Nonlocal Vibration Analysis for Micro/Nano Beam on Winkler Foundation via DTM

Year 2016, Volume: 8 Issue: 4, 108 - 118, 26.12.2016

Çiğdem Demir

https://doi.org/10.24107/ijeas.281514

Cited By: 5

Abstract

In the present study, vibration of micro/nano beams on Winkler
foundation is studied using Eringen's nonlocal elasticity theoy. Hamilton’s
principle is employed to derive the governing equations. Differential transform
method is used to obtain result. Simply supported and clamped–clamped boundary
conditions are used to study natural frequencies. The effect of nonlocal
parameter and Winkler elastic foundation modulus on the natural frequencies of
the nonlocal Euler-Bernoulli beam is investigated and tabulated. The
differential transform method is applicable for micro/nano beams and gives high
accuracy results.

Keywords

Natural frequency, differential transforms method, nonlocal elasticity theory, Winkler elastic foundation

References

• [1] Civalek, Ö., Korkmaz, A., Demir, Ç., Discrete singular convolution approach for buckling analysis of rectangular Kirchhoff plates subjected to compressive loads on two-opposite edges. Advances in Engineering Software, 41(4), 557-560, 2010.
• [2] Baltacıoğlu, A., Civalek, Ö., Akgöz, B., Demir, F., Large deflection analysis of laminated composite plates resting on nonlinear elastic foundations by the method of discrete singular convolution. International Journal of Pressure Vessels and Piping, 88(8), 290-300, 2011.
• [3] Avcar, M., Free vibration analysis of beams considering different geometric characteristics and boundary conditions. International Journal of Mechanics and Applications, 4(3), 94-100, 2014.
• [4] Avcar, M., Elastic buckling of steel columns under axial compression. American Journal of Civil Engineering, 2(3), 102-108, 2014.
• [5] Attia, A., Tounsi, A., Bedia, E.A., Mahmoud, S., Free vibration analysis of functionally graded plates with temperature-dependent properties using various four variable refined plate theories. Steel and composite structures, 18(1), 187-212, 2015.
• [6] Emsen, E., Mercan, K., Akgöz, B., Civalek, Ö., Modal Analysis Of Tapered Beam-Column Embedded In Winkler Elastic Foundation. International Journal of Engineering & Applied Sciences, 7(1), 25-35, 2015.
• [7] Mahi, A., Tounsi, A., A new hyperbolic shear deformation theory for bending and free vibration analysis of isotropic, functionally graded, sandwich and laminated composite plates. Applied Mathematical Modelling, 39(9), 2489-2508, 2015.
• [8] Mercan, K., Demir, Ç., Akgöz, B., Civalek, Ö., Coordinate Transformation for Sector and Annular Sector Shaped Graphene Sheets on Silicone Matrix. International Journal of Engineering & Applied Sciences, 7(2), 56-73, 2015.
• [9] Panda, S., Kumar, R., Ramachandra, L., Post-buckled vibration characteristic of composite cylindrical shell panels under parabolic in-plane edge compression. International Journal of Applied Mechanics, 7(03), 1550035, 2015.
• [10] Avcar, M., Effects of Material Non-Homogeneity and Two Parameter Elastic Foundation on Fundamental Frequency Parameters of Timoshenko Beams. Acta Physica Polonica A, 130(1), 375-378, 2016.
• [11] Civalek, O., Ersoy, H., Mercan, K., Free vibration of annular plates by discrete singular convolution and differential quadrature methods. Journal of Applied and Computational Mechanics, 2016.
• [12] Demir, Ç., Mercan, K., Civalek, Ö., Determination of critical buckling loads of isotropic, FGM and laminated truncated conical panel. Composites Part B: Engineering, 94, 1-10, 2016.
• [13] Ersoy, H., Mercan, K., Civalek, Ö., Frequencies of FGM shells and annular plates by the methods of discrete singular convolution and differential quadrature methods. Composite Structures, 2016.
• [14] Kandasamy, R., Dimitri, R., Tornabene, F., Numerical study on the free vibration and thermal buckling behavior of moderately thick functionally graded structures in thermal environments. Composite Structures, 157, 207-221, 2016.
• [15] Mercan, K., Demir, Ç., Civalek, Ö., Vibration analysis of FG cylindrical shells with power-law index using discrete singular convolution technique. Curved and Layered Structures, 3(1), 2016.
• [16] Eringen, A.C., Edelen, D., On nonlocal elasticity. International Journal of Engineering Science, 10(3), 233-248, 1972.
• [17] Reddy, J., Nonlocal theories for bending, buckling and vibration of beams. International Journal of Engineering Science, 45(2), 288-307, 2007.
• [18] Aydogdu, M., A general nonlocal beam theory: its application to nanobeam bending, buckling and vibration. Physica E: Low-dimensional Systems and Nanostructures, 41(9), 1651-1655, 2009.
• [19] Aydogdu, M., Axial vibration of the nanorods with the nonlocal continuum rod model. Physica E: Low-dimensional Systems and Nanostructures, 41(5), 861-864, 2009.
• [20] Civalek, Ö., Akgöz, B., Static analysis of single walled carbon nanotubes (SWCNT) based on Eringen’s nonlocal elasticity theory. International Journal of Engineering and Applied Sciences, 47-56, 2009.
• [21] Baltacıoglu, A.K., Akgöz, B., Civalek, Ö., Nonlinear static response of laminated composite plates by discrete singular convolution method. Composite Structures, 93(1), 153-161, 2010.
• [22] Phadikar, J., Pradhan, S., Variational formulation and finite element analysis for nonlocal elastic nanobeams and nanoplates. Computational materials science, 49(3), 492-499, 2010.
• [23] Civalek, Ö., Demir, Ç., Bending analysis of microtubules using nonlocal Euler–Bernoulli beam theory. Applied Mathematical Modelling, 35(5), 2053-2067, 2011.
• [24] Mahmoud, F., Eltaher, M., Alshorbagy, A., Meletis, E., Static analysis of nanobeams including surface effects by nonlocal finite element. Journal of Mechanical Science and Technology, 26(11), 3555-3563, 2012.
• [25] Alshorbagy, A.E., Eltaher, M., Mahmoud, F., Static analysis of nanobeams using nonlocal FEM. Journal of Mechanical Science and Technology, 27(7), 2035-2041, 2013.
• [26] Akgöz, B., Civalek, Ö., A novel microstructure-dependent shear deformable beam model. International Journal of Mechanical Sciences, 99, 10-20, 2015.
• [27] Eltaher, M., Khater, M., Emam, S.A., A review on nonlocal elastic models for bending, buckling, vibrations, and wave propagation of nanoscale beams. Applied Mathematical Modelling, 40(5), 4109-4128, 2016.
• [28] Wang, Q., Varadan, V., Vibration of carbon nanotubes studied using nonlocal continuum mechanics. Smart Materials and Structures, 15(2), 659, 2006.
• [29] Benzair, A., Tounsi, A., Besseghier, A., Heireche, H., Moulay, N., Boumia, L., The thermal effect on vibration of single-walled carbon nanotubes using nonlocal Timoshenko beam theory. Journal of Physics D: Applied Physics, 41(22), 225404, 2008.
• [30] Murmu, T., Pradhan, S., Thermo-mechanical vibration of a single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity theory. Computational Materials Science, 46(4), 854-859, 2009.
• [31] Şimşek, M., Vibration analysis of a single-walled carbon nanotube under action of a moving harmonic load based on nonlocal elasticity theory. Physica E: Low-dimensional Systems and Nanostructures, 43(1), 182-191, 2010.
• [32] Ansari, R., Ramezannezhad, H., Nonlocal Timoshenko beam model for the large-amplitude vibrations of embedded multiwalled carbon nanotubes including thermal effects. Physica E: Low-dimensional Systems and Nanostructures, 43(6), 1171-1178, 2011.
• [33] Ansari, R., Sahmani, S., Small scale effect on vibrational response of single-walled carbon nanotubes with different boundary conditions based on nonlocal beam models. Communications in Nonlinear Science and Numerical Simulation, 17(4), 1965-1979, 2012.
• [34] Gürses, M., Akgöz, B., Civalek, Ö., Mathematical modeling of vibration problem of nano-sized annular sector plates using the nonlocal continuum theory via eight-node discrete singular convolution transformation. Applied Mathematics and Computation, 219(6), 3226-3240, 2012.
• [35] Amirian, B., Hosseini-Ara, R., Moosavi, H., Thermal vibration analysis of carbon nanotubes embedded in two-parameter elastic foundation based on nonlocal Timoshenko's beam theory. Archives of Mechanics, 64(6), 581-602, 2013.
• [36] Ghannadpour, S., Mohammadi, B., Fazilati, J., Bending, buckling and vibration problems of nonlocal Euler beams using Ritz method. Composite Structures, 96, 584-589, 2013.
• [37] Rahmani, O., Pedram, O., Analysis and modeling the size effect on vibration of functionally graded nanobeams based on nonlocal Timoshenko beam theory. International Journal of Engineering Science, 77, 55-70, 2014.
• [38] Akgöz, B., Civalek, Ö., A microstructure-dependent sinusoidal plate model based on the strain gradient elasticity theory. Acta Mechanica, 226(7), 2277-2294, 2015.
• [39] Wang, C., Zhang, Y., Ramesh, S.S., Kitipornchai, S., Buckling analysis of micro-and nano-rods/tubes based on nonlocal Timoshenko beam theory. Journal of Physics D: Applied Physics, 39(17), 3904, 2006.
• [40] Wang, Q., Varadan, V., Quek, S., Small scale effect on elastic buckling of carbon nanotubes with nonlocal continuum models. Physics Letters A, 357(2), 130-135, 2006.
• Balkaya, Müge, Metin O. Kaya, and Ahmet Sağlamer,Analysis of the vibration of an elastic beam supported on elastic soil using the differential transform method,135-146,Archive of Applied Mechanics,79(2)