Abstract
Purpose
Rotating components such as rotating beams and plates are working in complex physical environments subjected to nonuniform distribution of their own materials, vibration caused by external excitation may result in work instability, structural fatigue and the like, so the dynamic analysis of rotating structures is of great significance.
Methods
Based on the nonlocal elasticity theory and Timoshenko beam model, the vibration of a rotating functionally graded piezoelectric nanobeam is investigated. First, the dynamic governing equations and corresponding boundary conditions are derived by the Hamiltonian principle. Then, the governing equations and boundary conditions are discretized by the differential quadrature method. Subsequently, the vibration characteristics of nanobeams are analyzed after detailed numerical calculations.
Results
The effects of various parameters including rotational velocity, nonlocal parameter, functional gradient index, length–height geometry ratio and external voltage on the vibration characteristics under different boundary conditions are examined. Finally, the numerical results show that these parameters have non-negligible effects on the natural frequencies and modal shapes.
Conclusions
The dynamics of rotating functionally graded nanobeams are affected by both external kinematic and voltage factors and inherent scale factor and their coupling effects. In particular, a nonlinear small-scale effect is observed for a rotating functionally graded piezoelectric nanobeam, which may be useful to the design and optimization of the nano-electro-mechanical system including the rotating structures.
Similar content being viewed by others
References
Wang ZL (2009) ZnO nanowire and nanobelt platform for nanotechnology. Mater Sci Eng R 64(3–4):33–71
Park KI, Xu S, Liu Y, Hwang GT, Kang SJ, Wang ZL et al (2010) Piezoelectric BaTiO3 thin film nanogenerator on plastic substrates. Nano Lett 10(12):4939–4943
Guo J, Kim K, Lei KW, Fan D (2015) Ultra-durable rotary micromotors assembled from nanoentities by electric fields. Nanoscale 7(26):11363–11370
Lim CW, Wang CM (2007) Exact variational nonlocal stress modeling with asymptotic higher-order strain gradients for nanobeams. J Appl Phys 101(5):054312–054317
De M et al (2017) New Insights on the deflection and internal forces of a bending nanobeam. Chin Phys Lett 34(9):096201
Yan JW, Lai SK (2019) Nonlinear dynamic behavior of single-layer graphene under uniformly distributed loads. Compos B 165:473–490
Yan JW, Lai SK (2018) Superelasticity and wrinkles controlled by twisting circular graphene. Comput Methods Appl Mech Eng 338:634–656
Eringen AC (1983) On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J Appl Phys 54(9):4703–4710
Lam DCC, Yang F, Chong ACM et al (2003) Experiments and theory in strain gradient elasticity. J Mech Phys Solids 51(8):1477–1508
Lim CW, Zhang G, Reddy JN (2015) A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation. J Mech Phys Solids 78(5):298–313
Li C, Liu JJ, Cheng M, Fan XL (2017) Nonlocal vibrations and stabilities in parametric resonance of axially moving viscoelastic piezoelectric nanoplate subjected to thermo-electro-mechanical forces. Compos Part B Eng 116:153–169
Shen JP, Wang PY, Gan WT et al (2020) Stability of vibrating functionally graded nanoplates with axial motion based on the nonlocal strain gradient theory. Int J Struct Stab Dyn 20(2):651–657
Li C (2016) On vibration responses of axially travelling carbon nanotubes considering nonlocal weakening effect. J Vib Eng Technol 4(2):175–181
Wang PY, Li C, Li S (2020) Bending vertically and horizontally of compressive nano-rods subjected to nonlinearly distributed loads using a continuum theoretical approach. J Vib Eng Technol 8(6):947–957
Li C, Lim CW, Yu JL (2011) Dynamics and stability of transverse vibrations of nonlocal nanobeams with a variable axial load. Smart Mater Struct 20(1):015023
Li C et al (2011) Analytical solutions for vibration of simply supported nonlocal nanobeams with an axial force. Int J Struct Stab Dyn 11:257–271
Li C, Lim CW, Yu JL (2011) Twisting statics and dynamics for circular elastic nanosolids by nonlocal elasticity theory. Acta Mech Solida Sin 24(6):484–494
Zhao Z, Ni Y, Zhu S et al (2020) Thermo-electro-mechanical size-dependent buckling response for functionally graded graphene platelet reinforced piezoelectric cylindrical nanoshells. Int J Struct Stab Dyn 20(9):2050100
Yu YM, Lim CW (2013) Nonlinear constitutive model for axisymmetric bending of annular graphene-like nanoplate with gradient elasticity enhancement effects. J Eng Mech 139(8):1025–1035
Lim CW, Yang Q, Zhang JB (2012) Thermal buckling of nanorod based on non-local elasticity theory. Int J Non-Linear Mech 47(5):496–505
Lim CW, Xu R (2012) Analytical solutions for coupled tension-bending of nanobeam-columns considering nonlocal size effects. Acta Mech 223(4):789–809
Yang Q, Lim CW (2012) Thermal effects on buckling of shear deformable nanocolumns with von Kármán nonlinearity based on nonlocal stress theory. Nonlinear Anal Real World Appl 13(2):905–922
Lim CW, Niu JC, Yu YM (2010) Nonlocal stress theory for buckling instability of nanotubes: new predictions on stiffness strengthening effects of nanoscales. J Comput Theor Nanosci 7(10):2104–2111
Wang CM, Kitipornchai S, Lim CW et al (2008) Beam bending solutions based on nonlocal Timoshenko beam theory. J Eng Mech 134(6):475–481
Yang Y, Lim CW (2012) Non-classical stiffness strengthening size effects for free vibration of a nonlocal nanostructure. Int J Mech Sci 54(1):57–68
Rahmani O, Hosseini SAH, Moghaddam MHN et al (2015) Torsional vibration of cracked nanobeam based on nonlocal stress theory with various boundary conditions: an analytical study. Int J Appl Mech 07(03):1550036
Li C, Sui SH, Chen L et al (2018) Nonlocal elasticity approach for free longitudinal vibration of circular truncated nanocones and method of determining the range of nonlocal small scale. Smart Struct Syst 21(3):279–286
Lim CW, Islam MZ, Zhang G (2015) A nonlocal finite element method for torsional statics and dynamics of circular nanostructures. Int J Mech Sci 94:232–243
Islam ZM, Jia P, Lim CW (2014) Torsional wave propagation and vibration of circular nanostructures based on nonlocal elasticity theory. Int J Appl Mech 6(2):1450011
Lim CW, Yang Q (2011) Nonlocal thermal-elasticity for nanobeam deformation: exact solutions with stiffness enhancement effects. J Appl Phys 110(1):5055–5476
Lim CW (2010) Is a nanorod (or nanotube) with a lower Young’s modulus stiffer? Is not Young’s modulus a stiffness indicator? Sci China 2010(04):712–724
Lim CW, Yang Y (2010) Wave propagation in carbon nanotubes: nonlocal elasticity-induced stiffness and velocity enhancement effects. J Mech Mater Struct 5(3):459–476
Lim CW (2010) On the truth of nanoscale for nanobeams based on nonlocal elastic stress field theory: equilibrium, governing equation and static deflection. Acta Mech Sin 31(001):37–54
Lim CW (2009) Equilibrium and static deflection for bending of a nonlocal nanobeam. Adv Vib Eng 8(4):277–300
Yang XD, Lim CW (2009) Nonlinear vibrations of nano-beams accounting for nonlocal effect using a multiple scale method. Sci China Ser E 52:617–621
Muraoka T, Kinbara K, Aida T (2006) Mechanical twisting of a guest by a photoresponsive host. Nature 440(7083):512–515
Serreli V, Lee CF, Kay ER et al (2007) A molecular information ratchet. Nature 445(7127):523–527
Carlone A, Goldup SM, Lebrasseur N et al (2012) A three-compartment chemically-driven molecular information ratchet. J Am Chem Soc 134(20):8321–8323
Ye Q, Takahashi K, Hoshino N et al (2015) Huge dielectric response and molecular motions in paddle-wheel [Cu(Adamantylcarboxylate)(DMF)]. Chem Eur J 17(51):14442–14449
Guo P, Noji H, Yengo CM et al (2016) Biological nanomotors with a revolution, linear, or rotation motion mechanism. Microbiol Mol Biol Rev Mmbr 80(1):161–186
Erbas-Cakmak S, Fielden SDP, Karaca U et al (2017) Rotary and linear molecular motors driven by pulses of a chemical fuel. Science 358(6361):340–343
Azimi M, Mirjavadi SS, Shafiei N et al (2017) Thermo-mechanical vibration of rotating axially functionally graded nonlocal Timoshenko beam. Appl Phys A 123(1):104–119
Mahinzare M, Barooti MM, Ghadiri M (2018) Vibrational investigation of the spinning bi-dimensional functionally graded (2-FGM) micro plate subjected to thermal load in thermal environment. Microsyst Technol 24(3):1695–1711
Ghadiri M, Shafiei N (2016) Vibration analysis of a nano-turbine blade based on Eringen nonlocal elasticity applying the differential quadrature method. J Vib Control 23(19):1077546315627723
Farzad E, Ali D (2017) Nonlocal strain gradient based wave dispersion behavior of smart rotating magneto-electro-elastic nanoplates. Mater Res Express 4(2):025003
Ebrahimi F, Barati MR (2016) A nonlocal higher-order shear deformation beam theory for vibration analysis of size-dependent functionally graded nanobeams. Arab J Sci Eng 41(5):1679–1690
Asemi SR, Farajpour A (2014) Thermo-electro-mechanical vibration of coupled piezoelectric-nanoplate systems under non-uniform voltage distribution embedded in Pasternak elastic medium. Curr Appl Phys 14(5):814–832
Eltaher MA, Emam SA, Mahmoud FF (2012) Free vibration analysis of functionally graded size-dependent nanobeams. Appl Math Comput 218(14):7406–7420
Li C, Lai SK, Yang X (2019) On the nano-structural dependence of nonlocal dynamics and its relationship to the upper limit of nonlocal scale parameter. Appl Math Model 69(5):127–141
Wang Q (2002) On buckling of column structures with a pair of piezoelectric layers. Eng Struct 24(2):199–205
Ebrahimi F, Barati MR (2017) Vibration analysis of parabolic shear-deformable piezoelectrically actuated nanoscale beams incorporating thermal effects. Mech Adv Mater Struct 25(2):917–929
Li J, Wang X, Zhao L et al (2014) Rotation motion of designed nano-turbine. Sci Rep 4:5846–5853
Kim K, Xu X, Guo J et al (2014) Ultrahigh-speed rotating nanoelectromechanical system devices assembled from nanoscale building blocks. Nat Commun 5:3632
Jandaghian AA, Rahmani O (2016) An analytical solution for free vibration of piezoelectric nanobeams based on a nonlocal elasticity theory. J Mech 32(02):143–151
Kaghazian A, Hajnayeb A, Foruzande H (2017) Free vibration analysis of a Piezoelectric nanobeam using nonlocal elasticity theory. Struct Eng Mech 61(5):617–624
Acknowledgements
This work was supported by the National Natural Science Foundation of China (Grant Nos. 11572210 and 11972240), Guangxi Key Laboratory of Cryptography and Information Security (Grant No. GCIS201905).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Hao-nan, L., Cheng, L., Ji-ping, S. et al. Vibration Analysis of Rotating Functionally Graded Piezoelectric Nanobeams Based on the Nonlocal Elasticity Theory. J. Vib. Eng. Technol. 9, 1155–1173 (2021). https://doi.org/10.1007/s42417-021-00288-9
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s42417-021-00288-9