An Analytical Solution For Free Vibrations Of A Cantilever Nanobeam With A Spring Mass System

Year 2015, Volume: 7 Issue: 4, 10 - 18, 01.12.2015

Mustafa Özgür Yaylı

https://doi.org/10.24107/ijeas.251255

Cited By: 4

Abstract

An analytical solution for the title problem is presented using the nonlocal elasticity theory based on Euler-Bernoulli beam theory. Fourier sine series is used to represent lateral displacement of the nanobeam. Stokes’ transformation is applied to derive the coefficient matrix of the corresponding systems of linear equations. This matrix also contains the relationship between spring and mass parameters. A convergence study is provided to show how the first three frequency parameter of the nanobeam would converge by an increase of series terms in the literature. The results are given in a series of figures and tables for various combinations of boundary conditions

Keywords

Nonlocal elasticity theory, nanobeam, Fourier sine series, Stokes’ transformation

References

• [1] Eringen, A. C., Nonlocal polar elastic continua. International Journal of Engineering Science, 10, 1-16, 1972.
• [2] Narendar, S., Buckling analysis of micro-/nano-scale plates based on two variable refined plate theory incorporating nonlocal scale effects, Compos. Struct., 93, 3093-3103, 2011
• [3] Pradhan, S.C., Phadikar, J.K., Nonlocal elasticity theory for vibration of nanoplates. J. Sound Vib., 325, 206-223, (2009).
• [4] Shen, L., Shen, H.S., Zhang, C.L., Nonlocal plate model for nonlinear vibration of single layer graphene sheets in thermal environments, Comput. Mater. Sci., 48, 680-685, 2010.
• [5] Aydogdu, M., A general nonlocal beam theory: its application to nanobeam bending, buckling and vibration, Physica E 41, 1651-1655, 2009.
• [6] Liu, T., Hai, M., Zhao, M., Delaminating buckling model based on nonlocal Timoshenko beam theory for microwedge indentation of a film/substrate system, Eng. Fract. Mech. 75, 4909-4919, 2008.
• [7] Reddy, J.N., Nonlocal theories for bending, buckling and vibration of beams, Int. J. Eng. Sci., 45, 288-307, 2007.
• [8] Civalek, Ö., Demir, Ç., Bending analysis of microtubules using nonlocal Euler–Bernoulli beam theory, Appl. Math. Model., 35, 2053-2067, 2011.
• [9] Civalek, Ö., Akgöz, B., Free vibration analysis of microtubules as cytoskeleton components: nonlocal Euler–Bernoulli beam modeling, Sci. Iranica Trans. B: Mech. Eng., 17, 367-375, 2010.
• [10] Lu, P., Lee, H.P., Lu, C., Zhang, P.Q., Dynamic properties of flexural beams using a nonlocal elasticity model, J. Appl. Phys., 99, 73510-73518, 2006.
• [11] Wang, C.M., Kitipornchai, S., Lim, C.W., Eisenberger, M., Beam bending solutions based on nonlocal Timoshenko beam theory, J. Eng. Mech., 134, 475-481, 2008.
• [12] Rahmani, O., Pedram, O., Analysis and modeling the size effect on vibration of functionally graded nanobeams based on nonlocal Timoshenko beam theory, Int. J. Eng. Sci, 77, 55-70, 2014.
• [13] Murmu, T., Pradhan, S.C., Small-scale effect on the vibration of nonuniform nanocantilever based on nonlocal elasticity theory, Physica E, 41, 1451-1456, 2009.
• [14] Setoodeh, A.R., Khosrownejad, M., Malekzadeh, P., Exact nonlocal solution for post buckling of single-walled carbon nanotubes. Physica E, 43, 1730-1737, 2011.
• [15] Eltaher, M.A., Emam, S.A., Mahmoud, F.F., Static and stability analysis of nonlocal functionally graded nanobeams. Compos. Struct, 96, 82-88, 2013.
• [16] Thai, H.T., A nonlocal beam theory for bending, buckling, and vibration of nanobeams. Int. J. Eng. Sci., 52, 56-64, 2012.
• [17] Yayli, M.Ö., Buckling Analysis of a Rotationally Restrained Single Walled Carbon Nanotube, Acta Physica Polonica A, 127, 3, 678-683, 2015.
• [18] Reddy J. N., Pang, S. D., Nonlocal continuum theories of beam for the analysis of carbon nanotubes,. Journal of Applied Physics, 103, 1-16, 2008.
• [19] Yayli, M.Ö., Stability analysis of gradient elastic microbeams with arbitrary boundary conditions, Journal of Mechanical Science and Technology, 29, 8, 3373-3380, 2015.