Analytical solution and energy harvesting from nonlinear vibration of an asymmetric bimorph piezoelectric plate and optimizing the plate parameters by genetic algorithm

2017 ◽  
Vol 29 (6) ◽  
pp. 1120-1138 ◽  
Author(s):  
Hamed Shorakaei ◽  
Alireza Shooshtari
Keyword(s):  
Genetic Algorithm ◽  
Nonlinear Vibration ◽  
Piezoelectric Plate ◽  
Electrical Power ◽  
Electric Displacement ◽  
Primary Resonance ◽  
Load Resistance ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Piezoelectric Layers

In this article, harvesting of electrical power from nonlinear vibration of an asymmetric bimorph piezoelectric plate is presented based on the classical plate theory with von Kármán strain–displacement nonlinear relations in the presence of temperature change effect. Two piezoelectric layers with different thicknesses cover top and bottom layers of a substructure. The structure has been excited under harmonic transverse forces in case of primary resonance. Two coupled ordinary differential equations for displacement and voltage have been derived and solved via multiple time scales method. The voltage equation has been defined by electric displacement and Gauss’s law. Analytic relations for voltage and harvested electrical power have been derived. The analytic relation for power is based on different parameters of the plate such as thicknesses of the layers, dimensions of plate, load resistance, frequency of harmonic excitation, and mechanical properties of structure. The parameters are optimized using genetic algorithm method with consideration of power relation as cost function. To harvest the maximum energy from the plate, the thicknesses of two piezoelectric layers, load resistance, and detuning parameter have been optimized. To illustrate the effectiveness of the optimization, results are depicted in three different simulations.

scholarly journals On the Application of the Multiple Scales Method on Electrostatically Actuated Resonators

2019 ◽  
Vol 14 (4) ◽  
Author(s):  
Saad Ilyas ◽  
Feras K. Alfosail ◽  
Mohammad I. Younis
Keyword(s):  
Analytical Solution ◽  
Multiple Scales ◽  
Excitation Frequency ◽  
Primary Resonance ◽  
Equation Of Motion ◽  
Higher Order ◽  
Resonance Excitation ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Electrostatically Actuated

We investigate modeling the dynamics of an electrostatically actuated resonator using the perturbation method of multiple time scales (MTS). First, we discuss two approaches to treat the nonlinear parallel-plate electrostatic force in the equation of motion and their impact on the application of MTS: expanding the force in Taylor series and multiplying both sides of the equation with the denominator of the forcing term. Considering a spring–mass–damper system excited electrostatically near primary resonance, it is concluded that, with consistent truncation of higher-order terms, both techniques yield same modulation equations. Then, we consider the problem of an electrostatically actuated resonator under simultaneous superharmonic and primary resonance excitation and derive a comprehensive analytical solution using MTS. The results of the analytical solution are compared against the numerical results obtained by long-time integration of the equation of motion. It is demonstrated that along with the direct excitation components at the excitation frequency and twice of that, higher-order parametric terms should also be included. Finally, the contributions of primary and superharmonic resonance toward the overall response of the resonator are examined.

Non-Stationary Oscillations at Slow Transitions Across Instabilities

2007 ◽  
Author(s):  
Rudolf R. Pusˇenjak ◽  
Maks M. Oblak ◽  
Jurij Avsec
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Time Scales ◽  
Primary Resonance ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Excitation Parameter ◽  
Linear Dynamical Systems ◽  
Weakly Nonlinear ◽  
Non Linear

The paper presents the study of non-stationary oscillations, which is based on extension of Lindstedt-Poincare (EL-P) method with multiple time scales for non-linear dynamical systems with cubic non-linearities. The generalization of the method is presented to discover the passage of weakly nonlinear systems through the resonance as a control or excitation parameter varies slowly across points of instabilities corresponding to the appearance of bifurcations. The method is applied to obtain non-stationary resonance curves of transition across points of instabilities during the passage through primary resonance of harmonically excited oscillators of Duffing type.

Parametric analysis of multilayered unimorph piezoelectric vibration energy harvesters

2015 ◽  
Vol 23 (15) ◽  
pp. 2538-2553 ◽  
Author(s):  
Ahmed Jemai ◽  
Fehmi Najar ◽  
Moez Chafra
Keyword(s):  
Energy Harvester ◽  
Electrical Power ◽  
Closed Form Solution ◽  
Beam Theory ◽  
Load Resistance ◽  
Form Solution ◽  
Energy Harvesters ◽  
Piezoelectric Energy ◽  
Piezoelectric Cantilever ◽  
Piezoelectric Layers

The use of a multilayer piezoelectric cantilever beam for vibration-based energy harvesting applications has been investigated as an effective technique to increase the harvested electrical power. It has been shown that the multilayered energy harvester performance is very sensitive to the number of layers and their electrical connection due to impedance variations. The objective of this work is to suggest a comprehensive mathematical model of multilayered unimorph piezoelectric energy harvester allowing analytical solution for the harvested voltage and electrical power. The model is used to deeply investigate the influence of different parameters on the harvested power. A distributed-parameter model of the harvester using the Euler–Bernoulli beam theory and Hamilton's principle is derived. Gauss's law is used to derive the electrical equations for parallel and series connections. A closed-form solution is proposed based on the Galerkin procedure and the obtained results are validated with a finite element 3D model. A parametric study is performed to ascertain the influence of the load resistance, the thickness ratio, the number of piezoelectric layers on the tip displacement and the electrical harvested power. It is shown that this model can be easily used to adjust the geometrical and electrical parameters of the energy harvester in order to improve the system's performances. In addition, it is proven that if one of the system's parameter is not correctly tuned, the harvested power can decrease by several orders of magnitude.

Vibration Analysis of a Shape Memory Oscillator by Harmonic Balance Method Verified Numerically

2019 ◽  
Vol 29 (03) ◽  
pp. 1930007 ◽  
Author(s):  
Rafal Rusinek ◽  
Joanna Rekas ◽  
Krzysztof Kecik
Keyword(s):  
Shape Memory ◽  
Periodic Solutions ◽  
Harmonic Balance ◽  
Primary Resonance ◽  
Harmonic Balance Method ◽  
Balance Method ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
System Parameters ◽  
Irregular Motion

This paper focuses on periodic solutions for a one-degree-of-freedom oscillator with a spring made of shape memory alloy (SMA). However, when periodic solutions are unstable, irregular motion is identified numerically. The shape memory spring is described by a polynomial characteristic in this model. The harmonic balance method (HBM) is employed to find periodic solutions near the primary resonance. The solutions are confronted with results obtained by the multiple time scales method and numerical simulations. Finally, the effect of system parameters and temperature on the system dynamics is discussed.

Nonlinear Forced Vibration of Curved Carbon Nanotube Resonators

2016 ◽  
Author(s):  
Hassan Askari ◽  
Ebrahim Esmailzadeh
Keyword(s):  
Carbon Nanotube ◽  
Forced Vibration ◽  
Primary Resonance ◽  
Beam Theory ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Nonlinear Forced Vibration ◽  
Resonator System ◽  
The Galerkin Method ◽  
Euler Bernoulli Beam

Nonlinear forced vibration of the nonlocal curved carbon nanotubes is investigated. The governing equation of vibration of a nonlocal curved carbon nanotube is developed. The nonlinear Winkler and Pasternak type foundations are chosen for the nanotube resonator system. Furthermore, the shape of the carbon nanotube system is assumed to be of a sinusoidal curvature form and different types of the boundary conditions are postulated for the targeted system. The Euler-Bernoulli beam theory in conjunction with the Eringen theory are implemented to obtain the partial differential equation of the system. The Galerkin method is applied to obtain the nonlinear ordinary differential equations of the system. For the sake of obtaining the primary resonance of the considered system the multiple time scales method is utilized. The influences of different parameters, namely, the position of the applied force, different forms of boundary condition, amplitude of curvature, and the coefficient of the Pasternak foundation, on the frequency response of the system were fully investigated.

Low- and High-Temperature Primary Resonance in Shape Memory Oscillator Observed by Multiple Time Scales and Harmonic Balance Method

2019 ◽  
Vol 14 (11) ◽  
Author(s):  
Andrzej Weremczuk ◽  
Joanna Rekas ◽  
Rafal Rusinek
Keyword(s):  
Shape Memory ◽  
Time Scales ◽  
Harmonic Balance ◽  
Primary Resonance ◽  
Harmonic Balance Method ◽  
Balance Method ◽  
Multiple Time Scales ◽  
Practical Implementation ◽  
Multiple Time ◽  
Fifth Order

Abstract This paper focuses on the primary resonance of a one degree-of-freedom (1DOF) oscillator with a spring made of shape memory alloy (SMA). The primary resonance is analyzed using the multiple time scales method (MTSM) and the harmonic balance method (HBM). The shape memory spring is described by a fifth-order polynomial function. The solutions are analyzed along with the results reported by another authors, and compared with numerical simulations. Three ranges of temperature are analyzed. Finally, the practical implementation aspect of the harmonic balance and MTSMs are discussed.

scholarly journals Dynamics of SMA micro-actuator in biomechanical system

2018 ◽  
Vol 148 ◽  
pp. 09001
Author(s):  
Rafal Rusinek ◽  
Andrzej Weremczuk ◽  
Marcin Szymanski ◽  
Jerzy Warminski
Keyword(s):  
Middle Ear ◽  
Degrees Of Freedom ◽  
Primary Resonance ◽  
Resonance Curve ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Relative Temperature ◽  
Micro Actuator ◽  
Middle Ear Prosthesis ◽  
Increasing Temperature

In this paper the polynomial model of shape memory alloy is used to characterise properties of a micro-actuator which is applied as a new middle ear prosthesis. A two degrees of freedom model of the reconstructed middle ear is solve by means of multiple time scales method. The system has various behaviours near the primary resonance depending on ambient temperature. The special case when relative temperature θ = 1.0 characterises untypical resonance curve. Increasing temperature to the normal human body one the resonance curves are typical. Then the system has only one periodic solution if the excitation is not too strong.

scholarly journals Internal Resonance and Energy Transfer of a Cable-stayed Beam With a Tuned Mass Damper

2021 ◽  
Author(s):  
Xiaoyang Su ◽  
Hou Jun Kang ◽  
Tieding Guo ◽  
Yunyue Cong
Keyword(s):  
Energy Transfer ◽  
Internal Resonance ◽  
Primary Resonance ◽  
Tuned Mass Damper ◽  
Phase Portraits ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Response Curves ◽  
Mass Damper ◽  
One To One

Abstract This study considers a novel nonlinear system, namely, a cable-stayed beam with a tuned mass damper (cable-beam-TMD model), allowing the description of energy transfer among the beam, cable and TMD. In this system, the vibration of the TMD is involved and one-to-one-to-one internal resonance among the modes of the beam, cable and TMD is investigated when external primary resonance of the beam occurs. Galerkin’s method is utilized to discretize the equations of motion of the beam and cable. In this way, a set of ordinary differential equations (ODEs) are derived, which are solved by the method of multiple time scales (MTS). Then the steady state solutions of the system are obtained by suing Newton-Raphson method and continued by pseudo arclength algorithm. The response curves, time histories and phase portraits are provided to explore the effect of the TMD on the nonlinear behaviours of the model. Meanwhile, a partially coupled system, namely, a cable-beam-TMD model ignoring the vibration of the TMD, is also studied. The nonlinear characteristics of the two cases are compared with each other. The results reveal the occurrence of energy transfer among the beam, cable and TMD.

scholarly journals Vibration Control of a Compressor Blade Using Position and Velocity Feedback

2019 ◽  
Vol 24 (No 1) ◽  
Author(s):  
Ali Kandil ◽  
Magdy Kamel
Keyword(s):  
Primary Resonance ◽  
Time History ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Nonlinear Behaviour ◽  
Response Curves ◽  
Velocity Feedback ◽  
Rotating Speed ◽  
Coupled Mode ◽  
Different Response

Position and velocity feedback controllers are applied in this work to reduce the oscillations of a rotating blade dynamical system running at an unsteady rotating speed. Both the primary resonance and the principal parametric resonance are controlled as they are the worst cases that were verified numerically. The two modes of vibrations are found to be powerfully linearly coupled, so we have applied the controller to only one mode and the other, coupled mode follows it. The overall nonlinear behaviour of the system with and without control is investigated through the multiple time scales method. Time history and different response curves of the controlled system are included to show the controller effect.

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