Nonlinear Oscillations Of Cantilevers With Magnetic Interactions

Cantilevered structures are of great use in many fields, includingmicro-electromechanical systems (MEMS). The main advantages of metalliccantilevers are their large-deformation strains and ease of application. Most ofvibration-based piezoelectric generators have one or more micro-cantilevers forharvesting power from ambient energy sources. The harvester with sole piezoelectriccantilever is mainly efficient near the first resonant frequency. To broaden thefrequency bandwidth, the multi-modal energy harvesting systems are increasinglygaining notice. Nonlinear techniques have also been proposed to enhance the powerharvesting efficiency over the last decade. By virtue of permanent magnets orferromagnetic structures close to the free ends of cantilevers, the power harvestingperformance improved at low driving frequencies for a simple harmonic excitation.Recent efforts have been made on designing a cantilevered device with magnetsto potentially improve the energy harvesting performance. The device comprised twoidentical viscoelastic cantilevers whose free ends are connected by a linear spring.Two identical cylindrical permanent magnets are used to produce nonlinearity here.One of them is fixed at the free end of the concerned cantilever. The other magnet isfixed at the host device and perpendicularly aligned with the previous one. Thestructural damping of cantilever is modeled by the Kelvin viscoelastic relationship.At present, the mechanical vibration of the harvesting appliance has beeninvestigated, with a special emphasis on the distributed-parameter model.The magnetic force, serving as the nonlinear boundary, is modeled as afractional function with respect to the distance between two magnets. The additionalstiffness induced from the nonlinear magnetic force is positive for a repulsive forceand is negative for an attractive force, respectively. By applying a repulsive magneticforce, the cantilevered dievice has sole stable static equilibrium configuration. On the contrary is the case of an attractive force. There might be zero or two staticequilibria. Only if the initial distance is large enough, there are two equilibria, one ofwhich is stable and the other is unstable. Based on the stable equilibrium, themodeling, analysis and numerical simulation on the nonlinear oscillation of thecantilever arrays under a small sinusoidal base excitation are to be carried out. Thegoverning equations are derived for the free and the forced oscillations. The naturalfrequency and modal function can be obtained from the undamping free vibration.The method of two time scales is developed to analyze the forced vibration inexternal and2:1internal resonances simutaneously, taking account of the effects ofthe exciting amplitude and the viscoelasticity on the steady-state response. Themethod of three time scales is dedicated to the single-modal analysis on thesteady-state response of the forced vibration without any internal resonance. Thenumerical calculation is following each of these analytical results.Taking2:1internal resonance into account, the second natural frequency isnearly twice the first natural frequency of the linear unperturbated system.Continuous systems actually have an infinite number of degrees of freedom. As aresult of the truncation into a finite-demensional system, an error on the naturalfrequency is inescapable. In hopes of a quantitative verification, it is necessary todirectly calculate the governing equation based on the original model which is nottruncated. The current paper investigated the dynamic response of the continuummodel possessing2:1internal resonance. It is analytically predicted for the saturationphenomenon occurring at the second main resonance. In addition, a double-jumpingphenomenon occurs at the first main resonance. Then the frequency-response curvehas two peaks bending to the opposite directions. The emergence and the vanishingof the double-jumping phenomenon have been revealed. With a little lack of tuningon the2:1internal resonance, one peak will dominate and the other will be allayed,even vanish. A consequent evolution exists among the hardening-type, thesoftening-type, and the double-jumping behaviors. Meanwhile, for certain parameters, a steady-state response may not exist at the first main resonance in spiteof the positive viscoelastic damping. It was illustrated that the presence of2:1internal resonance can cause a quasi-periodic motion under a large excitation.In the absence of2:1internal resonance, the typical jumping and the streamingphenomenon are both predicted by the method of three time scales. Particularattention has been placed on providing a view of the evolution phenomenon. Due tothe softening effect of quadratic nonlinearity on the system, the evolution process ofthe double-jumping phenomenon will be imperfect. It does not continue with a greatamount of deviation between the second natural frequency and twice the first linearfrequency. A nonlinear index is proposed to determine the behavior of afrequency-response curve. It can be remarked that2:1internal resonance plays asignificant role in the above-mentioned conversion. The double-jumpingfrequency-response curve can be converted into hardening-type, softening-type andeven linear by adjusting the distance between two magnets.Considered the spring’s stiffness is equal to zero, the model will become asingle cantilever with nonlinear boundary conditions. The multi-scale method is stilldeveloped to solve the natural frequencies and modes, to determine the relationshipequation between the exciting frequency and the steady-state response, and toestimate the effects of other modes on the primary resonance without any internalresonance. The analytical study has shown that the modes uninvolved in the primaryresonance can be negligible. By applying an attractive magnetic force, all thefrequency-response curves exhibit the softening-type behavior. On the other hand, ifthe two magnets are repulsive each other, there will be a positive cubic nonlinearityand a negative quadratic. Then the frequency-response curves will belong to thehardening-type with small quadratic nonlinearity. It can also be softened byaugmenting the magnetic force. This type of conversion is different from theprevious. It is determined by the nonlinear coefficients, not the natural frequencies. The finite difference method is used to examine numerically the analyticalresults so that these theoretical perspectives can actually work into a practicalapplication. The finite differences are simultaneously used in both time and space,although the purely numerical technique is extremely costly for the forcedoscillations of the cantilevered model. All the analytical results are supportedqualitatively and quantitatively, such as the streaming, the typical jumping, thedouble-jumping and the saturation phenomenon.The present methodology has also been partially extended to tackle othercomplex continuous dynamical systems, such as buckled beams, axially movingbeams, pipes conveying fluid flowing at a supercritical speed, as well as energyharvesters.