Nonlocal axial vibration of the multiple Bishop nanorod system

Abstract

Construction of reliable dynamic models of nanostructures is an important task for design procedures of different nanoresonator devices. Such theoretical models allow as to perform different numerical experiments, which is the key point in the development of advanced nanodevices. This paper presents a new nanoresonator model based on the axial vibration of the elastic multi-nanorod system. It is assumed that the system of multiple nanorods is embedded in an elastic medium. The governing equations of motion of a coupled multi-nanorod system are derived using the Hamilton’s principle, the nonlocal elastic constitutive relation, and Bishop’s rod theory, where effects of inertia of the lateral motion and the shear stiffness are considered. Exact closed form solutions for natural frequencies are obtained for one and multiple nanorod systems with different boundary conditions. Then, results for natural frequencies obtained by the finite difference method are compared with the results obtained analytically. Effects of nonlocal parameter, different rod theories, number of nanorods and stiffness coefficient of an elastic medium on natural frequencies are examined through several numerical examples.

Axial dynamic analysis of a Bishop nanorod with arbitrary boundary conditions

“…The following values of parameters of the Bishop nanorod (5,5) are used to calculate the the natural frequencies given in Tables 1-2 (ρ = 9517 kg/m 3 , E = 6.85 TPa, and L = 24.4 nm). It can be seen from Table 1, the presented natural frequencies of the proposed model are in excellent agreement with those computed in ref [26]. To obtain the natural frequencies of clamped-free boundary conditions, axial spring coefficients are √ √ √ √…”

A consistent variational formulation of Bishop nonlocal rods

“…The physically unmotivated definition of stress resultant (Eq. (10)2 in [30]), as typically adopted in literature, has been thus shown to be unnecessary.…”

“…In case of a uniformly loaded nano-rod with fixed-free ends, the classical BC are expressed as (29) The non-dimensional axial displacement of the nano-rod can be then detected exploiting the aforementioned solution technique while imposing the classical BC and CBC as (30) The non-dimensional axial displacement consistent with LBM is also detected as (31) ( ) 2 2 2 2 2 2 2 1 exp exp 2 1 exp 1 e 2 2 1 2 2 2 2 1 xp 1 x x x x u x l l r l l r l l r r r r r ae ö ç ÷ ae ö ae ö ç The maximum axial displacement is additionally determined for numerical presentation (32) The maximum axial deformation of the nonlocal Bishop nano-rod at the free end is independent of the non-dimensional radius of gyration . Accordingly, to examine the effects of non-dimensional gyration radius on the axial displacement filed, the value of the axial displacement field at the mid-span of the rod is considered (33)…”

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

“…Moreover, Zhang et al [13] applied Eringen's elasticity theory to present the scale parameter coefficients for the natural vibrations and buckling behavior of simply supported rectangular plates. Further studies about this topic were examined by other authors [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31].…”