Non-local vibration of simply supported nano-beams: Higher-order modes

Abstract

In this article, the effects of some parameters, including rotary inertia, non-local parameter, and length-to-thickness ratio, on natural frequencies are studied for both classical and non-local theories. For Timoshenko beam, the equations of motion and the boundary conditions are derived from Hamilton's principle and then non-local constitutive equations of Eringen are employed to altogether formulate the problem. Afterward, obtained governing equations are used to study the free vibrations of a Timoshenko's simply supported nano-beam. And finally, the effects of above-mentioned parameters on estimated frequencies in classical and non-local elasticity theories are investigated. Results show that the discrepancy between the frequencies of higher-order vibration modes obtained from two theories increases and also significant reductions in natural frequencies occur when the rotary inertia is considered in the computations.

A single-variable first-order shear deformation nonlocal theory for the flexure of isotropic nanobeams

“…Considerable attention in the literature has been given to predict the mechanical behaviour of shear deformable nanobeams by utilizing the nonlocal Timoshenko beam theory based on the nonlocal elasticity theory of Eringen (NTBT). Some of the notable references in the above-mentioned field are as follows: Wang et al [17] investigated the stability of micro-and nanorods or tubes; Wang and Varadan [18] studied the free vibration of single-walled and double-walled carbon nanotubes; Reddy [19] investigated the flexure, stability and free vibration of nanobeams; Wang and Liew [20] studied the flexure of micro-and nanorods or tubes; Reddy and Pang [21] studied the flexure, stability and free vibration of carbon nanotubes; Wang et al [22] investigated the free vibration of microor nanobeams; Wang et al [23] investigated the flexure of micro-and nanobeams; Benzair et al [24] studied the free vibration of single-walled carbon nanotubes by incorporating temperature effects; Murmu and Pradhan [25] employed the Winkler-type and Pasternak-type foundation models to study the stability of single-walled carbon nanotubes embedded in an elastic medium by utilizing the differential quadrature technique; Reddy [26] reformulated the nonlocal Timoshenko beam model to account for the geometric nonlinearity; Roque et al [27] utilized the global and local collocation techniques to obtain numerical solutions for the flexure, stability and free vibration of nanobeams by utilizing the mesh-less method; Hosseini-Ara et al [28] studied the stability of carbon nanotubes; Şimşek and Yurtcu [29] investigated the flexure and stability of functionally graded nanobeams; Eltaher et al [30] employed the finite element method to study the free vibration of functionally graded nanobeams; Eltaher et al [31] employed the finite element method to investigate the flexure and stability of functionally graded nanobeams; Rahmani and Pedram [32] obtained the closed-form solution for the free vibration of functionally graded nanobeams; Semmah et al [33] studied effects of small-scale and chirality on the thermal buckling characteristics of zigzag single-walled carbon nanotubes; Preethi et al [34] utilized the Gurtin-Murdoch surface elasticity theory to take into account surface stress effects and employed the finite element method to study the nonlinear flexure and free vibration of nanobeams; Kadioğlu and Yayli [35] employed the Fourier series to study the stability of nanosized beams; Masoumi and Masoumi [36] investigated effects of the small-scale and rotary inertia on the free vibration of nanobeams with emphasis on...…”

“…4. The work reported by references [17][18][19][20][21][22][23][24][25][26][27][28][33][34][35][36] pertains to beams made of isotropic material, the work reported by references [29][30][31][32] pertains to beams made of functionally graded material with the variation of material properties along the beam thickness direction, and the work reported by reference [38] pertains to beams made of functionally graded material with the variation of material properties along the beam axial direction. 5.…”

“…5. The work reported by references [18,19,24,25,27,29,32,33,36] presents illustrative examples pertaining to beams having simply supported boundary conditions, the work reported by reference [28] presents illustrative examples pertaining to beams having clamped boundary conditions, the work reported by references [30,31,35] presents illustrative examples pertaining to beams having simply supported, propped-cantilever and clamped boundary conditions, the work reported by references [20,34] presents illustrative examples pertaining to beams having simply supported, clamped and cantilever boundary conditions, the work reported by references [17,[21][22][23]38] presents illustrative examples pertaining to beams having simply supported, propped-cantilever, clamped and cantilever boundary conditions.…”

“…As evident through the recent work reported by references [33][34][35][36][37][38], the research in the field of predicting the mechanical behaviour of first-order shear deformable nanobeams is active. The present work needs to be construed in this context.…”

An analytical approach for vibration analysis of laminated orthotropic beam based on nonlocal theory

“…Sudak 2 used classical theory to present a nonlocal model for column buckling of multilayer carbon nanotubes considering the van der Waals interactions. Masoumi 3 investigated free vibration of simply supported nanobeam using Timoshenko beam theory and Thai 4 proposed a nonlocal shear deformation beam theory for bending, buckling, and free vibration analysis of simply supported nanobeams. They applied analytical methods to the solution.…”