Resonant frequency and flutter instability of a nanocantilever with the surface effects

Abstract

Nanorods and nanotubes are widely used as biosensors and reinforced fibers. Surface effects should be considered due to their nano-meter order dimension in at least one direction. This paper studies vibration and stability of a nanocantilever carrying mass under non-conservative loading. Special emphasis is placed on determination of flutter loads and resonant frequencies of a nanocantilever-mass system subjected to a generalized follower force. An exact characteristic equation is derived. For various concrete cases, the load–frequency interaction curves are displayed and the influences of surface elasticity and residual surface tension are analyzed for both conservative and non-conservative forces. The resonant frequencies are calculated for bending vibration of a sensor with attached mass. Flutter loads are obtained under a generalized follower force. Beck’s column with the surface effects is a special case of the present. Surface influences from residual surface tension and surface elasticity on the resonant frequencies and critical loads are examined. It is found that there exists the lowest flutter load independent of the tip mass through which all force–frequency interaction curves passes at a given frequency for any generalized follower force in possible direction.

Introduction

With the development of micro/nanoelectromechanical system, beam-like carbon nanotubes (CNTs), nanowires, nanorods, etc. have been widely used to fabricate nanomechanical resonators and biosensors [1], [2]. For such a class of one-dimensional structures, the determination of the natural frequencies is a fundamental issue. Based on the analysis of frequency shift, additional mass being carried by sensor can be identified by solving an inverse problem of mass-frequency interaction [3]. Up to now, nanocantilever or bridge nanotube are recognized as ultrasensitive sensors for ultrafine resolution applications up to atomic-resolution [4] and they may detect a range of 10 GHz–1.5 THz frequencies or zeptogram-scale nanoparticle [5], [6]. A simple theoretical model for nanomechanical mass sensors is based on the classical beam theory. For example, the simplest Euler beam theory has been applied to determine the resonant frequencies and frequency shift, which give a good coincidence with other simulation results to some extent [7], [8].

On the other hand, a large number of experimental evidences show size-dependent material properties of CNTs or other 1D nanostructures. Therefore, the above-mentioned approach through the classical beam theory exists an evident shortcoming, lying in that the classical beam theory does not contain the scale parameter and the size-dependence of material properties cannot be characterized. So far, researchers have proposed some size-dependent beam theories to deal with the problem under consideration. Adopting the size-dependent beam theory, the natural frequencies of nanocantilevers have been determined by analytic and numerical approaches such as the nonlocal beam theory [9]. Murmu and Adhikari [10] employed the nonlocal beam theory to analyze resonance of a nanocantilever biosensor. Shen et al. [11] studied a carbon nanotube-based nanomechanical sensor using the nonlocal Timoshenko beam theory. Elishakoff et al. [12] investigated the resonance frequencies based on the nonlocal beam theory incorporating with the surface effect. Li et al. [13] proposed an approximate explicit expression for the dependence of mass identification on the frequency shift using the nonloal beam theory.

Although many researches of nanoscale sensors have been conducted, most of the above-mentioned works are focused on the absence of axial forces. In fact, axial force or generalized follower force arises in practical applications due to heat effect, end constraint, initial stress, etc. The axial load plays a dominant role in piezo patch sensors and actuators [14]. Singh et al. [15] gave an approach to implement axial and follower forces in microelectromechanical systems. For nanomechanical sensor composed of a double-walled carbon nanotube, Shen et al. [16] studied the influence of initial stress using the nonlocal beam theory. Xiang et al. [17] dealt with dynamic stability of a nonlocal nanocolumn. Recently, based on the classical beam theory without the size effects, the natural frequencies and frequency shift of nanomechanical sensors in the presence of axial force has been analyzed by some researchers [18], [19].

In addition, Cuenot et al. [20] applied resonant-contact atomic force microscopy to measure the elastic properties of silver and lead nanowires and found that the increase of apparent elastic modulus for smaller diameters is attributed to surface tension effects. Li et al. [21] showed that the surface effects are responsible for the measured size-dependent Young’s modulus and obtained results coincide with the previous simulation results of direct-atomistic model. A large number of theoretical papers related to the surface effects have been reported to interpret how the surface stress and surface elasticity affect the mechanical behaviors of micro or nanobeams. For example, Zhang et al. [22] suggested three typical theoretical models for a rectangular beam to account for the effect of surface stress. For an V-shaped or generally shaped cantilever, the effect of the surface stress has been also analyzed [23]. For dynamic analysis, the surface stress also affects the resonant frequencies of microcantilever sensors [24]. The effect of surface stress on the stiffness of a cantilever plate has been formulated based on a three-dimensional model with neglect of surface stress difference and the corresponding couple [25]. Taking account of both surface stress and surface elasticity, static bending of the theories of Euler and Timoshenko beams by analyzed by He and Lilley [26] and by Li et al. [27], respectively. For dynamic response within the framework of the Euler–Bernoulli theory, transverse vibration of nanomechanical cantilevers has been investigated by [28]. When considering the shear deformation and rotational moment of inertia of cross-section, free vibration [29], [30] and forced vibration [31] of a nanobeam were formulated. Using various beam theories, governing equations were established and critical buckling loads of a simply-supported nanobeam-column with the surface effects were computed [32]. Although a great amount of work has been conducted, there is no information on the study of vibration and dynamic stability of nanobeams with the surface effects included when axial or generalized follower force is present.

In this article, vibration analysis and dynamic stability of a nanocantilever carrying additional mass under a non-conservative force are studied. The surface elasticity and surface stress are taken into account. Using Hamilton’s principle, a governing equation is derived and an exact characteristic equation for determining the resonant frequencies and critical loads is obtained. The load–frequency interaction curves are presented graphically for various cases of interest. Some novel findings on the resonant frequencies and flutter loads are given.