Restricted accessResearch articleFirst published online 2018-5
Geometric nonlinear finite element and genetic algorithm based optimal vibration energy harvesting from nonprismatic axially functionally graded piezolaminated beam
The present article deals with optimal vibration energy harvesting from an axially functionally graded (FG) non-prismatic piezolaminated beam using genetic algorithm (GA). Geometric nonlinear based finite element (FE) formulation using Newmark method in conjunction with Newton-Raphson method is formulated to solve the obtained governing equation. A two noded beam element with two degrees of freedom (DOF) at each node is used in the present FE formulation. The FG material (i.e. non-homogeneity) in the axial direction is considered which varies (continuously decreasing from root to tip of such cantilever beam) using a proposed power law formula. The various cross section profiles (such as linear, parabolic and cubic) are modelled using the Euler-Bernoulli beam theory. The effects of nonlinearity on the responses (such as displacement, voltage and output power) are deliberated with arbitrary power gradient index. The effects of tapers (both width and height in length directions) on the output power and voltage are analysed. A real-coded genetic algorithm based constrained optimization technique is also proposed to determine the best possible design variables for optimal power harvesting within the allowable limits of ultimate stress of beam and breakdown voltage of PZT sensor.
AbdelkefiA (2016) Aeroelastic energy harvesting: A review. International Journal of Engineering Science100: 112–135.
2.
AbdelkefiABarsalloN (2016) Nonlinear analysis and power improvement of broadband low-frequency piezomagnetoelastic energy harvesters. Nonlinear Dynamics83: 41–56.
3.
AbdelkefiAHajjMNayfehA (2013) Piezoelectric energy harvesting from transverse galloping of bluff bodies. Smart materials and structures22: 015014.
4.
AbdelkefiANajarFNayfehAet al. (2011) An energy harvester using piezoelectric cantilever beams undergoing coupled bending–torsion vibrations. Smart materials and structures20: 115007.
5.
AbdelkefiANayfehAHajjM (2012) Effects of nonlinear piezoelectric coupling on energy harvesters under direct excitation. Nonlinear Dynamics67: 1221–1232.
6.
AbdelmoulaHAbdelkefiA (2016) Piezoaeroelastic investigation on the control and energy harvesting of galloping systems. 57th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference. 0209.
7.
AjitsariaJChoeS-YShenDet al. (2007) Modeling and analysis of a bimorph piezoelectric cantilever beam for voltage generation. Smart materials and structures16: 447.
8.
AlshorbagyAEEltaherMMahmoudF (2011) Free vibration characteristics of a functionally graded beam by finite element method. Applied Mathematical Modelling35: 412–425.
9.
AntonSRSodanoHA (2007) A review of power harvesting using piezoelectric materials (2003–2006). Smart materials and structures16: R1–R21.
10.
AroraRTulshyanRDebK (2010) Parallelization of binary and real-coded genetic algorithms on GPU using CUDA. Evolutionary Computation (CEC), 2010 IEEE Congress on. IEEE, pp. 1–8.
11.
AydogduM (2008) Semi-inverse method for vibration and buckling of axially functionally graded beams. Journal of Reinforced Plastics and Composites27: 683–691.
12.
AyedSBAbdelkefiANajarFet al. (2013) Design and performance of variable-shaped piezoelectric energy harvesters. Journal of intelligent material systems and structures25: 174–186.
13.
BenasciuttiDMoroLZelenikaSet al. (2010) Vibration energy scavenging via piezoelectric bimorphs of optimized shapes. Microsystem Technologies16: 657–668.
14.
BiboAAbdelkefiADaqaqMF (2015) Modeling and characterization of a piezoelectric energy harvester under Combined Aerodynamic and Base Excitations. Journal of Vibration and Acoustics137: 031017.
15.
BiboADaqaqMF (2013) Energy harvesting under combined aerodynamic and base excitations. Journal of sound and vibration332: 5086–5102.
16.
BichiouYAbdelkefiAHajjMR (2016) Nonlinear aeroelastic characterization of wind turbine blades. Journal of Vibration and Control22: 621–631.
17.
BruchJJrSlossJAdaliSet al. (2000) Optimal piezo-actuator locations/lengths and applied voltage for shape control of beams. Smart materials and structures9: 205.
18.
CastagnettiD (2013) A wideband fractal-inspired piezoelectric energy converter: design, simulation and experimental characterization. Smart materials and structures22: 094024.
19.
CastagnettiD (2014) Comparison Between a Wideband Fractal-Inspired and a Traditional Multicantilever Piezoelectric Energy Converter. Journal of Vibration and Acoustics137: 011006.
20.
ChuangY-CChenC-THwangC (2016) A simple and efficient real-coded genetic algorithm for constrained optimization. Applied Soft Computing38: 87–105.
21.
DaiHAbdelkefiAWangL (2014) Piezoelectric energy harvesting from concurrent vortex-induced vibrations and base excitations. Nonlinear Dynamics77: 967–981.
22.
de PaulaASBalthazarJMFelixJLP (2013) Some comments on the nonlinear dynamics of a portal frame under base excitation. Shock and Vibration20: 1093–1101.
23.
DebK (2001) Multi-objective optimization using evolutionary algorithms, New York: John Wiley & Sons.
24.
Eggborn T. (2003) Analytical models to predict power harvesting with piezoelectric materials. MS Thesis, Virginia Polytechnic Institute and State University.
25.
ElishakoffICandanS (2001) Apparently first closed-form solution for vibrating: inhomogeneous beams. International Journal of Solids and Structures38: 3411–3441.
26.
ElishakoffIGuedeZ (2004) Analytical polynomial solutions for vibrating axially graded beams. Mechanics of Advanced Materials and Structures11: 517–533.
27.
ElvinNElvinAChoiD (2003) A self-powered damage detection sensor. The Journal of Strain Analysis for Engineering Design38: 115–124.
28.
ElvinNGElvinAASpectorM (2001) A self-powered mechanical strain energy sensor. Smart materials and structures10: 293–299.
29.
ErturkAInmanDJ (2009) An experimentally validated bimorph cantilever model for piezoelectric energy harvesting from base excitations. Smart materials and structures18: 025009.
30.
FlynnAMSandersSR (2002) Fundamental limits on energy transfer and circuit considerations for piezoelectric transformers. Power Electronics, IEEE Transactions on17: 8–14.
31.
GereJMTimoshenkoS (2001) Mechanics of materials Brooks. Cole, Pacific Grove, CA, 815–839.
32.
HerreraFLozanoMSánchezAM (2003) A taxonomy for the crossover operator for real‐coded genetic algorithms: An experimental study. International Journal of Intelligent Systems18: 309–338.
33.
HuangYLiX-F (2010) A new approach for free vibration of axially functionally graded beams with non-uniform cross-section. Journal of sound and vibration329: 2291–2303.
34.
HwangW-SParkHC (1993) Finite element modeling of piezoelectric sensors and actuators. AIAA journal31: 930–937.
35.
IliukIBalthazarJMTussetAet al. (2013a) A non-ideal portal frame energy harvester controlled using a pendulum. The European Physical Journal Special Topics222: 1575–1586.
36.
IliukIBalthazarJMTussetAMet al. (2013b) Application of passive control to energy harvester efficiency using a nonideal portal frame structural support system. Journal of intelligent material systems and structures25: 417–429.
37.
JavedUAbdelkefiAAkhtarI (2016) An improved stability characterization for aeroelastic energy harvesting applications. Communications in Nonlinear Science and Numerical Simulation36: 252–265.
38.
JiangSLiXGuoSet al. (2005) Performance of a piezoelectric bimorph for scavenging vibration energy. Smart materials and structures14: 769–774.
39.
KorlaSLeonRTanselINet al. (2011) Design and testing of an efficient and compact piezoelectric energy harvester. Microelectronics Journal42: 265–270.
40.
Leung A, Zhou W, Lim C, Yuen R and Lee U (2001) Dynamic stiffness for piecewise non-uniform Timoshenko column by power series – part I: Conservative axial force. International Journal for Numerical Methods in Engineering 51: 505–529.
41.
MasanaRDaqaqMF (2011) Electromechanical modeling and nonlinear analysis of axially loaded energy harvesters. Journal of Vibration and Acoustics133: 011007.
42.
MehraeenSJagannathanSCorzineK (2010) Energy harvesting from vibration with alternate scavenging circuitry and tapered cantilever beam. Industrial Electronics, IEEE Transactions on57: 820–830.
43.
MukherjeeAChaudhuriAS (2002) Piezolaminated beams with large deformations. International Journal of Solids and Structures39: 4567–4582.
44.
MurinJAminbaghaiMKutišV (2010) Exact solution of the bending vibration problem of FGM beams with variation of material properties. Engineering structures32: 1631–1640.
45.
ReddyJN (2014) An Introduction to Nonlinear Finite Element Analysis: with applications to heat transfer, fluid mechanics, and solid mechanics, Oxford: OUP.
46.
RosaMDe Marqui JuniorC (2014) Modeling and analysis of a piezoelectric energy harvester with varying cross-sectional area. Shock and Vibration. 2014.
47.
RoundySLelandESBakerJet al. (2005) Improving power output for vibration-based energy scavengers. Pervasive Computing, IEEE4: 28–36.
48.
RoundySWrightPK (2004) A piezoelectric vibration based generator for wireless electronics. Smart materials and structures13: 1131.
49.
RoyTChakrabortyD (2008) GA-LQR based optimal vibration control of smart FRP composite structures with bonded PZT patches. Journal of Reinforced Plastics and Composites28: 1383–1404.
50.
RoyTChakrabortyD (2009a) Genetic algorithm based optimal control of smart composite shell structures under mechanical loading and thermal gradient. Smart materials and structures18: 115006.
51.
RoyTChakrabortyD (2009b) Optimal vibration control of smart fiber reinforced composite shell structures using improved genetic algorithm. Journal of sound and vibration319: 15–40.
52.
RoyTManikandanPChakrabortyD (2010) Improved shell finite element for piezothermoelastic analysis of smart fiber reinforced composite structures. Finite elements in analysis and design46: 710–720.
53.
SankarB (2001) An elasticity solution for functionally graded beams. Composites Science and Technology61: 689–696.
54.
ShahbaAAttarnejadRHajilarS (2011a) Free vibration and stability of axially functionally graded tapered Euler-Bernoulli beams. Shock and Vibration18: 683–696.
55.
ShahbaAAttarnejadRMarviMTet al. (2011b) Free vibration and stability analysis of axially functionally graded tapered Timoshenko beams with classical and non-classical boundary conditions. Composites Part B: Engineering42: 801–808.
56.
ShahbaARajasekaranS (2012) Free vibration and stability of tapered Euler–Bernoulli beams made of axially functionally graded materials. Applied Mathematical Modelling36: 3094–3111.
ShenckNSParadisoJA (2001) Energy scavenging with shoe-mounted piezoelectrics. IEEE micro21: 30–42.
59.
TussetAMPiccirilloVBuenoAMet al. (2015) Chaos control and sensitivity analysis of a double pendulum arm excited by an RLC circuit based nonlinear shaker. Journal of Vibration and Control22: 3621–3637.
60.
TzouHYeR (1996) Analysis of piezoelastic structures with laminated piezoelectric triangle shell elements. AIAA journal34: 110–115.
61.
Wahrhaftig AdMBrasilRMBalthazarJM (2013) The first frequency of cantilevered bars with geometric effect: a mathematical and experimental evaluation. Journal of the Brazilian Society of Mechanical Sciences and Engineering35: 457–467.
62.
WankhadeRLBajoriaKM (2013) Free vibration and stability analysis of piezolaminated plates using the finite element method. Smart materials and structures22: 125040.
63.
WuLWangQ-sElishakoffI (2005) Semi-inverse method for axially functionally graded beams with an anti-symmetric vibration mode. Journal of sound and vibration284: 1190–1202.
