Nonlinear Vibration and Stability Analysis of Viscoelastic Rayleigh Beams Axially Moving on a Flexible Intermediate Supp
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RESEARCH PAPER
Nonlinear Vibration and Stability Analysis of Viscoelastic Rayleigh Beams Axially Moving on a Flexible Intermediate Support Rana Farshbaf Zinati1 · Mousa Rezaee1 · Saeed Lotfan1 Received: 18 July 2018 / Accepted: 2 July 2019 © Shiraz University 2019
Abstract In this study, the nonlinear vibration and stability of a simply supported axially moving Rayleigh viscoelastic beam equipped with an intermediate nonlinear support are investigated. The type of considered nonlinearity is geometric and is due to the axial stretching. The Kelvin–Voigt model is used to regard the beam internal damping. The Hamilton’s principle is employed to derive the governing equations and corresponding boundary conditions. The multiple scales method is applied to the dimensionless form of the governing equations and the nonlinear frequencies, time response of the system for two cases of the axial velocity fluctuation frequency are obtained. The stability of the system is investigated via solvability condition and Routh–Hurwitz criterion. Some case studies are accomplished to demonstrate the effect of rotary inertia, axial velocity and parameters of intermediate support on the system response, critical velocity and the system stability. Furthermore, the vari‑ ation of the first two resonance frequencies with respect to mean axial velocities for different locations of the intermediate support are investigated. It is found that by moving the intermediate support from one end of the beam to its midpoint, the region in which the first mode undergoes static instability, shrinks. Moreover, although rotary inertia impressively decreases the natural frequencies, intermediate support has the dominant effect on increasing the natural frequencies. Keywords Axially moving viscoelastic beam · Intermediate support · Rayleigh’s beam theory · Nonlinear vibration · Stability analysis
1 Introduction Many engineering structures such as rotating blades of aircraft, high-rise buildings, wings of aircraft, spacecraft antenna, etc. can be modeled as a beam. Studying the vibration of a beam received a great deal of attention due to its usage in design and control of the mechanical systems. For instance, Woinow‑ sky–Krieger (1950), Burgreen (1951), (Eisley 1964) and Srini‑ vasan (1965) studied the nonlinear vibration of a simply sup‑ ported beam. They showed that at large amplitude vibrations, the stretching of the beam’s mid-plane affects its dynamic behavior. In the engineering systems, to suppress vibrations due to external excitation, it is common to use an absorber, spring–mass system attached to the main system. Özkaya * Mousa Rezaee [email protected] Rana Farshbaf Zinati r‑[email protected] 1
Department of Mechanical Engineering, University of Tabriz, P.O. Box 51665315, Tabriz, Iran
et al. (1997) and Karlik et al. (1998) studied the nonlinear vibrations of beam–mass system under different boundary conditions. They demonstrated the effect of mass and its location as well as the effect of boundary conditions on the beam vibr
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