Free vibration behavior of rotating bidirectional functionally graded nano-beams is studied based on Eringen’s nonlocal theory. The beam material consists of ceramic and metal constituents, and the material is graded across the length and thickness directions. The mathematical formulation is framed on Euler–Bernoulli beam theory, and the system of governing equations is derived in variational form using Hamilton’s principle. The governing equations are discretized and transformed to an eigen value problem using Ritz method. The model is formulated to study the flapping and lead-lag motions due to free vibration. The model is verified with the available numerical results. The numerical results are presented in non-dimensional frequency-speed plane to study the influence of normalized nonlocal parameter, length gradient parameter, thickness gradient parameter, root radius parameter, and section aspect ratio. Some normalized mode shapes are presented to illustrate the mode switching phenomenon. The mathematical model of a nonlocal rotating bidirectional functionally graded material nano-beam is presented for the first time through this work and the reported results are new of its kind.
LamDCCYangFChongACM, et al.
Experiments and theory in strain gradient elasticity. J Mech Phys Solid2003;
51: 1477–1508.
2.
LiuHPanCHLiuP.Dimension effect on mechanical behavior of silicon micro-cantilever beams. Measurement2008;
41: 885–895.
3.
LiJWangXZhaoL, et al.
Rotation motion of designed nano-turbine.Sci Rep2014;
4: 5846.
4.
WangLWuHWangF.Design of nano screw pump for water transport and its mechanisms.Sci Rep2017;
7: 41717.
5.
WangLWuHWangF.Water desalination using nano screw pumps with a considerable processing rate. RSC Adv2017; 7: 20360–20368.
6.
EringenAC.Nonlocal polar elastic continua. Int J Eng Sci1972;
10: 1–16.
7.
EringenACEdelenDGB.On nonlocal elasticity. Int J Eng Sci1972;
10: 233–248.
8.
EringenAC.On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J Appl Phys1983;
54: 4703–4710.
9.
BirmanVByrdLW.Modeling and analysis of functionally graded materials and structures. Appl Mech Rev Trans ASME2007;
60: 195–2006.
10.
LinSCHsiaoKM.Vibration analysis of a rotating Timoshenko beam. J Sound Vibr2001;
240: 303–322.
11.
YooHHParkJHParkJ.Vibration analysis of rotating pre-twisted blades. Comput Struct2001;
79: 1811–1819.
12.
BanerjeeJRSuHJacksonDR.Free vibration of rotating tapered beams using the dynamic stiffness method. J Sound Vibr2006;
298: 1034–1054.
13.
AvramovKVPierreCShyriaievaN.Flexural-flexural-torsional nonlinear vibrations of pre-twisted rotating beams with asymmetric cross-sections. J Vibr Control2007;
13: 329–364.
14.
GundaJBGanguliR.New rational interpolation functions for finite element analysis of rotating beams. Int J Mech Sci2008;
50: 578–588.
15.
StoykovSRibeiroP.Vibration analysis of rotating 3D beams by the p-version finite element method. Finite Elem Anal Des2013;
65: 76–88.
16.
BanerjeeJRKennedyD.Dynamic stiffness method for inplane free vibration of rotating beams including Coriolis effects. J Sound Vibr2014;
333: 7299–7312.
17.
LibrescuLOhSYSongO.Thin-walled beams made of functionally graded materials and operating in a high temperature environment: vibration and stability. J Therm Stresses2005;
28: 649–712.
18.
PiovanMTSampaioR.A study on the dynamics of rotating beams with functionally graded properties. J Sound Vibr2009;
327: 134–143.
19.
RajasekaranS.Differential transformation and differential quadrature methods for centrifugally stiffened axially functionally graded tapered beams. Int J Mech Sci2013;
74: 15–31.
20.
EbrahimiFHashemiM.On vibration behavior of rotating functionally graded double-tapered beam with the effect of porosities. Proc IMechE, Part G: J Aerospace Engineering2016;
230: 1903–1916.
21.
PalSDasD.A tangent stiffness-based approach to study free vibration of shear-deformable functionally graded material rotating beam through a geometrically non-linear analysis. J Strain Anal Eng Des2017;
52: 310–332.
22.
DasD.Free vibration and buckling analyses of geometrically non-linear and shear-deformable FGM beam fixed to the inside of a rotating rim. Compos Struct2017;
179: 628–645.
23.
Dehrouyeh-SemnaniAM.The influence of size effect on flapwise vibration of rotating microbeams. Int J Eng Sci2015;
94: 150–163.
24.
Dehrouyeh-SemnaniAMBehboodiJouybariMDehrouyehM.On size-dependent lead-lag vibration of rotating microcantilevers. Int J Eng Sci2016;
101: 50–63.
25.
ArvinH.Free vibration analysis of micro rotating beams based on the strain gradient theory using the differential transform method: Timoshenko versus Euler-Bernoulli beam models. Eur J Mech A/Solids2017;
65: 336–348.
26.
ArvinH.The flapwise bending free vibration analysis of micro-rotating Timoshenko beams using the differential transform method. J Vibr Control2018;
24: 4868–4884.
27.
ShafieiNKazemiMGhadiriM.On size-dependent vibration of rotary axially functionally graded microbeam. Int J Eng Sci2016;
101: 29–44.
28.
ShenasAGZiaeeSMalekzadehP.Vibrational behavior of rotating pre-twisted functionally graded microbeams in thermal environment. Compos Struct2016;
157: 222–235.
29.
MazanogluKGulerS.Flap-wise and chord-wise vibrations of axially functionally graded tapered beams rotating around a hub. Mech Syst Signal Process2017;
89: 97–107.
30.
FangJGuJWangH.Size-dependent three-dimensional free vibration of rotating functionally graded microbeams based on a modified couple stress theory. Int J Mech Sci2018;
136: 188–199.
31.
BhattacharyaSDasD.Free vibration analysis of bidirectional-functionally graded and double-tapered rotating micro-beam in thermal environment using modified couple stress theory. Compos Struct2019;
215: 471–492.
32.
BhattacharyaSDasD. Modified couple stress-based free vibration behavior of pre-twisted tapered BFGM rotating micro beam considering spin-softening and Coriolis effects. Proc IMechE, Part L: J Materials, Design and Applications 2020; 234: 21--47.
33.
PeddiesonJBuchananGRMcNittRP.Application of nonlocal continuum models to nanotechnology. Int J Eng Sci2003;
41: 305–312.
34.
ReddyJN.Nonlocal theories for bending, buckling and vibration of beams. Int J Eng Sci2007;
45: 288–307.
35.
NiuJCLimCWLeungAYT.Third-order non-local beam theories for the analysis of symmetrical nanobeams. Proc IMechE, Part C: J Mechanical Engineering Science2009;
223: 2451–2463.
36.
ThaiSThaiHTVoTP, et al.
A simple shear deformation theory for nonlocal beams. Compos Struct2018;
183: 262–270.
37.
AlvesMRibeiroP.Non-linear modes of vibration of Timoshenko nanobeams under electrostatic actuation. Int J Mech Sci2017;
130: 188–202.
38.
YangTTangYLiQ, et al.
Nonlinear bending, buckling and vibration of bi-directional functionally graded nanobeams. Compos Struct2018;
204: 313–319.
39.
NiknamHAghdamMM.A semi analytical approach for large amplitude free vibration and buckling of nonlocal FG beams resting on elastic foundation. Compos Struct2015;
119: 452–462.
40.
EltaherMAAlshorbagyAEMahmoudFF.Determination of neutral axis position and its effect on natural frequencies of functionally graded macro/nanobeams. Compos Struct2013;
99: 193–201.
41.
TrabelssiMEl-BorgiSFernandesR, et al.
Nonlocal free and forced vibration of a graded Timoshenko nanobeam resting on a nonlinear elastic foundation. Compos Struct2018;
184: 821–838.
42.
PradhanSCMurmuT.Application of nonlocal elasticity and DQM in the flapwise bending vibration of a rotating nanocantilever. Physica E2010;
42: 1944–1949.
43.
Aranda-RuizJLoyaJFernández-SáezJ.Bending vibrations of rotating nonuniform nanocantilevers using the Eringen nonlocal elasticity theory. Compos Struct2012;
94: 2990–3001.
44.
NarendarSGopalakrishnanS.Nonlocal wave propagation in rotating nanotube. Results Phys2011;
1: 17–25.
45.
NarendarS.Differential quadrature based nonlocal flapwise bending vibration analysis of rotating nanotube with consideration of transverse shear deformation and rotary inertia. Appl Math Comput2012;
219: 1232–1243.
46.
PourasgharAHomauniMKamarianS.Differential quadrature based nonlocal flapwise bending vibration analysis of rotating nanobeam using the Eringen nonlocal elasticity theory under axial load. Polym Compos2016; 37: 3175–3180.
47.
GhadiriMHosseiniSHSShafieiN.A power series for vibration of a rotating nanobeam with considering thermal effect. Mech Adv Mater Struct2016;
23: 1414–1420.
48.
GhafarianMAriaeiA.Free vibration analysis of a multiple rotating nano beams system based on the Eringen nonlocal elasticity theory. J Appl Phys2016;
120: 054301.
49.
GhadiriMShafieiN.Nonlinear bending vibration of a rotating nanobeam based on nonlocal Eringen’s theory using differential quadrature method. Microsyst Technol2016;
22: 2853–2867.
50.
ShafieiNKazemiMGhadiriM.Nonlinear vibration behavior of a rotating nanobeam under thermal stress using Eringen’s nonlocal elasticity and DQM. Appl Phys A2016;
122: 728.
51.
AzimiMMirjavadiSSShafieiN, et al.
Thermomechanical vibration of rotating axially functionally graded nonlocal Timoshenko beam. Appl Phys A2017;
123: 104.
52.
AzimiMMirjavadiSSShafieiN, et al.
Vibration of rotating functionally graded Timoshenko nano-beams with nonlinear thermal distribution. Mech Adv Mater Struct2018;
25: 467–480.
53.
KhanikiHB.Vibration analysis of rotating nanobeam systems using Eringen’s two-phase local/nonlocal model. Physica E2018;
99: 310–319.
54.
AtanasovMSStojanovićV.Nonlocal forced vibrations of rotating cantilever nano-beams. Eur J Mech A Solids2020;
79: 103850.
55.
KianiKSoltaniS.Nonlocal longitudinal, flapwise, and chordwise vibrations of rotary doubly coaxial/non-coaxial nanobeams as nanomotors. Int J Mech Sci2020;
168: 105291.
56.
MalikMDasD. Free flexural vibration analysis of rotating nano-beam using nonlocal theory. In: Proceedings of 64th Congress of Indian Society of Theoretical and Applied Mechanics (ISTAM), Bhubanesawar, India, 9–12 December 2019.
57.
GhadiriMShafieiNAkbarshahiA.Influence of thermal and surface effects on vibration behavior of nonlocal rotating Timoshenko nanobeam. Appl Phys A2016;
122: 673.
58.
BaghaniMMohammadiMFarajpourA.Dynamic and stability analysis of the rotating nanobeam in a nonuniform magnetic field considering the surface energy. Int J Appl Mech2016;
8: 1650048.
59.
MohammadiMSafarabadiMRastgooA, et al.
Hygro-mechanical vibration analysis of a rotating viscoelastic nanobeam embedded in a visco-Pasternak elastic medium and in a nonlinear thermal environment. Acta Mech2016;
227: 2207–2232.
60.
PreethiKRaghuPRajagopalA, et al.
Nonlocal nonlinear bending and free vibration analysis of a rotating laminated nano cantilever beam. Mech Adv Mater Struct2018;
25: 439–450.
