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Fixed-free single-walled carbon nanotubes (SWCNTs) have attracted a lot of interest in recent years due to their suitability for a wide range of applications, such as field emission and vacuum microelectronic devices, nanosensors, and nanoactuators. Based on a cantilever beam-bending model with a rigid mass at the free end and mode analysis, an analytical solution is developed in the present study to deal with the resonant frequency and mode shapes of a SWCNT- based mass sensor. The resonant frequency shift and mode shape of the fixed-free SWCNTs caused by the addition of a nanoscale particle to the beam tip are examined in order to explore the suitability of SWCNTs as a mass detector device. The simulation results reveal that the volume of the added particle has little effect on the first resonant frequency. In contrast, the second resonant frequency decreases with increasing the volume of the added particle. Furthermore, the resonant frequency shift of the first mode is very obvious for the amount of added mass, and the second resonant frequency decreases rapidly with increasing volume of added particle. Therefore, the first and second resonant frequencies can be used in the measurement of the mass of added particle and its volume, respectively.

Since their discovery in 1991 by Ijima, carbon nanotubes (CNTs) have demonstrated potential for use in a diverse range of applications, such as nanobiological devices and nanomechanical systems. Due to their remarkable mechanical, physical, and chemical properties, carbon nanotubes may be used as fluid conveyers or potential reinforcements in nanocomposite materials [1-3]. Since experiments at the nanoscale are extremely difficult to conduct, theoretical modeling of the mechanical response of CNTs has been carried out [4,5]. CNTs have been utilized as nanoactuators [

Several studies have investigated the use of CNTs as a mass sensor [7,12,13]. Compared to piezoelectric sensors, nanotubes provide better precision [

Resonant frequency shift-based mass sensors are explored using a tip mass in the form of a nanoscale particle, which is attached to fixed-free SWCNTs. Techniques that facilitate the development of smaller, faster, cheaper, and more sensitive mass sensor devices are required. Using a hierarchical modeling scheme, the equivalentcontinuum modeling technique [17,18] can be used to predict the bulk mechanical behavior of nanostructured materials, such as the beam shown in

Recently, the continuum mechanics method has been pplied to analyze the dynamic responses of individual CNTs. Based on the Euler-Bernoulli beam model [

where is the transversal displacement response, is the flexural stiffness, and is the mass per unit length. The natural mechanical resonant frequency is induced in a cantilever carbon nanotube when the applied frequency approaches the resonant frequency. In this study, a SWCNT-based mass sensor is simulated as a cantilever beam with a rigid mass at the free end. The continuum mechanics method is used to obtain the resonant frequency and the mode shapes of a sensor analytically by mode analysis method. One form of solution of Equation (1) can be obtained easily by the separation of variables:

where is a specific shape of the free-vibration motion with time-dependent amplitude. can be expressed as:

where, , , and are real constants that can be determined using the boundary condition.

Consider the cantilever beam with a rigid mass at the free end shown in

where is the resonant frequency of the SWCNT. Moreover, force and moment equilibrium of the rigid mass requires that the following four boundary conditions be satisfied:

Substituting Equation (3) and its derivative expressions into these equations gives

Making use of, , , and, Equations (10) and (11) yield and. Substituting these equalities into Equations (12) and (13), changing all signs, and placing the resulting expressions in matrix form, one obtains: (see (14))where and are the radius and the added mass of the particle, respectively. and can be defined as followed:

For coefficients and to be nonzero, the determinant of the square matrix in this equation must equal zero, thus giving the frequency equation:

The solution of this transcendental equation provides the values of, which represent the frequencies of vibration of the cantilever beam with a rigid mass at the free end. Either form of Equation (14) can now be employed to express coefficient in terms of; the first gives:

or

This result along with the previously obtained conditions that and allows the mode-shape expression of Equation (3) to be written in the form of Equation (19).

Substituting separately the frequency-equation roots for into this expression, one obtains the corresponding mode-shape functions.

In this study, a tip mass in the form of a nanoscale particle is attached to a fixed-free SWCNT, whose behavior of the nanotube is investigated using mass sensor mode analysis. A resonant frequency shift-based mass sensor is made using the fixed-free SWCNT. The dimensions of the SWCNT are as follows: inner radius 9.9 nm, outer radius 16nm, stiffness, mass per unit length and length 6.8 μm. In order to investigate the effects of attached mass on the resonant frequency, and were set as the dimensionless values of radius and added mass, respectively.