0
Research Papers

Numerical Continuation in a Physical Experiment: Investigation of a Nonlinear Energy Harvester

[+] Author and Article Information
David A. W. Barton1

Department of Engineering Mathematics, University of Bristol, Queen’s Building, Bristol BS8 1TR, UKdavid.barton@bristol.ac.uk

Stephen G. Burrow

Department of Aerospace Engineering, University of Bristol, Queen’s Building, Bristol BS8 1TR, UKstephen.burrow@bristol.ac.uk

1

Corresponding author.

J. Comput. Nonlinear Dynam 6(1), 011010 (Oct 04, 2010) (6 pages) doi:10.1115/1.4002380 History: Received May 23, 2009; Revised July 28, 2010; Published October 04, 2010; Online October 04, 2010

In this paper, we demonstrate the use of control-based continuation within a physical experiment: a nonlinear energy harvester, which is used to convert vibrational energy into usable electrical energy. By employing the methodology of Sieber (2008, “Experimental Continuation of Periodic Orbits Through a Fold,” Phys. Rev. Lett., 100(24), p. 244101), a branch of periodic orbits is continued through a saddle-node bifurcation and along the associated branch of unstable periodic orbits using a modified time-delay controller. At each step in the continuation, the pseudo-arclength equation is appended to a set of equations that ensure that the controller is noninvasive. The resulting nonlinear system is solved using a quasi-Newton iteration, where each evaluation of the nonlinear system requires changing the excitation parameters of the experiment and measuring the response. We present the continuation results for the energy harvester in a number of different configurations.

FIGURES IN THIS ARTICLE
<>
Copyright © 2011 by American Society of Mechanical Engineers
Your Session has timed out. Please sign back in to continue.

References

Figures

Grahic Jump Location
Figure 1

Left: A photograph of the harvester. Right: A schematic diagram of the harvester. The harvester consists of a cantilever beam with a tip mass. Magnets mounted on the tip mass provide a nonlinear spring force and a transduction mechanism.

Grahic Jump Location
Figure 2

A schematic diagram of how the components are connected

Grahic Jump Location
Figure 3

Experimental results from the heavily damped energy harvester: (a) a frequency sweep, (b) continuation, and (c) a frequency sweep and continuation combined. The harvester is excited with sinusoidal forcing with a fixed amplitude of 0.14 mm. The saddle-node bifurcation is marked with a circle.

Grahic Jump Location
Figure 4

Experimental results from the lightly damped energy harvester: (a) a frequency sweep, (b) continuation, and (c) a frequency sweep and continuation combined. In comparison with Fig. 3, the continuation is able to track out more of the branch due to a retuning of the controller gains part way around the branch. Near the tip of the resonance peak, the PD controller is again unable to sufficiently stabilize the periodic orbit.

Grahic Jump Location
Figure 5

Two coexisting periodic orbits at a forcing frequency of 30 Hz taken from the continuation run shown in Fig. 4. The dashed curve is the unstable orbit, the solid curve is the stable orbit, and the dotted curve is the forcing (a sinusoid with an amplitude of 0.14 mm).

Grahic Jump Location
Figure 6

Experimental results from the energy harvester with a small air-gap: (a) a frequency sweep, (b) continuation, and (c) a frequency sweep and continuation combined. Here, the asymmetries in the harvester are accentuated by the small air-gap and appear as a “bend” in the frequency response. The saddle-node bifurcations are marked with a circle.

Grahic Jump Location
Figure 7

Time profiles of three coexisting periodic orbits at 18.5 Hz taken from the continuation run shown in Fig. 6. The dotted curve shows the forcing (a sinusoid of amplitude 0.1 mm) of the three periodic orbits, the large and small amplitude solutions are stable (solid curves) and the medium amplitude solution is unstable (dashed curves).

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In