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)