Abstract
This study present frequency parameters and mode shapes of nonuniform Single-Walled Carbon Nanotube (SWCNT) placed on Winkler elastic foundation. Eringen’s nonlocal theory is implemented in the Euler–Bernoulli beam to inquire size-dependent behavior of single-walled carbon nanotube. Here flexural stiffness is assumed to vary exponentially which is responsible for making it nonuniform since many nanoelectromechanical systems acquire geometrically nonuniform model. Differential Quadrature Method (DQM) is adopted and MATLAB code has been developed to explore the tabular and graphical results for different scaling parameters. All the standard boundary condition, viz, S-S, C-S, C-C, and C-F are taken into consideration, and obtained results are compared with the well-known results available in the literature showing excellent agreement. Also, the effects of various scaling parameters like nonuniform parameter, the nonlocal parameter, aspect ratio, and Winkler modulus parameter on frequency parameters are demonstrated using numerical as well as graphical results.
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Dai H, Hafner JH, Rinzler AG, Colbert DT, Smalley RE (1996) Nanotubes as nanoprobes in scanning probe microscopy. Nature 384:147–150
Tornabene F, Fantuzzi N, Bacciocchi M, Viola E (2016) Effect of agglomeration on the natural frequencies of functionally graded carbon nanotube-reinforced laminated composite doubly-curved shells. Compos B 89:187–218
Fantuzzi N, Tornabene F, Bacciocchi M, Dimitri R (2017) Free vibration analysis of arbitrarily shaped functionally graded carbon nanotube-reinforced plates. Compos B 115:384–408
Tornabene F, Fantuzzi N, Bacciocchi M (2017) Linear static response of nanocomposite plates and shells reinforced by agglomerated carbon nanotubes. Compos B 115:449–476
Bani´c D, Bacciocchi M, Tornabene F, Ferreira AJM (2017) Influence of Winkler-Pasternak foundation on the vibrational behavior of plates and shells reinforced by agglomerated carbon nanotubes. Appl Sci 7:1–55
Eringen AC (1972) Nonlocal polar elastic continua. Int J Eng Sci 10:1–16
Reddy JN (2007) Nonlocal theories for bending, buckling and vibration of beams. Int J Eng Sci 45:288–307
Aydogdu M (2009) A general nonlocal beam theory: its application to nanobeam bending, buckling and vibration. Phys E 41:1651–1655
Eltaher MA, Alshorbagy AE, Mahmoud FF (2013) Vibration analysis of Euler-Bernoulli nanobeams by using finite element method. Appl Math Model 37:4787–4797
Zhou D (1993) A General solution to vibrations of beams on variable Winkler elastic foundation. Comput Struct 47:83–90
Eisenberger M (1994) Vibration frequencies for beams on variable one- and two-parameter elastic foundations. J Sound Vibr 176:577–584
Auersch L (2008) Dynamic interaction of various beams with the underlying soil–finite and infinite, half-space and Winkler models. Eur J Mech A/Solids 27:933–958
Ma X, Butterworth JW, Clifton GC (2009) Static analysis of an infinite beam resting on a tensionless Pasternak foundation. Eur J Mech A/Solids 28:697–703
Civalek O (2007) Nonlinear analysis of thin rectangular plates on Winkler-Pasternak elastic foundations by DSC-HDQ methods. Appl Math Model 31:606–624
Kacar A, Tan HT, Kaya MO (2011) Free vibration analysis of beams on variable Winkler elastic foundation by using the differential transform method. Math Comput Appl 16:773–783
Chakraverty S, Behera L (2015) Vibration and buckling analyses of nanobeams embedded in an elastic medium. Chin Phys B 24(1–8):097305
Civalek Ö (2004) Application of differential quadrature (DQ) and harmonic differential quadrature (HDQ) for buckling analysis of thin isotropic plates and elastic columns. Eng Struct 26:171–186
Civalek Ö (2005) Geometrically nonlinear dynamic analysis of doubly curved isotropic shells resting on elastic foundation by a combination of harmonic differential quadrature-finite difference methods. Int J Press Vessels Pip 82:470–479
Chakraverty S, Jena SK (2018) Free vibration of single walled carbon nanotube resting on exponentially varying elastic foundation. Curved Layer Struct 5:260–272
Chakraverty S, Behera L (2017) Buckling analysis of nanobeams with exponentially varying stiffness by differential quadrature method. Chin Phys B 26(7):074602
Jena SK, Chakraverty S (2018) Free vibration analysis of Euler–Bernoulli nanobeam using differential transform method. Int J Comput Mater Sci Eng 7(1–21):1850020
Jena SK, Chakraverty S (2018) Free vibration analysis of variable cross-section single layered graphene nano-ribbons (SLGNRs) Using differential quadrature method. Front Built Environ 4:63
Jena SK, Chakraverty S (2018) Free vibration analysis of single walled carbon nanotube with exponentially varying stiffness. Curved Layer Struct 5:201–212
Jena RM, Chakraverty S (2018) Residual power series method for solving time-fractional model of vibration equation of large membranes. J Appl Comput Mech 5:603–615
Jena SK, Chakraverty S (2019) Differential quadrature and differential transformation methods in buckling analysis of nanobeams. Curved Layer Struct 6:68–76
Jena SK, Chakraverty S, Jena RM, Tornabene F (2019) A novel fractional nonlocal model and its application in buckling analysis of Euler-Bernoulli nanobeam. Mater Res Express 6:1–17
Jena SK, Chakraverty S, Tornabene F (2019) Vibration characteristics of nanobeam with exponentially varying flexural rigidity resting on linearly varying elastic foundation using differential quadrature method. Mater Res Express 6:1–13
Jena SK, Chakraverty S, Tornabene F (2019) Dynamical behavior of nanobeam embedded in constant, linear, parabolic and sinusoidal types of winkler elastic foundation using first-order nonlocal strain gradient model. Mater Res Express 6:1–23
Jena RM, Chakraverty S, Jena SK (2019) Dynamic response analysis of fractionally damped beams subjected to external loads using homotopy analysis method. J Appl Comput Mech 5:355–366
Jena SK, Chakraverty S (2018) Solving fuzzy static structural problems using symmetric group method. Recent advances in applications of computational and fuzzy mathematics. Springer, Singapore, pp 95–107
Quan J, Chang C (1989) New insights in solving distributed system equations by the quadrature method—I. Analysis. Comput Chem Eng 13:779–788
Shu C (2000) Differential quadrature and its application in engineering. Springer, Singapore
Acknowledgements
The authors would like to thank Defence Research & Development Organization (DRDO), Ministry of Defence, New Delhi, India (Sanction Code: DG/TM/ERIPR/GIA/17-18/0129/020) for the funding to carry out the present research work smoothly.
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Jena, S.K., Chakraverty, S. (2020). Vibration Analysis of Nonuniform Single-Walled Carbon Nanotube Resting on Winkler Elastic Foundation Using DQM. In: Chakraverty, S., Biswas, P. (eds) Recent Trends in Wave Mechanics and Vibrations. Lecture Notes in Mechanical Engineering. Springer, Singapore. https://doi.org/10.1007/978-981-15-0287-3_27
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DOI: https://doi.org/10.1007/978-981-15-0287-3_27
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