Eigensolution of piezoelectric energy harvesters with geometric discontinuities: Analytical modeling and validation

Abstract

Although cantilevered beams are the most prolific design for resonant piezoelectric energy harvesters, other topologies have been studied for their compactness or conformability to their host structures’ geometry. These more complex structures have been analyzed using custom analytical models developed from the first principles or finite-element methods to compute their eigensolutions and piezoelectric coupling effects. This article discusses the use of the transfer matrix method to derive analytical solutions to beam structures with pointwise discontinuities, bends, or lumped inertias between members or at the tip. Euler–Bernoulli beam theory is used to derive transfer matrices for the uniform beam segments, and point transfer matrices are derived to handle discontinuities in the structure between beam segments. The eigensolution of the transfer matrix is shown to produce the natural frequencies and mode shapes for these structures. Subsequently, the electromechanical coupling effects are incorporated, and the base excitation problem is considered. Parametric case studies are provided for beam structures with varying piezoelectric layer coverage and angle between members. Finally, these results are compared to finite-element solutions using COMSOL, and the modeling discrepancies are discussed. Based on the favorable comparison between these two methods, the utility and accuracy of the transfer matrix method are proven.

Transfer matrix modeling of a tensioned piezo-solar hybrid energy harvesting ribbon

“…From the FTM provided in equation 8 it can be easily observed that in the absence of applied tension ( 0) T = the matrix collapses to yield the familiar form of the FTM as previously shown by Wickenheiser 14 .…”

“…Considering a differential Euler-Bernoulli beam segment, the balances of forces and moments yield 14,15 …”

Parametric design-based modal damped vibrational piezoelectric energy harvesters with arbitrary proof mass offset: Numerical and analytical validations

“…So far, two damping coefficients have been introduced into the system as presented in Eqs. (1), (10) and (18). The power dissipation due to air damping occurs due to the kinetic energy of the structure at particular times creating air friction.…”

“…This includes the comprehensive analytical studies of the optimal power harvesting behaviour with the load resistance using the electromagnetic system [1] and the piezoelectric materials [2]. Moreover, the majority of piezoelectric power harvesters using laminated beams with broad ranges of case studies have been investigated using various analytical techniques such as electromechanical lumped parameter models and electrical equivalent system [3]- [4], analytical approach using weak form techniques [5]- [8], assumed-mode methods [9], transfer matrix method [10] and closed form techniques [11]- [13].…”

Analysis of Energy Harvesting Using Frequency Up-Conversion by Analytic Approximations

“…Varying the cross sections along the beam length (Dietl & Garcia, 2010;Reissman et al, 2007;Roundy et al, 2005) and the ratio of tip mass to beam mass (Dietl & Garcia, 2010;Wickenheiser, 2011) have been shown to improve the electromechanical coupling (a factor in the energy conversion rate) over a uniform cantilever beam design. Multi-beam structures can reduce the overall dimensions of the design by folding it in on itself while retaining a similar natural frequency to the original, straight configuration (Karami & Inman, 2011;Erturk et al, 2009); however, this requires a more complex analysis of the natural frequencies and mode shapes (Wickenheiser, 2012).…”