A piezoelectric beam model with geometric, material and damping nonlinearities for energy harvesting

Abstract

To predict electrical generation in piezoelectric small-scale beam energy harvesting devices, it is important to have a complete mathematical model that captures the different associated phenomena. In the literature, some authors propose several alternatives of nonlinear mathematical formulations, with non-linearities coming from different physical aspects. All these formulations present good aptitudes to predict the nonlinear behavior of the system under different values of accelerations, geometry and boundary conditions. At the same time, they do not represent a unified general proposal for modeling multimodal energy harvesting devices of any type of mode generation and boundary conditions at large excitations. In this sense, this paper presents a mathematical description of inextensional nonlinear Euler-Bernoulli piezoelectric beams that combines the best contributions of the literature to the voltage generation of multimodal nonlinear piezoelectric energy harvesters (geometric, material and damping non-linearities). The developed analytical model yields a total set of N + 1 ordinary differential equations for the first N modes and for the output voltage. However, direct solution of this ordinary nonlinear differential system of N equations is computationally costly. Instead, a reduced algebraic system of 2(𝑁 + 1) algebraic equations is proposed applying the method of averaging. Its main advantage is that it makes more suitable and computationally economical for the implementation of a parameter identification process involving any number of piezoelectric inserts (unimorph or bimorph) and mode of generation (d33 or d31). Two types of validations are presented for some selected physical systems to test the validity of the assumptions: a numerical one, by the direct integration of the equations of motion and an experimental one. A final comparison between the results demonstrates the importance of the having a unified nonlinear model to predict the generated voltage in multimodal energy harvesters.

Electromechanical model and simple numerical analysis for a piezoelectric vibration energy harvester considering nonlinear piezoelectricity, nonlinear damping, and self-powered synchronized switch circuit

“…Unlike the polynomial model proposed by Wagner and Hagedorn (2002), the nonlinear terms of the enthalpy density based on the ferroelastic hysteresis are dependent on the absolute value (or the signum function) of the strain, which is characterized by the hysteresis loop. Recent studies of a multimodal PVEH (Machado et al, 2020), a PVEH with segmented electrodes (Hu et al, 2021) connected to a single resistive circuit, and a PVEH connected to a diode bridge rectifier circuit (Leadenham and Erturk, 2020) validated the model proposed by Leadenham and Erturk (2015) by replicating the experimental results well. In addition to nonlinear stiffness softening, some studies also reported the importance of including nonlinear damping in the modeling process to explain the experimentally observed result that the displacement magnitude decreased when the excitation level increased.…”

Dynamic Characteristics and Experimental Research of Linear-Arch Bi-Stable Piezoelectric Energy Harvester

“…Shahruz [ 7 ] arranged cantilever beam arrays with different resonant frequencies: research shows that under the action of external excitation, only one cantilever beam can match the vibration frequency of the environment, so the energy collection efficiency is low. In order to improve the energy harvesting efficiency of piezoelectric energy harvesters, researchers have carried out a lot of research on the introduction of nonlinear methods to broaden the working frequency band of energy harvesters [ 8 , 9 , 10 , 11 ]. Wang [ 12 ] proposed a cantilever beam with variable section thickness and established its distributed parameter model.…”

Impact of Series and Parallel Connection of Macro Fiber Composite Patches in Piezoelectric Harvester on Energy Storage

“…A prismatic beam with a rectangular cross-section is one commonly used mechanical structure of the harvester. The beam, which is constructed from a carrying substrate and a piezoelectric material, is applied for energy harvesting from various processes: mechanical vibration [1,2], rotational motion [3,4] and wind energy [5,6]. Mechanical energy is converted into electric energy in a piezoelectric material, which is squeezed or stretched during beam bending.…”

“…A carrying substrate is made from steel, brass, aluminum, copper or various composites. Ceramics, e.g., lead titanate zirconate [4,5]; composites made from piezoelectric ceramics and polymer matrix, e.g., Macro Fiber Composite [1,2]; and piezoelectric polymers, e.g., polyvinylidene fluoride [7], are the typical piezoelectric material used in beam manufacturing. Composites have lower efficiency in energy conversion than piezoelectric ceramics, but are more resistant than monolithic ceramic to destruction due to deformations [8].…”

“…The beam structure is made by gluing the piezoelectric material to one side (unimorph) [1,9] or both sides (bimorph) [2,10] of the carrying substrate. Two piezoelectric layers, which are glued on both sides of the substrate, may be electrically parallel or series connected [11][12][13].…”