Pendulum energy converter excited by random loads

Edwin  Kreuzer

Pendulum energy converter excited by random loads

Edwin Kreuzer

2017, ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik

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Abstract

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.

Figures (14)

Fig. 1 Contour lines of H(x, y) for & = 1. perturbed Hamiltonian system (5) excited by the stationary random process &,. Contour lines of the Hamiltonian are sho in Fig. 1. The fixed points of system (5) without dissipation and random perturbation, i.e. ¢ = 0, are

exist, and the process H converges, as € —> 0, weakly on the time interval of order O(1/e) to a diffusion Markov proces. H satisfying the Ité stochastic differential equation

Conveniently, we can use the same equations to solve for stochastic averaging outside the separatrix. In this regime, we can use Egs. (19) as well as a series of transformations in order to arrive at the analytical and semi-analytical results. While solving over the motion of a period, we can now use the bounds —z <x <z. 3.2.1 White noise case For the white noise case, we use Eqs. (26). We can use the transformation applied in the libration section that allows us to integrate over space, which transforms the equation for drift into

Fig. 2 Pierson-Moskowitz spectrum and approximation by a CARMA(2, 1) spectrum for w,, = 1 and H, = 91 process very efficiently [8]. For reliable results it is sufficient to use a CARMA(2,1) process given by the following differential equation

Fig. 5 Probability density of the pendulum system under white noise excitation for 1 =0.6, a=1, 7, = 0.01, and ~ = 0.3. Fig. 4 Heave acceleration spectrum in a Pierson-Moskowitz sea state and CARMA(2,1) approximation for significant wave height H, = 1 mand modal frequency w,, = 1.8 rad/s.

Fig. 3 Heave acceleration response amplitude operator R AO ; of a vertical cylinder. Table 1 Parameter specifications of the tri-pendulum experimental rig.

Fig. 7 Probability density of the pendulum wave energy con- verter for A = 0.25, a = 1, Yi = 0.01, and ~ = 0.3. Heave ac- celeration spectrum due to a Pierson-Moskowitz sea state with On = 1.8. Fig. 6 Probability density of the pendulum system for 1 from 0.3 to 0.6, @ = 1, 7; = 0.01, and % = 0.3.

Fig.8 Probability density of the pendulum wave energy converter for A from 0.2 to 0.35, @ = 1, 7, = 0.01, and 7) = 0.3. Heave acceleration spectrum due to a Pierson-Moskowitz sea state with w,, = 1.8.

Fig.9 First passage probability of the pendulum system for reaching the scaled rotational energy H, = 2 before reaching the low energy H, = 0.01, starting at the energy Ho, with a = 1, y, = 0.01, and ~ = 0, for a Pierson-Moskowitz sea state with X = H, /g, different modal frequencies w,,, and RAO; according to Fig. 3.

Fig. 10 First passage probability of the pendulum system for reaching the scaled rotational energy H, = 2 before reaching the low energy H; = 0.01, starting at the energy Ho, with a = 1, y, = 0.01, and ~ = 0, for a JONSWAP sea state with 2 = H,/g, different modal frequencies w,,, and RAO ; according to Fig. 3.

Fig. 11 First passage probability of the pendulum system for reaching the scaled rotational energy H, = 2 before reaching the low energy H; = 0.01, starting at the energy Ho, witha = 1, y, = 0.1, and % = 0, for a Pierson-Moskowitz sea state with A = H, /g, different modal frequencies w,,, and RAO; according to Fig. 3.

Sze-Hong Teh

Journal of Theoretical and Applied Mechanics, 2016

A method to extract energy from an excitation which is stochastic in nature is presented. The experimental rig comprises a pendulum, and a vertical excitation is provided by a solenoid. The control input assumed in the form of a direct current motor, and another motor, used in reverse, acts as a generator. The stochastic excitation has been achieved by varying the time interval between switching the RLC circuit on and off according to a random distribution. Such non-linear vertical excitations act on an oscillatory system from which a pendulum is pivoted. The Pierson-Moskowitz spectrum has been chosen as the random distribution while an inverse transform technique has been used for generation of the random excitation signal in LabVIEW environment. Moreover, a bang-bang control algorithm has been implemented to facilitate rotational motion of the pendulum. Experimental observations have been made for various noise levels of vertical excitations, and their implication on energy generation has been discussed. A positive amount of energy has been extracted for a minimal amount of the control input.

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