Regular and chaotic vibration in a piezoelectric energy harvester

Meccanica, Sep 2015

We examine regular and chaotic responses of a vibrational energy harvester composed of a vertical beam and a tip mass. The beam is excited horizontally by a harmonic inertial force while mechanical vibrational energy is converted to electrical power through a piezoelectric patch. The mechanical resonator can be described by single or double well potentials depending on the gravity force from the tip mass. By changing the tip mass we examine bifurcations from single well oscillations, to regular and chaotic vibrations between the potential wells. The appearance of chaotic responses in the energy harvesting system is illustrated by the bifurcation diagram, the corresponding Fourier spectra, the phase portraits, and is confirmed by the 0–1 test. The appearance of chaotic vibrations reduces the level of harvested energy.

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Regular and chaotic vibration in a piezoelectric energy harvester

Regular and chaotic vibration in a piezoelectric energy harvester Grzegorz Litak . Michael I. Friswell . Sondipon Adhikari 0 1 0 M. I. Friswell S. Adhikari College of Engineering, Swansea University Bay Campus , Fabian Way, Crymlyn Burrows, Swansea SA1 8EN , UK 1 G. Litak (&) Faculty of Mechanical Engineering, Lublin University of Technology , Nadbystrzycka 36, Lublin 20-618 , Poland We examine regular and chaotic responses of a vibrational energy harvester composed of a vertical beam and a tip mass. The beam is excited horizontally by a harmonic inertial force while mechanical vibrational energy is converted to electrical power through a piezoelectric patch. The mechanical resonator can be described by single or double well potentials depending on the gravity force from the tip mass. By changing the tip mass we examine bifurcations from single well oscillations, to regular and chaotic vibrations between the potential wells. The appearance of chaotic responses in the energy harvesting system is illustrated by the bifurcation diagram, the corresponding Fourier spectra, the phase portraits, and is confirmed by the 0-1 test. The appearance of chaotic vibrations reduces the level of harvested energy. Nonlinear dynamics; Chaotic solutions; 0-1 test; Energy harvesting 1 Introduction Broadband energy harvesting systems for many applications are often nonlinear, exhibiting such nonlinear phenomena as material nonlinearities, geometrical nonlinearities, multi-scale responses and the appearance of multiple solutions. These phenomena can be observed from the nonlinear time series analysis of simulated mathematical models or from measured system responses in experiments. It is well known that the efficiency of many engineered systems may be enhanced by operation in a nonlinear regime. Nonlinear vibrational energy harvesting shows such an advantage, making possible broadband frequency vibration energy accumulation and transduction into useful electrical power output. A range of vibration energy harvesting devices have been proposed [1–6]. The key aspect of nonlinear harvesters is the use of a double well potential function, so that the device will have two equilibrium positions [7–12]. Gammaitoni et al. [8] and Masana and Daqaq [13] highlighted the advantages of a double well potential for energy harvesting, particularly when inter well dynamics were excited. The Duffing oscillator model has been used for many energy harvesting simulations, with the addition of electromechanical coupling for the harvesting circuit [14, 15]. Electromagnetic harvesters with a cubic force nonlinearity have also been considered [16]. Litak et al. [17] and Ali et al. [18] investigated nonlinear piezomagnetoelastic energy harvesting under random broadband excitation. McInnes et al. [19] investigated the stochastic resonance phenomena for a nonlinear system with a double potential well. Gravitationally induced double potential wells in a system with a vertical elastic beam with a tip mass have been studied extensively [20, 23–25]. Recently, in the context of broad-band energy harvesting, bifurcations and chaotic vibrations have been studied in several papers. Cao et al. [26] studied chaos in the fractionally damped broadband piezoelectric energy generator in the system with additional magnets. Syta et al. [27] analysed the dynamic response of a piezoelectric material attached to a bistable laminate plate. It is worth to note that chaotic vibrations are, in most cases, characterized by moderate amplitude of vibrations and simultaneously give continuous spectrum of frequency, which can be useful to increase mechanical resonator durability. In this article we discuss different solutions appearing in that system. Especially, intra- and interwell oscillations as well as periodic and chaotic vibrations lead to different efficiency in the energy harvesting. Therefore, we identify the properties of given solutions by using nonlinear methods. 2 Mathematical model and equations of motion For nonlinear energy harvesting an inverted elastic beam is considered with a tip mass and the base is harmonically excited in the transverse direction. Only a summary of the equations is provided here; Friswell et al. [20] give a full derivation of the equations. Figure 1 shows the beam as a vertical cantilever of length L with harmonic base excitation zðtÞ ¼ z0 cos xt. The beam carries a concentrated tip mass, Mt, with moment of inertia It, at the tip of the beam. The horizontal and vertical elastic displacements at the tip mass are v and u respectively, and s represents the distance along the neutral axis of the beam. In the following analysis the beam is assumed to have uniform inertia and stiffness properties; a nonuniform beam is easily modeled by including the mechanical beam properties in the energy integrals. The beam has cross sectional area A, mass density q, equivalent Young’s modulus E, and second moment of area I. Fig. 1 Schematic of the inver (...truncated)


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Grzegorz Litak, Michael I. Friswell, Sondipon Adhikari. Regular and chaotic vibration in a piezoelectric energy harvester, Meccanica, 2016, pp. 1017-1025, Volume 51, Issue 5, DOI: 10.1007/s11012-015-0287-9