Good morning, everyone! It is my pleasure to present our recent work on nonlinear energy harvesting. The work was done by Dr Ze-Qi Lu, a Postdoc fellow, and me. We come from Shanghai University. First of all, I’ll show you the outline of the talk.

The talk is divided into 5 parts. It begins with an introduction to explain the background and the motivation of the investigation. Then the physical model and the mathematical model are presented and the frequency response equation is derived from the method of harmonic balance. It follows some numerical results to demonstrate the forming and the disappearance of bubble shaped frequency response curves. Then some numerical simulations are done to examine the harvesting performance under Gaussian random excitation. Finally, some concluding remarks end the talk.

Now turn to the Introduction.  

As we well know, energy harvesting is a significant issue. Especially, vibratory energy harvesting is to transform the kinetic energy of waste vibration into electricity.

Linear models have been widely used to design, to analyze, and to simulate energy harvesters. However, there is an essential limitation of linear energy harvesting. As a resonator, a linear energy harvester works only in a narrow frequency range near the resonance. Actually, its reason can be found in any textbooks on vibration.

To overcome the limitation, several approaches have been proposed. For example, multi-modal energy harvesters have been designed so that the working frequency range contains several resonant frequencies. Anyway, our talk is not on the topic. Another promising approach is to introduce nonlinearity intentionally. Our work belongs to this aspect.

Among many features of nonlinear oscillation, the jumping is a striking one. As we all know, an amplitude-frequency response curve of a linear oscillator is with a symmetric peak. The introduction of nonlinearity may destroy the symmetry. Hardening nonlinearity bends the peak to the increasing frequency direction, while softening nonlinearity the decreasing frequency direction. As you may observe, the bending of the frequency response curve makes the amplitude sufficiently large in a wider frequency range. Therefore, the jumping provides with a possibility of developing broadband vibration-based energy harvesters. The detailed discussions can be found in a recent review paper published in Applied Mechanics Reviews.  

If jumping can enhance energy harvesting, it is a natural idea that double-jumping, jumping in both sides, does the job better. Internal resonance leads to double-jumping. Our group constructed conceptually an electromagnetic energy harvester with internal resonance. The method of multiple scales was developed to reveal the double-jumping in the power frequency response curves. The analytical outcome was supported by the numerical integrations. Numerical results also showed that the internal resonance designproduces more power than other designs under the Gaussian white noise. The results were published in ASME Journal of Applied Mechanics. By the way, our analytical and numerical prediction on enlarged band width has been experimentally supported. For example, an experiment was reported in recent Applied PhysicsLetters. Our group’s experimental work would probably published in ASME Journalof Vibration and Acoustics. In double-jumping, bending is back-to-back. That is, the right branch of the response curve bends to the right, while the leftone to the left. Now, the problem is what happens if the bending is face-to-face. That is the right branch of the response curve bends to the left, and the lift right. In this case, there may be a bubble shaped response curve. It is the motivation of the work.

In other words, the objective of the work is to explore the possibility of using bubble shaped response curves to enhance vibration-based energy harvesting. To do so, a system should be conceptually designed with the bubble shaped response curve. Fortunately, previous works on vibration isolations shed some lights on the issue. An electromagnetic energy harvester is proposed as a linear-nonlinear coupled system. In addition, numerical simulations are done to examine the harvester performance under Gaussian white noise.