skip to main content

Title: Multiple-Scale Analysis of a Tunable Bi-Stable Piezoelectric Energy Harvester
Abstract This paper presents the theoretical modeling and multiple-scale analysis of a novel piezoelectric energy harvester composed of a metal cantilever beam, piezoelectric films, and an axial preload spring at the moveable end. The harvester experiences mono- and bi-stable regimes as the stiffness of preload spring increases. The governing equations are derived with two high-order coupling terms induced by the axial motion. The literature shows that these high-order coupling terms lead to tedious calculations in the stability analysis of solutions. This work introduces an analytical strategy and the implementation of the multiple-scale method for the harvester in either the mono- or bi-stable status. Numerical simulations are performed to verify the analytical solutions. The influence of the electrical resistance, excitation level, and the spring pre-deformation on the voltage outputs and dynamics are investigated. The spring pre-deformation has a slight influence on the energy harvesting performance of the mono-stable system, but a large effect on that of the bi-stable system.
; ;
Award ID(s):
Publication Date:
Journal Name:
ASME Letters in Dynamic Systems and Control
Sponsoring Org:
National Science Foundation
More Like this
  1. Mathematical analysis of the well known model of a piezoelectric energy harvester is presented. The harvester is designed as a cantilever Timoshenko beam with piezoelectric layers attached to its top and bottom faces. Thin, perfectly conductive electrodes are covering the top and bottom faces of the piezoelectric layers. These electrodes are connected to a resistive load. The model is governed by a system of three partial differential equations. The first two of them are the equations of the Timoshenko beam model and the third one represents Kirchhoff’s law for the electric circuit. All equations are coupled due to the piezoelectric effect. We represent the system as a single operator evolution equation in the Hilbert state space of the system. The dynamics generator of this evolution equation is a non-selfadjoint matrix differential operator with compact resolvent. The paper has two main results. Both results are explicit asymptotic formulas for eigenvalues of this operator, i.e., the modal analysis for the electrically loaded system is performed. The first set of the asymptotic formulas has remainder terms of the order O ( 1 n ) , where n is the number of an eigenvalue. These formulas are derived for the model with variable physicalmore »parameters. The second set of the asymptotic formulas has remainder terms of the order O ( 1 n 2 ) , and is derived for a less general model with constant parameters. This second set of formulas contains extra term depending on the electromechanical parameters of the model. It is shown that the spectrum asymptotically splits into two disjoint subsets, which we call the α -branch eigenvalues and the θ -branch eigenvalues. These eigenvalues being multiplied by “i” produce the set of the vibrational modes of the system. The α -branch vibrational modes are asymptotically located on certain vertical line in the left half of the complex plane and the θ -branch is asymptotically close to the imaginary axis. By having such spectral and asymptotic results, one can derive the asymptotic representation for the mode shapes and for voltage output. Asymptotics of vibrational modes and mode shapes is instrumental in the analysis of control problems for the harvester.« less
  2. The chief objective of this paper is to explore energy transfer mechanism between the sub-systems that are coupled by a nonlinear elastic path. In the proposed model (via a minimal order, two degree of freedom system), both sub-systems are defined as damped harmonic oscillators with linear springs and dampers. The first sub-system is attached to the ground on one side but connected to the second sub-system on the other side. In addition, linear elastic and dissipative characteristics of both oscillators are assumed to be identical, and a harmonic force excitation is applied only on the mass element of second oscillator. The nonlinear spring (placed in between the two sub-systems) is assumed to exhibit cubic, hardening type nonlinearity. First, the governing equations of the two degree of freedom system with a nonlinear elastic path are obtained. Second, the nonlinear differential equations are solved with a semi-analytical (multi-term harmonic balance) method, and nonlinear frequency responses of the system are calculated for different path coupling cases. As such, the nonlinear path stiffness is gradually increased so that the stiffness ratio of nonlinear element to the linear element is 0.01, 0.05, 0.1, 0.5 and 1.0 while the absolute value of linear spring stiffness ismore »kept intact. In all solutions, it is observed that the frequency response curves at the vicinity of resonant frequencies bend towards higher frequencies as expected due to the hardening effect. However, at moderate or higher levels of path coupling (say 0.1, 0.5 and 1.0), additional branches emerge in the frequency response curves but only at the first resonant frequency. This is due to higher displacement amplitudes at the first resonant frequency as compared to the second one. Even though the oscillators move in-phase around the first natural frequency, high amplitudes increase the contribution of the stored potential energy in the nonlinear spring to the total mechanical energy. The out-of-phase motion around the second natural frequency cannot significantly contribute due to very low motion amplitudes. Finally, the governing equations are numerically solved for the same levels of nonlinearity, and the motion responses of both sub-systems are calculated. Both in-phase and out-of-phase motion responses are successfully shown in numerical solutions, and phase portraits of the system are generated in order to illustrate its nonlinear dynamics. In conclusion, a better understanding of the effect of nonlinear elastic path on two damped harmonic oscillators is gained.« less
  3. Fiber-based flexible piezoelectric composites with interdigitated electrodes, namely Macro-Fiber Composite (MFC) structures, strike a balance between the deformation and actuation force capabilities for effective underwater bio-inspired locomotion. These materials are also suitable for vibration-based energy harvesting toward enabling self-powered electronic components. In this work, we design, fabricate, and experimentally characterize an MFC-based bio-inspired swimmer-energy harvester platform. Following in vacuo and in air frequency response experiments, the proposed piezoelectric robotic fish platform is tested and characterized under water for its swimming performance both in free locomotion (in a large water tank) and also in a closed-loop water channel under imposed flow. In addition to swimming speed characterization under resonant actuation, hydrodynamic thrust resultant in both quiescent water and under imposed flow are quantified experimentally. We show that the proposed design easily produces thrust levels on the order of biological fish with similar dimensions. Overall it produces thrust levels higher than other smart material-based designs (such as soft material-based concepts), while offering geometric scalability and silent operation unlike large scale robotic fish platforms that use conventional and bulky actuators. The performance of this untethered swimmer platform in piezoelectric energy harvesting is also quantified by underwater base excitation experiments in a quiescent watermore »and via vortex induced-vibration (VIV) experiments under imposed flow in a water channel. Following basic resistor sweep experiments in underwater base excitation experiments, VIV tests are conducted for cylindrical bluff body configurations of different diameters and distances from the leading edge of the energy harvesting tail portion. The resulting concept and design can find use for underwater swimmer and sensor applications such as ecological monitoring, among others.« less
  4. Abstract Controlling and manipulating elastic/acoustic waves via artificially structured metamaterials, phononic crystals, and metasurfaces have gained an increasing research interest in the last decades. Unlike others, a metasurface is a single layer in the host medium with an array of subwavelength-scaled patterns introducing an abrupt phase shift in the wave propagation path. In this study, an elastic metasurface composed of an array of slender beam resonators is proposed to control the elastic wavefront of low-frequency flexural waves. The phase gradient based on Snell’s law is achieved by tailoring the thickness of thin beam resonators connecting two elastic host media. Through analytical and numerical models, the phase-modulated metasurfaces are designed and verified to accomplish three dynamic wave functions, namely, deflection, non-paraxial propagation, and focusing. An oblique incident wave is also demonstrated to show the versatility of the proposed design for focusing of wave energy incident from multiple directions. Experimentally measured focusing metasurface has nearly three times wave amplification at the designed focal point which validates the design and theoretical models. Furthermore, the focusing metasurface is exploited for low-frequency energy harvesting and the piezoelectric harvester is improved by almost nine times in terms of the harvested power output as compared to themore »baseline harvester on the pure plate without metasurface.« less
  5. Abstract Thin-walled corrugated tubes that have a bending multistability, such as the bendy straw, allow for variable orientations over the tube length. Compared to the long history of corrugated tubes in practical applications, the mechanics of the bending stability and how it is affected by the cross sections and other geometric parameters remain unknown. To explore the geometry-driven bending stabilities, we used several tools, including a reduced-order simulation package, a simplified linkage model, and physical prototypes. We found the bending stability of a circular two-unit corrugated tube is dependent on the longitudinal geometry and the stiffness of the crease lines that connect separate frusta. Thinner shells, steeper cones, and weaker creases are required to achieve bending bi-stability. We then explored how the bending stability changes as the cross section becomes elongated or distorted with concavity. We found the bending bi-stability is favored by deep and convex cross sections, while wider cross sections with a large concavity remain mono-stable. The different geometries influence the amounts of stretching and bending energy associated with bending the tube. The stretching energy has a bi-stable profile and can allow for a stable bent configuration, but it is counteracted by the bending energy which increases monotonically.more »The findings from this work can enable informed design of corrugated tube systems with desired bending stability behavior.« less