Analytical Solution of Vibration Analysis on Fixed-Free Single-Walled Carbon Nanotube-Based Mass Sensor ()
1. Introduction
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 [6] and as nanosensors [7]. CNTs are extremely thin tubes whose diameters are on the order of a few nanometers, but whose lengths may be thousands of times larger. The use of vertically aligned single-walled CNTs (SWCNTs) for field emission and vacuum microelectronic devices, and as nanosensors and nanoactuators, is being actively explored [8,9]. Furthermore, Chen et al. [10] studied the effects of the geometric structure and an electric field on the electronic and optical properties of quasi-zero-dimensional finite CNTs using the tight-binding model coupled with curvature effects. Hsu et al. [11] developed a model that analyzes the resonant frequency of chiral SWCNTs subjected to a thermal vibration using the Timoshenko beam model that includes the effect of rotary inertia and shear deformation.
Several studies have investigated the use of CNTs as a mass sensor [7,12,13]. Compared to piezoelectric sensors, nanotubes provide better precision [12]. Wu et al. [13] simulated the mechanical responses of individual CNTs treated as cylindrical beams or thin shells using the continuum mechanics method with commercial FEM software. However, these studies adopted either experimental or numerical approaches, which are inherently time consuming and expensive. Saether et al. [14] proposed a simple formula of the resonant frequency of the fixedfree beam that uses its spring constant, and found that a change in the mass of the CNT resonator is indicated by a shift in the resonant frequency. The sensitivity and spatial resolution of the device can be varied by changing the dimensions of the resonator. In previous studies [15, 16], the current authors obtained accurate analytical solutions of vibration responses of atomic force microscope (AFM) and nanoscale processing using the modal superposition method. In the present study, a fixed-free SWCNT-based mass sensor is simulated as a cantilever beam-bending model with a rigid mass at the free end.The continuum mechanics method is used to obtain analytical solutions of vibration analysis, including the resonant frequency and mode shapes. The results show 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 attached mass of added mass and its volume, respectively.
2. Analysis
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 Figure 1. A previous study [12] presented a cantilever-type CNT-based mass sensor. The present study considers the case of a fixed-free SWCNT with a nanoscale particle attached to its tip, as shown in Figure 2. The operation of a cantilever-based mass sensor is based on the fact that mass added to the tip causes a measurable shift in the resonant frequency of the fixed-free beam.
(a)
(b)
Figure 1. Computation model of SWCNT: (a) discrete model and (b) continuum model.
(a)
(b)
Figure 2. Fixed-free SWCNT with nanoscale particle at its tip: (a) discrete model and (b) continuum model.
Recently, the continuum mechanics method has been pplied to analyze the dynamic responses of individual CNTs. Based on the Euler-Bernoulli beam model [19], it is well known that the equation of motion of a free-vibration rod in the limited to a small amplitude is governed by the fourth-order wave equation:
(1)
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:
(2)
where 
is a specific shape of the free-vibration motion with time-dependent amplitude
. 
can be expressed as:
(3)
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 Figure 2 to which a rigid lumped mass 
with a rotary moment of inertia 
is attached. The moment and shear are no longer equal to zero at the other end due to the presence of the lumped mass. These internal force components are shown on the free-body diagram in Figure 3 along with the translational and rotary inertial force components 
and
, respectively. Under free-vibration conditions with resonant frequency
, acceleration 
and its derivative are:
(4)
(5)
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:
(6)
(7)
(a)
(b)
Figure 3. Fixed-free SWCNT with nanoscale particle at its tip: (a) SWCNT properties; (b) Forces acting on the nanoscale particle.
(8)
(9)
Substituting Equation (3) and its derivative expressions into these equations gives
(10)
(11)
(12)
(13)
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:
(15)
For coefficients 
and 
to be nonzero, the determinant of the square matrix in this equation must equal zero, thus giving the frequency equation:
(16)
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:
(17)
or
(18)
(14)
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.
3. Results and Discussion
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.
Figure 4 and Figure 5 show the variation in the first and second resonant frequencies for fixed-free SWCNT with various added mass for various radii, respectively. The simulation results in the two figures indicate that the first and second resonant frequencies decrease with increasing attached mass. Furthermore, for a constant attached mass, Figure 4 reveals that the volume of added mass has little effect on the first resonant frequency. In contrast, Figure 5 shows that the second resonant frequency decreases with increasing the volume of added mass.
Figure 6 and Figure 7 show the variation in the first and second resonant frequencies for a fixed-free SWCNT for various radii and added mass of the particle. The two figures show that added mass significantly affects the first and second resonant frequencies, especially the former. Thus, the first resonant frequency can be used in the measurement of attached mass, Figure 7 shows that the second resonant frequency decreases rapidly with increasing radius of added mass. This is due to the rotating effect having a larger effect on the second resonant frequency. Therefore, the second resonant frequency can be