1. Introduction

Energy harvesting is the conversion of ambient energy present in the environment into
electrical energy. It is identical in principle to large-scale renewable energy
generation, for example, solar or wind power, but very different in scale. While
large-scale power generation is concerned with megawatts of power, energy harvesting
typically refers to micro- and/or milli-watts, *i.e*. much smaller
power generation systems. The development of energy harvesting has been driven by
the proliferation of technologies such as autonomous wireless electronic systems, a
classic example being wireless sensor nodes which combine together to form wireless
sensor networks; for this type of systems local power supply is an attractive
option. By converting ambient energy in the environment the energy harvester can
provide the required electric power for the lifetime of the wireless system which is
also free to be embedded or placed wherever it is best suited to perform its
function. Energy harvesting typically exploit kinetic, thermal, solar, or
electromagnetic radiation sources. Thermal gradients can be exploited by using
thermoelectric generators whereas solar energy is harvested using photovoltaics.
However, the most prominent type of devices are mechanical vibration energy
harvesters that convert kinetic energy via electromagnetic, electrostatic, or
piezoelectric transduction into electrical energy^{1}-^{5}.

Mechanical energy harvesters, also known as vibration power generators, are
typically, although not exclusively, inertial spring-mass systems where electrical
power is extracted by employing one or a combination of different transduction
mechanisms. Early studies considered linear springs and harmonic oscillators and
treated the external vibrations as sinusoidal vibrations. As most vibration power
generators are resonant systems, they generate maximum power when the resonant
frequency of the generator matches the ambient vibration frequency, known as
resonant energy harvesting^{1}. Adaptive
generators try to minimize the difference between these two frequencies in order to
maximize the amount of generated power^{2}, ^{4}, ^{6}.
Nearly all current vibration transducers operate in this regime^{7}. This approach presents numerous drawbacks, being one of the
most important ones that the linear harvester resonant peak is necessarily very
narrow^{8}. Thus, if the environmental
vibration frequency deviates from the harvester resonance very little power is
generated. To overcome this limitation various groups have begun to study
mass-spring systems with nonlinear springs and nonlinear oscillators^{9}-^{12}.
Many important results have been obtained if the broadband ambient vibrations are
modeled by Gaussian white noise. For example, it has been determined, using the
Fokker-Planck equation to describe Duffing-type energy harvesters, that the mean
power output of the device is not affected by the nonlinearity of the spring^{13}-^{14}.
Also the upper bounds on the power output of both linear and nonlinear energy
harvesters driven by Gaussian white noise have been obtained^{15}. The latter ones can be advantageous since the size of the
device can be reduced without affecting the power output^{14}. Now, while some environmental excitations exhibit the
characteristics of broadband white noise, many others have most of their energy
trapped within certain frequency bandwidths, *i.e*. external colored
noise. After some early experimental and simulation studies^{12}-^{16 }the power output
of both a monostable^{13} and a bistable Duffing
oscillator with a symmetric potential^{17}
driven by Ornstein-Uhlenbeck noise was determined by approximate methods, and the
exact analytical expressions for the net electrical power and efficiency of the
conversion of the power supplied by exponentially correlated noise into electrical
power was derived for a linear electromechanical oscillator employed as an energy
harvester^{18}.

In almost all theoretical models mentioned above, the coupling between the mechanical
oscillator and the ambient noise were supposed to be linear. This is because the
energy dissipation (damping) in previous studied systems were supposed to be linear
in general. However, damping has an important impact on the dynamic behavior of
submicrometre mechanical resonators, and recent researches have revealed that
nanostructures with high aspect ratio such as nanotubes and graphene nanoribbons can
be easily driven into nonlinear dissipation regime^{19}. This nonlinear damping has been so far studied mainly within the
field of thermal transport in a spin-boson nanojunction^{20}, in nonlinear two-level molecular junctions^{21}, in a monomodal harmonic molecular junction
(a single harmonic oscillator)^{22}, and in a
one-dimensional lattice of coupled nearest-neighbor harmonic oscillators^{23}.

In this work we propose to study the resonator driven by colored noise studied in
Ref.^{18} with a nonlinear damping between
the mechanical oscillator and the source of ambient noise to explore its influence
on the performance of the device as an energy harvester. The proposed nonlinear
damping has been previously considered within the context of a micromechanical
oscillator model in Ref.^{19} to properly
account for various experimental observations made in mechanical resonators based on
carbon nanotubes and graphene sheets, as mentioned above.

The rest of the paper is organized as follows: in Sec. 2 we present the model as well as our methodology. Results for both a linear and nonlinear oscillator are reported in Sec. 3. Finally, in Sec. 4 we discuss the results so far obtained and propose ways to further improve them.

2. Model and methodology

The herein considered energy harvester is a device that converts the power supplied
by external noise into electrical energy. This process begins with the damped
oscillator being driven by the external noise. Its kinetic energy is then converted
via a piezoelectric transducer mechanism into electrical energy that is then stored
in a capacitor. The mechanical part of the device is described by the equation for
the momentum of the stochastically driven damped oscillator of mass

with the dot standing for temporal derivative. In this equation ^{24}, but is out of the scope of the objectives
of the present work. Since our goal is to understand the effect of the nonlinear
damping on the performance of the device we have chosen simplest,
*i.e*. linear, expression for the connecting function in the
analysis below.

In this work we are considering a Ornstein-Uhlenbeck (OU) random force, with mean

with a Gaussian white noise correlation of

The simulations are performed by solving numerically the Langevin equations () by
using the so-called Heun algorithm; trajectories are computed over an interval of

Being ^{18})

where ^{18}.

3. Results

In Fig. 1(a) we present the comparison of the
correlations *i.e*. *i.e*. for moderate
displacements, is very similar to the one corresponding to

Since in both considered instances the maximum efficiency does not occur when the net
electrical power is maximal and vice versa, we studied how the maximum net
electrical power *i.e*. both are weakly dependent on the frequency for

The variation with respect to

Summarizing the previous discussion, the characteristic frequency of the noise at the
maximum power

4. Discussion and conclusions

Our results on the performance of a nonlinear electromechanical oscillator with a nonlinear coupling with an external finite-bandwidth ambient noise seem to indicate that the performance of the system as an energy harvester is only weakly affected by the presence of the aforementioned nonlinear coupling, since the constants that define the latter have to take exceedingly large values in order to affect in a mensurable way the correlations that characterize the performance of the considered energy harvester. In particular, the correlations wherewith both the maximum power and efficiency are defined present a very weak dependence on the nonlinear parameters of the interaction with the source of the ambient noise.

A constant among the proposed energy-harvesting devices is that the combined goals of
both maximum power and efficiency cannot be attained at the same time, being the
present one no exception. A compromise in the value of all parameters has to be
made, and thus we determined that the best performance of the energy harvester is
obtained by taking low values for the characteristic charging time of the capacitor ^{9}-^{12}. We intend to investigate this particular possibility in a future
work.