The objective of the present paper is to comprehensively study the nonlinear frequency response of viscoelastic piezoelectric nanoplates exposed to dual harmonic external excitation and thermo-electro-mechanical loads. To achieve this goal, firstly, a piezoelectric nanoplate resting on a viscoelastic foundation is modeled. Secondly, using the nonlocal piezoelectricity theory, Kelvin–Voigt model, von Karman nonlinear relations and Hamilton’s principle, the nonlinear governing differential equation of motion is derived. In the next step, employing the Galerkin technique and multiple time scales method, the partial differential equation is transformed to an ordinary one and solved. Finally, the modulation equation of viscoelastic piezoelectric nanoplates for combinational excitation is obtained. Emphasizing the effect of dual harmonic excitation and thermo-electro-mechanical loads on nonlinear frequency response of the system, jump and resonance phenomena are discussed. A detailed parametric study is conducted to examine the effect of nonlinearity, damping coefficient, nonlocal parameter, combinational excitation, electric voltage, initial stress and thermal environment.
AraniA-GKolahdouzanFAbdollahianM (2018) Nonlocal buckling of embedded magnetoelectroelastic sandwich nanoplate using refined zigzag theory. Applied Mathematics and Mechanics39(4): 529–546.
2.
AraniA-GShiravandARahiM, et al. (2012) Nonlocal vibration of coupled DLGS systems embedded on visco-Pasternak foundation. Physica B: Condensed Matter407(21): 4123–4131.
3.
ArefiMPourjamshidianMGhorbanpour AraniA (2018) Nonlinear free and forced vibration analysis of embedded functionally graded sandwich micro beam with moving mass. Journal of Sandwich Structures & Materials20(4): 462–492.
4.
AsemiSRFarajpourAMohammadiM (2014) Nonlinear vibration analysis of piezoelectric nanoelectromechanical resonators based on nonlocal elasticity theory. Composite Structures116: 703–712.
5.
BarrettaRFeoLLucianoR, et al. (2015) Variational formulations for functionally graded nonlocal Bernoulli-Euler nanobeams. Composite Structures129: 80–89.
6.
DaiSGharbiMSharmaP, et al. (2011) Surface piezoelectricity: size effects in nanostructures and the emergence of piezoelectricity in non-piezoelectric materials. Journal of Applied Physics110(10): 104305.
7.
EbrahimiFHosseiniSHS (2016) Thermal effects on nonlinear vibration behavior of viscoelastic nanosize plates. Journal of Thermal Stresses39(5): 606–625.
8.
EbrahimiFZiaM (2015) Large amplitude nonlinear vibration analysis of functionally graded Timoshenko beams with porosities. Acta Astronautica116: 117–125.
9.
EftekhariMRahmatabadiADMazidiA (2018) Nonlinear vibration of in-extensional rotating shaft under electromagnetic load. Mechanism and Machine Theory121: 42–58.
10.
EhyaeiJAkbarshahiAShafieiN (2017) Influence of porosity and axial preload on vibration behavior of rotating FG nanobeam. Advances in Nano Research5(2): 141–169.
11.
EringenAC (1972) Nonlocal polar elastic continua. International Journal of Engineering Science10(1): 1–16.
12.
EringenAC (1983) On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. Journal of Applied Physics54(9): 4703–4710.
13.
EringenACEdelenDGB (1972) On nonlocal elasticity. International Journal of Engineering Science10(3): 233–248.
14.
GhadiriMRajabpourAAkbarshahiA (2017) Non-linear forced vibration analysis of nanobeams subjected to moving concentrated load resting on a viscoelastic foundation considering thermal and surface effects. Applied Mathematical Modelling50: 676–694.
15.
GhadiriMShafieiNAkbarshahiA (2016) Influence of thermal and surface effects on vibration behavior of nonlocal rotating Timoshenko nanobeam. Applied Physics A122(7): 673.
16.
JeongSChoJYSungTH, et al. (2017) Electromechanical modeling and power performance analysis of a piezoelectric energy harvester having an attached mass and a segmented piezoelectric layer. Smart Materials and Structures26(3): 035035.
17.
JiYChoeMChoB, et al. (2012) Organic nonvolatile memory devices with charge trapping multilayer graphene film. Nanotechnology23(10): 105202.
18.
JomehzadehESaidiAR (2011) The small scale effect on nonlinear vibration of single layer graphene sheets. World Academy of Science Engineering Technology5: 06–29.
19.
KaramiBShahsavariDLiL, et al. (2018) Thermal buckling of embedded sandwich piezoelectric nanoplates with functionally graded core by a nonlocal second-order shear deformation theory. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science233(1): 287–301.
20.
KeL-LWangY-S (2012) Thermoelectric-mechanical vibration of piezoelectric nanobeams based on the nonlocal theory. Smart Materials and Structures21: 025018.
21.
KeL-LWangY-S (2014) Free vibration of size-dependent magneto-electro-elastic nanobeams based on the nonlocal theory. Physica E: Low-Dimensional Systems and Nanostructures63: 52–61.
22.
KeL-LWangY-SYangJ, et al. (2014a) Free vibration of size-dependent magneto-electro-elastic nanoplates based on the nonlocal theory. Acta Mechanica Sinica30(4): 516–525.
23.
KeL-LWangY-SYangJ, et al. (2014b) The size-dependent vibration of embedded magneto-electro-elastic cylindrical nanoshells. Smart Materials and Structures23(12): 125036.
24.
KhooSYRadeefZSOngZC, et al. (2017) Structural dynamics effect on voltage generation from dual coupled cantilever based piezoelectric vibration energy harvester system. Measurement107: 41–52.
25.
KolahchiR (2017) A comparative study on the bending, vibration and buckling of viscoelastic sandwich nano-plates based on different nonlocal theories using DC, HDQ and DQ methods. Aerospace Science and Technology66: 235–248.
26.
KolahchiRZareiMSHajmohammadMH, et al. (2017) Visco-nonlocal-refined Zigzag theories for dynamic buckling of laminated nanoplates using differential cubature-Bolotin methods. Thin-Walled Structures113: 162–169.
27.
LakesRLakesRS (2009) Viscoelastic Materials. Cambridge: Cambridge University Press.
28.
LiuCKeL-LWangY-S, et al. (2015) Nonlinear vibration of nonlocal piezoelectric nanoplates. International Journal of Structural Stability and Dynamics15(08): 1540013.
29.
LiuCKeL-LWangY-S, et al. (2013) Thermo-electro-mechanical vibration of piezoelectric nanoplates based on the nonlocal theory. Composite Structures106: 167–174.
30.
MahmoudMA (2016) Validity and accuracy of resonance shift prediction formulas for microcantilevers: a review and comparative study. Critical Reviews in Solid State and Materials Sciences41(5): 386–429.
31.
MurmuTAdhikariS (2013) Nonlocal mass nanosensors based on vibrating monolayer graphene sheets. Sensors and Actuators B: Chemical188: 1319–1327.
32.
MurmuTMcCarthyMAAdhikariS (2013) In-plane magnetic field affected transverse vibration of embedded single-layer graphene sheets using equivalent nonlocal elasticity approach. Composite Structures96: 57–63.
33.
NayfehAHMookDT (2008) Nonlinear Oscillations. Hoboken: John Wiley & Sons.
34.
NourbakhshHMohammadzadehRRafieeM, et al. (2012) Nonlinear effects on resonance behaviour of beams in micro scale. In Applied Mechanics and Materials110: 4178–4186.
35.
PotekinRKimSMcFarlandDM, et al. (2018) A micromechanical mass sensing method based on amplitude tracking within an ultra-wide broadband resonance. Nonlinear Dynamics92(2): 287–304.
36.
PradhanSCPhadikarJK (2009) Nonlocal elasticity theory for vibration of nanoplates. Journal of Sound and Vibration325(1–2): 206–223.
37.
RajuISRaoGVRajuKK (1976) Effect of longitudinal or inplane deformation and inertia on the large amplitude flexural vibrations of slender beams and thin plates. Journal of Sound and Vibration49(3): 415–422.
ReddyJN (2006) Theory and Analysis of Elastic Plates and Shells. Florida: CRC Press.
40.
ReddyJN (2013) An Introduction to Continuum Mechanics. Cambridge: Cambridge University Press.
41.
ShenLShenH-SZhangC-L (2010) Nonlocal plate model for nonlinear vibration of single layer graphene sheets in thermal environments. Computational Materials Science48(3): 680–685.
42.
ShenZ-BTangH-LLiD-K, et al. (2012) Vibration of single-layered graphene sheet-based nanomechanical sensor via nonlocal Kirchhoff plate theory. Computational Materials Science61: 200–205.
SudakLJ (2003) Column buckling of multiwalled carbon nanotubes using nonlocal continuum mechanics. Journal of Applied Physics94(11): 7281–7287.
45.
WangX (2012) Piezoelectric nanogenerators-harvesting ambient mechanical energy at the nanometer scale. Nano Energy1(1): 13–24.
46.
WangYLiF-MWangY-Z (2015) Nonlinear vibration of double layered viscoelastic nanoplates based on nonlocal theory. Physica E: Low-dimensional Systems and Nanostructures67: 65–76.
47.
WangZLSongJ (2006) Piezoelectric nanogenerators based on zinc oxide nanowire arrays. Science312(5771): 242–246.
48.
ZhangCChenWZhangC (2013) Two-dimensional theory of piezoelectric plates considering surface effect. European Journal of Mechanics – A/Solids41: 50–57.
49.
ZhangCZhuJChenW, et al. (2014) Two-dimensional theory of piezoelectric shells considering surface effect. European Journal of Mechanics – A/Solids43: 109–117.
50.
ZhangDPLeiYJShenZB (2017) Thermo-electro-mechanical vibration analysis of piezoelectric nanoplates resting on viscoelastic foundation with various boundary conditions. International Journal of Mechanical Sciences131–132: 1001–1015.
51.
ZhangJWangC (2012) Vibrating piezoelectric nanofilms as sandwich nanoplates. Journal of Applied Physics111(9): 094303.
52.
ZhangYQLiuGRWangJS (2004) Small-scale effects on buckling of multiwalled carbon nanotubes under axial compression. Physical Review B70(20): 205430.
53.
ZhouZ-GWuL-ZDuS-Y (2006) Non-local theory solution for a mode I crack in piezoelectric materials. European Journal of Mechanics – A/Solids25(5): 793–807.
