLOW; RESONANCE; STABILITY; SUSPENSIONS (COMPONENTS); VISCOELASTICITY; VISCOSITY;

AXIALLY MOVING STRING; CLOSED-FORM EXPRESSION; CONSTITUTIVE LAW; EULERIAN EQUATIONS; EXISTENCE CONDITIONS; GOVERNING EQUATIONS; KIRCHHOFF MODEL; LINEARIZED STABILITY; LYAPUNOV; METHOD OF MULTIPLE SCALE; METHOD OF MULTIPLE SCALES; NON-TRIVIAL; NONLINEAR PARAMETRIC VIBRATION; NUMERICAL CALCULATION; NUMERICAL RESULTS; PRINCIPAL PARAMETRIC RESONANCE; SPEED FLUCTUATIONS; STEADY-STATE RESPONSE; TRANSVERSE MOTIONS; TRANSVERSE RESPONSE; VISCO-ELASTIC PARAMETERS; VISCOELASTIC STRINGS;

EQUATIONS OF MOTION;

EID: 33750715682     PISSN: 13645021     EISSN: 14712946     Source Type: Journal    
DOI: 10.1098/rspa.2005.1471     Document Type: Article

References (32)

1. Abrate, A. S. 1992 Vibration of belts and belt drives. Mech. Mach. Theory 27, 645-659.
2. Andrianov, I. V., Awrejcewicz, J. and Barantsev, R. G. 2003 Asymptotic approaches in mechanics: new parameters and procedures. Appl. Mech. Rev. 56, 87-110.
3. Cartmell, M. P., Ziegler, S. W., Khanin, R. and Forehand, D. I. M. 2003 Multiple scales analyses of the dynamics of weakly nonlinear mechanical systems. Appl. Mech. Rev. 56, 455-492.
4. Chen, L. Q. 2005 Analysis and control of transverse vibrations of axially moving strings. Appl. Mech. Rev. 58, 91-116.
5. Chen, L. Q. and Zhao, W. J. 2005 A computation method for nonlinear vibration of axially accelerating viscoelstic strings. Appl. Math. Comput. 162, 305-310.
6. Chen, L. Q. and Zhao, W. J. 2005 The energetics and the stability of axially moving Kirchhoff strings. J. Acoust. Soc. Am. 117, 55-58.
7. Chen, L. Q. and Zu, J. W. 2001 Transverse vibration of axially moving strings and its control. Adv. Mech. 31, 535-546.
8. Chen, L. Q. and Zu, J. W. 2003 Parametrical resonance of the excited axially moving string with an integral constitutive law. Int. J. Nonlin. Sci. Numer. Simulat. 4, 169-177.
9. Chen, L. Q., Zu, J. W. and Wu, J. 2003 Dynamic response of the parametrically excited axially moving string constituted by the Boltzmann superposition principle. Acta Mech. 162, 143-155.
10. Chen, L. Q., Zu, J. W., Wu, J. and Yang, X. D. 2004a Transverse vibrations of an axially accelerating viscoelastic string with geometric nonlinearity. J. Eng. Math. 48, 171-182.
11. Chen, L. Q., Wu, J. and Zu, J. W. 2004b Asymptotic nonlinear behaviours in transverse vibration of an axially accelerating viscoelastic string. Nonlin. Dyn. 35, 347-360.
12. Chen, L. Q., Zhao, W. J. and Zu, J. W. 2004c Transient responses of an axially accelerating viscoelastic string constituted by a fractional differentiation law. J. Sound Vib. 278, 861-871.
13. Christensen, R. M. 1982 Theory of viscoelasticity. New York: Academic Press.
14. Chung, J., Han, C. S. and Yi, K. 2001 Vibration of an axially moving string with geometric nonlinearity and translating acceleration. J. Sound Vib. 240, 733-746.
15. Fung, Y. C. 1965 Foundations of solid mechanics. New York: Prentice-Hall.
16. Fung, R. F., Huang, J. S. and Chen, Y. C. 1997 The transient amplitude of the viscoelastic traveling string: an integral constitutive law. J. Sound Vib. 201, 153-167.
17. Kirchhoff, G. 1877 Vorlesungen uber mathematische Physik: Mechanik. Leipzig: B. G. Teubner.
18. Koivurova, H. and Salonen, E. M. 1999 Comments on nonlinear formulations for travelling string and beam problems. J. Sound Vib. 225, 845-856.
19. Miranker, W. L. 1960 The wave equation in a medium in motion. IBM J. Res. Dev. 4, 36-42.
20. Moon, J. and Wickert, J. A. 1997 Nonlinear vibration of power transmission belts. J. Sound Vib. 200, 419-431.
21. Mote Jr, C. D. 1966 On the nonlinear oscillation of an axially moving string. J. Appl. Mech. 33, 463-464.
22. Nayfeh, A. H. and Mook, D. T. 1979 Nonlinear oscillations. New York: Wiley.
23. Pakdemirli, M. and Batan, H. 1993 Dynamic stability of a constantly accelerating string. J. Sound Vib. 168, 371-378.
24. Pakdemirli, M. and Ulsoy, A. G. 1997 Stability analysis of an axially accelerating string. J. Sound Vib. 203, 815-832.
25. Pakdemirli, M., Ulsoy, A. G. and Ceranoglu, A. 1994 Transverse vibration of an axially accelerating string. J. Sound Vib. 169, 179-196.
26. Ozkaya, E. and Pakdemirli, M. 2000 Lie group theory and analytical solutions for the axially accelerating string problem. J. Sound Vib. 230, 729-742.
27. Thurman, A. L. and Mote Jr, C. D. 1969 Free, periodic, nonlinear oscillation of an axially moving strip. J. Appl. Mech. 36, 83-91.
28. Washizu, K. 1982 Variational methods in elasticity and plasticity. New York: Pergamon Press.
29. Wickert, J. A. and Mote Jr, C. D. 1988 Current research on the vibration and stability of axially moving materials. Shock Vib. Dig. 20, 3-13.
30. Wickert, J. A. and Mote Jr, C. D. 1990 Classical vibration analysis of axially moving continua. J. Appl. Mech. 57, 738-744.
31. Zhang, L. and Zu, J. W. 1999 Nonlinear vibration of parametrically excited moving belts. J. Appl. Mech. 66, 396-409.
32. Zhang, L., Zu, J. W. and Zhong, Z. 2002 Transient response of viscoelastic moving belts using blockby-block method. Int. J. Struct. Stab. Dyn. 2, 265-280.