Bending vibrations of rotating nonuniform nanocantilevers using the Eringen nonlocal elasticity theory
Introduction
Many micro- or nanoelectromechanical systems (MEMS or NEMS) devices incorporate structural elements such as beams and plates in micro- (or nano-) length scale. Size effects are significant in the mechanical behavior of these structures in which dimensions are small and comparable to molecular distances. Since the atomic and molecular models require a great computational effort, simplified models are useful for analyzing the mechanical behavior of such devices.
The nanostructures that undergoing rotation are system with a promise future to be used in nanomachines [1], [2] which include shaft of nanomotor [3], [4] devices such as fullerene gears and carbon nanotube gears [5].
Classical continuum mechanics cannot predict the size effect, due to its scale-free character. Despite some sporadic efforts in the 19th century and in the first half of the 20th century to capture the effects of microstructure using the continuum equations of elasticity with additional higher-order derivatives, it was not until the 1960s that a major revival took place. From this time are the works of Toupin [6], [7], Mindlin and Tiersten [3], Kröner [4], Green and Rivlin [8], Mindlin [9], [10], Mindlin and Eshel [11].
More recently, Eringen derived a simple stress-gradient theory from his earlier integral nonlocal theories [12].
In the early 1990s, Aifantis and coworkers suggested to extend the linear elastic constitutive relations with the Laplacian of the strain [1], [13], [14].
Askes and Gitman [15] show that both Eringen and Aifantes theories can be unified. An excellent overview on the historical development of theses theories, as well as its mining and implementation can be found in the paper by Askes and Aifantis [2].
Among the size-dependent continuum theories, the theory of nonlocal continuum mechanics initiated by Eringen and coworkers [16], [5], [12] has been widely used to analyze many problems, such as wave propagation, dislocation, and crack singularities and, from the pioneer work of Peddieson et al. [17], for problems involving nanostructures. Thus, the nonlocal theory of elasticity has been used to address the behavior of beams [18], [19], [20], [21], [22], [23], rods [24], [25], [26], [27], [28], [29], plates [30], as well as carbon nanotubes (CNTS) [31], [32], [33], [34], [35], [36], [37].
Nowadays, a great effort is devoted to the vibration analysis of nanobeams and CNTS under rotation using the Eringen nonlocal elasticity theory [38], [39], [40].
Pradhan and Murmu [38], applied a nonlocal beam model to investigate the flap wise bending-vibration characteristics of a uniform rotating nanocantilever. The nonlocal natural frequencies were obtained using the Differential Quadrature Method (DQM). They also discussed the effects of the nonlocal small-scale, angular velocity and hub radius on vibration characteristics of the nanocantilever.
Murmu and Adhikari [39], investigated the same problem, but now considering an initially prestressed single-walled carbon to analyze the effect on the initial preload in the vibration characteristics.
In both papers the nonlocal boundary conditions related to the free end of the nanobeam are not properly considered.
Narendar and Gopalakrishnan [40] analyzed the wave dispersion behavior of a uniform rotating nanotube modeled as an nonlocal Euler–Bernoulli beam. They consider the spatial variation of the centrifugal force in and average sense, replacing the variable axial effort by the maximum force (at the root of the nanocantilever).
In this paper we investigate the flap wise bending-vibration characteristics of a nonuniform rotating nanocantilever considering the true (quadratic) spatial variation of the axial force due to the rotation. The area of the nanobeam cross-section of the nanocantilever is assumed to change linearly. The solution method of the corresponding nonlocal equations of motion are solved using a pseudo-spectral collocation method based on Chebyshev polynomials. The effects of the nonlocal small-scale, angular velocity, nonuniformity of the section and hub radius on vibration characteristics of the nanocantilever are discussed.
The paper ir organized as follow: in Section 2 a brief resume of the constitutive equations of the Eringen nonlocal elasticity theory is given. In Section 3 the equation for the flapwise vibrations of the rotating nanocantilever is derived as well as the proper nonlocal boundary conditions. In Section 4 the pseudo-spectral collocation method based on Chebyshev polynomials used to find the solution is briefly exposed. The numerical results and discussion of the effect of different variables in the nonlocal frequencies of the nanostructures appear in Section 5. Finally some conclusions are given in Section 6.
Section snippets
Nonlocal constitutive relations
The theory of nonlocal elasticity, developed by Eringen [16], [41] and Eringen and Edelen [42] states that the nonlocal stress-tensor components σij at any point x in a body can be expressed as:where tij(x) are the components of the classical local stress tensor at point x, which are related to the components of the linear strain tensor εkl by the conventional constitutive relations for a Hookean material, i.e:
The meaning of Eq. (1) is that the nonlocal
Problem formulation
Let us consider a beam of length L along the axial coordinate x, of constant thickness b and variable height t(x). The beam is clamped at section O (x = 0) located at distance r from the axes around which rotates at constant angular velocity Ω as shown in Fig. 1(a). Let v(x, t) be the transverse deflection along the coordinate y.
The equation of motion in the vertical direction for a beam slice of length dx (see Fig. 1(b)) can be written as:where ρ is the density of the material, A(
Pseudo-spectral collocation solution based on Chebyshev polynomials
In order to obtain the natural frequencies of the problem defined by Eqs. (23), (24), (25), (26), (27), a pseudo-spectral collocation method based on Chebyshev polynomials will be used.
Chebyshev polynomials are recursive orthogonal polynomials defined as:where m is an integer. Due to their recursive nature and fast convergence characteristics, Chebyshev polynomials have been used in the literature for the solution of boundary-value problems [43], [44], [45], [46]
Results and discussion
In this article the small-scale effects on the vibration response of rotating nanocantilever beams are shown with respect to the angular velocity of rotating and to the cross-section variation for different hub radius.
Fig. 2, Fig. 3, Fig. 4, Fig. 5, Fig. 6, Fig. 7 show the variation of the three first modes nondimensional frequencies, λ2, with nondimensional angular velocity parameter, γ2, for different values of the nonlocal parameter, h, and different values of β, which represents the
Conclusions
In this work, The effects of the nonlocal small-scale, angular velocity, nonuniformity of the section and hub radius on the three first flapwise vibration frequencies have been considered. By means of a pseudo-spectral collocation method, based on Chebyshev polynomials, the nonlocal equations of motion for the beam are solved.
It is observed that the nondimensional frequencies increase with the rotating angular velocity for both local and nonlocal elastic models. For the same geometry and
Acknowledgments
The authors thank the Comisión Interministerial de Ciencia y Tecnología of the Spanish Government and to the Comunidad Autónoma de Madrid for partial support of this work through the Research Projects DPI2011-23191 and CCG10-UC3M-DPI-5596, respectively.
References (49)
- et al.
Gradient elasticity in statics and dynamics: an overview of formulations, length scale identification procedures, finite element implementations and new results
Int J Solids Struct
(2011) On the physical reality of torque stresses in continuum mechanics
Int J Eng Sci
(1963)Linear theory of nonlocal elasticity and dispersion of plane-waves
Int J Eng Sci
(1972)Second gradient of strain and surface-tension in linear elasticity
Int J Solids Struct
(1965)- et al.
On first strain-gradient theories in linear elasticity
Int J Solids Struct
(1968) On the role of gradients in the localization of deformation and fracture
Int J Eng Sci
(1992)- et al.
On the structure of the mode iii crack-tip in gradient elasticity
Scripta Metall Mater
(1992) Nonlocal polar elastic continua
Int J Eng Sci
(1972)- et al.
Application of nonlocal continuum models to nanotechnology
Int J Eng Sci
(2003) - et al.
buckling and vibration of beams
Int J Eng Sci
(2007)