From: Subharmonic Resonance Cascades in a Class of Coupled Resonators Date of download: 11/1/2017 Copyright © ASME. All rights reserved. From: Subharmonic Resonance Cascades in a Class of Coupled Resonators J. Comput. Nonlinear Dynam. 2013;8(4):041015-041015-7. doi:10.1115/1.4024542 Figure Legend: 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.

From: Subharmonic Resonance Cascades in a Class of Coupled Resonators Date of download: 11/1/2017 Copyright © ASME. All rights reserved. From: Subharmonic Resonance Cascades in a Class of Coupled Resonators J. Comput. Nonlinear Dynam. 2013;8(4):041015-041015-7. doi:10.1115/1.4024542 Figure Legend: 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.

From: Subharmonic Resonance Cascades in a Class of Coupled Resonators Date of download: 11/1/2017 Copyright © ASME. All rights reserved. From: Subharmonic Resonance Cascades in a Class of Coupled Resonators J. Comput. Nonlinear Dynam. 2013;8(4):041015-041015-7. doi:10.1115/1.4024542 Figure Legend: 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.

From: Subharmonic Resonance Cascades in a Class of Coupled Resonators Date of download: 11/1/2017 Copyright © ASME. All rights reserved. From: Subharmonic Resonance Cascades in a Class of Coupled Resonators J. Comput. Nonlinear Dynam. 2013;8(4):041015-041015-7. doi:10.1115/1.4024542 Figure Legend: 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.

From: Subharmonic Resonance Cascades in a Class of Coupled Resonators Date of download: 11/1/2017 Copyright © ASME. All rights reserved. From: Subharmonic Resonance Cascades in a Class of Coupled Resonators J. Comput. Nonlinear Dynam. 2013;8(4):041015-041015-7. doi:10.1115/1.4024542 Figure Legend: 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.

From: Subharmonic Resonance Cascades in a Class of Coupled Resonators Date of download: 11/1/2017 Copyright © ASME. All rights reserved. From: Subharmonic Resonance Cascades in a Class of Coupled Resonators J. Comput. Nonlinear Dynam. 2013;8(4):041015-041015-7. doi:10.1115/1.4024542 Figure Legend: 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).

From: Subharmonic Resonance Cascades in a Class of Coupled Resonators Date of download: 11/1/2017 Copyright © ASME. All rights reserved. From: Subharmonic Resonance Cascades in a Class of Coupled Resonators J. Comput. Nonlinear Dynam. 2013;8(4):041015-041015-7. doi:10.1115/1.4024542 Figure Legend: 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.