Pendulum Wave Energy Converters
Here we provide insight on how nonlinear dynamics techniques can be applied to design and analyse the response of wave energy systems. This is now a high priority at both national and international levels in order to achieve net-zero emissions targets as soon as possible.
Clean energy growth continues to be one of the grand challenges identified by the UK’s industrial strategy, predominantly aimed at the rapid development of efficient and cost-effective low carbon technologies, and to achieve a future mix of energy solutions that is more harmonious with the environment. Still today, around 80% of the world energy consumption is supplied from fossil fuels and so the energy transition phase to a cleaner energy mix will most likely take a few decades to properly realise its ultimate goal.
In this context, and for many years now there have been developments in Wave Energy Converter (WEC) technology, and these technology development efforts have resulted in several WEC designs of which there are four main types: (i) point absorbers, (ii) wave attenuators, (iii) oscillating water column mechanisms and (iv) overtopping devices. However, many of these designs have been deemed to be unsuitable for long-term reliable operation for a number of reasons that include low mechanical robustness and reliability of energy extraction; issues relating to the ability to deploy these systems offshore; high in-service maintenance requirements in addition to challenging seabed footprint issues. Another drawback is the typically large size-to-energy extraction ratio required, which makes some of these devices uneconomically viable. Lessons learned from past designs indicate that any effective wave energy convertor must possess a high efficiency, have low structural weight, a minimum number of moving parts and have a direct mechanism to convert mechanical energy into electrical energy.
In the past few years, a detailed dynamic analysis of the energy extraction via pendula mechanisms and supported by experimental studies on electrodynamic shakers and in wave tanks have been undertaken by scholars at the Centre for Applied Dynamics Research (CADR) at the University of Aberdeen. This research and associated experimental work have demonstrated the significant potential of pendula-based WECs (see, for example, references [1-6] below). Specifically, pendula-based WECs satisfy all of the four criteria listed above, and if the pendula are encapsulated in a vessel in the form of a floating buoy or submarine, the WEC is not exposed to the harsh marine environment. A schematic of such a buoy-type WEC is shown in Fig. 1(c).
This new paradigm for extraction of wave energy is proposed by exploring the complex nonlinear dynamics of pendula-based systems operating predominantly in a rotary mode. The novelty comes from utilising the natural but often overlooked rotary motion of a driven pendulum, which is very stable and from the WEC point of view is characterised by direct energy conversion with no stop points. This concept was demonstrated by a dual pendula system that was developed by CADR, which is shown in Fig. 1(a), and for which a computed experimentally validated response map is shown in Fig. 1(b). Pendula-based WEC designs based on this concept can lead to highly effective solutions in terms of simplicity, efficiency, stability, reliability, robustness and cost. Fig. 1(c) shows a conceptual design of a point absorber type of such a WEC that has two pairs of synchronised counter-rotating pendula in the form of four discs (only two are visible in the figure).
At the heart of this new approach is the optimisation of the complex dynamics of pendula systems that are forced by an arbitrary 3D oscillatory motion, and that is realised by encapsulating such pendula systems within a subsea vessel as shown on Fig. 1(c). During the direct conversion from arbitrary oscillatory to controlled rotary motion / energy, each pendulum on its own acts as a planar pendulum. For a wide range of excitations, the natural response of a pendulum is to rotate whenever the effective amplitude of the oscillatory forcing is above a critical value, known as the critical amplitude. Thus, if we attach a pendulum to an electrical generator, we create an elegant and controlled mechanism for energy extraction from irregular sea waves. The nonlinearities of the pendula system, which is mainly geometrical, can naturally convert a wide range of excitation patterns (ranging from regular to random) into nearly uniform rotations of the generator that the pendulum system drives.
This active project is a continuation of extensive research [1-6] on efficient ways to harvest and convert mechanical energy and where strongly nonlinear interactions are being exploited. The pendula-based WECs proposed here, in comparison to existing WECs, can offer much higher efficiency and power-to-mass ratio and can operate effectively over a wide range of offshore environmental conditions.
Figure 1. (a) Two-pendula system developed at the University of Aberdeen. (b) Experimentally calibrated map of the dynamic responses of a pendula based system showing a large area of rotary responses. (c) Schematic view of a potential design of a buoy-type WEC where all the moving parts, power take-off system, sensors, instrumentation and control systems are protected by a robust buoy shell.
 Wiercigroch, M., Najdecka, A., Vaziri, V. (2011). Nonlinear dynamics of pendulums system for energy harvesting. Plenary lecture at 10th Intl Conf. Vibration Problems, 5-8 Sept 2011, Prague. In ICOVP 2011 Proc., 35-42, Springer. (https://abdn.pure.elsevier.com/en/publications/nonlinear-dynamics-of-pendulums-system-for-energy-harvesting)
 Horton, B., Sieber, J., Thompson, J. M. T., Wiercigroch, M. (2011) Dynamics of the nearly parametric pendulum. Int. J. Non Linear Mech. 46, 436-442. (https://abdn.pure.elsevier.com/en/publications/dynamics-of-the-nearly-parametric-pendulum)
 Nandakumar, K., Wiercigroch, M., Chatterjee, A. (2012) Optimum energy extraction from rotational motion in parametrically excited pendulum. Mech. Res. Commun. 43, 7-14. (https://abdn.pure.elsevier.com/en/publications/optimum-energy-extraction-from-rotational-motion-in-a-parametrica)
 Strzalko, J., Grabski, J., Wojewoda, J., Wiercigroch, M., Kapitaniak, T. (2012) Synchronous rotation of the set of double pendula: Experimental observations. Chaos 22, 4047503. (https://abdn.pure.elsevier.com/en/publications/synchronous-rotation-of-the-set-of-double-pendula-experimental-ob)
 Najdecka, A., Kapitaniak, T., Wiercigroch, M. (2015) Synchronous rotational motion of parametric pendulums. Int. J. Non Linear Mech. 70, 84-94. (https://abdn.pure.elsevier.com/en/publications/synchronous-rotational-motion-of-parametric-pendulums)
 Vaziri, V., Najdecka, A., Wiercigroch, M. (2014) Experimental control for initiating and maintaining rotation of parametric pendulum. Eur. Phys. J. Spec. Top. 223(4), 795-812. (https://abdn.pure.elsevier.com/en/publications/experimental-control-for-initiating-and-maintaining-rotation-of-p)
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