ZAMM · Z. Angew. Math. Mech. 98, No. 3, 349–366 (2018) / DOI 10.1002/zamm.201700007
Pendulum energy converter excited by random loads
, Kevin Korner
, Edwin Kreuzer
, and Daniil Yurchenko
Institute of Mechanics and Ocean Engineering, Hamburg University of Technology, Hamburg, Germany
Department of Mechanical and Civil Engineering, California Institute of Technology, Pasadena, California,
Institute of Mechanical, Process & Energy Engineering, Heriot-Watt University, Edinburgh, United Kingdom
Received 11 January 2017, revised 6 July 2017, accepted 22 September 2017
Published online 8 November 2017
Key words Parametric excitation, pendulum dynamics, stochastic averaging, energy harvesting, random seas.
MSC (2010) 60H35
We present new solutions for the dynamics of a pendulum energy converter which is vertically excited at its suspension
point. Thereby, we deal with a random excitation by a non-white Gaussian stochastic process. We formulate the pendulum
energy converter as a weakly perturbed Hamiltonian system. The random process across the energy levels of the
Hamiltonian system is then approximated by a Markov process, which is obtained by stochastic averaging. This procedure
leads to analytical results for the energy of the pendulum motion, which are used for analyzing the required probability
of reaching higher energy states of the pendulum energy converter in order to maximize the harvested energy.
2017 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
Energy harvesting from a source, whose nature is random, is a fundamental problem in the ﬁeld of renewable energy.
Several new concepts of ocean wave energy converters were recently developed and studied, for example [2, 17, 24, 32].
Recently, the rotational motion of a pendulum has become a topic of interest in developing wave energy converters, since
one possible method of energy generation is to use ocean gravity waves to excite the pivot of a pendulum in order to
induce and maintain rotational motion, as described in [2, 32]. A detailed study on the stochastic excitation of a pendulum
based energy converter resulting from random wave motion can be found in . The induced motion can be converted to
electrical energy. This type of wave energy converter also has the added beneﬁt of a low environmental impact. Because it
is a point absorber of energy, there is great ﬂexibility in the layout of energy farms. This means that this type of system
can have a very small effect on the local ecosystem and offshore economic activity. As stated in , wave energy has
been recognized as one of the most promising resources of renewable energy due to the notably high power density of 9.42
compared with wind and solar energy with power density of 0.58 and 0.17 kW/m
Possible sources of excitation are not limited to ocean waves. Moreover, the parametric pendulum has a very rich
dynamical behavior  and has applications in various other problems, such as mechanical, electrical, MEM, optical,
and other systems, cf. [16, 23]. Therefore, our analysis deals with the general dynamics of a randomly excited pendulum.
In the case of harmonic excitation, approximate analytical solutions for oscillatory and rotational motion of a parametric
pendulum were obtained in  by using the method of multiple scales. In  it was also shown that a controller based
on the Time-Delay Feedback control method can maintain rotational motion of the parametric pendulum, even in case of a
harmonic excitation which is disturbed by white noise. However, new results are necessary in order to control the pendulum
motion if the excitation of the pendulum is a non-white random process. Such excitation is encountered for example in real
sea states. Due to the signiﬁcant nonlinearity and the involved random excitation, researchers relied on numerical methods
and Monte Carlo simulation in order to obtain new results for large amplitude oscillations of a randomly excited pendulum
In the context of energy harvesting, it is important that the energy level of a randomly excited energy converter will
reach higher energy levels with high probability. The main purpose of this article is to obtain analytical results for the
energy of the pendulum motion and use these results for analyzing the required probability of reaching higher energy states.
This is fundamental for the development of appropriate control methods for energy generation. In order to achieve such
a result, we reduce the dimension of the original process to a one-dimensional process by stochastic averaging. Previous
results on stochastic averaging involved asymptotic methods with respect to a small parameter in order to determine the
Corresponding author, e-mail: email@example.com, Phone: +49 40 428 78 2209, Fax: +49 40 428 78 2028
2017 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim