Closed-form solutions for forced vibrations of piezoelectric energy harvesters by means of Green’s functions

2017 ◽  
Vol 28 (17) ◽  
pp. 2372-2387 ◽  
Author(s):  
X Zhao ◽  
EC Yang ◽  
YH Li ◽  
W Crossley
Keyword(s):  
Closed Form ◽  
Piezoelectric Materials ◽  
Forced Vibrations ◽  
Rotational Inertia ◽  
Closed Form Solutions ◽  
Energy Harvesters ◽  
Piezoelectric Energy ◽  
Laplace Transform Technique ◽  
Damping Effects ◽  
Tip Mass

In this article, the closed-form solutions are obtained for the forced vibrations of cantilevered unimorph piezoelectric energy harvesters. A tip mass is attached at the free end, and the moment of its inertia to the fixed end is considered. Timoshenko beam assumptions are used to establish a coupled electromechanical model for the harvester. Two damping effects, transverse and rotational damping effects, are taken into account. Green’s function method and Laplace transform technique are used to solve the coupled electromechanical vibration system. The conventional case of a harmonic base excitation is considered, and numerical calculations are performed. The present model is validated by comparing its predictions with the existing data, the experimental results, and the finite element method solutions. The influences of shear deformation and rotational inertia on the predictions are discussed. The effect of load resistance on the electrical power is studied, and the optimal load resistances are obtained. Ultimately, the optimal schemes are proposed to improve electricity generation performance for the soft piezoelectric materials: PZT-5A/5H.

Closed-form solutions of bending-torsion coupled forced vibrations of a piezoelectric energy harvester under a fluid vortex

2021 ◽  
pp. 1-31
Author(s):  
Xiang Zhao ◽  
Weidong Zhu ◽  
Ying-hui Li
Keyword(s):  
Closed Form ◽  
Torsional Vibration ◽  
Energy Harvester ◽  
High Energy ◽  
Forced Vibrations ◽  
Vibration Energy ◽  
Beam Model ◽  
Vibration Energy Harvester ◽  
Closed Form Solutions ◽  
Piezoelectric Energy

Abstract Vibration energy harvesting problems have strongly developed in recent years. However, many researchers just consider bending vibration models of energy harvesters. As a matter of fact, torsional vibration is also important and cannot be ignored in many cases. In this work, closed-form solutions of bending-torsion coupled forced vibrations of a piezoelectric energy harvester subjected to a fluid vortex are derived. Timoshenko beam model is used for modeling the energy harvester, and the extended Hamilton's principle is used in the modeling process. Since piezoelectric effects in both bending and torsional directions are considered, two kinds of electric coupling effects appear in forced vibration equations, and a new model for the electric circuit equation is developed. Lamb-Oseen vortex model is considered in this study. Both the external aerodynamic force and moment are simple harmonic loads. Three damping coefficients are considered in the present model. Based on Green's function method, closed-form solutions of the piezoelectric energy harvester subjected to the water vortex are derived. Some published results are used to verify the present solutions. It can be concluded through analysis that when torsional vibration is considered, the bandwidth of the high energy area of the voltage becomes large, and the bending-torsion coupled vibration energy harvester can produce more power than a transverse vibration energy harvester.

scholarly journals Piezoelectric Energy Generators Based on Spring and Inertial Mass

Materials ◽  
2018 ◽  
Vol 11 (11) ◽  
pp. 2163 ◽  
Author(s):  
Sanghyun Yoon ◽  
Jinhwan Kim ◽  
Kyung-Ho Cho ◽  
Young-Ho Ko ◽  
Sang-Kwon Lee ◽  
...  
Keyword(s):  
Piezoelectric Materials ◽  
Electric Energy ◽  
Inertial Mass ◽  
Design Parameters ◽  
Mechanical Shock ◽  
Generator System ◽  
Energy Harvesters ◽  
Shock Energy ◽  
Piezoelectric Energy ◽  
And Performance

In this study, inertial mass-based piezoelectric energy generators with and without a spring were designed and tested. This energy harvesting system is based on the shock absorber, which is widely used to protect humans or products from mechanical shock. Mechanical shock energies, which were applied to the energy absorber, were converted into electrical energies. To design the energy harvester, an inertial mass was introduced to focus the energy generating position. In addition, a spring was designed and tested to increase the energy generation time by absorbing the mechanical shock energy and releasing a decreased shock energy over a longer time. Both inertial mass and the spring are the key design parameters for energy harvesters as the piezoelectric materials, Pb(Mg1/3Nb2/3)O3-PbTiO3 piezoelectric ceramics were employed to store and convert the mechanical force into electric energy. In this research, we will discuss the design and performance of the energy generator system based on shock absorbers.

A Novel Multi-Directional Nonlinear Piezoelectric Energy Harvester Coupled With Nonlinear Conditioning Circuits

2015 ◽  
Author(s):  
Zhengbao Yang ◽  
Jean Zu
Keyword(s):  
Energy Harvester ◽  
Piezoelectric Materials ◽  
Electrical Energy ◽  
Elastic Rods ◽  
Energy Harvesters ◽  
Piezoelectric Energy ◽  
High Power Output ◽  
Transduction Mechanisms ◽  
Self Powered ◽  
Aluminum Plates

Energy harvesting from vibrations has become, in recent years, a recurring target of a quantity of research to achieve self-powered operation of low-power electronic devices. However, most of energy harvesters developed to date, regardless of different transduction mechanisms and various structures, are designed to capture vibration energy from single predetermined direction. To overcome the problem of the unidirectional sensitivity, we proposed a novel multi-directional nonlinear energy harvester using piezoelectric materials. The harvester consists of a flexural center (one PZT plate sandwiched by two bow-shaped aluminum plates) and a pair of elastic rods. Base vibration is amplified and transferred to the flexural center by the elastic rods and then converted to electrical energy via the piezoelectric effect. A prototype was fabricated and experimentally compared with traditional cantilevered piezoelectric energy harvester. Following that, a nonlinear conditioning circuit (self-powered SSHI) was analyzed and adopted to improve the performance. Experimental results shows that the proposed energy harvester has the capability of generating power constantly when the excitation direction is changed in 360. It also exhibits a wide frequency bandwidth and a high power output which is further improved by the nonlinear circuit.

Equivalent impedance and power analysis of monostable piezoelectric energy harvesters

2020 ◽  
Vol 31 (14) ◽  
pp. 1697-1715
Author(s):  
Chunbo Lan ◽  
Yabin Liao ◽  
Guobiao Hu ◽  
Lihua Tang
Keyword(s):  
Equivalent Circuit ◽  
Electromechanical Coupling ◽  
Energy Harvester ◽  
Performance Enhancement ◽  
Damping Ratio ◽  
Power Performance ◽  
Closed Form Solutions ◽  
Energy Harvesters ◽  
Piezoelectric Energy ◽  
Power Limit

Nonlinearity has been successfully introduced into piezoelectric energy harvesting for power performance enhancement and bandwidth enlargement. While a great deal of emphasis has been placed by researchers on the structural design and broadband effect, this article is motivated to investigate the maximum power of a representative type of nonlinear piezoelectric energy harvesters, that is, monostable piezoelectric energy harvester. An equivalent circuit is proposed to analytically study and explain system behaviors. The effect of nonlinearity is modeled as a nonlinear stiffness element mechanically and a nonlinear capacitive element electrically. Facilitated by the equivalent circuit, closed-form solutions of power limit and critical electromechanical coupling, that is, minimum coupling to reach the power limit, of monostable piezoelectric energy harvesters are obtained, which are used for a clear explanation of the system behavior. Several important conclusions have been drawn from the analytical analysis and validated by numerical simulations. First, given the same level of external excitation, the monostable piezoelectric energy harvester and its linear counterpart are subjected to the same power limit. Second, while the critical coupling of linear piezoelectric energy harvesters depends on the mechanical damping ratio only, it also depends on the vibration excitation and magnetic field for monostable piezoelectric energy harvesters, which can be used to adjust the power performance of the system.

Orthotropy Rescaling for the Fracture Problem in Anisotropic Piezoelectric Materials

2002 ◽  
Author(s):  
William S. Oates ◽  
Christopher S. Lynch
Keyword(s):  
Closed Form ◽  
Piezoelectric Materials ◽  
Stress Singularity ◽  
Closed Form Solution ◽  
Form Solution ◽  
Governing Equations ◽  
Closed Form Solutions ◽  
Order Polynomial ◽  
Work Done ◽  
Elastic Dielectric

To date, much of the work done on ferroelectric fracture assumes the material is elastically isotropic, yet there can be considerable polarization induced anisotropy. More sophisticated solutions of the fracture problem incorporate anisotropy through the Stroh formalism generalized to the piezoelectric material. This gives equations for the stress singularity, but the characteristic equation involves solving a sixth order polynomial. In general this must be accomplished numerically for each composition. In this work it is shown that a closed form solution can be obtained using orthotropy rescaling. This technique involves rescaling the coordinate system based on certain ratios of the elastic, dielectric, and piezoelectric coefficients. The result is that the governing equations can be reduced to the biharmonic equation and solutions for the isotropic material utilized to obtain solutions for the anisotropic material. This leads to closed form solutions for the stress singularity in terms of ratios of the elastic, dielectric, and piezoelectric coefficients. The results of the two approaches are compared and the contribution of anisotropy to the stress intensity factor discussed.

Assumed-Modes Formulation of Piezoelectric Energy Harvesters: Euler-Bernoulli, Rayleigh and Timoshenko Models With Axial Deformations

2010 ◽  
Author(s):  
Alper Erturk ◽  
Daniel J. Inman
Keyword(s):  
Analytical Solution ◽  
Piezoelectric Materials ◽  
Axial Direction ◽  
Vibration Energy ◽  
Power Requirement ◽  
Large Power ◽  
Energy Harvesters ◽  
Piezoelectric Energy ◽  
Cantilevered Beam ◽  
Transduction Mechanisms

Harvesting of vibration energy has been investigated by numerous researchers over the last decade. The research motivation in this field is due to the reduced power requirement of small electronic components such as wireless sensor networks used in monitoring applications. The ultimate goal is to power such devices by using the waste vibration energy available in their environment so that the maintenance requirement for battery replacement is minimized. Among the basic transduction mechanisms that can be used for vibration-to-electricity conversion, piezoelectric transduction has received the most attention due to the large power densities and ease of application of piezoelectric materials. Typically, a piezoelectric energy harvester is a cantilevered beam with one or two piezoceramic layers and the source of excitation is the base motion in the transverse direction. This paper presents general formulations for electromechanical modeling of base-excited piezoelectric energy harvesters with symmetric and asymmetric laminates. The electromechanical derivations are given using the assumed-modes method under the Euler-Bernoulli, Rayleigh and Timoshenko beam assumptions in three sections. The formulations account for an independent axial displacement variable in all cases. Comparisons are provided against the analytical solution given by the authors for symmetric laminates and convergence of the assumed-modes solution to the analytical solution with the increasing number of modes is shown. Experimental validations are also presented by comparing the electromechanical frequency response functions derived here against the experimentally obtained ones. The electromechanical assumed-modes formulations given here can be used for modeling of piezoelectric energy harvesters with asymmetric laminates as well as those with moderate thickness and varying geometry in the axial direction.

Exact Eigensolutions for a Family of Nonuniform Rods With End Point Masses

2017 ◽  
Vol 139 (5) ◽  
Author(s):  
Lourdes Rubio ◽  
José Fernández-Sáez ◽  
Antonino Morassi
Keyword(s):  
Closed Form ◽  
Canonical Form ◽  
Longitudinal Vibration ◽  
Countable Family ◽  
Darboux Transformations ◽  
Closed Form Solutions ◽  
End Point ◽  
Sturm Liouville ◽  
Tip Mass

In this paper, new exact closed-form solutions for free longitudinal vibration of a one-parameter countable family of cantilever rods with one end tip mass are obtained. The analysis is based on the reduction of the equation governing the longitudinal vibration to the Sturm–Liouville canonical form and on the use of double Darboux transformations. The rods for which exact eigensolutions are provided are explicitly determined in terms of an initial rod with known closed-form eigensolutions. The method can be also extended to include longitudinally vibrating rods with tip mass at both ends.

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