According to the present literature review, the axial vibration of the axially functionally graded (FG) tapered nanorod with attached nonlinear spring has not been addressed so far. In this study, the axial vibration of the FG tapered nanorod is studied based on Eringen’s nonlocal theory, in which one end of the nanorod is clamped and the other end is attached to a nonlinear spring and a nanoparticle. The influence of different parameters such as the nonlinear spring constant, mass of the nanoparticle, and the nonlocal parameter on the natural frequencies is presented in detail. The solutions via Tables can be utilized as a reliable reference for evaluating the validity of future researches. The presented nonlocal solutions can be helpful for those who are interested in designing micro/nano electromechanical systems.
AdhikariSMurmuTMcCarthyMA (2013) Dynamic finite element analysis of axially vibrating nonlocal rod. Finite Elements in Analysis and Design63: 42–50.
2.
AkgözBCivalekÖ (2011) Buckling analysis of cantilever carbon nanotubes using the strain gradient elasticity and modified couple stress theories. Journal of Computational and Theoretical Nanoscience8(9): 1821–1827.
3.
AkgözBCivalekÖ (2012) Investigation of size effects on static response of single-walled carbon nanotubes based on strain gradient elasticity. International Journal of Computational Methods9(2): 1240032.
4.
AkgözBCivalekÖ (2013) Longitudinal vibration analysis of strain gradient bars made of functionally graded materials (FGM). Composites Part B: Engineering55: 263–268.
5.
AkgözBCivalekÖ (2014) Longitudinal vibration analysis for microbars based on strain gradient elasticity theory. Journal of Vibration and Control20(4): 606–616.
6.
AkgozBCivalekÖ (2017) Effects of thermal and shear deformation on vibration response of functionally graded thick composite microbeams. Composites Part B: Engineering129: 77–87.
7.
AkgözBCivalekÖ (2018) Vibrational characteristics of embedded microbeams lying on a two-parameter elastic foundation in thermal environment. Composites Part B: Engineering150: 68–77.
8.
AsghariMTaatiE (2013) A size-dependent model for functionally graded micro-plates for mechanical analyses. Journal of Vibration and Control19(11): 1614–1632.
9.
AydogduM (2009a) Axial vibration of the nanorods with the nonlocal continuum rod model. Physica E: Low-dimensional Systems and Nanostructures41(5): 861–864.
10.
AydogduM (2009b) A general nonlocal beam theory: Its application to nanobeam bending, buckling and vibration. Physica E: Low-dimensional Systems and Nanostructures41(9): 1651–1655.
11.
AydogduM (2012a) Axial vibration analysis of nanorods (carbon nanotubes) embedded in an elastic medium using nonlocal elasticity. Mechanics Research Communications43: 34–40.
12.
AydogduM (2012b) Longitudinal wave propagation in nanorods using a general nonlocal unimodal rod theory and calibration of nonlocal parameter with lattice dynamics. International Journal of Engineering Science56: 17–28.
13.
AydogduMElishakoffI (2014) On the vibration of nanorods restrained by a linear spring in-span. Mechanics Research Communications57: 90–96.
14.
BahramiA (2017a) Free vibration, wave power transmission and reflection in multi-cracked nanorods. Composites Part B: Engineering127: 53–62.
15.
BahramiA (2017b) A wave-based computational method for free vibration, wave power transmission and reflection in multi-cracked nanobeams. Composites Part B: Engineering120: 168–181.
16.
BahramiATeimourianA (2015) Nonlocal scale effects on buckling, vibration and wave reflection in nanobeams via wave propagation approach. Composite Structures134: 1061–1075.
17.
BahramiATeimourianA (2016) Study on the effect of small scale on the wave reflection in carbon nanotubes using nonlocal Timoshenko beam theory and wave propagation approach. Composites Part B: Engineering91: 492–504.
18.
BarrettaRČanadijaMde SciarraFM (2016) A higher-order Eringen model for Bernoulli–Euler nanobeams. Archive of Applied Mechanics86(3): 483–495.
19.
ChangTP (2012) Small scale effect on axial vibration of non-uniform and non-homogeneous nanorods. Computational Materials Science54: 23–27.
20.
DaneshMFarajpourAMohammadiM (2012) Axial vibration analysis of a tapered nanorod based on nonlocal elasticity theory and differential quadrature method. Mechanics Research Communications39(1): 23–27.
21.
Dehrouyeh-SemnaniAMBahramiA (2016) On size-dependent Timoshenko beam element based on modified couple stress theory. International Journal of Engineering Science107: 134–148.
22.
DemirÇCivalekÖ (2017) A new nonlocal FEM via Hermitian cubic shape functions for thermal vibration of nano beams surrounded by an elastic matrix. Composite Structures168: 872–884.
23.
DemirÇCivalekÖAkgözB (2010) Free vibration analysis of carbon nanotubes based on shear deformable beam theory by discrete singular convolution technique. Mathematical and Computational applications15(1): 57–65.
24.
EringenAC (1972) Theory of micromorphic materials with memory. International Journal of Engineering Science10(7): 623–641.
25.
EringenAC (2002) Nonlocal Continuum Field Theories, Berlin: Springer Science & Business Media.
26.
GüvenU (2014) Love–Bishop rod solution based on strain gradient elasticity theory. Comptes Rendus Mécanique342(1): 8–16.
27.
HeLLouJZhangE, et al.(2015) A size-dependent four variable refined plate model for functionally graded microplates based on modified couple stress theory. Composite Structures130: 107–115.
28.
HuangZ (2012) Nonlocal effects of longitudinal vibration in nanorod with internal long-range interactions. International Journal of Solids and Structures49(15–16): 2150–2154.
29.
KahrobaiyanMHAsghariMRahaeifardM, et al.(2010) Investigation of the size-dependent dynamic characteristics of atomic force microscope microcantilevers based on the modified couple stress theory. International Journal of Engineering Science48(12): 1985–1994.
30.
KaramiHFaridM (2015) A new formulation to study in-plane vibration of curved carbon nanotubes conveying viscous fluid. Journal of Vibration and Control21(12): 2360–2371.
31.
KarličićDCajićMMurmuT, et al.(2015) Nonlocal longitudinal vibration of viscoelastic coupled double-nanorod systems. European Journal of Mechanics – A/Solids49: 183–196.
32.
Li L, Hu Y and Li X (2016) Longitudinal vibration of size-dependent rods via nonlocal strain gradient theory. International Journal of Mechanical Sciences 115-116: 135–144.
33.
LiCLiSYaoL, et al.(2015) Nonlocal theoretical approaches and atomistic simulations for longitudinal free vibration of nanorods/nanotubes and verification of different nonlocal models. Applied Mathematical Modelling39(15): 4570–4585.
34.
LimCWZhangGReddyJN (2015) A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation. Journal of the Mechanics and Physics of Solids78: 298–313.
35.
MercanKCivalekÖ (2016) DSC method for buckling analysis of boron nitride nanotube (BNNT) surrounded by an elastic matrix. Composite Structures143: 300–309.
36.
MercanKCivalekÖ (2017) Buckling analysis of Silicon carbide nanotubes (SiCNTs) with surface effect and nonlocal elasticity using the method of HDQ. Composites Part B: Engineering114: 34–45.
37.
MindlinRTierstenH (1962) Effects of couple-stresses in linear elasticity. Archive for Rational Mechanics and Analysis11(1): 415–448.
38.
MindlinRD (1964) Micro-structure in linear elasticity. Archive for Rational Mechanics and Analysis16(1): 51–78.
39.
MurmuTAdhikariS (2010) Nonlocal effects in the longitudinal vibration of double-nanorod systems. Physica E: Low-dimensional Systems and Nanostructures43(1): 415–422.
40.
NaghinejadNOvesyHR (2019) Viscoelastic free vibration behavior of nano-scaled beams via finite element nonlocal integral elasticity approach. Journal of Vibration and Control25(2): 445–459.
41.
NarendarSGopalakrishnanS (2011) Axial wave propagation in coupled nanorod system with nonlocal small scale effects. Composites Part B: Engineering42(7): 2013–2023.
42.
NumanoğluHMAkgözBCivalekÖ (2018) On dynamic analysis of nanorods. International Journal of Engineering Science130: 33–50.
43.
ShenasAGZiaeeSMalekzadehP (2017) Nonlinear vibration analysis of pre-twisted functionally graded microbeams in thermal environment. Thin-Walled Structures118: 87–104.
44.
ŞimşekM (2012) Nonlocal effects in the free longitudinal vibration of axially functionally graded tapered nanorods. Computational Materials Science61: 257–265.
45.
WangKWangB (2014) Vibration modeling of carbon nanotube-based biosensors incorporating thermal and nonlocal effects. Journal of Vibration and Control22(5): 1405–1414.
46.
Xu XJ, Zheng ML and Wang XC (2017) On vibrations of nonlocal rods: Boundary conditions, exact solutions and their asymptotics. International Journal of Engineering Science 119: 217–231.
47.
YangFChongACMLamDCC, et al.(2002) Couple stress based strain gradient theory for elasticity. International Journal of Solids and Structures39(10): 2731–2743.
48.
YayliMÖ (2018) An efficient solution method for the longitudinal vibration of nanorods with arbitrary boundary conditions via a hardening nonlocal approach. Journal of Vibration and Control24(11): 2230–2246.
