Abstract
The Hamilton principle is applied to deduce the free vibration frequencies of a cantilever single-walled carbon nanotube (SWCNT) in the presence of an added mass, which can be distributed along an arbitrary part of the span. The nonlocal elasticity theory by Eringen has been employed, in order to take into account the nanoscale effects. An exact formulation leads to the equations of motion, which can be solved to give the frequencies and the corresponding vibration modes. Moreover, two approximate semianalytical methods are also illustrated, which can provide quick parametric relationships. From a more practical point of view, the problem of detecting the mass of the attached particle has been solved by calculating the relative frequency shift due to the presence of the added mass: from it, the mass value can be easily deduced. The paper ends with some numerical examples, in which the nonlocal effects are thoroughly investigated.
1. Introduction
Carbon nanotubes (CNT)—as discovered by Iijima in 1991 (see [1])—have unique electrical, mechanical, and thermal properties, so that they are widely used in a large range of technical areas: nanoelectronics, scanning probes, nanoscale sensors, biomedical devices, and others. From a theoretical point of view, the nanoscale of these structures suggests an atomistic model, but this approach turns out to be very expensive. On the other hand, the usual beam theories (Euler-Bernoulli, Timoshenko, or even higher-order theories [2]) do not capture the influence of the size-effects, because they are inherently scale-free, so that it is usual to adopt the nonlocal elasticity theory, as developed by Eringen in [3, 4]. One of the most important goals of the nanomechanics is to use biosensors in order to detect external deposited masses, and a good mechanical model can be assumed to be a cantilever beam with an attached added mass along the span. It is important to note that the usual hypothesis of a “point mass” is not always justified [5–11], whereas a more realistic model [12] should assume a distributed added mass along a finite portion of the span. For this model, the free vibration frequencies can be calculated according to the classical energy method: the equations of motion and the boundary conditions are derived by applying the Hamilton principle, and the resulting boundary value problem is solved, to give the secular equation, which in turn permits deducing the frequencies.
Sometimes it is necessary to deduce the influence of some control parameter on the free vibration frequencies, so that some parametric curves must be sketched: in these cases the so-called semianalytical (SAN) methods become unvaluable, because they lead to approximate closed-form formulae for the frequencies as functions of the control parameter. In this paper, the added mass will be treated as a control parameter, and the use of two approximate approaches will permit us to examine the variation of the first frequency as a function of the added mass. In the first approach, we generalize a Meirovitch suggestion [13], starting from the equations of motion, in the spirit of Galerkin, whereas in the second method we start from the energies, following a Ritz-like approach. Both methods give close approximations to the true results, so that it is possible to use the parametric curve in order to find the added mass in terms of the frequency shift.
2. Analysis of the Problem
Let us consider the cantilevered nanotube in Figure 1, with span , cross-sectional area , second moment of area , and mass density . The well-known Euler-Bernoulli theory for slender beams will be used, so that the Young modulus suffices to define the material properties. Finally, an attached distributed mass is located, between the abscissae and . In order to take into account the nanoscale effects, the nonlocal elasticity theory has been adopted, as suggested by Eringen.
According to Hamilton principle it is possible to write where is the sum of the kinetic energies of the nanotube in the three sections, between , between , and between , respectively, and of the kinetic energy of the added mass. Moreover, is the total potential energy, is the strain energy of the nanotube, and is the potential energy of the inertial force due to the additional displacement [14]. Here , where is a material constant, which has to be defined through experimental results, and is the internal characteristic length of the nanotube.
The first variation of these two energies can be easily calculated so that (1) gives
Integrations by part (see Appendix) lead to a system of three equations of motion: together with the following general boundary conditions at the ends:
The boundary conditions for are given by and finally, at ,
The solutions of (5) can be expressed as If the nondimensional abscissa can be introduced, the system of three equations of motion (5) becomes where the following nondimensional parameters have been introduced:
The boundary conditions for clamped-free nanotube are given by
The general solutions of (10) are given by with
The twelve constants can be found by imposing the boundary conditions (12). The resulting homogeneous system has nontrivial solutions if and only if the coefficient determinant is zero and the corresponding secular equation has infinite solutions . The circular frequencies can be easily deduced, as well as the natural frequencies .
2.1. Nonlocal Fundamental Natural Frequency on CNT with Attached Mass-First Method
Starting from the equations of motion (5), it is possible to integrate each of them in their domain, and the resulting integrals can be summed up:
It is now possible to insert a trial function [14], leading to
Two successive integrations by part can be performed: so that (16) becomes
The boundary conditions at the right end permit simplifying the previous equation: whereas the free end will be subjected to the following equilibrium conditions:
Finally, (19) reduces to and the frequency can be written down, putting , as or, in terms of the nondimensional abscissa ,
In order to obtain a satisfactory approximation of the fundamental frequency, we use as approximating function the exact displacement of the cantilever beam without added mass:
The following integrals can be defined [15]:
For the case of distributed added mass between the abscissae and , the integral can be defined as and finally, in order to take into account the nonlocal effects, we define the fourth integral:
The fundamental natural frequency can be deduced from (23) in terms of these four integrals as
Finally, it is usual to cancel out the first integral, so arriving to the natural frequency, where and , , and are the so-called calibration constants.
2.2. Nonlocal Fundamental Natural Frequency on CNT with Attached Mass-Second Method
In this approach, let us start from the energy terms: and let us assume the separation of variables so that the energies read
The maximum kinetic energy will be equal to the maximum total potential energy, so that and the frequency can be deduced as or, in terms of the nondimensional abscissa ,
Let us assume the same approximating function equation (24), so that the following integrals can be calculated: and finally if the added mass is placed between the abscissae and , the integral can be obtained as
Therefore, an alternative version of the natural frequency can be obtained as with the three calibration constants:
3. Nonlocal Sensor Equations
The sensor equations in the presence of nonlocal elasticity can now be deduced, and the added mass of a biomolecule can be detected by calculating the corresponding CNT frequency shift. In fact, let us start from the natural frequency of the CNT without the added mass: and let us express the natural frequency, in the presence of the added mass, as (cf. (29) and (39))
The frequency shift of the biosensor can be defined as and finally the relative frequency shift is given by from which the value of the added mass can be easily obtained: and for the first approach and for the second approach.
4. Numerical Examples
4.1. First Example
Table 1 shows properties of the cantilever nanotube, which will be used throughout this section. The added distributed mass will be placed from the section to the free end, so that , and will vary from 0.9 to 0.1. In Table 2 the fundamental natural frequency is given for various values of the parameter and for increasing values of the nondimensional coefficient (see (11)). The first column gives the fundamental natural frequency in the absence of nonlocal effects. The table has been obtained by solving the system of three differential equations of motion (13), so that the results can be considered “exact.” As can be easily observed, the first fundamental natural frequency increases for increasing values of the parameter, whereas it decreases for increasing values of the parameter.
The fundamental natural frequency is reported in Table 3, as obtained by means of (29) and with . A numerical comparison with the exact values in Table 1 shows that the relative error is greater for , whereas the results for almost coincide. At , the relative error varies between 0.035% for , 0.34% for , and finally 2.36% for . Of course, this value can be considered as a limiting case, whereas and are more realistic choices. For example (see [17]), is adopted. In Table 4 the fundamental natural frequency is given, as obtained by means of (39) and with . A numerical comparison with the exact values in Table 2 shows that the relative error is greater than the previous case. More particularly, for it varies between 0.36% for , 2.83% for , and finally 8.45% for . Therefore, the first method seems to be more reliable than the second one.
4.2. Second Example
As a second example, let us suppose that the added mass is distributed along a fixed length, so that , but its real placement along the nanotube is unknown. In Table 5 the first fundamental natural frequency is reported, for different placements of the added mass and for four parameters. The frequencies have been obtained by solving the equations of motion, so that the results can be considered exact, and the nonlocal parameter has been allowed to vary between 0 and 0.3. The fundamental natural frequency increases for increasing values of the nonlocal parameter, and higher values correspond to added masses nearer to the clamped end. The same example is illustrated in Tables 6 and 7, using the approximate formula (29) and the approximate formula (39), respectively. As in the first example, the first method gives better results.
4.3. Third Example
Finally, let us address the practical problem of the added mass detection. In order to solve this problem, it is necessary to plot the relationship between the added mass equation (47) and the relative frequency shift equation (44). More precisely, in Figure 2 the nondimensional mass ratio is plotted against the relative frequency shift equation (44), with , and the four curves refer to four different values, (without nonlocal effects), , , and . The added mass is placed at the tip of the cantilever nanotube, so that , whereas is allowed to vary between 0.05 and 0.6. The geometrical data of the nanotube are given in Table 1. It is interesting to note that, according to our results, the relative frequency shift decreases for increasing values of the nonlocal coefficient . This should be compared with the different behaviour exhibited by the results given in [17]. The curves in Figure 2 have been drawn using the first approach, because it gives better approximations to the true values. Actually, in Figure 3 we have compared the exact method with the two proposed approaches, for , but the curve describing the first approach is undistinguishable from the exact curve.
5. Conclusions
The frequency shift between the free vibration frequencies of a cantilever nanotube with, and without, an attached distributed mass has been used, in order to detect the added mass value. It is shown that the size-effects must be taken into account, and the frequencies have to be calculated according to the nonlocal elasticity theory. Three different approaches have been proposed to solve the problem, and their results have been compared for a couple of examples. Moreover, the relative frequency shift decreases, for increasing values of the nonlocal coefficient , so that careful calibration of this coefficient becomes necessary, in order to obtain reliable values of the added mass.
Appendix
Equation (4), which is reported here for the sake of readability, has to be integrated by part:
The first three terms of (A.1) can be treated as follows: where we considered the fact that at and .
Quite similarly, the single term of the distributed mass becomes
The nonlocal effects are contained into three integrals, which can be integrated as follows:
Finally, the first variation of the strain energy can be treated as usual:
All the previous integrated terms can be collected together, leading to
Finally, from (A.6) the following equations can be deduced (cf. (4)): whereas the boundary conditions can be deduced from (A.7), for and for ,
The boundary conditions for are and those for are
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.