A numerical study on parametric resonance of intrinsic localized modes in coupled cantilever arrays

Abstract

ySchool of Engineering, The University of Shiga Prefecture2500 Hassaka-cho, Hikone, Shiga 522-8533, JapanzDepartment of Electrical Engineering, Kyoto UniversityKatsura, Nishikyo, Kyoto 615-8510, JapanEmail: kimura.m@e.usp.ac.jp, matsushita.y@e.usp.ac.jp, hikihara.takashi.2n@kyoto-u.ac.jpAbstract —In a coupled cantilever array modeled as acoupled ordinary di erential equation, symmetric and an-tisymmetric intrinsic localized modes (ILMs) exist. Thesymmetric ILM is stable while the other is unstable underthe regime in which the ratio in nonlinearities of inter-siteand on-site potentials is less than the critical value at whichthe stability change occurs. This paper shows that a sta-ble ILM loses its stability when the system is parametri-cally excited. If the amplitude of parametric excitation islarge enough, the destabilized ILM wanders in the wholesystem. The parameter region where the instability occursare numerically investigated and compared with that in theMathieu equation. The similarity of the shape of the re-gions strongly suggests that the instability is caused by theparametric resonance.1. IntroductionSpatially localized and temporary periodic vibrations of-ten appear in nonlinear coupled oscillators [1]. The energylocalized vibration in discrete media which is rst discov-ered by A. J. Sievers and S. Takeno [2] is called intrinsic lo-calized mode(ILM) or discrete breather(DB). Experimentalobservations of ILM have been reported for a variety ofphysical system in this decade as well as theoretical andnumerical studies. In particular of them, the observationin micro-mechanical cantilever array allow us to expectthe realization of applications using ILM in micro/nano-engineering [3], because it was also observed that ILM canmove without decaying its energy concentration and can bemanipulated by an extraneous stimulus [4].For the realization of such application, the controlscheme for the ILM should be established. The capture andrelease manipulation using the stability change is proposedas an alternative method to manipulate ILM in a micro-cantilever array [5]. For the manipulation method, it isutilized that a stable ILM begins to move after it loses itsstability. Therefore, the manipulation is a way to generatemoving ILM. In this paper, we propose another method togenerate moving ILM by parametric excitation of the ratioin nonlinearities. We rst investigate how moving ILMsbehave around a stable ILM. The motion of moving ILM isapproximated by a simple equation of pendulum. Then, be-haviors of moving ILM created by parametric excitation areshown. Finally, the region where a stable solution loses itsstability by parametric excitation is computed for an ILM,and is compared with that of the Mathieu equation.2. Coupled Cantilever ArrayCantilever array, which is often used as a coupled me-chanical resonator in nano/micro-engineering, is one ofnonlinear coupled oscillators in which ILM can be ob-served experimentally [3]. The motion of cantilevers canapproximately be described by ordinary di erential equa-tion [4–6],u¨