A study on feedback control of intrinsic localized modes in a micro-mechanical cantilever array

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 —Intrinsic localized mode (ILM), which isalso called discrete breather (DB), is an energy localizedvibration in nonlinear coupled oscillators. It is well knownthat the ILM can move in the system without decay of itsenergy concentration. This paper shows that the positionof ILM can be controlled by proportional-derivative con-trol. To create a force to the ILM, linear on-site coe cientsare modulated linearly with respect to the lattice number.Namely, value of the linear on-site coe cients linearly in-crease/decrease as the lattice number increases. Magnitudeof the tilt is adjusted with PD control scheme. As a result ofnumerical simulations, a standing ILM is successfully con-trolled toward a reference position with keeping its energyconcentration.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. In our previ-ous research, it has been shown that a standing ILM losesits stability by parametric excitation [5]. This result impliesthat appropriately adjusting parameter can creates a force tostanding ILM, because the parametric resonance is usuallycaused by changing a potential shape. Therefore, if the pa-rameter of the system is changed appropriately based on thedistance from the reference position, the position of ILMcan be controlled by using an ordinary control method. Inthis paper, proportional-derivative control is applied to thecontrol of the position of ILM. First, an approximate equa-tion describing the motion of traveling ILM is derived. Sec-ond, behavior of moving ILM is shown when on-site linearcoe cients are gradually changed with respect to the lat-tice number. Finally, the proportional-derivative control ofILM is demonstrated and discussed.2. Coupled Cantilever Array and Standing ILMMicro-cantilever array is one of nonlinear coupled oscil-lators having ILM [3]. By focusing on the rst mode ofbeam vibrations, motions of each cantilever's tip are ap-proximately described by the following ordinary di eren-tial equation [4,6,7],¨u