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Research Papers

Subharmonic Resonance Cascades in a Class of Coupled Resonators

[+] Author and Article Information
B. Scott Strachan

e-mail: strach20@msu.edu

Steven W. Shaw

e-mail: shawsw@msu.edu
Department of Mechanical Engineering,
Michigan State University,
East Lansing, MI 48823

Oleg Kogan

Laboratory of Atomic and Solid State Physics,
Cornell University,
Ithaca, NY 14853
e-mail: oleg.kogan@cornell.edu

One can use direct excitation of the first resonator, but the present form is taken for convenience.

The 4:1 ratio for resonator cubic nonlinearity holds for fixed-fixed beams with identical uniform cross sections, where the lengths are varied to account for the 2:1 change of modal frequency.

Contributed by the Design Engineering Division of ASME for publication in the Journal of Computational and Nonlinear Dynamics. Manuscript received February 10, 2013; final manuscript received May 4, 2013; published online June 10, 2013. Assoc. Editor: Eric A. Butcher.

J. Comput. Nonlinear Dynam 8(4), 041015 (Jun 10, 2013) (7 pages) Paper No: CND-13-1031; doi: 10.1115/1.4024542 History: Received February 10, 2013; Revised May 04, 2013

We consider a chain of N nonlinear resonators with natural frequency ratios of approximately 2:1 along the chain and weak nonlinear coupling that allows energy to flow between resonators. Specifically, the coupling is such that the response of one resonator parametrically excites the next resonator in the chain, and also creates a resonant back-action on the previous resonator in the chain. This class of systems, which is a generic model for passive frequency dividers, is shown to have rich dynamical behavior. Of particular interest in applications is the case when the high frequency end of the chain is resonantly excited, and coupling results in a cascade of subharmonic bifurcations down the chain. When the entire chain is activated, that is, when all N resonators have nonzero amplitudes, if the input frequency on the first resonator is Ω, the terminal resonator responds with frequency Ω/2N. The details of the activation depend on the strength and frequency of the input, the level of resonator dissipation, and the frequency mistuning in the chain. In this paper we present analytical results, based on perturbation methods, which provide useful predictions about these responses in terms of system and input parameters. Parameter conditions for activation of the entire chain are derived, along with results about other phenomena, such as the period doubling accumulation to full activation, and regions of multistability. We demonstrate the utility of the predictive results by direct comparison with simulations of the equations of motion, and we also present a sample mechanical system that embodies the desired properties. These results are useful for the design and operation of mechanical frequency dividers that are based on subharmonic resonances.

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Figures

Grahic Jump Location
Fig. 1

A mechanical implementation of the subharmonic cascade, consisting of rigid bars with elastic hinges at their bases, and coupled by springs with linear stiffness. The energy is down-converted from the high frequency beam, parametrically driven by a source at approximately twice its natural frequency, down the chain to a frequency of Ω/2N at the terminal beam. For small stiffness of the coupling springs the bar displacements ui are roughly equal to the system modal coordinates qi, that is, the system modes are localized in the individual beams.

Grahic Jump Location
Fig. 2

Analytical predictions for the response regimes in (α,F) space; the destabilization boundaries αi and Fi for i=1,2 and ∞ are shown. The fully active regime is shaded grey and the partially activated regime is shaded light grey. Parameter values for all simulations are as follows, unless specified otherwise: ζ=0.03, γ=0.075, δ=0.064, and β=0.008.

Grahic Jump Location
Fig. 3

Simulation of a six resonator chain, showing the sequential activation of the six elements when started with small initial conditions. Resonators 1-6 are shown from top to bottom. The thick lines indicate results of the amplitudes obtained by simulating the averaged equations, and the underlying fast oscillations are from simulations of the full equations of motion. The settling time of resonator j is proportional to Q/2j-1.

Grahic Jump Location
Fig. 4

Steady state amplitudes of a six resonator cascade for α=0(>α∞) for various forcing amplitudes. The fully activated response is achieved when F>F∞=req-, which is depicted as the grey shaded region. The top two examples (circles and triangles) converge to the infinite lattice amplitude req+, which is denoted by the row of asterisks, with the amplitude of the final resonator given by a slightly lower value, rN+. The partially activated (diamonds) solutions occur for F1<F<F∞ and is the light grey region. The trivial response is for F<F1 and is the unshaded region.

Grahic Jump Location
Fig. 5

Frequency response at F=0.6 for six resonators with near zero initial conditions, computed from the averaged equations. The resonator amplitudes are the thick curves whose color darkens for each resonator, for example, the first resonator is light grey and the sixth resonator is black. The vertical lines are the predicted region boundaries, and the equal amplitude solution is shown as a black dashed line. The light dashed line is the activation amplitude for rj given in terms of rj-1, given in Eq. (11).

Grahic Jump Location
Fig. 6

Activation boundaries in (α,F) space computed from the averaged equations. The black lines correspond to the activation boundaries determined from simulations the original equations of motion for a six resonator cascade. The red dotted lines correspond to the analytical approximations for the boundaries of the first two resonators and the infinite lattice. In general, there is good agreement except near the bottom of the Arnold tongue where more interesting dynamics occur.

Grahic Jump Location
Fig. 7

Activation boundaries with small forcing in the limit of zero damping. While the case with zero back coupling, β = 0, has the infinite chain solution as a subset for every other region, the case with back coupling, β≠0, suggests an aphysical scenario in which the entire chain is activated before the second resonator activates. At very low forcing (F < 0.1), this incorrectly suggests that the infinite chain will activate before the first resonator activates, indicating a breakdown of the applicability of the infinite chain results.

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