# Free Vibrations of a Cantilevered SWCNT with Distributed Mass in the Presence of Nonlocal Effect

Authors: M. A. De Rosa; M. Lippiello; H. D. Martin
Journal: The Scientific World Journal
 (2015)

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## 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.


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## Body

## 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 Figure1, with span L, cross-sectional area A, second moment of area I, and mass density ρ. The well-known Euler-Bernoulli theory for slender beams will be used, so that the Young modulus E suffices to define the material properties. Finally, an attached distributed mass m- is located, between the abscissae γ1L and γ2L. In order to take into account the nanoscale effects, the nonlocal elasticity theory has been adopted, as suggested by Eringen.Figure 1
Geometrical properties of the nanotube.According to Hamilton principle it is possible to write(1)∫t1t2‍δTt-δETtdt=0,
where
(2)T=12∫0γ1L‍ρA∂v1(z,t)∂t2dz+12∫γ1Lγ2L‍ρA∂v2(z,t)∂t2dz+12∫γ2LL‍ρA∂v3(z,t)∂t2dz+12∫γ1Lγ2L‍m-∂v2(z,t)∂t2dz
is the sum of the kinetic energies of the nanotube in the three sections, v1 between (0,γ1L), v2 between (γ1L,γ2L), and v3 between (γ2L,L), respectively, and of the kinetic energy of the added mass. Moreover,
(3)ET=Le-P=12∫0γ1L‍EI∂2v1(z,t)∂z22dz+12∫γ1Lγ2L‍EI∂2v2z,t∂z22dz+12∫γ2LL‍EI∂2v3(z,t)∂z22dz-∫0γ1L‍ρA∂2v1(z,t)∂t2μ2∂2v1(z,t)∂z2dz-∫γ1Lγ2L‍ρA∂2v2(z,t)∂t2μ2∂2v2(z,t)∂z2dz-∫γ2LL‍ρA∂2v3(z,t)∂t2μ2∂2v3(z,t)∂z2dz
is the total potential energy, Le is the strain energy of the nanotube, and P is the potential energy of the inertial force ρA∂2vz,t/∂t2 due to the additional displacement μ2∂2vz,t/∂z2 [14]. Here μ2=(e0a)2, where e0 is a material constant, which has to be defined through experimental results, and a is the internal characteristic length of the nanotube.The first variation of these two energies can be easily calculated so that (1) gives
(4)∫t1t2‍∫0γ1L‍ρA∂v1(z,t)∂tδ∂v1(z,t)∂tdz+∫γ1Lγ2L‍ρA∂v2z,t∂tδ∂v2z,t∂tdz+∫γ2LL‍ρA∂v3(z,t)∂tδ∂v3(z,t)∂tdz+∫γ1Lγ2L‍m-∂v2(z,t)∂tδ∂v2(z,t)∂tdz-∫0γ1L‍EI∂2v1z,t∂z2δ∂2v1z,t∂z2llllllllllllllllllll-μ2ρA∂2v1z,t∂t2δ∂2v1z,t∂z2dz-∫γ1Lγ2L‍EI∂2v2z,t∂z2δ∂2v2z,t∂z2llllllllllllllllllll-μ2ρA∂2v2z,t∂t2δ∂2v2z,t∂z2dz-∫γ2LL‍EI∂2v3(z,t)∂z2δ∂2v3(z,t)∂z2llllllllllllllllllll-μ2ρA∂2v3z,t∂t2δ∂2v3z,t∂z2dzdt=0.Integrations by part (see Appendix) lead to a system of three equations of motion:(5)EI∂4v1z,t∂z4-μ2ρA∂4v1z,t∂z2∂t2+ρA∂2v1z,t∂t2=0,gggggggggggggggggggggggggggggg10<z<γ1L,EI∂4v2z,t∂z4-μ2ρA∂4v2z,t∂z2∂t2+(ρA+M)∂2v2z,t∂t2=0,gggggggggggggggggggggggggggggggγ1L<z<γ2L,EI∂4v3z,t∂z4-μ2ρA∂4v3z,t∂z2∂t2+ρA∂2v3z,t∂t2=0,ggggggggggggggggggggggggggglγ2L<z<L,
together with the following general boundary conditions at the ends:
(6)v1(0,t)=0,∂v10,t∂z=0,EI∂3v3L,t∂z3-μ2ρA∂3v3L,t∂t2∂z=0,-EI∂2v3L,t∂z2+μ2ρA∂2v3L,t∂t2=0.The boundary conditions forz=γ1L are given by
(7)v1γ1L,t=v2γ1L,t,∂v1γ1L,t∂z=∂v2γ1L,t∂z,μ2ρA∂3v1γ1L,t∂t2∂z-EI∂3v1γ1L,t∂z3-μ2ρA∂3v2γ1L,t∂t2∂z+EI∂3v2γ1L,t∂z3=0,μ2ρA∂2v1γ1L,t∂t2-EI∂2v1γ1L,t∂z2-μ2ρA∂2v2γ1L,t∂t2+EI∂2v2γ1L,t∂z2=0,
and finally, at z=γ2L,
(8)v2γ2L,t=v3γ2L,t,∂v2γ2L,t∂z=∂v3γ2L,t∂z,μ2ρA∂3v2γ2L,t∂t2∂z-EI∂3v2γ2L,t∂z3-μ2ρA∂3v3γ2L,t∂t2∂z+EI∂3v3γ2L,t∂z3=0,μ2ρA∂2v2γ2L,t∂t2-EI∂2v2γ2L,t∂z2-μ2ρA∂2v3γ2L,t∂t2+EI∂2v3γ2L,t∂z2=0.The solutions of (5) can be expressed as
(9)vh(z,t)=vh(z)eiωt,h=1,2,3.
If the nondimensional abscissa ζ=z/L can be introduced, the system of three equations of motion (5) becomes
(10)∂4v1ζ∂ζ4+η2Ω4∂2v1ζ∂ζ2-Ω4v1ζ=0for0<ζ<γ1,∂4v2ζ∂ζ4+η2Ω4∂2v2ζ∂ζ2-1+λΩ4v2ζ=0ggggggggggggggggggggggggforγ1<ζ<γ2,∂4v3ζ∂ζ4+η2Ω4∂2v3ζ∂ζ2-Ω4v3ζ=0forγ2<ζ<1,
where the following nondimensional parameters have been introduced:
(11)λ=m-ρA;η=μL;Ω=ρAL4ω2EI.The boundary conditions for clamped-free nanotube are given by(12)v1(0)=0,∂v1(0)∂ζ=0,v1γ1=v2γ1,∂v1γ1∂ζ=∂v2γ1∂ζ,η2Ω4∂v1γ1∂ζ+∂3v1γ1∂ζ3-η2Ω4∂v2γ1∂ζ-∂3v2γ1∂ζ3=0,η2Ω4v1γ1+∂2v1γ1∂ζ2-η2Ω4v2γ1-∂2v2γ1∂ζ2=0,v2γ2=v3γ2,∂v2γ2∂ζ=∂v3γ2∂ζ,η2Ω4∂v2γ2∂ζ+∂3v2γ2∂ζ3-η2Ω4∂v3γ2∂ζ-∂3v3γ2∂ζ3=0,η2Ω4v2γ2+∂2v2γ2∂ζ2-η2Ω4v3γ2-∂2v3γ2∂ζ2=0,∂3v3(1)∂ζ3+η2Ω4∂v3(1)∂ζ=0,-∂2v31∂ζ2-η2Ω4v3(1)=0.The general solutions of (10) are given by
(13)v1ζ=A1cos⁡⁡(αζ)+A2sin⁡(αζ)+A3cos⁡⁡h(αζ)+A4sin⁡hαζ,v2ζ=B1cos⁡⁡α1ζ+B2sin⁡α1ζ+B3cos⁡⁡hβ1ζ+B4sin⁡hβ1ζ,v3ζ=C1cos⁡⁡(αζ)+C2sin⁡(αζ)+C3cos⁡⁡h(βζ)+C4sin⁡h(βζ)
with
(14)α=12η2Ω4+Ω24+η4Ω4;β=12-η2Ω4+Ω24+η4Ω4;α1=12η2Ω4+Ω24+4λ+η4Ω4;β1=12-η2Ω4+Ω24+4λ+η4Ω4.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 Ωj. The circular frequencies ωj can be easily deduced, as well as the natural frequencies fj=ωj/2π.


### 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:
(15)∫0L‍EI∂4v(z)∂z4dz+ω2∫0L‍μ2ρA∂2v(z)∂z2dz-ω2∫0L‍ρAv(z)dz-ω2∫γ1Lγ2L‍m-v(z)dz=0.It is now possible to insert a trial functiony(z) [14], leading to
(16)∫0L‍EI∂4v(z)∂z4y(z)dz+ω2∫0L‍μ2ρA∂2v(z)∂z2y(z)dz-ω2∫0L‍ρAv(z)y(z)dz-ω2∫γ1Lγ2L‍m-v(z)y(z)dz=0.Two successive integrations by part can be performed:(17)∫0L‍EI∂4vz∂z4yzdz=EI∂3vz∂z3yz0L-∫0L‍EI∂3v(z)∂z3∂y(z)∂zdz=EI∂3vz∂z3yz0L-EI∂2vz∂z2∂yz∂z0L+∫0L‍EI∂2vz∂z2∂2yz∂z2dz,ω2∫0L‍μ2ρA∂2vz∂z2yzdz=ω2μ2ρA∂v(z)∂zy(z)0L-ω2∫0L‍μ2ρA∂v(z)∂z∂y(z)∂zdz=ω2μ2ρA∂v(z)∂zy(z)0L-ω2μ2ρAv(z)∂y(z)∂z0L+ω2∫0L‍μ2ρAv(z)∂2y(z)∂z2dz
so that (16) becomes
(18)∫0L‍EI∂2v(z)∂z2∂2y(z)∂z2dz-ω2∫0L‍ρAv(z)y(z)dz-ω2∫γ1Lγ2L‍m-v(z)y(z)dz+ω2∫0L‍μ2ρAv(z)∂2y(z)∂z2dz+EI∂3v(z)∂z3y(z)0L-EI∂2v(z)∂z2∂y(z)∂z0L+ω2μ2ρA∂vz∂zyz0L-ω2μ2ρAvz∂yz∂z0L=0.The boundary conditions at the right end permit simplifying the previous equation:(19)∫0L‍EI∂2vz∂z2∂2yz∂z2dz+ω2∫0Lμ2ρAv(z)∂2y(z)∂z2dz‍-ω2∫0L‍ρAv(z)y(z)dz-ω2∫γ1Lγ2L‍m-v(z)y(z)dz+EI∂3v(L)∂z3y(L)-EI∂2v(L)∂z2∂y(L)∂z+ω2μ2ρA∂vL∂zy(L)-ω2μ2ρAv(L)∂yL∂z=0,
whereas the free end will be subjected to the following equilibrium conditions:
(20)ω2μ2ρA∂v(L)∂z+EI∂3v(L)∂z3=0,-ω2μ2ρAv(L)-EI∂2vL∂z2=0.Finally, (19) reduces to
(21)∫0L‍EI∂2v(z)∂z2∂2y(z)∂z2dz+ω2∫0L‍μ2ρAv(z)∂2y(z)∂z2dz-ω2∫0L‍ρAv(z)y(z)dz-ω2∫γ1Lγ2L‍m-v(z)y(z)dz=0
and the frequency ω2 can be written down, putting y(z)=v(z), as
(22)ω2=∫0L‍EI∂2v(z)∂z2∂2v(z)∂z2dz×∫0L‍μ2ρAv(z)∂2v(z)∂z2dz∫0L‍ρAv2zdz+∫γ1Lγ2L‍m-v2zdz-∫0L‍μ2ρAv(z)∂2v(z)∂z2dz-1
or, in terms of the nondimensional abscissa ζ=z/L,
(23)ω2=EIL3∫01‍∂2v(ζ)∂ζ2∂2v(ζ)∂ζ2dζ×η2ρAL∫01‍v(ζ)∂2v(ζ)∂ζ2dζρAL∫01‍v2(ζ)dζ+m-L∫γ1γ2‍v2(ζ)dζ-η2ρAL∫01‍v(ζ)∂2v(ζ)∂ζ2dζ-1.In order to obtain a satisfactory approximation of the fundamental frequency, we use as approximating functionv(ζ) the exact displacement of the cantilever beam without added mass:
(24)vζ=Cosh1.8751ζ-Cos⁡1.8751ζ-Sinh1.8751-Sin1.8751Cosh1.8751+Cos⁡1.8751×Sinh1.8751ζ-Sin1.8751ζ.The following integrals can be defined [15]:
(25)I1=∫01‍v2ζdζ=1,I3=∫01‍∂2v(ζ)∂ζ2∂2v(ζ)∂ζ2dζ=12.3623.For the case of distributed added mass between the abscissaeγ1L and γ2L, the integral I2 can be defined as
(26)I2=∫γ1γ2‍v2ζdζ0≤γ1≤1;0≤γ2≤1,
and finally, in order to take into account the nonlocal effects, we define the fourth integral:
(27)I4=∫01‍v(ζ)∂2v(ζ)∂ζ2dζ=0.858264.The fundamental natural frequency can be deduced from (23) in terms of these four integrals as
(28)fn1=ω2π=β2πI3I1+λI2-η2I4.Finally, it is usual to cancel out the first integral, so arriving to the natural frequency,(29)fn1=β2πCk1+λCm-η2Cn1,
where
(30)β=EImL4;Ck=I3I1=3.5160;Cn1=I4I1=0.858264;Cm=I2I1
and Ck, Cn1, and Cm 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:(31)T=12∫0L‍ρA∂v(z,t)∂t2dz+12∫γ1Lγ2L‍m-∂v(z,t)∂t2dz,ET=Le-P=12∫0L‍EI∂2vz∂z22dz-∫0L‍μ2ρA∂2v(z,t)∂t2∂2v(z,t)∂z2dz
and let us assume the separation of variables
(32)v(z,t)=v(z)Cos⁡(ωt)
so that the energies read
(33)T=ω22∫0L‍ρAvz2dz+∫γ1Lγ2L‍m-vz2dzCos⁡2ωt,ET=Le-P=12∫0L‍EI∂2v(z)∂z22dzSin2(ωt)+ω2∫0L‍μ2ρAvz∂2vz∂z2dzSin2ωt.The maximum kinetic energy will be equal to the maximum total potential energy, so that(34)12∫0L‍EI∂2v(z)∂z22dz+ω2∫0L‍μ2ρAv(z)∂2v(z)∂z2dz=ω22∫0L‍ρAvz2dz+ω22∫γ1Lγ2L‍m-vz2dz
and the frequency ω2 can be deduced as
(35)ω2=12∫0L‍EI∂2vz∂z22dz×-∫0L‍μ2ρAvz∂2vz∂z2dz12∫0L‍ρAvz2dz+12∫γ1Lγ2L‍m-vz2dz-∫0L‍μ2ρAvz∂2vz∂z2dz-1
or, in terms of the nondimensional abscissa ζ=z/L,
(36)ω2=12EIL3∫01‍∂2v(ζ)∂ζ2∂2v(ζ)∂ζ2dζ×η2ρAL∫01‍v(ζ)∂2v(ζ)∂ζ2dζ12ρAL∫01‍v2ζdζ+12m-L∫γ1γ2‍v2ζdζ-η2ρAL∫01‍v(ζ)∂2v(ζ)∂ζ2dζ-1.Let us assume the same approximating function equation (24), so that the following integrals can be calculated:
(37)I1=∫01‍v2ζdζ=1,I3=∫01‍∂2v(ζ)∂ζ2∂2v(ζ)∂ζ2dζ=12.3623;I4=∫01‍v(ζ)∂2v(ζ)∂ζ2dζ=0.858264,
and finally if the added mass is placed between the abscissae γ1L and γ2L, the integral I2 can be obtained as
(38)I2=∫γ1γ2‍v2ζdζ;0≤γ1≤1;0≤γ2≤1.Therefore, an alternative version of the natural frequency can be obtained as(39)fn2=β2πCk1+λCm-η2Cn2
with the three calibration constants:
(40)Ck=I3I1=3.51601;Cn2=2I4I1=1.71653;Cm=I2I1.


## 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:
(15)∫0L‍EI∂4v(z)∂z4dz+ω2∫0L‍μ2ρA∂2v(z)∂z2dz-ω2∫0L‍ρAv(z)dz-ω2∫γ1Lγ2L‍m-v(z)dz=0.It is now possible to insert a trial functiony(z) [14], leading to
(16)∫0L‍EI∂4v(z)∂z4y(z)dz+ω2∫0L‍μ2ρA∂2v(z)∂z2y(z)dz-ω2∫0L‍ρAv(z)y(z)dz-ω2∫γ1Lγ2L‍m-v(z)y(z)dz=0.Two successive integrations by part can be performed:(17)∫0L‍EI∂4vz∂z4yzdz=EI∂3vz∂z3yz0L-∫0L‍EI∂3v(z)∂z3∂y(z)∂zdz=EI∂3vz∂z3yz0L-EI∂2vz∂z2∂yz∂z0L+∫0L‍EI∂2vz∂z2∂2yz∂z2dz,ω2∫0L‍μ2ρA∂2vz∂z2yzdz=ω2μ2ρA∂v(z)∂zy(z)0L-ω2∫0L‍μ2ρA∂v(z)∂z∂y(z)∂zdz=ω2μ2ρA∂v(z)∂zy(z)0L-ω2μ2ρAv(z)∂y(z)∂z0L+ω2∫0L‍μ2ρAv(z)∂2y(z)∂z2dz
so that (16) becomes
(18)∫0L‍EI∂2v(z)∂z2∂2y(z)∂z2dz-ω2∫0L‍ρAv(z)y(z)dz-ω2∫γ1Lγ2L‍m-v(z)y(z)dz+ω2∫0L‍μ2ρAv(z)∂2y(z)∂z2dz+EI∂3v(z)∂z3y(z)0L-EI∂2v(z)∂z2∂y(z)∂z0L+ω2μ2ρA∂vz∂zyz0L-ω2μ2ρAvz∂yz∂z0L=0.The boundary conditions at the right end permit simplifying the previous equation:(19)∫0L‍EI∂2vz∂z2∂2yz∂z2dz+ω2∫0Lμ2ρAv(z)∂2y(z)∂z2dz‍-ω2∫0L‍ρAv(z)y(z)dz-ω2∫γ1Lγ2L‍m-v(z)y(z)dz+EI∂3v(L)∂z3y(L)-EI∂2v(L)∂z2∂y(L)∂z+ω2μ2ρA∂vL∂zy(L)-ω2μ2ρAv(L)∂yL∂z=0,
whereas the free end will be subjected to the following equilibrium conditions:
(20)ω2μ2ρA∂v(L)∂z+EI∂3v(L)∂z3=0,-ω2μ2ρAv(L)-EI∂2vL∂z2=0.Finally, (19) reduces to
(21)∫0L‍EI∂2v(z)∂z2∂2y(z)∂z2dz+ω2∫0L‍μ2ρAv(z)∂2y(z)∂z2dz-ω2∫0L‍ρAv(z)y(z)dz-ω2∫γ1Lγ2L‍m-v(z)y(z)dz=0
and the frequency ω2 can be written down, putting y(z)=v(z), as
(22)ω2=∫0L‍EI∂2v(z)∂z2∂2v(z)∂z2dz×∫0L‍μ2ρAv(z)∂2v(z)∂z2dz∫0L‍ρAv2zdz+∫γ1Lγ2L‍m-v2zdz-∫0L‍μ2ρAv(z)∂2v(z)∂z2dz-1
or, in terms of the nondimensional abscissa ζ=z/L,
(23)ω2=EIL3∫01‍∂2v(ζ)∂ζ2∂2v(ζ)∂ζ2dζ×η2ρAL∫01‍v(ζ)∂2v(ζ)∂ζ2dζρAL∫01‍v2(ζ)dζ+m-L∫γ1γ2‍v2(ζ)dζ-η2ρAL∫01‍v(ζ)∂2v(ζ)∂ζ2dζ-1.In order to obtain a satisfactory approximation of the fundamental frequency, we use as approximating functionv(ζ) the exact displacement of the cantilever beam without added mass:
(24)vζ=Cosh1.8751ζ-Cos⁡1.8751ζ-Sinh1.8751-Sin1.8751Cosh1.8751+Cos⁡1.8751×Sinh1.8751ζ-Sin1.8751ζ.The following integrals can be defined [15]:
(25)I1=∫01‍v2ζdζ=1,I3=∫01‍∂2v(ζ)∂ζ2∂2v(ζ)∂ζ2dζ=12.3623.For the case of distributed added mass between the abscissaeγ1L and γ2L, the integral I2 can be defined as
(26)I2=∫γ1γ2‍v2ζdζ0≤γ1≤1;0≤γ2≤1,
and finally, in order to take into account the nonlocal effects, we define the fourth integral:
(27)I4=∫01‍v(ζ)∂2v(ζ)∂ζ2dζ=0.858264.The fundamental natural frequency can be deduced from (23) in terms of these four integrals as
(28)fn1=ω2π=β2πI3I1+λI2-η2I4.Finally, it is usual to cancel out the first integral, so arriving to the natural frequency,(29)fn1=β2πCk1+λCm-η2Cn1,
where
(30)β=EImL4;Ck=I3I1=3.5160;Cn1=I4I1=0.858264;Cm=I2I1
and Ck, Cn1, and Cm 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:(31)T=12∫0L‍ρA∂v(z,t)∂t2dz+12∫γ1Lγ2L‍m-∂v(z,t)∂t2dz,ET=Le-P=12∫0L‍EI∂2vz∂z22dz-∫0L‍μ2ρA∂2v(z,t)∂t2∂2v(z,t)∂z2dz
and let us assume the separation of variables
(32)v(z,t)=v(z)Cos⁡(ωt)
so that the energies read
(33)T=ω22∫0L‍ρAvz2dz+∫γ1Lγ2L‍m-vz2dzCos⁡2ωt,ET=Le-P=12∫0L‍EI∂2v(z)∂z22dzSin2(ωt)+ω2∫0L‍μ2ρAvz∂2vz∂z2dzSin2ωt.The maximum kinetic energy will be equal to the maximum total potential energy, so that(34)12∫0L‍EI∂2v(z)∂z22dz+ω2∫0L‍μ2ρAv(z)∂2v(z)∂z2dz=ω22∫0L‍ρAvz2dz+ω22∫γ1Lγ2L‍m-vz2dz
and the frequency ω2 can be deduced as
(35)ω2=12∫0L‍EI∂2vz∂z22dz×-∫0L‍μ2ρAvz∂2vz∂z2dz12∫0L‍ρAvz2dz+12∫γ1Lγ2L‍m-vz2dz-∫0L‍μ2ρAvz∂2vz∂z2dz-1
or, in terms of the nondimensional abscissa ζ=z/L,
(36)ω2=12EIL3∫01‍∂2v(ζ)∂ζ2∂2v(ζ)∂ζ2dζ×η2ρAL∫01‍v(ζ)∂2v(ζ)∂ζ2dζ12ρAL∫01‍v2ζdζ+12m-L∫γ1γ2‍v2ζdζ-η2ρAL∫01‍v(ζ)∂2v(ζ)∂ζ2dζ-1.Let us assume the same approximating function equation (24), so that the following integrals can be calculated:
(37)I1=∫01‍v2ζdζ=1,I3=∫01‍∂2v(ζ)∂ζ2∂2v(ζ)∂ζ2dζ=12.3623;I4=∫01‍v(ζ)∂2v(ζ)∂ζ2dζ=0.858264,
and finally if the added mass is placed between the abscissae γ1L and γ2L, the integral I2 can be obtained as
(38)I2=∫γ1γ2‍v2ζdζ;0≤γ1≤1;0≤γ2≤1.Therefore, an alternative version of the natural frequency can be obtained as(39)fn2=β2πCk1+λCm-η2Cn2
with the three calibration constants:
(40)Ck=I3I1=3.51601;Cn2=2I4I1=1.71653;Cm=I2I1.


## 3. Nonlocal Sensor Equations

The sensor equations in the presence of nonlocal elasticity can now be deduced, and the added massm- 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:
(41)fn0=12πCkβ,
and let us express the natural frequency, in the presence of the added mass, as (cf. (29) and (39))
(42)fni=fn01+λCm-η2Cni,i=1,2.The frequency shift of the biosensor can be defined as(43)Δf=fn0-fni
and finally the relative frequency shift is given by
(44)Δffn0=1-1√1+λCm-η2Cni
from which the value of the added mass m- can be easily obtained:
(45)Δffn0-12=11+λCm-η2Cni,(46)λ=1CmΔf/fn0-12+Cniη2Cm-1Cm(47)m-=ρACmΔf/fn0-12+Cniη2ρACm-ρACm
and i=1 for the first approach and i=2 for the second approach.


## 4. Numerical Examples

### 4.1. First Example

Table1 shows properties of the cantilever nanotube, which will be used throughout this section. The added distributed mass will be placed from the section γ1L to the free end, so that γ2=1, and γ1 will vary from 0.9 to 0.1. In Table 2 the fundamental natural frequency is given for various values of the γ=γ2-γ1=1-γ1 parameter and for increasing values of the nondimensional η 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 η parameter, whereas it decreases for increasing values of the γ parameter.Table 1
Nanotube properties (see [16]).
SWCNT properties density
Symbol
Value
Unit
Cross section area
A
7.851 10-19
m2
Radius
R
0.5 10-9
m
Length
L
9 10-9
m
Moment of inertia
I
4.91 10-38
m4
Density
ρ
2300
Kg/m3
Young's modulus
E
1000 109
PaTable 2
The first exact natural frequency (×1010) f0 for various values of the nondimensional length of the added mass and for four increasing values of the nonlocal nondimensional coefficient η.
γ
η
=
0
η
=
0.1
η
=
0.3
η
=
0.5
0.1
3.10361
3.11245
3.20756
3.46665
0.2
2.84585
2.85418
2.9260
3.10933
0.3
2.7036
2.71057
2.77009
2.91661
0.4
2.62348
2.6297
2.68251
2.80985
0.5
2.58023
2.58605
2.63527
2.75263
0.6
2.55911
2.56473
2.6122
2.72473
0.7
2.55048
2.55602
2.60277
2.71335
0.8
2.54791
2.55342
2.59996
2.70995
0.9
2.54749
2.55301
2.59951
2.70941The fundamental natural frequencyfn1 is reported in Table 3, as obtained by means of (29) and with λ=1. A numerical comparison with the exact values in Table 1 shows that the relative error is greater for γ=0.1, whereas the results for γ=0.9 almost coincide. At γ=0.1, the relative error varies between 0.035% for η=0.1, 0.34% for η=0.3, and finally 2.36% for η=0.5. Of course, this η value can be considered as a limiting case, whereas η=0.1 and η=0.3 are more realistic choices. For example (see [17]), η=0.235 is adopted. In Table 4 the fundamental natural frequency fn2 is given, as obtained by means of (39) and with λ=1. A numerical comparison with the exact values in Table 2 shows that the relative error is greater than the previous case. More particularly, for γ=0.1 it varies between 0.36% for η=0.1, 2.83% for η=0.3, and finally 8.45% for η=0.5. Therefore, the first method seems to be more reliable than the second one.Table 3
First approximate fundamental natural frequency (×1010) fn1, as obtained using (29), for various values of the nondimensional length of the added mass and for four increasing values of the nonlocal nondimensional coefficient η.
γ
η
=
0
η
=
0.1
η
=
0.3
η
=
0.5
0.1
3.10361
3.11354
3.19659
3.38478
0.2
2.84803
2.85570
2.91937
3.06061
0.3
2.70498
2.71154
2.76586
2.88508
0.4
2.62411
2.63011
2.67959
2.78756
0.5
2.580453
2.586150
2.63315
2.73539
0.6
2.55916
2.56472
2.61054
2.71007
0.7
2.55050
2.5560
2.60135
2.69979
0.8
2.54792
2.55341
2.59861
2.69673
0.9
2.54750
2.55299
2.59817
2.69624Table 4
First exact fundamental natural frequency (×1010) fn2, as obtained using (39), for various values of the nondimensional length of the added mass and for four increasing values of the nonlocal nondimensional coefficient η.
γ
η
=
0
η
=
0.1
η
=
0.3
η
=
0.5
0.1
3.10361
3.12357
3.29845
3.75946
0.2
2.84803
2.86343
2.99634
3.32922
0.3
2.70498
2.71816
2.83106
3.10674
0.4
2.62411
2.63614
2.73875
2.98593
0.5
2.58045
2.59188
2.68921
2.92207
0.6
2.55916
2.57032
2.66515
2.89127
0.7
2.55050
2.56154
2.65536
2.87879
0.8
2.54792
2.55893
2.65246
2.87509
0.9
2.54750
2.55851
2.65199
2.87449


### 4.2. Second Example

As a second example, let us suppose that the added mass is distributed along a fixed length, so thatγ=γ2-γ1=0.3, 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 η 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 η 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.Table 5
First exact fundamental natural frequency (×1010) f0 for various values of the nondimensional length of the added mass (with γ=0.3) and for four increasing values of the nonlocal nondimensional coefficient η.
γ
1
γ
2
η
=
0
η
=
0.1
η
=
0.2
η
=
0.3
0.7
1
2.70498
2.71057
2.73205
2.77009
0.6
0.9
2.90531
2.91317
2.9375
2.98072
0.5
0.8
3.10026
3.10941
3.13780
3.18851
0.4
0.7
3.27557
3.28637
3.31997
3.38053
0.3
0.6
3.41695
3.42952
3.46879
3.54028
0.2
0.5
3.51481
3.52888
3.57304
3.65415
0.1
0.4
3.57016
3.58519
3.63251
3.71994
0.0
0.3
3.59421
3.60970
3.65848
3.74886Table 6
First exact fundamental natural frequency (×1010) f1, as obtained using (29), for various values of the nondimensional length of the added mass (with γ=0.3) and for four increasing values of the nonlocal nondimensional coefficient η.
γ
1
γ
2
η
=
0
η
=
0.1
η
=
0.2
η
=
0.3
0.7
1
2.70498
2.71154
2.73154
2.76586
0.6
0.9
2.90556
2.91370
2.93855
2.98142
0.5
0.8
3.10177
3.11168
3.14201
3.19457
0.4
0.7
3.27802
3.28973
3.32563
3.38815
0.3
0.6
3.41894
3.43224
3.47306
3.54445
0.2
0.5
3.51573
3.53019
3.57465
3.65264
0.1
0.4
3.57039
3.585549
3.63215
3.71405
0.0
0.3
3.59424
3.60969
3.65727
3.74092Table 7
First exact fundamental natural frequency (×1010) f2, as obtained using (39), for various values of the nondimensional length of the added mass (with γ=0.3) and for four increasing values of the nonlocal nondimensional coefficient η.
γ
1
γ
2
η
=
0
η
=
0.1
η
=
0.2
η
=
0.3
0.7
1
2.70498
2.71816
2.75890
2.83106
0.6
0.9
2.90556
2.92191
2.97270
3.06355
0.5
0.8
3.10177
3.12169
3.18385
3.29624
0.4
0.7
3.27802
3.30157
3.37536
3.51017
0.3
0.6
3.41894
3.44568
3.52982
3.68488
0.2
0.5
3.51573
3.54482
3.63663
3.80690
0.1
0.4
3.57039
3.60087
3.69723
3.87658
0.0
0.3
3.59424
3.62534
3.72373
3.90716


### 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 massm- equation (47) and the relative frequency shift equation (44). More precisely, in Figure 2 the nondimensional mass ratio M/ρAL⁡ is plotted against the relative frequency shift equation (44), with M=m-(γ2-γ1)L, and the four curves refer to four different η values, η=0 (without nonlocal effects), η=0.1, η=0.2, and η=0.3. The added mass is placed at the tip of the cantilever nanotube, so that η2=1, whereas γ2-γ1) 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 η. 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 η=0.2, but the curve describing the first approach is undistinguishable from the exact curve.Figure 2
The nondimensional mass ratioM/ρAL⁡ is plotted against the relative frequency shift—as obtained using (44)—with M=m-γ2-γ1L. The four curves refer to four different η values, η=0 (without nonlocal effects), η=0.1, η=0.2, and η=0.3.Figure 3
Numerical comparison between two proposed approaches.


## 4.1. First Example

Table1 shows properties of the cantilever nanotube, which will be used throughout this section. The added distributed mass will be placed from the section γ1L to the free end, so that γ2=1, and γ1 will vary from 0.9 to 0.1. In Table 2 the fundamental natural frequency is given for various values of the γ=γ2-γ1=1-γ1 parameter and for increasing values of the nondimensional η 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 η parameter, whereas it decreases for increasing values of the γ parameter.Table 1
Nanotube properties (see [16]).
SWCNT properties density
Symbol
Value
Unit
Cross section area
A
7.851 10-19
m2
Radius
R
0.5 10-9
m
Length
L
9 10-9
m
Moment of inertia
I
4.91 10-38
m4
Density
ρ
2300
Kg/m3
Young's modulus
E
1000 109
PaTable 2
The first exact natural frequency (×1010) f0 for various values of the nondimensional length of the added mass and for four increasing values of the nonlocal nondimensional coefficient η.
γ
η
=
0
η
=
0.1
η
=
0.3
η
=
0.5
0.1
3.10361
3.11245
3.20756
3.46665
0.2
2.84585
2.85418
2.9260
3.10933
0.3
2.7036
2.71057
2.77009
2.91661
0.4
2.62348
2.6297
2.68251
2.80985
0.5
2.58023
2.58605
2.63527
2.75263
0.6
2.55911
2.56473
2.6122
2.72473
0.7
2.55048
2.55602
2.60277
2.71335
0.8
2.54791
2.55342
2.59996
2.70995
0.9
2.54749
2.55301
2.59951
2.70941The fundamental natural frequencyfn1 is reported in Table 3, as obtained by means of (29) and with λ=1. A numerical comparison with the exact values in Table 1 shows that the relative error is greater for γ=0.1, whereas the results for γ=0.9 almost coincide. At γ=0.1, the relative error varies between 0.035% for η=0.1, 0.34% for η=0.3, and finally 2.36% for η=0.5. Of course, this η value can be considered as a limiting case, whereas η=0.1 and η=0.3 are more realistic choices. For example (see [17]), η=0.235 is adopted. In Table 4 the fundamental natural frequency fn2 is given, as obtained by means of (39) and with λ=1. A numerical comparison with the exact values in Table 2 shows that the relative error is greater than the previous case. More particularly, for γ=0.1 it varies between 0.36% for η=0.1, 2.83% for η=0.3, and finally 8.45% for η=0.5. Therefore, the first method seems to be more reliable than the second one.Table 3
First approximate fundamental natural frequency (×1010) fn1, as obtained using (29), for various values of the nondimensional length of the added mass and for four increasing values of the nonlocal nondimensional coefficient η.
γ
η
=
0
η
=
0.1
η
=
0.3
η
=
0.5
0.1
3.10361
3.11354
3.19659
3.38478
0.2
2.84803
2.85570
2.91937
3.06061
0.3
2.70498
2.71154
2.76586
2.88508
0.4
2.62411
2.63011
2.67959
2.78756
0.5
2.580453
2.586150
2.63315
2.73539
0.6
2.55916
2.56472
2.61054
2.71007
0.7
2.55050
2.5560
2.60135
2.69979
0.8
2.54792
2.55341
2.59861
2.69673
0.9
2.54750
2.55299
2.59817
2.69624Table 4
First exact fundamental natural frequency (×1010) fn2, as obtained using (39), for various values of the nondimensional length of the added mass and for four increasing values of the nonlocal nondimensional coefficient η.
γ
η
=
0
η
=
0.1
η
=
0.3
η
=
0.5
0.1
3.10361
3.12357
3.29845
3.75946
0.2
2.84803
2.86343
2.99634
3.32922
0.3
2.70498
2.71816
2.83106
3.10674
0.4
2.62411
2.63614
2.73875
2.98593
0.5
2.58045
2.59188
2.68921
2.92207
0.6
2.55916
2.57032
2.66515
2.89127
0.7
2.55050
2.56154
2.65536
2.87879
0.8
2.54792
2.55893
2.65246
2.87509
0.9
2.54750
2.55851
2.65199
2.87449


## 4.2. Second Example

As a second example, let us suppose that the added mass is distributed along a fixed length, so thatγ=γ2-γ1=0.3, 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 η 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 η 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.Table 5
First exact fundamental natural frequency (×1010) f0 for various values of the nondimensional length of the added mass (with γ=0.3) and for four increasing values of the nonlocal nondimensional coefficient η.
γ
1
γ
2
η
=
0
η
=
0.1
η
=
0.2
η
=
0.3
0.7
1
2.70498
2.71057
2.73205
2.77009
0.6
0.9
2.90531
2.91317
2.9375
2.98072
0.5
0.8
3.10026
3.10941
3.13780
3.18851
0.4
0.7
3.27557
3.28637
3.31997
3.38053
0.3
0.6
3.41695
3.42952
3.46879
3.54028
0.2
0.5
3.51481
3.52888
3.57304
3.65415
0.1
0.4
3.57016
3.58519
3.63251
3.71994
0.0
0.3
3.59421
3.60970
3.65848
3.74886Table 6
First exact fundamental natural frequency (×1010) f1, as obtained using (29), for various values of the nondimensional length of the added mass (with γ=0.3) and for four increasing values of the nonlocal nondimensional coefficient η.
γ
1
γ
2
η
=
0
η
=
0.1
η
=
0.2
η
=
0.3
0.7
1
2.70498
2.71154
2.73154
2.76586
0.6
0.9
2.90556
2.91370
2.93855
2.98142
0.5
0.8
3.10177
3.11168
3.14201
3.19457
0.4
0.7
3.27802
3.28973
3.32563
3.38815
0.3
0.6
3.41894
3.43224
3.47306
3.54445
0.2
0.5
3.51573
3.53019
3.57465
3.65264
0.1
0.4
3.57039
3.585549
3.63215
3.71405
0.0
0.3
3.59424
3.60969
3.65727
3.74092Table 7
First exact fundamental natural frequency (×1010) f2, as obtained using (39), for various values of the nondimensional length of the added mass (with γ=0.3) and for four increasing values of the nonlocal nondimensional coefficient η.
γ
1
γ
2
η
=
0
η
=
0.1
η
=
0.2
η
=
0.3
0.7
1
2.70498
2.71816
2.75890
2.83106
0.6
0.9
2.90556
2.92191
2.97270
3.06355
0.5
0.8
3.10177
3.12169
3.18385
3.29624
0.4
0.7
3.27802
3.30157
3.37536
3.51017
0.3
0.6
3.41894
3.44568
3.52982
3.68488
0.2
0.5
3.51573
3.54482
3.63663
3.80690
0.1
0.4
3.57039
3.60087
3.69723
3.87658
0.0
0.3
3.59424
3.62534
3.72373
3.90716


## 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 massm- equation (47) and the relative frequency shift equation (44). More precisely, in Figure 2 the nondimensional mass ratio M/ρAL⁡ is plotted against the relative frequency shift equation (44), with M=m-(γ2-γ1)L, and the four curves refer to four different η values, η=0 (without nonlocal effects), η=0.1, η=0.2, and η=0.3. The added mass is placed at the tip of the cantilever nanotube, so that η2=1, whereas γ2-γ1) 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 η. 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 η=0.2, but the curve describing the first approach is undistinguishable from the exact curve.Figure 2
The nondimensional mass ratioM/ρAL⁡ is plotted against the relative frequency shift—as obtained using (44)—with M=m-γ2-γ1L. The four curves refer to four different η values, η=0 (without nonlocal effects), η=0.1, η=0.2, and η=0.3.Figure 3
Numerical comparison between two proposed approaches.


## 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 coefficientη, so that careful calibration of this coefficient η becomes necessary, in order to obtain reliable values of the added mass.



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*Source: 825342-2015-03-01.xml*