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

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.\ud 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

Dynamic testing of nonlinear vibrating structures using nonlinear normal modes

“…The damped invariant manifold can therefore be approximated by the undamped invariant manifold. The free damped dynamics closely follows the NNM of the underlying undamped system [21,34,35].…”

Control of primary and subharmonic resonances of a Duffing oscillator via non-linear energy sink

“…Discussion on energy transfer between linear and essentially nonlinear oscillators is given in [14][15][16][17], while extension of the use of NES to control non-ideal main structures is presented in [18,19]. NES attached to continuum primary linear structures, such as string, beams or plates, is considered in [20][21][22][23][24][25][26][27]. Multi-d.o.f.…”

Enhancing the dynamic range of targeted energy transfer in acoustics using several nonlinear membrane absorbers

“…The dynamics of such a system differs dramatically from those of linear absorbers and have been described in details in [1,2,3,4,5] in terms of resonance capture and nonlinear normal modes. A lot of papers have been published about different cases of linear system (a wave guide [6], a rod [7,8,9], a beam [10], a plate [11], a chain of coupled linear oscillators [12]) and different applications (seismic mitigation [13,14], aeroelastic instabilities control [15,16] or drill-string systems stabilization [17]). For applications to Acoustics, the possibility of using a NES (composed by a thin circular viscoelastic membrane) coupled to an acoustic medium was demonstrated in [18,19] as a new technique of passive noise control in the low frequency domain where no efficient dissipative mechanism exists.…”