Zener internal damping in modelling of axially moving viscoelastic beam with time-dependent tension

Abstract

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Analytical Analysis on Nonlinear Parametric Vibration of an Axially Moving String with Fractional Viscoelastic Damping

“…In addition, some structures made of more complex viscoelastic materials are explored. Marynowski and Kapitaniak [26] presented a mathematical model of an axially moving viscoelastic beam with the three-parameter Zener element and investigated both regular and chaos motions using the Galerkin method. Chen et al [27] analytically studied nonlinear parametric responses of an axially moving string composed of the complicated viscoelastic material based on the Boltzmann superposition principle.…”

Nonlinear Vibrations of Axially Moving Beams

“…Inserting u=0 into equation (11) and then retaining lower order nonlinear terms only yield a nonlinear partial-differential equation (Chen & Zu, 2004) for uniformly moving elastic beams without external excitation, (Marynowski, 2002(Marynowski, , 2004 and (Marynowski & Kapitaniak, 2007) for axially moving viscoelastic beams without external excitation, and for axially accelerating viscoelastic beams, and for uniformly moving elastic beams without external excitation. The applications of the nonlinear integropartial-differential equation include (Wickert, 1991), (Pellicano & Zirilli, F., 1997), (Pellicano & Vestroni, 2000), (Chakraborty & Mallik, 2000a), (Pellicano, 2001), (Kong & Parker, 2004) and (Chen & Zhao, 2005) for uniformly moving elastic beams without external excitation, (Ghayesh, 2008) for uniformly moving viscoelastic beams without external excitation, (Pellicano & Vestroni, 2000), (Özhan & Pakdemirli, 2009) for uniformly moving elastic beams, , (Öz et al, 2001) and (Ravindra & Zhu, 1998) for axially accelerating elastic beams without external excitation, (Chakraborty & Mallik, 1998) , (Chakraborty & Mallik, 2000b) for axially moving elastic beams, (Parker & Lin, 2001) for axially accelerating elastic beams, and , ) for axially accelerating viscoelastic beams, and (Özhan & Pakdemirli, 2009) for uniformly moving viscoelastic beams.…”

Three-Dimensional Nonlinear Global Dynamics of Axially Moving Viscoelastic Beams

“…More advanced analytical techniques, such as the method of multiple scales and matched asymptotic expansion, were used by Pakdemirli and coworkers [11][12][13] in order to analyze the nonlinear dynamics of axially accelerating beams and strings. Marynowski and Kapitaniak [14] investigated the nonlinear dynamics of an axially moving viscoelastic beam by introducing a three-parameter Zener internal damping in the beam model; they used the Galerkin scheme along with Runge-Kutta method to solve the equations numerically. A finite element approach was used by Stylianou and Tabarrok [15], who analyzed the stability of an axially moving beam.…”