Chapter 23 Energy Harvesting Perspectives from Parametric Resonant Systems Maryam Ghandchi Tehrani, Elvio Bonisoli, and Matteo Scapolan Abstract Parametric resonances can occur in internally stressed systems due to the periodic variation of the stiffness in time. Parametric resonance can lead to unstable dynamic behaviour; however, their response is limited by existing nonlinearities in the system, thus resulting in limit cycle oscillations (LCOs). This phenomenon can be exploited in the design of energy harvesters. The amplitude and frequency of the parametric excitation can be adjusted so that the vibration response of internally stressed systems is close to instability. In this paper, a cantilever beam is considered in vertical position and an axial excitation is applied to the base of the beam. The imposed kinematics of the base leads to internal stress along the beam, which produces a variation of the bending stiffness. If the frequency of the base excitation is twice the first natural frequency of the beam, the principal parametric resonances can occur. A quasi-linear FEM approach is adopted, together with a simplified single-degree-of-freedom model of the beam, in order to numerically simulate its dynamic behaviour, to identify unstable conditions and to obtain the Floquet diagram. An analytical approach is developed as well, using a multi-degree-offreedom model of the beam, considering the system as autoparametric. Harmonic balance method is used to determine the Floquet diagram and to validate the numerical model. Principal parametric resonance is observed experimentally. Harvesting energy from parametric resonance is therefore potentially very efficient, especially if the external source is not directly exploitable. Parametric resonance in this case acts as a power amplification. Keywords Energy harvesting • Parametric resonance • Floquet diagram • Lupos 23.1 Introduction Faraday [1] first observed parametric resonance phenomena in a vertically oscillating fluid container, developing horizontal surface waves. The first model that described this behaviour was presented by Mathieu [2]. Parametric resonance can occur in systems with periodic-time-varying parameter(s) in the differential equation such as, m.t/ Rx.t/ Cc.t/ Px.t/ Ck.t/ x.t/ Df.t/ (23.1) where m(t), c(t), k(t) are the time varying mass, damping and stiffness coefficients respectively andf (t) is the external forces acting on the system. The excitation can be externally provided by a forcing function (external excitation) or can arise internally from the time variation of the parameters (self-excitation). In contrast to externally excited systems, self-excitation is linked with the homogeneous equation of motion (f.t/ D0). Thus, the presence of a periodic time varying parameter in the homogenous equation can act as an excitation, commonly referred to as parametric excitation. Floquet theory is used to solve the homogeneous equation of this kind. For undamped systems, Equation 1 can be simplified to the well-known Mathieu equation, Rx.t/ C.ı C"cos. t//x.t/ D0 (23.2) M.G. Tehrani Institute of Sound and Vibration Research, University of Southampton, Highfield, Southampton SO171BJ, United Kingdom E. Bonisoli ( ) • M. Scapolan Department of Management and Production Engineering, Politecnico di Torino, Corso Duca degli Abruzzi, Torino 24-10129, Italy e-mail: elvio.bonisoli@polito.it © The Society for Experimental Mechanics, Inc. 2015 A. Wicks (ed.), Shock & Vibration, Aircraft/Aerospace, and Energy Harvesting, Volume 9, Conference Proceedings of the Society for Experimental Mechanics Series, DOI 10.1007/978-3-319-15233-2_23 223
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