Titolo della tesi: Multidisciplinary modelling and parameter optimization of piezoelectric beams for energy harvesting from vibrations
The present thesis deals with numerical and experimental modelling of energy harvesting from ambient
vibrations with piezoelectric materials. The aim of the work is indeed to study the system with a multiphysics
approach, describing not only the electromechanical behaviour of the device, but also its interaction
with the electric conversion system closing the circuit on the external load. This holistic procedure fills
a gap in past literature, in which usually mechanical and electrical description are addressed separately
for this kind of devices, thus preventing from a global optimization. Indeed, finding the optimum set
of material, geometrical, and electric parameters is fundamental to maximise the overall efficiency and
make the power produced by the energy harvester suitable for real-life applications. Moreover, exploring
optimal combinations of mechanical and electrical parameters through optimization techniques, or even
new configurations, requires efficient numerical tools to evaluate a large number of times the concurrent
solutions. Thus, as a second main point of thesis, reduced-order models of the electro-mechanical system
are developed here to overcome problems usually occurring when to interfacing computational expensive
models, and are experimentally validated to demonstrate their effectiveness.
The investigated system, aimed at harvesting energy from basement vibrations, is a cantilever multilayer
structure with a unique piezoelectric lamina glued on the supporting material (unimorph configuration,
like in many commercial devices), the latter being in charge of several functions: facilitating the
device installation, ensuring structural integrity and mechanical stability, and avoiding charge cancellation.
The connection between support layer and piezoelectric lamina is supposed to be perfect, without
any loss in mechanical energy transmission. Since piezoceramics perform only if working in their narrow
resonance bandwidth, the cantilevered plate configuration allows for accurate prediction of its bending
frequencies with a simple and efficient layout. Indeed, the investigated device exhibits significant oscillation
amplitudes at resonance frequencies under vertical basement vibrations applied at the clamped end.
Nonetheless, this layout can be easily adapted for different energy sources, like wind (for fluttering flags)
or waves (for wave energy converters).
Furthermore, the conversion circuit connected to the device is modelled as well, being indispensable to
convert AC voltage produced by the piezoelectric harvester into a DC voltage suitable for electronic devices,
and so of primary concern in building realistic energy harvesters. The circuit can also play a role in
enhancing the power production, increasing the efficiency of the whole energy chain. Moreover, storage
capacitance, fundamental to decouple power demand and production, is introduced. Finally, segmentation
of the device electrodes in width or length is explored, showing the benefit in supporting with the conversion
circuit functionalities and in avoiding charge cancellation, respectively. Being the latter not a frequent
topic in a thesis of the PhD course of Theoretical and Applied Mechanics, the electric system is explained
in detail to clarify every aspect of the investigated circuits behaviour.
As said previously, the system has been described with a reduced-order model (ROM), allowing an easier
data exchange between piezoelectric plate and electric conversion system, and a simple interaction with
the optimization algorithm. The piezoelectric plate is described as a multi-layer composite cantilevered
Euler – Bernoulli beam model with non-uniform material distribution through its length. Though geometrically
approximated, the beam model captures the system response in design excitation conditions. The
electromechanical coupling has been introduced in the structural model by using the linear piezoelectric
constitutive equations. A tip mass is positioned on the free edge to tune harvester’s natural frequencies
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with lower excitation frequencies and to enhance oscillations. Although a concentrated mass model is initially
considered, the not-negligible mass extension and associated rotational inertia effect are later taken
into account to obtain a more accurate natural frequency estimation. Finally, to simulate a non-perfect
clamping, yielding in rotation, a torsional spring is added at x = 0. The elastic constant K will be defined
by tuning the eigenfrequencies of the structure, analytically found, with those measured in dedicating vibration
testing of the simulated device. Both tip mass and yielding clamp have been introduced to better
describe possible commercial and custom solutions.
By analytically developing Lagrange equations from extended Hamilton’s principle including also electrical
potential energy and electric load interactions, a partial differential equation system is found and
then projected on the exact bending modes of the structure. Thanks to the analytical determination of the
not-uniform beam modes, the developed ROM allows mechanically decoupling modal oscillators and then
easily neglecting those modes not contributing to energy production. The set of ordinary differential equations
is numerically solved both in MATLAB and Simulink. The electric conversion system is developed
in Simulink by Simscape’s blocks, allowing for a simple simulation of voltage rectifier, storage capacitance,
Maximum Power Point Tracking system, and resistive load, with an easy data exchange with the ROM
model.
Results on non-uniform bending modes, and thus resonance frequencies, are then compared with a
3D high-fidelity model of the harvester in Comsol Multiphysics, to show that geometrical and mechanical
hypothesis do not undermine the overall consistency of the ROM.
To enhance the accuracy of the model, an identification of the torsional spring and modal damping
coefficients is carried out. Then, the developed theory is compared with experiments on a prototype of
the investigated energy harvester. The device is tested under sinusoidal excitation, imposed by a shaker,
finding the acceleration - voltage frequency transfer function of the energy harvester for different resistive
load and tip mass conditions. Experimental data and numerical results of the ROM model are found in
good agreement and thus validating the developed theory.
Finally, some mechanical and electrical parameters describing the main features of the system are chosen
as design variables to be optimized so as to maximise the power output. Different optimization procedures
are carried out with the patternsearch algorithm in MATLAB, investigating the device sensitivity
to parameters change and underlying the crucial co-dependency of mechanical and electrical behaviour,
linked together by means of the piezoelectric effect. Moreover, it is demonstrated how an optimization
procedure comprehensive of both mechanical and electrical design variables leads to better results than
separate and single discipline optimizations. Finally, the optimization problem with duty cycle as design
variable is explored, finding a more efficient solution than Open Circuit Voltage MPPT method, and thus,
leading the way for further studies on Machine Learning MPPT implementation.
As a concluding remark, the combination of a multi-physics efficient and robust ROM model with an
optimization approach constitutes the novelty proposed in this thesis to provide a useful tool for improving
the design of piezoelectric energy harvesters for real life applications, paving the way for a significant
increase in the device performances.