Multi-scaled analysis of the damped dynamics of an elastic rod with an essentially nonlinear end attachment

Panagopoulos, Panayotis and Georgiades, Fotios and Tsakirtzis, Stylianos and Vakakis, Alexander, F. and Bergman, Lawrence, A. (2007) Multi-scaled analysis of the damped dynamics of an elastic rod with an essentially nonlinear end attachment. International Journal of Solids and Structures, 44 (18-19). pp. 6256-6278. ISSN 0020-7683

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Abstract

We study multi-frequency transitions in the transient dynamics of a viscously damped dispersive finite rod with an essentially nonlinear end attachment. The attachment consists of a small mass connected to the rod by means of an essentially nonlinear stiffness in parallel to a viscous damper. First, the periodic orbits of the underlying hamiltonian system with no damping are computed, and depicted in a frequency–energy plot (FEP). This representation enables one to clearly distinguish between the different types of periodic motions, forming back bone curves and subharmonic tongues. Then the damped dynamics of the system is computed; the rod and attachment responses are initially analyzed by the numerical Morlet wavelet transform (WT), and then by the empirical mode decomposition (EMD) or Hilbert–Huang transform (HTT), whereby, the time series are decomposed in terms of intrinsic mode functions (IMFs) at different characteristic time scales (or, equivalently, frequency scales). Comparisons of the evolutions of the instantaneous frequencies of the IMFs to the WT spectra of the time series enables one to identify the dominant IMFs of the signals, as well as, the time scales at which the dominant dynamics evolve at different time windows of the responses; hence, it is possible to reconstruct complex transient responses as superposition of the dominant IMFs involving different time scales of the dynamical response.
Moreover, by superimposing the WT spectra and the instantaneous frequencies of the IMFs to the FEPs of the underlying hamiltonian system, one is able to clearly identify the multi-scaled transitions that occur in the transient damped dynamics, and to interpret them as ‘jumps’ between different branches of periodic orbits of the underlying hamiltonian system. As a result, this work develops a physics-based, multi-scaled framework and provides the necessary computational tools for multi-scaled analysis of complex multi-frequency transitions of essentially nonlinear dynamical systems.

Keywords:Multi-scaled analysis, Nonlinear damped transitions, Essential nonlinearity, Georgiadis
Subjects:H Engineering > H340 Acoustics and Vibration
H Engineering > H143 Structural Mechanics
H Engineering > H342 Vibration
H Engineering > H300 Mechanical Engineering
H Engineering > H140 Mechanics
H Engineering > H142 Solid Mechanics
H Engineering > H310 Dynamics
H Engineering > H210 Structural Engineering
Divisions:College of Science > School of Engineering
ID Code:9473
Deposited On:17 May 2013 09:16

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