Extreme stiffness tunability through the excitation of nonlinear defect modes

Abstract

The incremental stiffness characterizes the variation of a material's force response to a small deformation change. In lattices with noninteracting vibrational modes, the excitation of localized states does not have any effect on material properties, such as the incremental stiffness. We report that, in nonlinear lattices, driving a defect mode introduces changes in the static force-displacement relation of the material. By varying the defect excitation frequency and amplitude, the incremental stiffness can be tuned continuously to arbitrarily large positive or negative values. Furthermore, the defect excitation parameters also determine the displacement region at which the force-displacement relation is being tuned. We demonstrate this phenomenon experimentally in a compressed array of spheres tuning its incremental stiffness from a finite positive value to zero and continuously down to negative infinity.

Turing-complete mechanical processor via automated nonlinear system design

“…Third, near the bifurcation point where the system transitions between having a single and multiple solutions, some of the effective time constants governing convergence to the steady-state become increasingly long (being infinite at the exact bifurcation point, where the system has no tendency to converge between the two stable solutions). While exploring this phenomenon is outside the scope of this work, a discussion on this issue for a related system can be found in the supplementary material of Reference [31]. Here, it is confirmed numerically that the speed of convergence is not significantly slowed down by nonlinearity as the system is essentially fully converged after 3Q C periods (Fig.…”

Tunable, synchronized frequency down-conversion in magnetic lattices with defects

“…In the electronic supplementary material, Information, we investigate the nonlinear coupling terms in the case where the defect is not centred. The interaction between modes can be understood in the following way: owing to nonlinearity, the vibration of the defect mode is rectified into a static force pushing against its neighbours, in a way that is analogous to thermal expansion of a crystal [18] or the optical pressure in an opto-mechanical system (figure 3c). For small amplitudes, this expansion is proportional to the square of the vibration amplitude, resulting in the term γ x 2 L in the extended mode equation.…”

“…These terms do not appear in our lattice due to the location of the defect, but they are not generally zero (See Supplementary Information for a study on the relation between nonlinear terms and defect location). The interaction between modes can be understood in the following way: Due to nonlinearity, the vibration of the defect mode pushes against its neighbors, in a way that is analogous to thermal expansion of a crystal 18 or the optical pressure in an optomechanical system (Fig. 3(c)).…”

“…Chains of nonlinearly interacting elements have been studied for decades, beginning in the FPU problem 9,10 . They present a wide variety of phenomena, including solitons 11,12 , band-gaps 13 , energy trapping 14 , breathers 15,16 , unidirectional wave propagation 17 , negative or extreme stiffness 18 , localized modes with tunable profile 19 , shocks and rarefaction waves 20 . These phenomena can be used to realize acoustic rectifiers 17 , logic gates 21 , lenses 22 , filters 23 , vibration-attenuation 24 and energy harvesting systems 16 .…”

Complete delocalization in a defective periodic structure

“…Defective lattices have been the subject of various investigations due to their interesting nonlinear wave dynamics. Examples include, among many others, stability and bifurcation analysis of nonlinear defect modes [11], interaction of solitons with defects in photonics [12,13], breather modes in defective granular chains [14][15][16], nonlinear wave characteristics in defective systems [17], nondestructive defect identification in granular media [18], mechanical systems with tunable stiffness [19], all-mechanical switch [20] based on the supratransmission phenomenon [21,22], and autonomous magnetomechanical frequency converters [23]. A dynamic response with a spatially localized profile is the centerpiece in all these examples.…”