Transverse Vibration of Functionally Graded Tapered Double Nanobeams Resting on Elastic Foundation

Abstract

The natural vibration behavior of axially functionally graded (AFG) double nanobeams is studied based on the Euler–Bernoulli beam and Eringen’s non-local elasticity theory. The double nanobeams are continuously connected by a layer of linear springs. The oscillatory differential equation of motion is established using the Hamilton’s principle and the constitutive relations. The Chebyshev spectral collocation method (CSCM) is used to transform the coupled governing differential equations of motion into algebraic equations. The discretized boundary conditions are used to modify the Chebyshev differentiation matrices, and the system of equations is put in the matrix-vector form. Then, the dimensionless transverse frequencies and the mode shapes are obtained by solving the standard eigenvalue problem. The effects of the coupling springs, Winkler stiffness, the shear stiffness parameter, the breadth and taper ratios, the small-scale parameter, and the boundary conditions on the natural transverse frequencies are carried out. Several numerical examples were conducted, and the authors believe that the results may be interesting in designing and analyzing double and multiple one-dimensional nano structures.

Nonlocalized thermal behavior of rotating micromachined beams under dynamic and thermodynamic loads

“…By inserting Equation ( 6) into Equation ( 2), the energy of the strain can be calculated as [21,26]…”

“…The nonlocal theory of elasticity includes a scale‐dependent parameter named non‐local parameter which provides a mechanism to reduce the rigidity due to the nonlocal effects [14]. In recent years, several researches were conducted to analyze the elastic, thermal, wave propagation, and vibration behavior of micro‐ and nano‐structures on the basis of various beam theories [15–39].…”

A Perturbation Approach for Lateral Excited Vibrations of a Beam-like Viscoelastic Microstructure Using the Nonlocal Theory

“…Consequently, the nonlocal hysteresis effect is found in the lateral excited vibration of a beam-like microstructure, which is mainly caused by the small-scale effect and viscoelastic effect. It is noted that the hysteresis effect has not been reported in the nonlocal free vibration of micro/nano-structures [6][7][8][9][10][11][12][13][14][20][21][22][23][24][25][26][27][28][29][30]. Resultingly, it is a special phenomenon in excited vibration.…”

“…In fact, the microstructures are often subjected to time-dependent external excitation, and, also, the fully clamped boundary is one of the most common end restrictions, both of which are easily seen in micro-electromechanical systems (MEMS) and other devices at the microscale [5]. However, in recent years, much attention has been paid to the free vibration of microscaled materials and structures [6][7][8][9][10][11][12][13][14]. For example, Lim et al [6] investigated the free vibration of circular nanotubes subjected to both the torsional deformation and axial motion.…”

“…On the other hand, most previous studies only focused on the mechanical properties of elastic microstructures [4,[6][7][8][9][10][11][12][13][14][20][21][22][23][24][25][26][27][28][29][30]. However, from a practical engineering standpoint, completely elastic structures are rare, and most of them should be modeled as viscoelastic structures [32][33][34].…”