# Dynamics of the N-pendulum and its application to a wave energy converter concept

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## Abstract

This paper considers a design of an N-pendulum, which represents a special case of a physical pendulum. The design of the N-pendulum not only allows uncoupling the natural frequency of the pendulum from its length, but also provides easy control of the frequency and torque. The proposed design is stimulated by the idea of developing a wave power take-off system based on the parametric pendulum. Different designs are being considered and their dynamic characteristics are investigated with respect to the feasibility of such an application. Due to the observed low frequency of ocean waves, the size of a heaving simple pendulum should span along unrealistic sizes in order to be parametrically resonant. Thus, the N-pendulum is considered and the configurations that would fulfill these frequency requirements are sought. Last, numerical simulation is conducted for an under development experimental rig aiming to test the functionality of the concept, modelling the response of the N-pendulum.

## Keywords

Parametric pendulum Parametric resonance Tri-pendulum Power take-off Wave energy Heaving motion## 1 Introduction

The mathematical pendulum—a lumped mass oscillating on an unstretchable massless string, is an important system in studying the theory of vibrations. This fruitful example introduces us to a sinus type nonlinearity and things related to nonlinear systems, such as dependence of the pendulum’s period on its amplitude, stability etc. Since it is difficult to deal with the nonlinear equation of motion, most often this equation is linearized around its low equilibrium point. Then, the squared natural frequency of the system is proportional to the acceleration of gravity and inversely proportional to the pendulum’s length. Thus, the length of the pendulum should be around 10 m to get frequency of about \(1\) rad/s, and it must be around 100 m long to lower the frequency to \(0.316\) rad/s. Obviously, such a bulky device is difficult to build and operate. The only well known example of such a long pendulum (\(67\) m) is related to Foucault, who built it in 1851 to demonstrate the rotation of the Earth.

In this paper another design is proposed, which may be considered as a special case of the physical pendulum. However, in view of the practical application, which the proposed design is intended to be used for, the N-pendulum was proven to be much more suitable, since it allows not only making the natural frequency of the pendulum independent of its length, but also adjusting the frequency and torque of the system independently. The later is especially important in developing a wave power take-off device.

## 2 N-pendulum and its natural frequency

Let’s introduce an N-pendulum—a system of \(N\) rigidly connected to a common ring pendulums, which are positioned at equally spaced intervals of \(2\pi /N\), so that the bi-pendulum and tri-pendulum will have two and three arms of equal masses \(m\) correspondingly. Although it is possible to create a pendulum with not-equally-spaced arms, this paper does not consider such a design. All arms except one carry a lumped mass M, positioned at distance \(L_2\) whereas the last arm carries the same mass at distance \(L_1\), so that \(L_1>L_2\). The cases of even and odd number of arms could be considered separately.

It should be stressed that such a design is very easy to calibrate and it is not sensitive to small descrepancies of the arms during the manufacturing process, since moving the lumped masses to a proper distance \(\bar{L}_1\) would eliminate them.

## 3 Application of the tri-pendulum for a wave energy power take-off system

Wave energy converters (WECs) are used to convert wave energy to electricity. An important part of any WEC device is the power take-off (PTO) system, which actually provides a mechanism for extracting wave energy. Almost 40 years have passed since Salter [1] published his pioneering work on a wave energy PTO. Since then, various devices to capture wave energy have been developed. All of them may be classified in different ways and most of them may be found in the recent report by RenewableUK [2].

There are devices such as the heaving buoy, which benefit from using the vertical motion of waves. Some of them use a direct drive linear generator [3], others employ hydraulic rams or equivalent devices. So far there has been no wave energy converter which would generate electricity in a conventional way by spinning the rotor of an electric generator. In the paper by Xu et al [4] the authors outlined the possibility of using the parametric resonance phenomenon of a lumped mass pendulum having its pivot point vertically excited by waves, to create a wave PTO system. This idea was investigated numerically using the stochastic approach in [5] and it has been shown that in random sea environment it is still possible to achieve sustainable rotational motion of the pendulum. Although it is possible to generate electricity from an oscillatory motion, it leads to a number of problems related to converting a variable speed and variable voltage outcome. Rotational motion is also preferable since the kinetic energy stored in the system is bigger than that of oscillations.

The system in Eq. (6) is a strongly nonlinear one. The subharmonic and homoclinic bifurcations of the parametric pendulum leading to unstable motion were studied in [8] using the Melnikov method. Bishop et al [6] showed the path from the symmetry-breaking of the stable oscillatory response to the rotational and Szemplinska-Stupnicka and Tyrkiel [7] conducted an extensive study on the oscillatory and rotatory attractors. The harmonic balance method and the critical velocity criterion were used in Clifford and Bishop [9] to approximate the escape zone of non-rotating orbits. In Xu and Wiercigroch [10] approximate analytical expressions of the resonance curves were presented utilizing the multiple scales method. Several studies [4, 11, 12] reported the existence of rotational response of the parametric pendulum with approximate analytical solutions of the rotating orbits having been obtained in [13] using the pertrubation method. Moreover, Yurchenko et al [5] showed that rotational response is possible when the ocean waves are considered and modelled as a narrow-band stochastic process as well. A number of laboratory experiments have verified this idea, providing another possible wave energy conversion principle [15, 16].

^{1}. However, the numerical simulations, conducted by Xu et al. [4] for the deterministic system (6) revealed that the rotational motion of the pendulum is possible for values of \(\lambda >1\).

Although in this work the authors consider strictly vertical motion of the buoy, in general the motion may be more complex, which incorporates inclinations from the vertical position (pitch angle) due to incoming and outgoing waves [18, 19, 20]. The tri-pendulum should be mounted onto a floating platform. For waves of slightly over 1 m in amplitude, one can select \(L_1=2\) m and \(L_2=1.89\) m.

Furthermore, it should be stressed that the current design opens a wide range of possibilities, including capturing large waves and adjusting to a wave frequency by moving the lumped masses along the arms, using a threaded rod for instance.

## 4 Energy harvesting by different configurations of the tri-pendulum

Obviously, the inertia of the rotor is proportional to the squared lengths of the arms so its value is increasing to the upper right of each figure. Let us now shortly discuss the impact of the inertia on the potentially produced energy. A main advantage of the proposed concept is the straightforward PTO system where a conventional generator is to be attached directly to the shaft of the rotor. This is important in a sense that no intermediate system is required for transforming the wave power to electrical other than a coupling and a gearbox in contrast with existing WEC technologies where it is rather common for hydraulic systems, linear generators or turbines to be used for ultimately producing electricity. Having the generator driven directly by the rotor’s shaft would definitely decrease the complexity of the WEC. In that case, the generator would add in Eq. (2) a resisting torque which would be normalised by the rotor’s inertia. So, the bigger the inertia that is excited by waves the bigger the generator that could be set to run by it.

Closing this section, two remarks could be made upon the behaviour of the frequency with respect to the given parameters. First, from both Figs. 6a and 7a it is noticed that for bigger values of the shortest arm’s length, the required ratios to achieve the same frequency increase as well. This means that the length increase is not proportional in between the arms and rather extreme values would be required should the shortest arm was to be heavily lengthened. Secondly, these extreme values could well enough introduce shock-like excitations of the pivot which would result in undesirable response of the rotor, let alone the thereafter structural concerns. In the same manner, it is fortunate that the sought frequency range is achievable for \(n,\gamma , \gamma _2, \gamma _3\) ratio values close to 1 and thus extreme phenomena are avoided.

Performing a rough estimation of the mechanical power harvested from ocean waves by use of the tri-pendulum, consider the design as described last with \(n=M_1/M_2\) and \(\gamma =L_1/L_2\) and suppose \(M_2=500\) kg and \(L_2=5\) m. That would direct us to Fig. 7d. In order to approach the necessary natural frequency one would have to choose \(n,\gamma \approx 1.1\) leading to \(I/M_2 \approx 83~\mathrm{m}^{-2}\). Taking into account that the mechanical power of the rotor could be expressed as \(P=I\ddot{\theta }\dot{\theta }\) and considering a period-1 rotational point of the parameters \(\left( \nu ,\lambda \right) \), then the average power extracted by the heaving motion of waves and transformed to mechanical power of the rotor would be around 4 kW.

## 5 Discussion of an experimental rig

Parametric excitation of a pendulum has attracted significant interest of the scientific community regarding the stability boundaries, chaotic attractors and the bifurcations that have been observed. It is only reasonable that experimental investigation has been part of the ongoing research with the scope varying from identification of the chaotic attractors [21, 22] to the bifurcations occurring in an inverted pendulum [23] or even a double pendulum [24]. In the frame of developing a pendulum-based WEC, rotational response of the parametric pendulum driven by a shaker was sought [15] and a benchmark—proof of concept—experimental study was conducted in a wave tank [16].

However, most of the previous experiments concentrated on considerably higher frequencies than those observed in ocean waves. At the time, an experimental rig of an ongoing investigation is under development aiming to demonstrate the functionality of a tri-pendulum as it was introduced in the previous sections. The exciting force is designed to simulate ocean waves by applying frequencies and amplitudes much closer to those occurring in nature.

Figure 10d assumes a size ratio \(j\rightarrow \infty \) practically devolving to the perfect harmonic excitation described in Eq. (6). This plot resembles as expected the ones previously created for the perfect harmonic signal [5] and is shown to facilitate comparison. Further on, three values for the size ratio \(j=2,3,4\) are considered and the corresponding plots are shown in Fig. 10a–c. A first observation would be that the boundaries separating oscillatory and asymptotically stable motion from the rotary and chaotic ones remain almost independent of the parameter \(j\) something that especially holds for the primary resonance zone around \(\nu =2\). However, the picture is rather different when it comes to rotational and chaotic regions. While in the purely harmonically excited case the two rotational regions merge for \(\lambda >3.2\), they remain completely distinct for \(j=2\) shown in Fig. 10a, separated by a long chaotic attractor. Increasing the size ratio reshapes the plots towards the limiting case of \(j\rightarrow \infty \) retaining though distinguishable differences in the internal structure of the map. Thus, an experimental investigation as the one described that would be based on Eq. (6) and not take into account the subharmonic excitation, would reflect significantly erroneous results depending on the selected size ratio \(j\).

## 6 Conclusions

This paper offers a design of the N-pendulum, which can be considered as a special case of a physical pendulum. Nevertheless, the proposed N-pendulum is a much better choice for some applications and in particular for the wave energy harvesting application, because it is capable of not only achieving low frequencies staying relatively small in size, but also being able to easily control its natural frequency and the created torque. It has been shown that the proposed design can be used as the first horizontal axis wave PTO system. Besides, other major advantages of the proposed design are its simplicity, possible automatic calibration and simple start-stop mechanism, all of which are important features of energy converters. Although the numerical results for the system’s response have been provided, their experimental validation is required. For this purpose the proposed design has been assembled and about to be tested at the Heriot-Watt University.

## Footnotes

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