Abstract
In the present work, the nonlinear response of a vertically moving viscoelastic beam subjected to a periodically varying contact load is investigated. The generalized Galerkin’s method is used to discretize the nonlinear partial differential equation of motion into the temporal equation of motion. The temporal equation of motion contains many nonlinear terms such as cubic geometric and inertial nonlinear terms, nonlinear damping term, and nonlinear parametric excitation terms in addition to forced excitation and parametric excitation terms. The first-order approximate solutions are obtained by using the method of multiple scales, and the stability and bifurcations of the obtained steady-state responses are studied. Extensive numerical simulations are presented to illustrate the influences of various types of system parameters for different resonance conditions. A significant amount of vibration reduction is obtained with the increase in the material loss factor. The results obtained by numerically solving the temporal equation of motion are found to be in good agreement with the results determined by the method of multiple scales. The obtained results are useful for reduction in the vibration of the viscoelastic flexible beam with prismatic joint or single-link viscoelastic Cartesian manipulator with payload subjected to a sinusoidally varying contact load.
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References
Saito H., Otomi K.: Parametric response of viscoelastically supported beams. J. Sound Vib. 63, 169–178 (1979)
Gurgoze M.: Parametric vibrations of a viscoelastic beam (Maxwell model) under steady axial load and transverse displacement excitation at one end. J. Sound Vib. 115, 329–338 (1987)
Shih Y.S., Yeh Z.F.: Dynamic stability of a viscoelastic beam with frequency-dependent modulus. Int. J. Solids Struct. 42, 2145–2159 (2005)
Fung R.F., Huang J.S., Chen W.H.: Dynamic stability of a viscoelastic beam subjected to harmonic and parametric excitations simultaneously. J. Sound Vib. 198, 1–16 (1998)
Parker R.G., Lin Y.: Parametric instability of axially moving media subjected to multifrequency tension and speed fluctuations. ASME J. Appl. Mech. 68, 49–57 (2001)
Chen L.Q., Yang X.D., Cheng C.J.: Dynamic stability of an axially accelerating viscoelastic beam. Eur. J. Mech. A Solids 23, 659–666 (2004)
Chen L.Q., Yang X.D.: Stability in parametric resonance of axially moving viscoelastic beams with time-dependent speed. J. Sound Vib. 284, 879–891 (2005)
Yang X.D., Chen L.Q.: Dynamic stability of axially moving viscoelastic beams with pulsating speed. Appl. Math. Mech. 26, 990–997 (2005)
Ding H., Chen L.Q.: Stability of axially accelerating viscoelastic beams: multi-scale analysis with numerical confirmations. Eur. J. Mech. A/Solids 27, 1108–1120 (2008)
Yang X.D., Chen L.Q.: Bifurcation and chaos of an axially accelerating viscoelastic beam. Chaos Solitons Fractals 23, 249–258 (2005)
Pellicano F., Vestroni F.: Complex dynamic of high-speed axially moving systems. J. Sound Vib. 258, 31–44 (2002)
Kargarnovin M.H., Younesian D., Thompson D.J., Jones C.J.C.: Response of beams on nonlinear viscoelastic foundations to harmonic moving loads. Comput. Struct. 83, 1865–1877 (2005)
Kocatürk T., Simsek M.: Vibration of viscoelastic beams subjected to an eccentric compressive force and a concentrated moving harmonic force. J. Sound Vib. 291, 302–322 (2006)
Gurgoze M., Dogruoglu A.N., Zeren S.: On the eigencharacteristics of a cantilevered visco-elastic beam carrying a tip mass and its representation by a spring-damper-mass system. J. Sound Vib. 301, 420–426 (2007)
Yang X.D., Chen L.Q.: Nonlinear forced vibration of axially moving viscoelastic beams. Acta Mech. Solida Sin. 19, 365–373 (2006)
Mahmoodi S.N., Khadem S.E., Kokabi M.: Non-linear free vibrations of Kelvin–Voigt visco-elastic beams. Int. J. Mech. Sci. 49, 722–732 (2007)
Marynowski K., Kapitaniak T.: Zener internal damping in modeling of axially moving viscoelastic beam with time- dependent tension. Int. J. Non Linear Mech. 42, 118–131 (2007)
Chen L.Q, Hu D.: Steady-state responses of axially accelerating viscoelastic beams: Approximate analysis and numerical confirmation. Sci. China Ser. G Phys. Mech. Astron. 51, 1707–1721 (2008)
Wang, L.H., Hu, Z.D., Zhong, Z.D., Ju, J.W.: Dynamic analysis of an axially translating viscoelastic beam with an arbitrarily varying length. Acta Mech. (2010). doi:10.1007/s00707-010-0287-8
Ansari, M., Esmailzadeh, E., Younesian, D.: Internal-external resonance of beams on non-linear viscoelastic foundation traversed by moving load. Nonlinear Dyn. (2010). doi:10.1007/s11071-009-9639-0
Pratiher B., Dwivedy S.K.: Non-linear vibration of a single link viscoelastic Cartesian manipulator. Int. J. Non linear Mech. 43, 683–696 (2008)
Nayfeh A.H., Mook D.T.: Nonlinear Oscillations. Wiley, Canada (1995)
Cartmell M.P.: Introduction to Linear, Parametric and Nonlinear Vibrations. Chapman & Hall, London (1990)
Nayfeh A.H., Balachandran B.: Applied Nonlinear Dynamics: Analytical, Computational and Experimental Methods. Wiley, Canada (1995)
Zavodney L.D., Nayfeh A.H.: The nonlinear response of a slender beam carrying lumped mass to a principal parametric excitation: theory and experiment. Int. J. Non Linear Mech. 24, 105–125 (1989)
Cuvalci O.: The effect of detuning parameters on the absorption region for a coupled system: a numerical and experimental study. J. Sound Vib. 229, 837–857 (2000)
Moon F.C., Pao Y.H.: Vibration and dynamic instability of a beam-plate in a transverse magnetic field. J. Appl. Mech. 36, 92–100 (1969)
Wu G.Y., Tsai R., Shih Y.S.: The analysis of dynamic stability and vibration motions of a cantilever beam with axial loads and transverse magnetic fields. J. Acoust. Soc. ROC 4, 40–55 (2000)
Pratiher B., Dwivedy S.K.: Parametric instability of a cantilever beam with magnetic field and periodic axial load. J. Sound Vib. 305, 904–917 (2007)
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Pratiher, B., Dwivedy, S.K. Nonlinear response of a vertically moving viscoelastic beam subjected to a fluctuating contact load. Acta Mech 218, 65–85 (2011). https://doi.org/10.1007/s00707-010-0397-3
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DOI: https://doi.org/10.1007/s00707-010-0397-3