TY - JOUR
T1 - Transient dynamics of a dispersive elastic wave guide weakly coupled to an essentially nonlinear end attachment
AU - Vakakis, Alexander F.
AU - Manevitch, Leonid I.
AU - Musienko, Andrey I.
AU - Kerschen, Gaetan
AU - Bergman, Lawrence A.
N1 - Funding Information:
AFV would like to acknowledge the influence of Professor T.K. Caughey of the California Institute of Technology on the modeling of wave effects through Laplace transform analysis. The authors would like to thank Professor Yu.V. Mikhlin of the Kharkov Polytechnical Institute, Ukraine, and Professor I. Andrianov of the Institute of General Mechanics, RWTH Aachen, Germany, for providing references on analytical approximations of Bessel functions. This work was funded in part by the Russian Foundation for Basic Research, project 04-03-32119, and the Russian Foundation for Support of National Science (AIM) (LIM and AIM), and by two grants from the Belgian National Fund for Scientific Research—FNRS and the Belgian Rotary District 1630 (GK).
PY - 2005/2
Y1 - 2005/2
N2 - We consider the dynamics of a dispersive semi-infinite linear rod weakly connected to an essentially nonlinear end attachment. Using a Green's function formulation we reduce the dynamics to an integro-differential equation, which we express in the form of an infinite set of ordinary differential equations using Neumann expansions. We examine resonant interactions of the attachment with incident traveling waves propagating in the pass band of the rod, as well as resonance capture phenomena where the attachment engages in transient 1:1 internal resonance with the in-phase mode of the rod, at the bounding frequency of its pass and stop bands. The studied resonant interactions of the nonlinear attachment with traveling waves are similar to resonance capture cascades occurring in finite-chain nonlinear attachment configurations; these are the transient dynamics of the attachment as it engages in a series of instantaneous resonances with a continuous spectrum of frequencies in the pass band of the medium. As the energy of the attachment decreases due to damping and energy radiation back to the rod, the attachment engages in 1:1 resonance capture with the in-phase mode of the rod; this resonance capture occurs due to the dispersion property in the linear medium. It is shown that the transient dynamics of resonance capture can be reduced to a set of two 'slow flow' amplitude and phase modulation equations. The analytical results are in agreement with direct numerical simulations.
AB - We consider the dynamics of a dispersive semi-infinite linear rod weakly connected to an essentially nonlinear end attachment. Using a Green's function formulation we reduce the dynamics to an integro-differential equation, which we express in the form of an infinite set of ordinary differential equations using Neumann expansions. We examine resonant interactions of the attachment with incident traveling waves propagating in the pass band of the rod, as well as resonance capture phenomena where the attachment engages in transient 1:1 internal resonance with the in-phase mode of the rod, at the bounding frequency of its pass and stop bands. The studied resonant interactions of the nonlinear attachment with traveling waves are similar to resonance capture cascades occurring in finite-chain nonlinear attachment configurations; these are the transient dynamics of the attachment as it engages in a series of instantaneous resonances with a continuous spectrum of frequencies in the pass band of the medium. As the energy of the attachment decreases due to damping and energy radiation back to the rod, the attachment engages in 1:1 resonance capture with the in-phase mode of the rod; this resonance capture occurs due to the dispersion property in the linear medium. It is shown that the transient dynamics of resonance capture can be reduced to a set of two 'slow flow' amplitude and phase modulation equations. The analytical results are in agreement with direct numerical simulations.
KW - Nonlinear resonant interactions of traveling waves
KW - Resonance capture
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U2 - 10.1016/j.wavemoti.2004.06.002
DO - 10.1016/j.wavemoti.2004.06.002
M3 - Article
AN - SCOPUS:7544243685
SN - 0165-2125
VL - 41
SP - 109
EP - 132
JO - Wave Motion
JF - Wave Motion
IS - 2
ER -