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

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 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 $\stackrel{-}{m}$ is located, between the abscissae γ ₁ L and γ ₂ L. 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

$$\begin{array}{c}\underset{t_{\mathrm{1}}}{\overset{t_{\mathrm{2}}}{{\displaystyle \int }}}\left(\delta T\left(t\right)-\delta E_T\left(t\right)\right)\text{dt}=\mathrm{0},\end{array}$$ (1)

where

$$\begin{array}{c}T=\frac{\mathrm{1}}{\mathrm{2}}\underset{\mathrm{0}}{\overset{{\gamma }_{\mathrm{1}}L}{{\displaystyle \int }}}\rho A{\left(\frac{\partial v_{\mathrm{1}}(z,t)}{\partial t}\right)}^{\mathrm{2}}\text{dz}+\frac{\mathrm{1}}{\mathrm{2}}\underset{{\gamma }_{\mathrm{1}}L}{\overset{{\gamma }_{\mathrm{2}}L}{{\displaystyle \int }}}\rho A{\left(\frac{\partial v_{\mathrm{2}}(z,t)}{\partial t}\right)}^{\mathrm{2}}\text{dz}\\ \hspace{1em}+\frac{\mathrm{1}}{\mathrm{2}}\underset{{\gamma }_{\mathrm{2}}L}{\overset{L}{{\displaystyle \int }}}\rho A{\left(\frac{\partial v_{\mathrm{3}}(z,t)}{\partial t}\right)}^{\mathrm{2}}\text{dz}+\frac{\mathrm{1}}{\mathrm{2}}\underset{{\gamma }_{\mathrm{1}}L}{\overset{{\gamma }_{\mathrm{2}}L}{{\displaystyle \int }}}\stackrel{-}{m}{\left(\frac{\partial v_{\mathrm{2}}(z,t)}{\partial t}\right)}^{\mathrm{2}}\text{dz}\end{array}$$ (2)

is the sum of the kinetic energies of the nanotube in the three sections, v ₁ between (0, γ ₁ L), v ₂ between (γ ₁ L, γ ₂ L), and v ₃ between (γ ₂ L, L), respectively, and of the kinetic energy of the added mass. Moreover,

$$\begin{array}{c}{E}_T=L_e-P=\frac{\mathrm{1}}{\mathrm{2}}\underset{\mathrm{0}}{\overset{{\gamma }_{\mathrm{1}}L}{{\displaystyle \int }}}\text{EI}{\left(\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{1}}(z,t)}{\partial z^{\mathrm{2}}}\right)}^{\mathrm{2}}\text{dz}\\ \hspace{1em}+\frac{\mathrm{1}}{\mathrm{2}}\underset{{\gamma }_{\mathrm{1}}L}{\overset{{\gamma }_{\mathrm{2}}L}{{\displaystyle \int }}}\text{EI}{\left(\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{2}}\left(z,t\right)}{\partial z^{\mathrm{2}}}\right)}^{\mathrm{2}}\text{dz}\\ \hspace{1em}+\frac{\mathrm{1}}{\mathrm{2}}\underset{{\gamma }_{\mathrm{2}}L}{\overset{L}{{\displaystyle \int }}}\text{EI}{\left(\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{3}}(z,t)}{\partial z^{\mathrm{2}}}\right)}^{\mathrm{2}}\text{dz}\\ \hspace{1em}-\underset{\mathrm{0}}{\overset{{\gamma }_{\mathrm{1}}L}{{\displaystyle \int }}}\rho A\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{1}}(z,t)}{\partial t^{\mathrm{2}}}{\mu }^{\mathrm{2}}\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{1}}(z,t)}{\partial z^{\mathrm{2}}}\text{dz}\\ \hspace{1em}-\underset{{\gamma }_{\mathrm{1}}L}{\overset{{\gamma }_{\mathrm{2}}L}{{\displaystyle \int }}}\rho A\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{2}}(z,t)}{\partial t^{\mathrm{2}}}{\mu }^{\mathrm{2}}\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{2}}(z,t)}{\partial z^{\mathrm{2}}}\text{dz}\\ \hspace{1em}-\underset{{\gamma }_{\mathrm{2}}L}{\overset{L}{{\displaystyle \int }}}\rho A\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{3}}(z,t)}{\partial t^{\mathrm{2}}}{\mu }^{\mathrm{2}}\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{3}}(z,t)}{\partial z^{\mathrm{2}}}\text{dz}\end{array}$$ (3)

is the total potential energy, L _e is the strain energy of the nanotube, and P is the potential energy of the inertial force (ρA(∂² v(z, t)/∂t ²)) due to the additional displacement μ ²(∂² v(z, t)/∂z ²) [14]. Here μ ² = (e ₀ a)², where e ₀ 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

$$\begin{array}{c}\underset{t_{\mathrm{1}}}{\overset{t_{\mathrm{2}}}{{\displaystyle \int }}}\left(\underset{\mathrm{0}}{\overset{{\gamma }_{\mathrm{1}}L}{{\displaystyle \int }}}\rho A\frac{\partial v_{\mathrm{1}}(z,t)}{\partial t}\delta \left(\frac{\partial v_{\mathrm{1}}(z,t)}{\partial t}\right)\text{dz}\right.\\ \hspace{1em}\hspace{1em}+\underset{{\gamma }_{\mathrm{1}}L}{\overset{{\gamma }_{\mathrm{2}}L}{{\displaystyle \int }}}\rho A\frac{\partial v_{\mathrm{2}}\left(z,t\right)}{\partial t}\delta \left(\frac{\partial v_{\mathrm{2}}\left(z,t\right)}{\partial t}\right)\text{dz}\\ \hspace{1em}\hspace{1em}+\underset{{\gamma }_{\mathrm{2}}L}{\overset{L}{{\displaystyle \int }}}\rho A\frac{\partial v_{\mathrm{3}}(z,t)}{\partial t}\delta \left(\frac{\partial v_{\mathrm{3}}(z,t)}{\partial t}\right)\text{dz}\\ \hspace{1em}\hspace{1em}+\underset{{\gamma }_{\mathrm{1}}L}{\overset{{\gamma }_{\mathrm{2}}L}{{\displaystyle \int }}}\stackrel{-}{m}\frac{\partial v_{\mathrm{2}}(z,t)}{\partial t}\delta \left(\frac{\partial v_{\mathrm{2}}(z,t)}{\partial t}\right)\text{dz}\\ \hspace{1em}\hspace{1em}-\underset{\mathrm{0}}{\overset{{\gamma }_{\mathrm{1}}L}{{\displaystyle \int }}}\left(\text{EI}\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{1}}\left(z,t\right)}{\partial z^{\mathrm{2}}}\delta \left(\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{1}}\left(z,t\right)}{\partial z^{\mathrm{2}}}\right)\right.\\ \phantom{llllllllllllllllllll}\left.-{\mu }^{\mathrm{2}}\rho A\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{1}}\left(z,t\right)}{\partial t^{\mathrm{2}}}\delta \left(\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{1}}\left(z,t\right)}{\partial z^{\mathrm{2}}}\right)\right)\text{dz}\\ \hspace{1em}\hspace{1em}-\underset{{\gamma }_{\mathrm{1}}L}{\overset{{\gamma }_{\mathrm{2}}L}{{\displaystyle \int }}}\left(\text{EI}\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{2}}\left(z,t\right)}{\partial z^{\mathrm{2}}}\delta \left(\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{2}}\left(z,t\right)}{\partial z^{\mathrm{2}}}\right)\right.\\ \phantom{llllllllllllllllllll}\left.-{\mu }^{\mathrm{2}}\rho A\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{2}}\left(z,t\right)}{\partial t^{\mathrm{2}}}\delta \left(\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{2}}\left(z,t\right)}{\partial z^{\mathrm{2}}}\right)\right)\text{dz}\\ \hspace{1em}\hspace{1em}-\underset{{\gamma }_{\mathrm{2}}L}{\overset{L}{{\displaystyle \int }}}\left(\text{EI}\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{3}}(z,t)}{\partial z^{\mathrm{2}}}\delta \left(\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{3}}(z,t)}{\partial z^{\mathrm{2}}}\right)\right.\\ \phantom{llllllllllllllllllll}\left.\left.-{\mu }^{\mathrm{2}}\rho A\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{3}}\left(z,t\right)}{\partial t^{\mathrm{2}}}\delta \left(\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{3}}\left(z,t\right)}{\partial z^{\mathrm{2}}}\right)\right)\text{dz}\right)\text{dt}=\mathrm{0}.\end{array}$$ (4)

Integrations by part (see Appendix) lead to a system of three equations of motion:

$$\begin{array}{c}\text{EI}\frac{{\partial }^{\mathrm{4}}{v}_{\mathrm{1}}\left(z,t\right)}{\partial z^{\mathrm{4}}}-{\mu }^{\mathrm{2}}\rho A\frac{{\partial }^{\mathrm{4}}{v}_{\mathrm{1}}\left(z,t\right)}{\partial z^{\mathrm{2}}\partial t^{\mathrm{2}}}+\rho A\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{1}}\left(z,t\right)}{\partial t^{\mathrm{2}}}=\mathrm{0},\\ \phantom{\text{gggggggggggggggggggggggggggggg}\mathrm{1}}\mathrm{0}<z<{\gamma }_{\mathrm{1}}L,\\ \text{EI}\frac{{\partial }^{\mathrm{4}}{v}_{\mathrm{2}}\left(z,t\right)}{\partial z^{\mathrm{4}}}-{\mu }^{\mathrm{2}}\rho A\frac{{\partial }^{\mathrm{4}}{v}_{\mathrm{2}}\left(z,t\right)}{\partial z^{\mathrm{2}}\partial t^{\mathrm{2}}}+(\rho A+M)\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{2}}\left(z,t\right)}{\partial t^{\mathrm{2}}}=\mathrm{0},\\ \phantom{ggggggggggggggggggggggggggggggg}{\gamma }_{\mathrm{1}}L<z<{\gamma }_{\mathrm{2}}L,\\ \text{EI}\frac{{\partial }^{\mathrm{4}}{v}_{\mathrm{3}}\left(z,t\right)}{\partial z^{\mathrm{4}}}-{\mu }^{\mathrm{2}}\rho A\frac{{\partial }^{\mathrm{4}}{v}_{\mathrm{3}}\left(z,t\right)}{\partial z^{\mathrm{2}}\partial t^{\mathrm{2}}}+\rho A\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{3}}\left(z,t\right)}{\partial t^{\mathrm{2}}}=\mathrm{0},\\ \phantom{gggggggggggggggggggggggggggl}{\gamma }_{\mathrm{2}}L<z<L,\end{array}$$ (5)

together with the following general boundary conditions at the ends:

$$\begin{array}{c}{v}_{\mathrm{1}}(\mathrm{0},t)=\mathrm{0},\\ \frac{\partial v_{\mathrm{1}}\left(\mathrm{0},t\right)}{\partial z}=\mathrm{0},\\ \text{EI}\frac{{\partial }^{\mathrm{3}}{v}_{\mathrm{3}}\left(L,t\right)}{\partial z^{\mathrm{3}}}-{\mu }^{\mathrm{2}}\rho A\frac{{\partial }^{\mathrm{3}}{v}_{\mathrm{3}}\left(L,t\right)}{\partial t^{\mathrm{2}}\partial z}=\mathrm{0},\\ -\text{EI}\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{3}}\left(L,t\right)}{\partial z^{\mathrm{2}}}+{\mu }^{\mathrm{2}}\rho A\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{3}}\left(L,t\right)}{\partial t^{\mathrm{2}}}=\mathrm{0}.\end{array}$$ (6)

The boundary conditions for z = γ ₁ L are given by

$$\begin{array}{c}{v}_{\mathrm{1}}\left({\gamma }_{\mathrm{1}}L,t\right)=v_{\mathrm{2}}\left({\gamma }_{\mathrm{1}}L,t\right),\\ \frac{\partial v_{\mathrm{1}}\left({\gamma }_{\mathrm{1}}L,t\right)}{\partial z}=\frac{\partial v_{\mathrm{2}}\left({\gamma }_{\mathrm{1}}L,t\right)}{\partial z},\\ {\mu }^{\mathrm{2}}\rho A\frac{{\partial }^{\mathrm{3}}{v}_{\mathrm{1}}\left({\gamma }_{\mathrm{1}}L,t\right)}{\partial t^{\mathrm{2}}\partial z}-\text{EI}\frac{{\partial }^{\mathrm{3}}{v}_{\mathrm{1}}\left({\gamma }_{\mathrm{1}}L,t\right)}{\partial z^{\mathrm{3}}}-{\mu }^{\mathrm{2}}\rho A\frac{{\partial }^{\mathrm{3}}{v}_{\mathrm{2}}\left({\gamma }_{\mathrm{1}}L,t\right)}{\partial t^{\mathrm{2}}\partial z}\\ \hspace{1em}\hspace{1em}+\text{EI}\frac{{\partial }^{\mathrm{3}}{v}_{\mathrm{2}}\left({\gamma }_{\mathrm{1}}L,t\right)}{\partial z^{\mathrm{3}}}=\mathrm{0},\\ {\mu }^{\mathrm{2}}\rho A\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{1}}\left({\gamma }_{\mathrm{1}}L,t\right)}{\partial t^{\mathrm{2}}}-\text{EI}\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{1}}\left({\gamma }_{\mathrm{1}}L,t\right)}{\partial z^{\mathrm{2}}}-{\mu }^{\mathrm{2}}\rho A\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{2}}\left({\gamma }_{\mathrm{1}}L,t\right)}{\partial t^{\mathrm{2}}}\\ \hspace{1em}\hspace{1em}+\text{EI}\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{2}}\left({\gamma }_{\mathrm{1}}L,t\right)}{\partial z^{\mathrm{2}}}=\mathrm{0},\end{array}$$ (7)

and finally, at z = γ ₂ L,

$$\begin{array}{c}{v}_{\mathrm{2}}\left({\gamma }_{\mathrm{2}}L,t\right)=v_{\mathrm{3}}\left({\gamma }_{\mathrm{2}}L,t\right),\\ \frac{\partial v_{\mathrm{2}}\left({\gamma }_{\mathrm{2}}L,t\right)}{\partial z}=\frac{\partial v_{\mathrm{3}}\left({\gamma }_{\mathrm{2}}L,t\right)}{\partial z},\\ {\mu }^{\mathrm{2}}\rho A\frac{{\partial }^{\mathrm{3}}{v}_{\mathrm{2}}\left({\gamma }_{\mathrm{2}}L,t\right)}{\partial t^{\mathrm{2}}\partial z}-\text{EI}\frac{{\partial }^{\mathrm{3}}{v}_{\mathrm{2}}\left({\gamma }_{\mathrm{2}}L,t\right)}{\partial z^{\mathrm{3}}}-{\mu }^{\mathrm{2}}\rho A\frac{{\partial }^{\mathrm{3}}{v}_{\mathrm{3}}\left({\gamma }_{\mathrm{2}}L,t\right)}{\partial t^{\mathrm{2}}\partial z}\\ \hspace{1em}\hspace{1em}+\text{EI}\frac{{\partial }^{\mathrm{3}}{v}_{\mathrm{3}}\left({\gamma }_{\mathrm{2}}L,t\right)}{\partial z^{\mathrm{3}}}=\mathrm{0},\\ {\mu }^{\mathrm{2}}\rho A\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{2}}\left({\gamma }_{\mathrm{2}}L,t\right)}{\partial t^{\mathrm{2}}}-\text{EI}\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{2}}\left({\gamma }_{\mathrm{2}}L,t\right)}{\partial z^{\mathrm{2}}}-{\mu }^{\mathrm{2}}\rho A\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{3}}\left({\gamma }_{\mathrm{2}}L,t\right)}{\partial t^{\mathrm{2}}}\\ \hspace{1em}\hspace{1em}+\text{EI}\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{3}}\left({\gamma }_{\mathrm{2}}L,t\right)}{\partial z^{\mathrm{2}}}=\mathrm{0}.\end{array}$$ (8)

The solutions of (5) can be expressed as

$$\begin{array}{c}{v}_h(z,t)=v_h\left(z\right)e^{i\omega t},\hspace{1em}h=\mathrm{1,2},\mathrm{3}.\end{array}$$ (9)

If the nondimensional abscissa ζ = z/L can be introduced, the system of three equations of motion (5) becomes

$$\begin{array}{c}\frac{{\partial }^{\mathrm{4}}{v}_{\mathrm{1}}\left(\zeta \right)}{\partial {\zeta }^{\mathrm{4}}}+{\eta }^{\mathrm{2}}{\Omega }^{\mathrm{4}}\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{1}}\left(\zeta \right)}{\partial {\zeta }^{\mathrm{2}}}-{\Omega }^{\mathrm{4}}{v}_{\mathrm{1}}\left(\zeta \right)=\mathrm{0}\hspace{1em}\text{for}\mathrm{\hspace{0.17em}\hspace{0.17em}}\mathrm{0}<\zeta <{\gamma }_{\mathrm{1}},\\ \frac{{\partial }^{\mathrm{4}}{v}_{\mathrm{2}}\left(\zeta \right)}{\partial {\zeta }^{\mathrm{4}}}+{\eta }^{\mathrm{2}}{\Omega }^{\mathrm{4}}\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{2}}\left(\zeta \right)}{\partial {\zeta }^{\mathrm{2}}}-\left(\mathrm{1}+\lambda \right){\Omega }^{\mathrm{4}}{v}_{\mathrm{2}}\left(\zeta \right)=\mathrm{0}\\ \phantom{\text{gggggggggggggggggggggggg}}\text{for}\mathrm{\hspace{0.17em}\hspace{0.17em}}{\gamma }_{\mathrm{1}}<\zeta <{\gamma }_{\mathrm{2}},\\ \frac{{\partial }^{\mathrm{4}}{v}_{\mathrm{3}}\left(\zeta \right)}{\partial {\zeta }^{\mathrm{4}}}+{\eta }^{\mathrm{2}}{\Omega }^{\mathrm{4}}\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{3}}\left(\zeta \right)}{\partial {\zeta }^{\mathrm{2}}}-{\Omega }^{\mathrm{4}}{v}_{\mathrm{3}}\left(\zeta \right)=\mathrm{0}\hspace{1em}\text{for}\mathrm{\hspace{0.17em}\hspace{0.17em}}{\gamma }_{\mathrm{2}}<\zeta <\mathrm{1},\end{array}$$ (10)

where the following nondimensional parameters have been introduced:

$$\begin{array}{c}\lambda =\frac{\stackrel{-}{m}}{\rho A};\hspace{1em}\hspace{1em}\eta =\frac{\mu }{L};\hspace{1em}\hspace{1em}\Omega =\sqrt{\sqrt{\frac{\rho A{L}^{\mathrm{4}}{\omega }^{\mathrm{2}}}{\text{EI}}}}.\end{array}$$ (11)

The boundary conditions for clamped-free nanotube are given by

$$\begin{array}{c}{v}_{\mathrm{1}}\left(\mathrm{0}\right)=\mathrm{0},\\ \frac{\partial v_{\mathrm{1}}\left(\mathrm{0}\right)}{\partial \zeta }=\mathrm{0},\\ v_{\mathrm{1}}\left({\gamma }_{\mathrm{1}}\right)=v_{\mathrm{2}}\left({\gamma }_{\mathrm{1}}\right),\\ \frac{\partial v_{\mathrm{1}}\left({\gamma }_{\mathrm{1}}\right)}{\partial \zeta }=\frac{\partial v_{\mathrm{2}}\left({\gamma }_{\mathrm{1}}\right)}{\partial \zeta },\\ {\eta }^{\mathrm{2}}{\Omega }^{\mathrm{4}}\frac{\partial v_{\mathrm{1}}\left({\gamma }_{\mathrm{1}}\right)}{\partial \zeta }+\frac{{\partial }^{\mathrm{3}}{v}_{\mathrm{1}}\left({\gamma }_{\mathrm{1}}\right)}{\partial {\zeta }^{\mathrm{3}}}-{\eta }^{\mathrm{2}}{\Omega }^{\mathrm{4}}\frac{\partial v_{\mathrm{2}}\left({\gamma }_{\mathrm{1}}\right)}{\partial \zeta }-\frac{{\partial }^{\mathrm{3}}{v}_{\mathrm{2}}\left({\gamma }_{\mathrm{1}}\right)}{\partial {\zeta }^{\mathrm{3}}}=\mathrm{0},\\ {\eta }^{\mathrm{2}}{\Omega }^{\mathrm{4}}{v}_{\mathrm{1}}\left({\gamma }_{\mathrm{1}}\right)+\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{1}}\left({\gamma }_{\mathrm{1}}\right)}{\partial {\zeta }^{\mathrm{2}}}-{\eta }^{\mathrm{2}}{\Omega }^{\mathrm{4}}{v}_{\mathrm{2}}\left({\gamma }_{\mathrm{1}}\right)-\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{2}}\left({\gamma }_{\mathrm{1}}\right)}{\partial {\zeta }^{\mathrm{2}}}=\mathrm{0},\\ v_{\mathrm{2}}\left({\gamma }_{\mathrm{2}}\right)=v_{\mathrm{3}}\left({\gamma }_{\mathrm{2}}\right),\\ \frac{\partial v_{\mathrm{2}}\left({\gamma }_{\mathrm{2}}\right)}{\partial \zeta }=\frac{\partial v_{\mathrm{3}}\left({\gamma }_{\mathrm{2}}\right)}{\partial \zeta },\\ {\eta }^{\mathrm{2}}{\Omega }^{\mathrm{4}}\frac{\partial v_{\mathrm{2}}\left({\gamma }_{\mathrm{2}}\right)}{\partial \zeta }+\frac{{\partial }^{\mathrm{3}}{v}_{\mathrm{2}}\left({\gamma }_{\mathrm{2}}\right)}{\partial {\zeta }^{\mathrm{3}}}-{\eta }^{\mathrm{2}}{\Omega }^{\mathrm{4}}\frac{\partial v_{\mathrm{3}}\left({\gamma }_{\mathrm{2}}\right)}{\partial \zeta }-\frac{{\partial }^{\mathrm{3}}{v}_{\mathrm{3}}\left({\gamma }_{\mathrm{2}}\right)}{\partial {\zeta }^{\mathrm{3}}}=\mathrm{0},\\ {\eta }^{\mathrm{2}}{\Omega }^{\mathrm{4}}{v}_{\mathrm{2}}\left({\gamma }_{\mathrm{2}}\right)+\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{2}}\left({\gamma }_{\mathrm{2}}\right)}{\partial {\zeta }^{\mathrm{2}}}-{\eta }^{\mathrm{2}}{\Omega }^{\mathrm{4}}{v}_{\mathrm{3}}\left({\gamma }_{\mathrm{2}}\right)-\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{3}}\left({\gamma }_{\mathrm{2}}\right)}{\partial {\zeta }^{\mathrm{2}}}=\mathrm{0},\\ \frac{{\partial }^{\mathrm{3}}{v}_{\mathrm{3}}\left(\mathrm{1}\right)}{\partial {\zeta }^{\mathrm{3}}}+{\eta }^{\mathrm{2}}{\Omega }^{\mathrm{4}}\frac{\partial v_{\mathrm{3}}\left(\mathrm{1}\right)}{\partial \zeta }=\mathrm{0},\\ -\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{3}}\left(\mathrm{1}\right)}{\partial {\zeta }^{\mathrm{2}}}-{\eta }^{\mathrm{2}}{\Omega }^{\mathrm{4}}{v}_{\mathrm{3}}\left(\mathrm{1}\right)=\mathrm{0}.\end{array}$$ (12)

The general solutions of (10) are given by

$$\begin{array}{c}{v}_{\mathrm{1}}\left(\zeta \right)=A_{\mathrm{1}}\cos \left(\alpha \zeta \right)+A_{\mathrm{2}}\sin \left(\alpha \zeta \right)+A_{\mathrm{3}}\cos h\left(\alpha \zeta \right)\\ \hspace{1em}+A_{\mathrm{4}}\sin h\left(\alpha \zeta \right),\\ v_{\mathrm{2}}\left(\zeta \right)=B_{\mathrm{1}}\cos \left({\alpha }_{\mathrm{1}}\zeta \right)+B_{\mathrm{2}}\sin \left({\alpha }_{\mathrm{1}}\zeta \right)+B_{\mathrm{3}}\cos h\left({\beta }_{\mathrm{1}}\zeta \right)\\ \hspace{1em}+B_{\mathrm{4}}\sin h\left({\beta }_{\mathrm{1}}\zeta \right),\\ v_{\mathrm{3}}\left(\zeta \right)=C_{\mathrm{1}}\cos \left(\alpha \zeta \right)+C_{\mathrm{2}}\sin \left(\alpha \zeta \right)+C_{\mathrm{3}}\cos h\left(\beta \zeta \right)\\ \hspace{1em}+C_{\mathrm{4}}\sin h\left(\beta \zeta \right)\end{array}$$ (13)

with

$$\begin{array}{c}\alpha =\sqrt{\frac{\mathrm{1}}{\mathrm{2}}\left({\eta }^{\mathrm{2}}{\Omega }^{\mathrm{4}}+{\Omega }^{\mathrm{2}}\sqrt{\mathrm{4}+{\eta }^{\mathrm{4}}{\Omega }^{\mathrm{4}}}\right)};\\ \beta =\sqrt{\frac{\mathrm{1}}{\mathrm{2}}\left(-{\eta }^{\mathrm{2}}{\Omega }^{\mathrm{4}}+{\Omega }^{\mathrm{2}}\sqrt{\mathrm{4}+{\eta }^{\mathrm{4}}{\Omega }^{\mathrm{4}}}\right)};\\ {\alpha }_{\mathrm{1}}=\sqrt{\frac{\mathrm{1}}{\mathrm{2}}\left({\eta }^{\mathrm{2}}{\Omega }^{\mathrm{4}}+{\Omega }^{\mathrm{2}}\sqrt{\mathrm{4}+\mathrm{4}\lambda +{\eta }^{\mathrm{4}}{\Omega }^{\mathrm{4}}}\right)};\\ {\beta }_{\mathrm{1}}=\sqrt{\frac{\mathrm{1}}{\mathrm{2}}\left(-{\eta }^{\mathrm{2}}{\Omega }^{\mathrm{4}}+{\Omega }^{\mathrm{2}}\sqrt{\mathrm{4}+\mathrm{4}\lambda +{\eta }^{\mathrm{4}}{\Omega }^{\mathrm{4}}}\right)}.\end{array}$$ (14)

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 Ω _j. The circular frequencies ω _j can be easily deduced, as well as the natural frequencies f _j = ω _j/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:

$$\begin{array}{c}\underset{\mathrm{0}}{\overset{L}{{\displaystyle \int }}}\text{EI}\frac{{\partial }^{\mathrm{4}}v\left(z\right)}{\partial z^{\mathrm{4}}}\text{dz}+{\omega }^{\mathrm{2}}\underset{\mathrm{0}}{\overset{L}{{\displaystyle \int }}}{\mu }^{\mathrm{2}}\rho A\frac{{\partial }^{\mathrm{2}}v\left(z\right)}{\partial z^{\mathrm{2}}}\text{dz}\\ \hspace{1em}\hspace{1em}-{\omega }^{\mathrm{2}}\underset{\mathrm{0}}{\overset{L}{{\displaystyle \int }}}\rho Av\left(z\right)\text{dz}-{\omega }^{\mathrm{2}}\underset{{\gamma }_{\mathrm{1}}L}{\overset{{\gamma }_{\mathrm{2}}L}{{\displaystyle \int }}}\stackrel{-}{m}v\left(z\right)\text{dz}=\mathrm{0}.\end{array}$$ (15)

It is now possible to insert a trial function y(z) [14], leading to

$$\begin{array}{c}\underset{\mathrm{0}}{\overset{L}{{\displaystyle \int }}}\text{EI}\frac{{\partial }^{\mathrm{4}}v\left(z\right)}{\partial z^{\mathrm{4}}}y\left(z\right)\text{dz}+{\omega }^{\mathrm{2}}\underset{\mathrm{0}}{\overset{L}{{\displaystyle \int }}}{\mu }^{\mathrm{2}}\rho A\frac{{\partial }^{\mathrm{2}}v\left(z\right)}{\partial z^{\mathrm{2}}}y\left(z\right)\text{dz}\\ \hspace{1em}\hspace{1em}-{\omega }^{\mathrm{2}}\underset{\mathrm{0}}{\overset{L}{{\displaystyle \int }}}\rho Av\left(z\right)y\left(z\right)\text{dz}-{\omega }^{\mathrm{2}}\underset{{\gamma }_{\mathrm{1}}L}{\overset{{\gamma }_{\mathrm{2}}L}{{\displaystyle \int }}}\stackrel{-}{m}v\left(z\right)y\left(z\right)\text{dz}=\mathrm{0}.\end{array}$$ (16)

Two successive integrations by part can be performed:

$$\begin{array}{c}\underset{\mathrm{0}}{\overset{L}{{\displaystyle \int }}}\text{EI}\frac{{\partial }^{\mathrm{4}}v\left(z\right)}{\partial z^{\mathrm{4}}}y\left(z\right)\text{dz}\\ \hspace{1em}\hspace{0.5em}={\left[\text{EI}\frac{{\partial }^{\mathrm{3}}v\left(z\right)}{\partial z^{\mathrm{3}}}y\left(z\right)\right]}_{\mathrm{0}}^L-\underset{\mathrm{0}}{\overset{L}{{\displaystyle \int }}}\text{EI}\frac{{\partial }^{\mathrm{3}}v\left(z\right)}{\partial z^{\mathrm{3}}}\frac{\partial y\left(z\right)}{\partial z}\text{dz}\\ \hspace{1em}\hspace{0.5em}={\left[\text{EI}\frac{{\partial }^{\mathrm{3}}v\left(z\right)}{\partial z^{\mathrm{3}}}y\left(z\right)\right]}_{\mathrm{0}}^L-{\left[\text{EI}\frac{{\partial }^{\mathrm{2}}v\left(z\right)}{\partial z^{\mathrm{2}}}\frac{\partial y\left(z\right)}{\partial z}\right]}_{\mathrm{0}}^L\\ \hspace{1em}\hspace{1em}\hspace{0.5em}+\underset{\mathrm{0}}{\overset{L}{{\displaystyle \int }}}\text{EI}\frac{{\partial }^{\mathrm{2}}v\left(z\right)}{\partial z^{\mathrm{2}}}\frac{{\partial }^{\mathrm{2}}y\left(z\right)}{\partial z^{\mathrm{2}}}\text{dz},\\ {\omega }^{\mathrm{2}}\underset{\mathrm{0}}{\overset{L}{{\displaystyle \int }}}{\mu }^{\mathrm{2}}\rho A\frac{{\partial }^{\mathrm{2}}v\left(z\right)}{\partial z^{\mathrm{2}}}y\left(z\right)\text{dz}\\ \hspace{1em}\hspace{0.5em}={\left[{\omega }^{\mathrm{2}}{\mu }^{\mathrm{2}}\rho A\frac{\partial v\left(z\right)}{\partial z}y\left(z\right)\right]}_{\mathrm{0}}^L-{\omega }^{\mathrm{2}}\underset{\mathrm{0}}{\overset{L}{{\displaystyle \int }}}{\mu }^{\mathrm{2}}\rho A\frac{\partial v\left(z\right)}{\partial z}\frac{\partial y\left(z\right)}{\partial z}\text{dz}\\ \hspace{1em}\hspace{0.5em}={\left[{\omega }^{\mathrm{2}}{\mu }^{\mathrm{2}}\rho A\frac{\partial v\left(z\right)}{\partial z}y\left(z\right)\right]}_{\mathrm{0}}^L-{\left[{\omega }^{\mathrm{2}}{\mu }^{\mathrm{2}}\rho Av\left(z\right)\frac{\partial y\left(z\right)}{\partial z}\right]}_{\mathrm{0}}^L\\ \hspace{1em}\hspace{1em}\hspace{0.5em}+{\omega }^{\mathrm{2}}\underset{\mathrm{0}}{\overset{L}{{\displaystyle \int }}}{\mu }^{\mathrm{2}}\rho Av\left(z\right)\frac{{\partial }^{\mathrm{2}}y\left(z\right)}{\partial z^{\mathrm{2}}}\text{dz}\end{array}$$ (17)

so that (16) becomes

$$\begin{array}{c}\underset{\mathrm{0}}{\overset{L}{{\displaystyle \int }}}‍\text{EI}\frac{{\partial }^{\mathrm{2}}v\left(z\right)}{\partial z^{\mathrm{2}}}\frac{{\partial }^{\mathrm{2}}y\left(z\right)}{\partial z^{\mathrm{2}}}\text{dz}-{\omega }^{\mathrm{2}}\underset{\mathrm{0}}{\overset{L}{{\displaystyle \int }}}‍\rho Av\left(z\right)y\left(z\right)\text{dz}\\ \hspace{1em}\hspace{0.5em}-{\omega }^{\mathrm{2}}\underset{{\gamma }_{\mathrm{1}}L}{\overset{{\gamma }_{\mathrm{2}}L}{{\displaystyle \int }}}‍\stackrel{-}{m}v\left(z\right)y\left(z\right)\text{dz}+{\omega }^{\mathrm{2}}\underset{\mathrm{0}}{\overset{L}{{\displaystyle \int }}}‍{\mu }^{\mathrm{2}}\rho Av\left(z\right)\frac{{\partial }^{\mathrm{2}}y\left(z\right)}{\partial z^{\mathrm{2}}}\text{dz}\\ \hspace{1em}\hspace{0.5em}+{\left[\text{EI}\frac{{\partial }^{\mathrm{3}}v\left(z\right)}{\partial z^{\mathrm{3}}}y\left(z\right)\right]}_{\mathrm{0}}^L-{\left[\text{EI}\frac{{\partial }^{\mathrm{2}}v\left(z\right)}{\partial z^{\mathrm{2}}}\frac{\partial y\left(z\right)}{\partial z}\right]}_{\mathrm{0}}^L\\ \hspace{1em}\hspace{0.5em}+{\left[{\omega }^{\mathrm{2}}{\mu }^{\mathrm{2}}\rho A\frac{\partial v\left(z\right)}{\partial z}y\left(z\right)\right]}_{\mathrm{0}}^L-{\left[{\omega }^{\mathrm{2}}{\mu }^{\mathrm{2}}\rho Av\left(z\right)\frac{\partial y\left(z\right)}{\partial z}\right]}_{\mathrm{0}}^L=\mathrm{0}.\end{array}$$ (18)

The boundary conditions at the right end permit simplifying the previous equation:

$$\begin{array}{c}\underset{\mathrm{0}}{\overset{L}{{\displaystyle \int }}}‍\text{EI}\frac{{\partial }^{\mathrm{2}}v\left(z\right)}{\partial z^{\mathrm{2}}}\frac{{\partial }^{\mathrm{2}}y\left(z\right)}{\partial z^{\mathrm{2}}}\text{dz}+{\omega }^{\mathrm{2}}\underset{\mathrm{0}}{\overset{L}{{\displaystyle \int }}}{\mu }^{\mathrm{2}}\rho Av\left(z\right)\frac{{\partial }^{\mathrm{2}}y\left(z\right)}{\partial z^{\mathrm{2}}}\text{dz}‍\\ \hspace{1em}\hspace{1em}-{\omega }^{\mathrm{2}}\underset{\mathrm{0}}{\overset{L}{{\displaystyle \int }}}‍\rho Av\left(z\right)y\left(z\right)\text{dz}-{\omega }^{\mathrm{2}}\underset{{\gamma }_{\mathrm{1}}L}{\overset{{\gamma }_{\mathrm{2}}L}{{\displaystyle \int }}}‍\stackrel{-}{m}v\left(z\right)y\left(z\right)\text{dz}\\ \hspace{1em}\hspace{1em}+\text{EI}\frac{{\partial }^{\mathrm{3}}v\left(L\right)}{\partial z^{\mathrm{3}}}y\left(L\right)-\text{EI}\frac{{\partial }^{\mathrm{2}}v\left(L\right)}{\partial z^{\mathrm{2}}}\frac{\partial y\left(L\right)}{\partial z}\\ \hspace{1em}\hspace{1em}+{\omega }^{\mathrm{2}}{\mu }^{\mathrm{2}}\rho A\frac{\partial v\left(L\right)}{\partial z}y\left(L\right)-{\omega }^{\mathrm{2}}{\mu }^{\mathrm{2}}\rho Av\left(L\right)\frac{\partial y\left(L\right)}{\partial z}=\mathrm{0},\end{array}$$ (19)

whereas the free end will be subjected to the following equilibrium conditions:

$$\begin{array}{c}{\omega }^{\mathrm{2}}{\mu }^{\mathrm{2}}\rho A\frac{\partial v\left(L\right)}{\partial z}+\text{EI}\frac{{\partial }^{\mathrm{3}}v\left(L\right)}{\partial z^{\mathrm{3}}}=\mathrm{0},\\ -{\omega }^{\mathrm{2}}{\mu }^{\mathrm{2}}\rho Av\left(L\right)-\text{EI}\frac{{\partial }^{\mathrm{2}}v\left(L\right)}{\partial z^{\mathrm{2}}}=\mathrm{0}.\end{array}$$ (20)

Finally, (19) reduces to

$$\begin{array}{c}\underset{\mathrm{0}}{\overset{L}{{\displaystyle \int }}}‍\text{EI}\frac{{\partial }^{\mathrm{2}}v\left(z\right)}{\partial z^{\mathrm{2}}}\frac{{\partial }^{\mathrm{2}}y\left(z\right)}{\partial z^{\mathrm{2}}}\text{dz}+{\omega }^{\mathrm{2}}\underset{\mathrm{0}}{\overset{L}{{\displaystyle \int }}}‍{\mu }^{\mathrm{2}}\rho Av\left(z\right)\frac{{\partial }^{\mathrm{2}}y\left(z\right)}{\partial z^{\mathrm{2}}}\text{dz}\\ \hspace{1em}\hspace{1em}-{\omega }^{\mathrm{2}}\underset{\mathrm{0}}{\overset{L}{{\displaystyle \int }}}‍\rho Av\left(z\right)y\left(z\right)\text{dz}-{\omega }^{\mathrm{2}}\underset{{\gamma }_{\mathrm{1}}L}{\overset{{\gamma }_{\mathrm{2}}L}{{\displaystyle \int }}}‍\stackrel{-}{m}v\left(z\right)y\left(z\right)\text{dz}=\mathrm{0}\end{array}$$ (21)

and the frequency ω ² can be written down, putting y(z) = v(z), as

$$\begin{array}{c}{\omega }^{\mathrm{2}}=\left(\underset{\mathrm{0}}{\overset{L}{{\displaystyle \int }}}‍\text{EI}\frac{{\partial }^{\mathrm{2}}v\left(z\right)}{\partial z^{\mathrm{2}}}\frac{{\partial }^{\mathrm{2}}v\left(z\right)}{\partial z^{\mathrm{2}}}\text{dz}\hspace{0.17em}\right)\\ \hspace{1em}\times \left(\phantom{\underset{\mathrm{0}}{\overset{L}{{\displaystyle \int }}}‍{\mu }^{\mathrm{2}}\rho Av\left(z\right)\frac{{\partial }^{\mathrm{2}}v\left(z\right)}{\partial z^{\mathrm{2}}}\text{dz}}\underset{\mathrm{0}}{\overset{L}{{\displaystyle \int }}}‍\rho A{v}^{\mathrm{2}}\left(z\right)\text{dz}+\underset{{\gamma }_{\mathrm{1}}L}{\overset{{\gamma }_{\mathrm{2}}L}{{\displaystyle \int }}}‍\stackrel{-}{m}{v}^{\mathrm{2}}\left(z\right)\text{dz}\right.\\ \hspace{1em}\hspace{1em}\hspace{0.5em}{\left.-\underset{\mathrm{0}}{\overset{L}{{\displaystyle \int }}}‍{\mu }^{\mathrm{2}}\rho Av\left(z\right)\frac{{\partial }^{\mathrm{2}}v\left(z\right)}{\partial z^{\mathrm{2}}}\text{dz}\right)}^{-1}\end{array}$$ (22)

or, in terms of the nondimensional abscissa ζ = z/L,

$$\begin{array}{c}{\omega }^{\mathrm{2}}=\left(\frac{\text{EI}}{L^{\mathrm{3}}}\underset{\mathrm{0}}{\overset{\mathrm{1}}{{\displaystyle \int }}}‍\frac{{\partial }^{\mathrm{2}}v\left(\zeta \right)}{\partial {\zeta }^{\mathrm{2}}}\frac{{\partial }^{\mathrm{2}}v\left(\zeta \right)}{\partial {\zeta }^{\mathrm{2}}}\text{d}\zeta \right)\\ \hspace{1em}\times \left(\phantom{\hspace{0.17em}{\eta }^{\mathrm{2}}\rho AL\underset{\mathrm{0}}{\overset{\mathrm{1}}{{\displaystyle \int }}}‍v\left(\zeta \right)\frac{{\partial }^{2}v\left(\zeta \right)}{\partial {\zeta }^2}\text{d}\zeta }\rho AL\underset{\mathrm{0}}{\overset{\mathrm{1}}{{\displaystyle \int }}}‍v^2\left(\zeta \right)\text{d}\zeta +\stackrel{-}{m}L\underset{{\gamma }_{\mathrm{1}}}{\overset{{\gamma }_{\mathrm{2}}}{{\displaystyle \int }}}‍v^2\left(\zeta \right)\text{d}\zeta \right.\\ \hspace{1em}\hspace{1em}\hspace{0.5em}{\left.-\hspace{0.17em}{\eta }^{\mathrm{2}}\rho AL\underset{\mathrm{0}}{\overset{\mathrm{1}}{{\displaystyle \int }}}‍v\left(\zeta \right)\frac{{\partial }^{2}v\left(\zeta \right)}{\partial {\zeta }^2}\text{d}\zeta \right)}^{-\mathrm{1}}.\end{array}$$ (23)

In order to obtain a satisfactory approximation of the fundamental frequency, we use as approximating function v(ζ) the exact displacement of the cantilever beam without added mass:

$$\begin{array}{c}v\left(\zeta \right)=\left(\text{Cosh}\left(\mathrm{1.8751}\zeta \right)-\mathrm{Cos}\left(\mathrm{1.8751}\zeta \right)\right)\\ \hspace{1em}-\frac{\left(\text{Sinh}\left(\mathrm{1.8751}\right)-\mathrm{Sin}\left(\mathrm{1.8751}\right)\right)}{\left(\text{Cosh}\left(\mathrm{1.8751}\right)+\mathrm{Cos}\left(\mathrm{1.8751}\right)\right)}\\ \hspace{1em}\times \left(\text{Sinh}\left(\mathrm{1.8751}\zeta \right)-\mathrm{Sin}\left(\mathrm{1.8751}\zeta \right)\right).\end{array}$$ (24)

The following integrals can be defined [15]:

$$\begin{array}{c}{I}_{\mathrm{1}}=\underset{\mathrm{0}}{\overset{\mathrm{1}}{{\displaystyle \int }}}{v}^{\mathrm{2}}\left(\zeta \right)\text{d}\zeta =\mathrm{1},\\ I_{\mathrm{3}}=\underset{\mathrm{0}}{\overset{\mathrm{1}}{{\displaystyle \int }}}‍\frac{{\partial }^{\mathrm{2}}v\left(\zeta \right)}{\partial {\zeta }^{\mathrm{2}}}\frac{{\partial }^{\mathrm{2}}v\left(\zeta \right)}{\partial {\zeta }^{\mathrm{2}}}\text{d}\zeta =\mathrm{12.3623}.\end{array}$$ (25)

For the case of distributed added mass between the abscissae γ ₁ L and γ ₂ L, the integral I ₂ can be defined as

$$\begin{array}{c}{I}_{\mathrm{2}}=\underset{{\gamma }_{\mathrm{1}}}{\overset{{\gamma }_{\mathrm{2}}}{{\displaystyle \int }}}{v}^{\mathrm{2}}\left(\zeta \right)\text{d}\zeta \hspace{1em}\mathrm{0}\le {\gamma }_{\mathrm{1}}\le \mathrm{1};\hspace{0.17em}\mathrm{0}\le {\gamma }_{\mathrm{2}}\le \mathrm{1},\end{array}$$ (26)

and finally, in order to take into account the nonlocal effects, we define the fourth integral:

$$\begin{array}{c}{I}_{\mathrm{4}}=\underset{\mathrm{0}}{\overset{\mathrm{1}}{{\displaystyle \int }}}v\left(\zeta \right)\frac{{\partial }^{\mathrm{2}}v\left(\zeta \right)}{\partial {\zeta }^{\mathrm{2}}}\text{d}\zeta =\mathrm{0.858264}.\end{array}$$ (27)

The fundamental natural frequency can be deduced from (23) in terms of these four integrals as

$$\begin{array}{c}{f}_{\text{n}1}=\frac{\omega }{\mathrm{2}\pi }=\frac{\beta }{\mathrm{2}\pi }\sqrt{\frac{I_{\mathrm{3}}}{I_{\mathrm{1}}+\lambda I_{\mathrm{2}}-{\eta }^{\mathrm{2}}{I}_{\mathrm{4}}}}.\end{array}$$ (28)

Finally, it is usual to cancel out the first integral, so arriving to the natural frequency,

$$\begin{array}{c}{f}_{\text{n}1}=\frac{\beta }{2\pi }\frac{C_k}{\sqrt{1+\lambda C_m-{\eta }^2{C}_{\text{n}1}}},\end{array}$$ (29)

where

$$\begin{array}{c}\beta =\sqrt{\frac{\text{EI}}{m{L}^{\mathrm{4}}}};\\ C_k=\sqrt{\frac{I_{\mathrm{3}}}{I_{\mathrm{1}}}}=\mathrm{3.5160};\\ C_{\text{n}1}=\frac{I_{\mathrm{4}}}{I_{\mathrm{1}}}=\mathrm{0.858264};\\ C_m=\frac{I_{\mathrm{2}}}{I_{\mathrm{1}}}\end{array}$$ (30)

and C _k, C _n1, and C _m 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:

$$\begin{array}{c}T=\frac{\mathrm{1}}{\mathrm{2}}\underset{\mathrm{0}}{\overset{L}{{\displaystyle \int }}}‍\rho A{\left(\frac{\partial v(z,t)}{\partial t}\right)}^{\mathrm{2}}\text{dz}+\frac{\mathrm{1}}{\mathrm{2}}\underset{{\gamma }_{\mathrm{1}}L}{\overset{{\gamma }_{\mathrm{2}}L}{{\displaystyle \int }}}‍\stackrel{-}{m}{\left(\frac{\partial v(z,t)}{\partial t}\right)}^{\mathrm{2}}\text{dz},\\ E_T=L_e-P\\ =\frac{1}{2}\underset{0}{\overset{L}{{\displaystyle \int }}}‍\text{EI}{\left(\frac{{\partial }^{2}v\left(z\right)}{\partial z^2}\right)}^2\text{dz}\\ \hspace{1em}-\underset{0}{\overset{L}{{\displaystyle \int }}}‍{\mu }^2\rho A\frac{{\partial }^{2}v(z,t)}{\partial t^2}\frac{{\partial }^{2}v(z,t)}{\partial z^2}\text{dz}\end{array}$$ (31)

and let us assume the separation of variables

$$\begin{array}{c}v(z,t)=v\left(z\right)\mathrm{Cos}\left(\omega \text{t}\right)\end{array}$$ (32)

so that the energies read

$$\begin{array}{c}T=\frac{{\omega }^{\mathrm{2}}}{\mathrm{2}}\left(\underset{\mathrm{0}}{\overset{L}{{\displaystyle \int }}}\rho Av{\left(z\right)}^{\mathrm{2}}\text{dz}+\underset{{\gamma }_{\mathrm{1}}L}{\overset{{\gamma }_{\mathrm{2}}L}{{\displaystyle \int }}}\stackrel{-}{m}v{\left(z\right)}^{\mathrm{2}}\text{dz}\right){\mathrm{Cos}}^{\mathrm{2}}\left(\omega t\right),\\ E_T=L_e-P\\ =\frac{\mathrm{1}}{\mathrm{2}}\left(\underset{\mathrm{0}}{\overset{L}{{\displaystyle \int }}}\text{EI}{\left(\frac{{\partial }^{\mathrm{2}}v\left(z\right)}{\partial z^{\mathrm{2}}}\right)}^{\mathrm{2}}\text{dz}\right){\mathrm{Sin}}^{\mathrm{2}}\left(\omega t\right)\\ \hspace{1em}+{\omega }^{\mathrm{2}}\underset{\mathrm{0}}{\overset{L}{{\displaystyle \int }}}\left({\mu }^{\mathrm{2}}\rho Av\left(z\right)\left(\frac{{\partial }^{\mathrm{2}}v\left(z\right)}{\partial z^{\mathrm{2}}}\right)\text{dz}\right){\mathrm{Sin}}^{\mathrm{2}}\left(\omega t\right).\end{array}$$ (33)

The maximum kinetic energy will be equal to the maximum total potential energy, so that

$$\begin{array}{c}\frac{\mathrm{1}}{\mathrm{2}}\underset{\mathrm{0}}{\overset{L}{{\displaystyle \int }}}\text{EI}{\left(\frac{{\partial }^{\mathrm{2}}v\left(z\right)}{\partial z^{\mathrm{2}}}\right)}^{\mathrm{2}}\text{dz}+{\omega }^{\mathrm{2}}\underset{\mathrm{0}}{\overset{L}{{\displaystyle \int }}}{\mu }^{\mathrm{2}}\rho Av\left(z\right)\left(\frac{{\partial }^{\mathrm{2}}v\left(z\right)}{\partial z^{\mathrm{2}}}\right)\text{dz}\\ \hspace{1em}\hspace{1em}=\frac{{\omega }^{\mathrm{2}}}{\mathrm{2}}\underset{\mathrm{0}}{\overset{L}{{\displaystyle \int }}}\rho Av{\left(z\right)}^{\mathrm{2}}\text{dz}+\frac{{\omega }^{\mathrm{2}}}{\mathrm{2}}\underset{{\gamma }_{\mathrm{1}}L}{\overset{{\gamma }_{\mathrm{2}}L}{{\displaystyle \int }}}\stackrel{-}{m}v{\left(z\right)}^{\mathrm{2}}\text{dz}\end{array}$$ (34)

and the frequency ω ² can be deduced as

$$\begin{array}{c}{\omega }^{\mathrm{2}}=\left(\frac{\mathrm{1}}{\mathrm{2}}\underset{\mathrm{0}}{\overset{L}{{\displaystyle \int }}}\text{EI}{\left(\frac{{\partial }^{\mathrm{2}}v\left(z\right)}{\partial z^{\mathrm{2}}}\right)}^{\mathrm{2}}\text{dz}\right)\\ \hspace{1em}\times \left(\phantom{-\underset{\mathrm{0}}{\overset{L}{{\displaystyle \int }}}{\mu }^{\mathrm{2}}\rho Av\left(z\right)\frac{{\partial }^{\mathrm{2}}v\left(z\right)}{\partial z^{\mathrm{2}}}\text{dz}}\frac{\mathrm{1}}{\mathrm{2}}\underset{\mathrm{0}}{\overset{L}{{\displaystyle \int }}}\rho Av{\left(z\right)}^{\mathrm{2}}\text{dz}+\frac{\mathrm{1}}{\mathrm{2}}\underset{{\gamma }_{\mathrm{1}}L}{\overset{{\gamma }_{\mathrm{2}}L}{{\displaystyle \int }}}\stackrel{-}{m}v{\left(z\right)}^{\mathrm{2}}\text{dz}\right.\\ \hspace{1em}\hspace{1em}\hspace{0.5em}{\left.-\underset{\mathrm{0}}{\overset{L}{{\displaystyle \int }}}{\mu }^{\mathrm{2}}\rho Av\left(z\right)\frac{{\partial }^{\mathrm{2}}v\left(z\right)}{\partial z^{\mathrm{2}}}\text{dz}\right)}^{-\mathrm{1}}\end{array}$$ (35)

or, in terms of the nondimensional abscissa ζ = z/L,

$$\begin{array}{c}{\omega }^{\mathrm{2}}=\left(\frac{\mathrm{1}}{\mathrm{2}}\frac{\text{EI}}{L^{\mathrm{3}}}\underset{\mathrm{0}}{\overset{\mathrm{1}}{{\displaystyle \int }}}\frac{{\partial }^{\mathrm{2}}v\left(\zeta \right)}{\partial {\zeta }^{\mathrm{2}}}\frac{{\partial }^{\mathrm{2}}v\left(\zeta \right)}{\partial {\zeta }^{\mathrm{2}}}\text{d}\zeta \right)\\ \hspace{1em}\times \left(\phantom{{\eta }^{\mathrm{2}}\rho AL\underset{\mathrm{0}}{\overset{\mathrm{1}}{{\displaystyle \int }}}v\left(\zeta \right)\frac{{\partial }^{\mathrm{2}}v\left(\zeta \right)}{\partial {\zeta }^{\mathrm{2}}}\text{d}\zeta }\frac{\mathrm{1}}{\mathrm{2}}\rho AL\underset{\mathrm{0}}{\overset{\mathrm{1}}{{\displaystyle \int }}}{v}^{\mathrm{2}}\left(\zeta \right)\text{d}\zeta +\frac{\mathrm{1}}{\mathrm{2}}\stackrel{-}{m}L\underset{{\gamma }_{\mathrm{1}}}{\overset{{\gamma }_{\mathrm{2}}}{{\displaystyle \int }}}{v}^{\mathrm{2}}\left(\zeta \right)\text{d}\zeta \right.\\ \hspace{1em}\hspace{1em}\hspace{0.5em}{\left.-{\eta }^{\mathrm{2}}\rho AL\underset{\mathrm{0}}{\overset{\mathrm{1}}{{\displaystyle \int }}}v\left(\zeta \right)\frac{{\partial }^{\mathrm{2}}v\left(\zeta \right)}{\partial {\zeta }^{\mathrm{2}}}\text{d}\zeta \right)}^{-\mathrm{1}}.\end{array}$$ (36)

Let us assume the same approximating function equation (24), so that the following integrals can be calculated:

$$\begin{array}{c}{I}_{\mathrm{1}}=\underset{\mathrm{0}}{\overset{\mathrm{1}}{{\displaystyle \int }}}‍v^2\left(\zeta \right)\text{d}\zeta =\mathrm{1},\\ I_{\mathrm{3}}=\underset{\mathrm{0}}{\overset{\mathrm{1}}{{\displaystyle \int }}}‍\frac{{\partial }^{2}v\left(\zeta \right)}{\partial {\zeta }^2}\frac{{\partial }^{2}v\left(\zeta \right)}{\partial {\zeta }^2}\text{d}\zeta =\mathrm{12.3623};\\ I_{\mathrm{4}}=\underset{\mathrm{0}}{\overset{\mathrm{1}}{{\displaystyle \int }}}‍v\left(\zeta \right)\frac{{\partial }^{2}v\left(\zeta \right)}{\partial {\zeta }^2}\text{d}\zeta =\mathrm{0.858264},\end{array}$$ (37)

and finally if the added mass is placed between the abscissae γ ₁ L and γ ₂ L, the integral I ₂ can be obtained as

$$\begin{array}{c}{I}_2=\underset{{\gamma }_1}{\overset{{\gamma }_2}{{\displaystyle \int }}}‍v^2\left(\zeta \right)\text{d}\zeta ;\hspace{1em}0\le {\gamma }_1\le 1;\hspace{0.17em}0\le {\gamma }_2\le 1.\end{array}$$ (38)

Therefore, an alternative version of the natural frequency can be obtained as

$$\begin{array}{c}{f}_{\text{n}2}=\frac{\beta }{2\pi }\frac{C_k}{\sqrt{1+\lambda C_m-{\eta }^2{C}_{\text{n}2}}}\end{array}$$ (39)

with the three calibration constants:

$$\begin{array}{c}{C}_k=\sqrt{\frac{I_{\mathrm{3}}}{I_{\mathrm{1}}}}=\mathrm{3.51601};\\ C_{\text{n}2}=\frac{2I_4}{I_1}=1.71653;\\ C_m=\frac{I_{\mathrm{2}}}{I_{\mathrm{1}}}.\end{array}$$ (40)

3. Nonlocal Sensor Equations

The sensor equations in the presence of nonlocal elasticity can now be deduced, and the added mass $\stackrel{-}{m}$ 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:

$$\begin{array}{c}{f}_{\text{n}0}=\frac{1}{2\pi }{C}_k\beta ,\end{array}$$ (41)

and let us express the natural frequency, in the presence of the added mass, as (cf. (29) and (39))

$$\begin{array}{c}{f}_{\text{ni}}=\frac{f_{\text{n}0}}{\sqrt{\left(1+\lambda C_m-{\eta }^2{C}_{\text{ni}}\right)}},\hspace{1em}\hspace{0.17em}\text{i}=\mathrm{1,2}.\end{array}$$ (42)

The frequency shift of the biosensor can be defined as

$$\begin{array}{c}\mathrm{\Delta }\text{f}=f_{\text{n}0}-f_{\text{ni}}\end{array}$$ (43)

and finally the relative frequency shift is given by

$$\begin{array}{c}\frac{\mathrm{\Delta }\text{f}}{f_{\text{n}\mathrm{0}}}=\left(1-\frac{1}{\surd \left(1+\lambda C_m-{\eta }^2{C}_{\text{ni}}\right)}\right)\end{array}$$ (44)

from which the value of the added mass $\stackrel{-}{m}$ can be easily obtained:

$$\begin{array}{c}{\left(\frac{\mathrm{\Delta }\text{f}}{f_{\text{n}\mathrm{0}}}-1\right)}^2=\frac{1}{\left(1+\lambda C_m-{\eta }^2{C}_{\text{ni}}\right)},\end{array}$$ (45)

$$\begin{array}{c}\lambda =\frac{1}{C_m{\left(\mathrm{\Delta }\text{f}/f_{\text{n}0}-1\right)}^2}+\frac{C_{\text{ni}}{\eta }^2}{C_m}-\frac{1}{C_m}\end{array}$$ (46)

$$\begin{array}{c}\stackrel{-}{m}=\frac{\rho A}{C_m{\left(\mathrm{\Delta }\text{f}/f_{\text{n}0}-1\right)}^2}+\frac{C_{\text{ni}}{\eta }^2\rho A}{C_m}-\frac{\rho A}{C_m}\end{array}$$ (47)

and i = 1 for the first approach and i = 2 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 γ ₁ L to the free end, so that γ ₂ = 1, and γ ₁ will vary from 0.9 to 0.1. In Table 2 the fundamental natural frequency is given for various values of the γ = γ ₂ − γ ₁ = 1 − γ ₁ 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	Pa

Table 2.

The first exact natural frequency (×10¹⁰) f ₀ 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	2.70941

The fundamental natural frequency f _n1 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 f _n2 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) f _n1, 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	2.69624

Table 4.

First exact fundamental natural frequency (×10¹⁰) f _n2, 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 that γ = γ ₂ − γ ₁ = 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) f ₀ 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.

γ1	γ2	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	3.74886

Table 6.

First exact fundamental natural frequency (×10¹⁰) f ₁, 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.

γ1	γ2	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	3.74092

Table 7.

First exact fundamental natural frequency (×10¹⁰) f ₂, 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.

γ1	γ2	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 mass $\stackrel{-}{m}$ 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 $M=\stackrel{-}{m}({\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}})L$, 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 η ₂ = 1, whereas (γ ₂−γ ₁) 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 ratio M/ρAL⁡ is plotted against the relative frequency shift—as obtained using (44)—with $M=\stackrel{-}{m}\left({\gamma }_{\mathrm{2}}-{\gamma }_{\mathrm{1}}\right)L$. 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 coefficient η, so that careful calibration of this coefficient η 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:

$$\begin{array}{c}\underset{t_{\mathrm{1}}}{\overset{t_{\mathrm{2}}}{{\displaystyle \int }}}\left(\underset{\mathrm{0}}{\overset{{\gamma }_{\mathrm{1}}L}{{\displaystyle \int }}}‍\rho A\frac{\partial v_{\mathrm{1}}(z,t)}{\partial t}\delta \left(\frac{\partial v_{\mathrm{1}}(z,t)}{\partial t}\right)\text{dz}\right.\\ \hspace{1em}\hspace{1em}+\underset{{\gamma }_{\mathrm{1}}L}{\overset{{\gamma }_{\mathrm{2}}L}{{\displaystyle \int }}}\rho A\frac{\partial v_{\mathrm{2}}(z,t)}{\partial t}\delta \left(\frac{\partial v_{\mathrm{2}}(z,t)}{\partial t}\right)\text{dz}‍\\ \hspace{1em}\hspace{1em}+\underset{{\gamma }_{\mathrm{2}}L}{\overset{L}{{\displaystyle \int }}}‍\rho A\frac{\partial v_{\mathrm{3}}(z,t)}{\partial t}\delta \left(\frac{\partial v_{\mathrm{3}}(z,t)}{\partial t}\right)\text{dz}\\ \hspace{1em}\hspace{1em}+\underset{{\gamma }_{\mathrm{1}}L}{\overset{{\gamma }_{\mathrm{2}}L}{{\displaystyle \int }}}‍\stackrel{-}{m}\frac{\partial v_{\mathrm{2}}(z,t)}{\partial t}\delta \left(\frac{\partial v_{\mathrm{2}}(z,t)}{\partial t}\right)\text{dz}\\ \hspace{1em}\hspace{1em}-\underset{\mathrm{0}}{\overset{{\gamma }_{\mathrm{1}}L}{{\displaystyle \int }}}‍\left(\text{EI}\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{1}}(z,t)}{\partial z^{\mathrm{2}}}\delta \left(\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{1}}(z,t)}{\partial z^{\mathrm{2}}}\right)\right.\\ \phantom{gggggggggg}\left.-{\mu }^{\mathrm{2}}\rho A\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{1}}\left(z,t\right)}{\partial t^{\mathrm{2}}}\delta \left(\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{1}}\left(z,t\right)}{\partial z^{\mathrm{2}}}\right)\right)\text{dz}\\ \hspace{1em}\hspace{1em}-\underset{{\gamma }_{\mathrm{1}}L}{\overset{{\gamma }_{\mathrm{2}}L}{{\displaystyle \int }}}\left(\text{EI}\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{2}}\left(z,t\right)}{\partial z^{\mathrm{2}}}\delta \left(\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{2}}\left(z,t\right)}{\partial z^{\mathrm{2}}}\right)\right.\\ \left.\phantom{gggggggggg}-{\mu }^{\mathrm{2}}\rho A\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{2}}\left(z,t\right)}{\partial t^{\mathrm{2}}}\delta \left(\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{2}}\left(z,t\right)}{\partial z^{\mathrm{2}}}\right)\right)\text{dz}\\ \hspace{1em}\hspace{1em}-\underset{{\gamma }_{\mathrm{2}}L}{\overset{L}{{\displaystyle \int }}}‍\left(\text{EI}\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{3}}\left(z,t\right)}{\partial z^{\mathrm{2}}}\delta \left(\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{3}}\left(z,t\right)}{\partial z^{\mathrm{2}}}\right)\right.\\ \phantom{gggggggggg}-{\mu }^{\mathrm{2}}\rho A\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{3}}\left(z,t\right)}{\partial t^{\mathrm{2}}}\\ \left.\left.\phantom{gggggggggl}\times \delta \left(\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{3}}\left(z,t\right)}{\partial z^{\mathrm{2}}}\right)\right)\text{dz}\right)\text{dt}=\mathrm{0}.\end{array}$$ (A.1)

The first three terms of (A.1) can be treated as follows:

$$\begin{array}{c}\underset{\mathrm{0}}{\overset{{\gamma }_{\mathrm{1}}L}{{\displaystyle \int }}}\underset{t_{\mathrm{1}}}{\overset{t_{\mathrm{2}}}{{\displaystyle \int }}}\rho A\frac{\partial v_{\mathrm{1}}(z,t)}{\partial t}\delta \left(\frac{\partial v_{\mathrm{1}}(z,t)}{\partial t}\right)\text{dt}\hspace{0.17em}\text{dz}‍\\ \hspace{1em}\hspace{1em}=\underset{\mathrm{0}}{\overset{{\gamma }_{\mathrm{1}}L}{{\displaystyle \int }}}{\left[\rho A\frac{\partial v_{\mathrm{1}}\left(z,t\right)}{\partial t}\delta \left(v_{\mathrm{1}}\left(z,t\right)\right)\right]}_{t_{\mathrm{1}}}^{t_{\mathrm{2}}}\text{dz}‍\\ \hspace{1em}\hspace{1em}\hspace{1em}-\underset{\mathrm{0}}{\overset{{\gamma }_{\mathrm{1}}L}{{\displaystyle \int }}}\underset{t_{\mathrm{1}}}{\overset{t_{\mathrm{2}}}{{\displaystyle \int }}}\rho A\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{1}}\left(z,t\right)}{\partial t^{\mathrm{2}}}\delta \left(v_{\mathrm{1}}\left(z,t\right)\right)\text{dt}\hspace{0.17em}\text{dz};‍\\ \underset{{\gamma }_{\mathrm{1}}L}{\overset{{\gamma }_{\mathrm{2}}L}{{\displaystyle \int }}}\underset{t_{\mathrm{1}}}{\overset{t_{\mathrm{2}}}{{\displaystyle \int }}}\rho A\frac{\partial v_{\mathrm{2}}\left(z,t\right)}{\partial t}\delta \left(\frac{\partial v_{\mathrm{2}}\left(z,t\right)}{\partial t}\right)\text{dt}\hspace{0.17em}\text{dz}‍\\ \hspace{1em}\hspace{1em}=\underset{{\gamma }_{\mathrm{1}}L}{\overset{{\gamma }_{\mathrm{2}}L}{{\displaystyle \int }}}‍{\left[\rho A\frac{\partial v_{\mathrm{2}}(z,t)}{\partial t}\delta \left(v_{\mathrm{2}}(z,t)\right)\right]}_{t_{\mathrm{1}}}^{t_{\mathrm{2}}}\text{dz}\\ \hspace{1em}\hspace{1em}\hspace{1em}-\underset{{\gamma }_{\mathrm{1}}L}{\overset{{\gamma }_{\mathrm{2}}L}{{\displaystyle \int }}}\underset{t_{\mathrm{1}}}{\overset{t_{\mathrm{2}}}{{\displaystyle \int }}}\rho A\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{2}}(z,t)}{\partial t^{\mathrm{2}}}\delta \left(v_{\mathrm{2}}(z,t)\right)\text{dt}\hspace{0.17em}\text{dz};‍\\ \underset{{\gamma }_{\mathrm{2}}L}{\overset{L}{{\displaystyle \int }}}\underset{t_{\mathrm{1}}}{\overset{t_{\mathrm{2}}}{{\displaystyle \int }}}\rho A\frac{\partial v_{\mathrm{3}}\left(z,t\right)}{\partial t}\delta \left(\frac{\partial v_{\mathrm{3}}\left(z,t\right)}{\partial t}\right)\text{dt}\hspace{0.17em}\text{dz}‍\\ \hspace{1em}\hspace{1em}=\underset{{\gamma }_{\mathrm{2}}L}{\overset{L}{{\displaystyle \int }}}{\left[\rho A\frac{\partial v_{\mathrm{3}}(z,t)}{\partial t}\delta \left(v_{\mathrm{3}}\left(z,t\right)\right)\right]}_{t_{\mathrm{1}}}^{t_{\mathrm{2}}}\text{dz}‍\\ \hspace{1em}\hspace{1em}\hspace{1em}-\underset{{\gamma }_{\mathrm{2}}L}{\overset{L}{{\displaystyle \int }}}\underset{t_{\mathrm{1}}}{\overset{t_{\mathrm{2}}}{{\displaystyle \int }}}\rho A\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{3}}\left(z,t\right)}{\partial t^{\mathrm{2}}}\delta \left(v_{\mathrm{3}}\left(z,t\right)\right)\text{dt}\hspace{0.17em}\text{dz},‍\end{array}$$ (A.2)

where we considered the fact that δ(v _i(z, t)) = 0 at t = t ₁ and t = t ₂.

Quite similarly, the single term of the distributed mass becomes

$$\begin{array}{c}\underset{{\gamma }_{\mathrm{1}}L}{\overset{{\gamma }_{\mathrm{2}}L}{{\displaystyle \int }}}\underset{t_{\mathrm{1}}}{\overset{t_{\mathrm{2}}}{{\displaystyle \int }}}\stackrel{-}{m}\frac{\partial v_{\mathrm{2}}\left(z,t\right)}{\partial t}\delta \left(\frac{\partial v_{\mathrm{2}}\left(z,t\right)}{\partial t}\right)\text{dt}\hspace{0.17em}\text{dz}‍\\ \hspace{1em}\hspace{1em}=\underset{{\gamma }_{\mathrm{1}}L}{\overset{{\gamma }_{\mathrm{2}}L}{{\displaystyle \int }}}{\left[\stackrel{-}{m}\frac{\partial v_{\mathrm{2}}\left(z,t\right)}{\partial t}\delta \left(v_{\mathrm{2}}\left(z,t\right)\right)\right]}_{t_{\mathrm{1}}}^{t_{\mathrm{2}}}\text{dz}‍\\ \hspace{1em}\hspace{1em}\hspace{1em}-\underset{{\gamma }_{\mathrm{1}}L}{\overset{{\gamma }_{\mathrm{2}}L}{{\displaystyle \int }}}\underset{t_{\mathrm{1}}}{\overset{t_{\mathrm{2}}}{{\displaystyle \int }}}\stackrel{-}{m}\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{2}}\left(z,t\right)}{\partial t^{\mathrm{2}}}\delta \left(v_{\mathrm{2}}\left(z,t\right)\right)\text{dt}\hspace{0.17em}\text{dz}.‍\end{array}$$ (A.3)

The nonlocal effects are contained into three integrals, which can be integrated as follows:

$$\begin{array}{c}\underset{t_{\mathrm{1}}}{\overset{t_{\mathrm{2}}}{{\displaystyle \int }}}\underset{\mathrm{0}}{\overset{{\gamma }_{\mathrm{1}}L}{{\displaystyle \int }}}{\mu }^{\mathrm{2}}\rho A\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{1}}\left(z,t\right)}{\partial t^{\mathrm{2}}}\delta \left(\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{1}}\left(z,t\right)}{\partial z^{\mathrm{2}}}\right)\text{dz}\hspace{0.17em}\text{dt}‍\\ \hspace{1em}\hspace{0.5em}=\underset{t_{\mathrm{1}}}{\overset{t_{\mathrm{2}}}{{\displaystyle \int }}}{\left[{\mu }^{\mathrm{2}}\rho A\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{1}}(z,t)}{\partial t^{\mathrm{2}}}\delta \left(\frac{\partial v_{\mathrm{1}}(z,t)}{\partial z}\right)\right]}_{\mathrm{0}}^{{\gamma }_{\mathrm{1}}L}\text{dt}‍\\ \hspace{1em}\hspace{1em}\hspace{0.5em}-\underset{t_{\mathrm{1}}}{\overset{t_{\mathrm{2}}}{{\displaystyle \int }}}\underset{\mathrm{0}}{\overset{{\gamma }_{\mathrm{1}}L}{{\displaystyle \int }}}{\mu }^{\mathrm{2}}\rho A\frac{{\partial }^{\mathrm{3}}{v}_{\mathrm{1}}(z,t)}{\partial t^{\mathrm{2}}\partial z}\delta \left(\frac{\partial v_{\mathrm{1}}(z,t)}{\partial z}\right)\text{dz}\hspace{0.17em}\text{dt}‍\\ \hspace{1em}\hspace{0.5em}=\underset{t_{\mathrm{1}}}{\overset{t_{\mathrm{2}}}{{\displaystyle \int }}}{\left[{\mu }^{\mathrm{2}}\rho A\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{1}}(z,t)}{\partial t^{\mathrm{2}}}\delta \left(\frac{\partial v_{\mathrm{1}}(z,t)}{\partial z}\right)\right]}_{\mathrm{0}}^{{\gamma }_{\mathrm{1}}L}\text{dt}‍\\ \hspace{1em}\hspace{1em}\hspace{0.5em}-\underset{t_{\mathrm{1}}}{\overset{t_{\mathrm{2}}}{{\displaystyle \int }}}{\left[{\mu }^{\mathrm{2}}\rho A\frac{{\partial }^{\mathrm{3}}{v}_{\mathrm{1}}\left(z,t\right)}{\partial t^{\mathrm{2}}\partial z}\delta \left(v_{\mathrm{1}}\left(z,t\right)\right)\right]}_{\mathrm{0}}^{{\gamma }_{\mathrm{1}}L}\text{dt}‍\\ \hspace{1em}\hspace{1em}\hspace{0.5em}+\underset{t_{\mathrm{1}}}{\overset{t_{\mathrm{2}}}{{\displaystyle \int }}}\underset{\mathrm{0}}{\overset{{\gamma }_{\mathrm{1}}L}{{\displaystyle \int }}}{\mu }^{\mathrm{2}}\rho A\frac{{\partial }^{\mathrm{4}}{v}_{\mathrm{1}}(z,t)}{\partial t^{\mathrm{2}}\partial z^{\mathrm{2}}}\delta \left(v_{\mathrm{1}}\left(z,t\right)\right)\text{dz}\hspace{0.17em}\text{dt};‍\\ \underset{t_{\mathrm{1}}}{\overset{t_{\mathrm{2}}}{{\displaystyle \int }}}\underset{{\gamma }_{\mathrm{1}}L}{\overset{{\gamma }_{\mathrm{2}}L}{{\displaystyle \int }}}‍{\mu }^{\mathrm{2}}\rho A\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{2}}(z,t)}{\partial t^{\mathrm{2}}}\delta \left(\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{2}}(z,t)}{\partial z^{\mathrm{2}}}\right)\text{dz}\hspace{0.17em}\text{dt}\\ \hspace{1em}\hspace{0.5em}=\underset{t_{\mathrm{1}}}{\overset{t_{\mathrm{2}}}{{\displaystyle \int }}}{\left[{\mu }^{\mathrm{2}}\rho A\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{2}}(z,t)}{\partial t^{\mathrm{2}}}\delta \left(\frac{\partial v_{\mathrm{2}}(z,t)}{\partial z}\right)\right]}_{{\gamma }_{\mathrm{1}}L}^{{\gamma }_{\mathrm{2}}L}\text{dt}‍\\ \hspace{1em}\hspace{1em}\hspace{0.5em}-\underset{t_{\mathrm{1}}}{\overset{t_{\mathrm{2}}}{{\displaystyle \int }}}\underset{{\gamma }_{\mathrm{1}}L}{\overset{{\gamma }_{\mathrm{2}}L}{{\displaystyle \int }}}{\mu }^{\mathrm{2}}\rho A\frac{{\partial }^{\mathrm{3}}{v}_{\mathrm{2}}(z,t)}{\partial t^{\mathrm{2}}\partial z}\delta \left(\frac{\partial v_{\mathrm{2}}(z,t)}{\partial z}\right)\text{dz}\hspace{0.17em}\text{dt}‍\\ \hspace{1em}\hspace{0.5em}=\underset{t_{\mathrm{1}}}{\overset{t_{\mathrm{2}}}{{\displaystyle \int }}}{\left[{\mu }^{\mathrm{2}}\rho A\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{2}}(z,t)}{\partial t^{\mathrm{2}}}\delta \left(\frac{\partial v_{\mathrm{2}}(z,t)}{\partial z}\right)\right]}_{{\gamma }_{\mathrm{1}}L}^{{\gamma }_{\mathrm{2}}L}\text{dt}‍\\ \hspace{1em}\hspace{1em}\hspace{0.5em}-\underset{t_{\mathrm{1}}}{\overset{t_{\mathrm{2}}}{{\displaystyle \int }}}{\left[{\mu }^{\mathrm{2}}\rho A\frac{{\partial }^{\mathrm{3}}{v}_{\mathrm{2}}(z,t)}{\partial t^{\mathrm{2}}\partial z}\delta \left(v_{\mathrm{2}}(z,t)\right)\right]}_{{\gamma }_{\mathrm{1}}L}^{{\gamma }_{\mathrm{2}}L}\text{dt}‍\\ \hspace{1em}\hspace{1em}\hspace{0.5em}+\underset{t_{\mathrm{1}}}{\overset{t_{\mathrm{2}}}{{\displaystyle \int }}}\underset{{\gamma }_{\mathrm{1}}L}{\overset{{\gamma }_{\mathrm{2}}L}{{\displaystyle \int }}}{\mu }^{\mathrm{2}}\rho A\frac{{\partial }^{\mathrm{4}}{v}_{\mathrm{2}}(z,t)}{\partial t^{\mathrm{2}}\partial z^{\mathrm{2}}}\delta \left(v_{\mathrm{2}}\left(z,t\right)\right)\text{dz}\hspace{0.17em}\text{dt};‍\\ \underset{t_{\mathrm{1}}}{\overset{t_{\mathrm{2}}}{{\displaystyle \int }}}\underset{{\gamma }_{\mathrm{2}}L}{\overset{L}{{\displaystyle \int }}}{\mu }^{\mathrm{2}}\rho A\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{3}}(z,t)}{\partial t^{\mathrm{2}}}\delta \left(\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{3}}(z,t)}{\partial z^{\mathrm{2}}}\right)\text{dz}\hspace{0.17em}\text{dt}‍\\ \hspace{1em}\hspace{0.5em}=\underset{t_{\mathrm{1}}}{\overset{t_{\mathrm{2}}}{{\displaystyle \int }}}{\left[{\mu }^{\mathrm{2}}\rho A\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{3}}(z,t)}{\partial t^{\mathrm{2}}}\delta \left(\frac{\partial v_{\mathrm{3}}(z,t)}{\partial z}\right)\right]}_{{\gamma }_{\mathrm{2}}L}^L\text{dt}‍\\ \hspace{1em}\hspace{1em}\hspace{0.5em}-\underset{t_{\mathrm{1}}}{\overset{t_{\mathrm{2}}}{{\displaystyle \int }}}\underset{{\gamma }_{\mathrm{2}}L}{\overset{L}{{\displaystyle \int }}}{\mu }^{\mathrm{2}}\rho A\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{3}}(z,t)}{\partial t^{\mathrm{2}}}\delta \left(\frac{\partial v_{\mathrm{3}}(z,t)}{\partial z}\right)\text{dz}\hspace{0.17em}\text{dt}‍\\ \hspace{1em}\hspace{0.5em}=\underset{t_{\mathrm{1}}}{\overset{t_{\mathrm{2}}}{{\displaystyle \int }}}{\left[{\mu }^{\mathrm{2}}\rho A\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{3}}(z,t)}{\partial t^{\mathrm{2}}}\delta \left(\frac{\partial v_{\mathrm{3}}(z,t)}{\partial z}\right)\right]}_{{\gamma }_{\mathrm{2}}L}^L\text{dt}‍\\ \hspace{1em}\hspace{1em}\hspace{0.5em}-\underset{t_{\mathrm{1}}}{\overset{t_{\mathrm{2}}}{{\displaystyle \int }}}{\left[{\mu }^{\mathrm{2}}\rho A\frac{{\partial }^{\mathrm{3}}{v}_{\mathrm{3}}(z,t)}{\partial t^{\mathrm{2}}\partial z}\delta \left(v_{\mathrm{3}}(z,t)\right)\right]}_{{\gamma }_{\mathrm{2}}L}^L\text{dt}‍\\ \hspace{1em}\hspace{1em}\hspace{0.5em}+\underset{t_{\mathrm{1}}}{\overset{t_{\mathrm{2}}}{{\displaystyle \int }}}\underset{{\gamma }_{\mathrm{2}}L}{\overset{L}{{\displaystyle \int }}}{\mu }^{\mathrm{2}}\rho A\frac{{\partial }^{\mathrm{4}}{v}_{\mathrm{3}}(z,t)}{\partial t^{\mathrm{2}}\partial z^{\mathrm{2}}}\delta \left(v_{\mathrm{3}}(z,t)\right)\text{dz}\hspace{0.17em}\text{dt}.‍\end{array}$$ (A.4)

Finally, the first variation of the strain energy can be treated as usual:

$$\begin{array}{c}-\underset{t_{\mathrm{1}}}{\overset{t_{\mathrm{2}}}{{\displaystyle \int }}}\underset{\mathrm{0}}{\overset{{\gamma }_{\mathrm{1}}L}{{\displaystyle \int }}}\text{EI}\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{1}}\left(z,t\right)}{\partial z^{\mathrm{2}}}\delta \left(\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{1}}\left(z,t\right)}{\partial z^{\mathrm{2}}}\right)\text{dz}\hspace{0.17em}\text{dt}‍\\ \hspace{1em}\hspace{0.5em}=-\underset{t_{\mathrm{1}}}{\overset{t_{\mathrm{2}}}{{\displaystyle \int }}}{\left[\text{EI}\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{1}}(z,t)}{\partial z^{\mathrm{2}}}\delta \left(\frac{\partial v_{\mathrm{1}}(z,t)}{\partial z}\right)\right]}_{\mathrm{0}}^{{\gamma }_{\mathrm{1}}L}\text{dt}‍\\ \hspace{1em}\hspace{1em}\hspace{0.5em}+\underset{t_{\mathrm{1}}}{\overset{t_{\mathrm{2}}}{{\displaystyle \int }}}\underset{\mathrm{0}}{\overset{{\gamma }_{\mathrm{1}}L}{{\displaystyle \int }}}\text{EI}\frac{{\partial }^{\mathrm{3}}{v}_{\mathrm{1}}(z,t)}{\partial z^{\mathrm{3}}}\delta \left(\frac{\partial v_{\mathrm{1}}(z,t)}{\partial z}\right)\text{dz}\hspace{0.17em}\text{dt}‍\\ \hspace{1em}\hspace{0.5em}=-\underset{t_{\mathrm{1}}}{\overset{t_{\mathrm{2}}}{{\displaystyle \int }}}{\left[\text{EI}\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{1}}(z,t)}{\partial z^{\mathrm{2}}}\delta \left(\frac{\partial v_{\mathrm{1}}(z,t)}{\partial z}\right)\right]}_{\mathrm{0}}^{{\gamma }_{\mathrm{1}}L}\text{dt}‍\\ \hspace{1em}\hspace{1em}\hspace{0.5em}+\underset{t_{\mathrm{1}}}{\overset{t_{\mathrm{2}}}{{\displaystyle \int }}}{\left[\text{EI}\frac{{\partial }^{\mathrm{3}}{v}_{\mathrm{1}}(z,t)}{\partial z^{\mathrm{3}}}\delta \left(v_{\mathrm{1}}\left(z,t\right)\right)\right]}_{\mathrm{0}}^{{\gamma }_{\mathrm{1}}L}\text{dt}‍\\ \hspace{1em}\hspace{1em}\hspace{0.5em}-\underset{t_{\mathrm{1}}}{\overset{t_{\mathrm{2}}}{{\displaystyle \int }}}\underset{\mathrm{0}}{\overset{{\gamma }_{\mathrm{1}}L}{{\displaystyle \int }}}\text{EI}\frac{{\partial }^{\mathrm{4}}{v}_{\mathrm{1}}(z,t)}{\partial z^{\mathrm{4}}}\delta \left(v_{\mathrm{1}}\left(z,t\right)\right)\text{dz}\hspace{0.17em}\text{dt};‍\\ -\underset{t_{\mathrm{1}}}{\overset{t_{\mathrm{2}}}{{\displaystyle \int }}}‍\underset{{\gamma }_{\mathrm{1}}L}{\overset{{\gamma }_{\mathrm{2}}L}{{\displaystyle \int }}}‍\text{EI}\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{2}}(z,t)}{\partial z^{\mathrm{2}}}\delta \left(\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{2}}(z,t)}{\partial z^{\mathrm{2}}}\right)\text{dz}\hspace{0.17em}\text{dt}\\ \hspace{1em}\hspace{0.5em}=-\underset{t_{\mathrm{1}}}{\overset{t_{\mathrm{2}}}{{\displaystyle \int }}}‍{\left[\text{EI}\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{2}}(z,t)}{\partial z^{\mathrm{2}}}\delta \left(\frac{\partial v_{\mathrm{2}}(z,t)}{\partial z}\right)\right]}_{{\gamma }_{\mathrm{1}}L}^{{\gamma }_{\mathrm{2}}L}\text{dt}\\ \hspace{1em}\hspace{1em}\hspace{0.5em}+\underset{t_{\mathrm{1}}}{\overset{t_{\mathrm{2}}}{{\displaystyle \int }}}\underset{{\gamma }_{\mathrm{1}}L}{\overset{{\gamma }_{\mathrm{2}}L}{{\displaystyle \int }}}‍\text{EI}\frac{{\partial }^{\mathrm{3}}{v}_{\mathrm{2}}(z,t)}{\partial z^{\mathrm{3}}}\delta \left(\frac{\partial v_{\mathrm{2}}(z,t)}{\partial z}\right)\text{dz}\hspace{0.17em}\text{dt}\\ \hspace{1em}\hspace{0.5em}=-\underset{t_{\mathrm{1}}}{\overset{t_{\mathrm{2}}}{{\displaystyle \int }}}{\left[\text{EI}\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{2}}(z,t)}{\partial z^{\mathrm{2}}}\delta \left(\frac{\partial v_{\mathrm{2}}(z,t)}{\partial z}\right)\right]}_{{\gamma }_{\mathrm{1}}L}^{{\gamma }_{\mathrm{2}}L}\text{dt}‍\\ \hspace{1em}\hspace{1em}\hspace{0.5em}+\underset{t_{\mathrm{1}}}{\overset{t_{\mathrm{2}}}{{\displaystyle \int }}}{\left[\text{EI}\frac{{\partial }^{\mathrm{3}}{v}_{\mathrm{2}}(z,t)}{\partial z^{\mathrm{3}}}\delta \left(v_{\mathrm{2}}\left(z,t\right)\right)\right]}_{{\gamma }_{\mathrm{1}}L}^{{\gamma }_{\mathrm{2}}L}\text{dt}‍\\ \hspace{1em}\hspace{1em}\hspace{0.5em}-\underset{t_{\mathrm{1}}}{\overset{t_{\mathrm{2}}}{{\displaystyle \int }}}\underset{{\gamma }_{\mathrm{1}}L}{\overset{{\gamma }_{\mathrm{2}}L}{{\displaystyle \int }}}\text{EI}\frac{{\partial }^{\mathrm{4}}{v}_{\mathrm{2}}(z,t)}{\partial z^{\mathrm{4}}}\delta \left(v_{\mathrm{2}}\left(z,t\right)\right)\text{dz}\hspace{0.17em}\text{dt};‍\\ -\underset{t_{\mathrm{1}}}{\overset{t_{\mathrm{2}}}{{\displaystyle \int }}}\underset{{\gamma }_{\mathrm{2}}L}{\overset{L}{{\displaystyle \int }}}\text{EI}\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{3}}(z,t)}{\partial z^{\mathrm{2}}}\delta \left(\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{3}}(z,t)}{\partial z^{\mathrm{2}}}\right)\text{dz}\hspace{0.17em}\text{dt}‍\\ \hspace{1em}\hspace{0.5em}=-\underset{t_{\mathrm{1}}}{\overset{t_{\mathrm{2}}}{{\displaystyle \int }}}{\left[\text{EI}\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{3}}(z,t)}{\partial z^{\mathrm{2}}}\delta \left(\frac{\partial v_{\mathrm{3}}(z,t)}{\partial z}\right)\right]}_{{\gamma }_{\mathrm{2}}L}^L\text{dt}‍\\ \hspace{1em}\hspace{1em}\hspace{0.5em}+\underset{t_{\mathrm{1}}}{\overset{t_{\mathrm{2}}}{{\displaystyle \int }}}\underset{{\gamma }_{\mathrm{2}}L}{\overset{L}{{\displaystyle \int }}}\text{EI}\frac{{\partial }^{\mathrm{3}}{v}_{\mathrm{3}}(z,t)}{\partial z^{\mathrm{3}}}\delta \left(\frac{\partial v_{\mathrm{3}}(z,t)}{\partial z}\right)\text{dz}\hspace{0.17em}\text{dt}‍\\ \hspace{1em}\hspace{0.5em}=-\underset{t_{\mathrm{1}}}{\overset{t_{\mathrm{2}}}{{\displaystyle \int }}}{\left[\text{EI}\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{3}}(z,t)}{\partial z^{\mathrm{2}}}\delta \left(\frac{\partial v_{\mathrm{3}}(z,t)}{\partial z}\right)\right]}_{{\gamma }_{\mathrm{2}}L}^L\text{dt}‍\\ \hspace{1em}\hspace{1em}\hspace{0.5em}+\underset{t_{\mathrm{1}}}{\overset{t_{\mathrm{2}}}{{\displaystyle \int }}}{\left[\text{EI}\frac{{\partial }^{\mathrm{3}}{v}_{\mathrm{3}}(z,t)}{\partial z^{\mathrm{3}}}\delta \left(v_{\mathrm{3}}\left(z,t\right)\right)\right]}_{{\gamma }_{\mathrm{2}}L}^L\text{dt}‍\\ \hspace{1em}\hspace{1em}\hspace{0.5em}-\mathrm{\hspace{0.17em}\hspace{0.17em}}\underset{t_{\mathrm{1}}}{\overset{t_{\mathrm{2}}}{{\displaystyle \int }}}\underset{{\gamma }_{\mathrm{2}}L}{\overset{L}{{\displaystyle \int }}}\text{EI}\frac{{\partial }^{\mathrm{4}}{v}_{\mathrm{3}}(z,t)}{\partial z^{\mathrm{4}}}\delta \left(v_{\mathrm{3}}\left(z,t\right)\right)\text{dz}\hspace{0.17em}\text{dt}‍.\end{array}$$ (A.5)

All the previous integrated terms can be collected together, leading to

$$\begin{array}{c}\underset{t_{\mathrm{1}}}{\overset{t_{\mathrm{2}}}{{\displaystyle \int }}}\left(-\underset{\mathrm{0}}{\overset{{\gamma }_{\mathrm{1}}L}{{\displaystyle \int }}}\rho A\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{1}}\left(z,t\right)}{\partial t^{\mathrm{2}}}\delta \left(v_{\mathrm{1}}\left(z,t\right)\right)\text{dz}‍\right.‍\\ \hspace{1em}\hspace{1em}\hspace{0.5em}-\underset{\mathrm{0}}{\overset{{\gamma }_{\mathrm{1}}L}{{\displaystyle \int }}}\text{EI}\frac{{\partial }^{\mathrm{4}}{v}_{\mathrm{1}}\left(z,t\right)}{\partial z^{\mathrm{4}}}\delta \left(v_{\mathrm{1}}\left(z,t\right)\right)\text{dz}‍\\ \hspace{1em}\hspace{1em}\hspace{0.5em}+\underset{\mathrm{0}}{\overset{{\gamma }_{\mathrm{1}}L}{{\displaystyle \int }}}{\mu }^{\mathrm{2}}\rho A\frac{{\partial }^{\mathrm{4}}{v}_{\mathrm{1}}(z,t)}{\partial t^{\mathrm{2}}\partial z^{\mathrm{2}}}\delta \left(v_{\mathrm{1}}\left(z,t\right)\right)\text{dz}‍\\ \hspace{1em}\hspace{1em}\hspace{0.5em}-\underset{{\gamma }_{\mathrm{1}}L}{\overset{{\gamma }_{\mathrm{2}}L}{{\displaystyle \int }}}\rho A\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{2}}\left(z,t\right)}{\partial t^{\mathrm{2}}}\delta \left(v_{\mathrm{2}}\left(z,t\right)\right)\text{dz}‍\\ \hspace{1em}\hspace{1em}\hspace{0.5em}-\underset{{\gamma }_{\mathrm{1}}L}{\overset{{\gamma }_{\mathrm{2}}L}{{\displaystyle \int }}}\text{EI}\frac{{\partial }^{\mathrm{4}}{v}_{\mathrm{2}}\left(z,t\right)}{\mathrm{4}}\delta \left(v_{\mathrm{2}}\left(z,t\right)\right)\text{dz}‍\\ \hspace{1em}\hspace{1em}\hspace{0.5em}+\underset{{\gamma }_{\mathrm{1}}L}{\overset{{\gamma }_{\mathrm{2}}L}{{\displaystyle \int }}}{\mu }^{\mathrm{2}}\rho A‍\frac{{\partial }^{\mathrm{4}}{v}_{\mathrm{2}}\left(z,t\right)}{\partial t^{\mathrm{2}}\partial z^{\mathrm{2}}}\delta \left(v_{\mathrm{2}}\left(z,t\right)\right)\text{dz}\\ \hspace{1em}\hspace{1em}\hspace{0.5em}-\underset{{\gamma }_{\mathrm{2}}L}{\overset{L}{{\displaystyle \int }}}\rho A\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{3}}\left(z,t\right)}{\partial t^{\mathrm{2}}}\delta \left(v_{\mathrm{3}}\left(z,t\right)\right)\text{dz}‍\\ \hspace{1em}\hspace{1em}\hspace{0.5em}-\underset{{\gamma }_{\mathrm{2}}L}{\overset{L}{{\displaystyle \int }}}‍\text{EI}\frac{{\partial }^{\mathrm{4}}{v}_{\mathrm{3}}\left(z,t\right)}{\partial z^{\mathrm{4}}}\delta \left(v_{\mathrm{3}}\left(z,t\right)\right)\text{dz}\\ \hspace{1em}\hspace{1em}\hspace{0.5em}+\underset{{\gamma }_{\mathrm{2}}L}{\overset{L}{{\displaystyle \int }}}{\mu }^{\mathrm{2}}\rho A\frac{{\partial }^{\mathrm{4}}{v}_{\mathrm{3}}\left(z,t\right)}{\partial t^{\mathrm{2}}\partial z^{\mathrm{2}}}\delta \left(v_{\mathrm{3}}\left(z,t\right)\right)\text{dz}‍\\ \hspace{1em}\hspace{1em}\hspace{0.5em}\left.-\underset{{\gamma }_{\mathrm{1}}L}{\overset{{\gamma }_{\mathrm{2}}L}{{\displaystyle \int }}}\stackrel{-}{m}\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{2}}(z,t)}{\partial t^{\mathrm{2}}}\delta \left(v_{\mathrm{2}}\left(z,t\right)\right)\text{dz}‍\right)\text{dt}=\mathrm{0};\end{array}$$ (A.6)

$$\begin{array}{c}\underset{t_{\mathrm{1}}}{\overset{t_{\mathrm{2}}}{{\displaystyle \int }}}\left({\left[{\mu }^{\mathrm{2}}\rho A\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{1}}(z,t)}{\partial t^{\mathrm{2}}}\delta \left(\frac{\partial v_{\mathrm{1}}(z,t)}{\partial z}\right)\right]}_{\mathrm{0}}^{{\gamma }_{\mathrm{1}}L}\right.‍\\ \hspace{1em}\hspace{1em}\hspace{0.5em}-{\left[{\mu }^{\mathrm{2}}\rho A\frac{{\partial }^{\mathrm{3}}{v}_{\mathrm{1}}(z,t)}{\partial t^{\mathrm{2}}\partial z}\delta \left(v_{\mathrm{1}}(z,t)\right)\right]}_{\mathrm{0}}^{{\gamma }_{\mathrm{1}}L}\\ \hspace{1em}\hspace{1em}\hspace{0.5em}-{\left[\text{EI}\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{1}}(z,t)}{\partial z^{\mathrm{2}}}\delta \left(\frac{\partial v_{\mathrm{1}}(z,t)}{\partial z}\right)\right]}_{\mathrm{0}}^{{\gamma }_{\mathrm{1}}L}\\ \hspace{1em}\hspace{1em}\hspace{0.5em}+{\left[\text{EI}\frac{{\partial }^{\mathrm{3}}{v}_{\mathrm{1}}(z,t)}{\partial z^{\mathrm{3}}}\delta \left(v_{\mathrm{1}}(z,t)\right)\right]}_{\mathrm{0}}^{{\gamma }_{\mathrm{1}}L}\\ \hspace{1em}\hspace{1em}\hspace{0.5em}+{\left[{\mu }^{\mathrm{2}}\rho A\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{2}}(z,t)}{\partial t^{\mathrm{2}}}\delta \left(\frac{\partial v_{\mathrm{2}}(z,t)}{\partial z}\right)\right]}_{{\gamma }_{\mathrm{1}}L}^{{\gamma }_{\mathrm{2}}L}\\ \hspace{1em}\hspace{1em}\hspace{0.5em}-{\left[{\mu }^{\mathrm{2}}\rho A\frac{{\partial }^{\mathrm{3}}{v}_{\mathrm{2}}(z,t)}{\partial t^{\mathrm{2}}\partial z}\delta \left(v_{\mathrm{2}}(z,t)\right)\right]}_{{\gamma }_{\mathrm{1}}L}^{{\gamma }_{\mathrm{2}}L}\\ \hspace{1em}\hspace{1em}\hspace{0.5em}-{\left[\text{EI}\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{2}}(z,t)}{\partial z^{\mathrm{2}}}\delta \left(\frac{\partial v_{\mathrm{2}}(z,t)}{\partial z}\right)\right]}_{{\gamma }_{\mathrm{1}}L}^{{\gamma }_{\mathrm{2}}L}\\ \hspace{1em}\hspace{1em}\hspace{0.5em}+{\left[\text{EI}\frac{{\partial }^{\mathrm{3}}{v}_{\mathrm{2}}(z,t)}{\partial z^{\mathrm{3}}}\delta \left(v_{\mathrm{2}}(z,t)\right)\right]}_{{\gamma }_{\mathrm{1}}L}^{{\gamma }_{\mathrm{2}}L}\\ \hspace{1em}\hspace{1em}\hspace{0.5em}+{\left[{\mu }^{\mathrm{2}}\rho A\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{3}}(z,t)}{\partial t^{\mathrm{2}}}\delta \left(\frac{\partial v_{\mathrm{3}}(z,t)}{\partial z}\right)\right]}_{{\gamma }_{\mathrm{2}}L}^L\\ \hspace{1em}\hspace{1em}\hspace{0.5em}-{\left[{\mu }^{\mathrm{2}}\rho A\frac{{\partial }^{\mathrm{3}}{v}_{\mathrm{3}}(z,t)}{\partial t^{\mathrm{2}}\partial z}\delta \left(v_{\mathrm{3}}(z,t)\right)\right]}_{{\gamma }_{\mathrm{2}}L}^L\\ \hspace{1em}\hspace{1em}\hspace{0.5em}-{\left[\text{EI}\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{3}}(z,t)}{\partial z^{\mathrm{2}}}\delta \left(\frac{\partial v_{\mathrm{3}}(z,t)}{\partial z}\right)\right]}_{{\gamma }_{\mathrm{2}}L}^L\\ \hspace{1em}\hspace{1em}\hspace{0.5em}\left.+{\left[\text{EI}\frac{{\partial }^{\mathrm{3}}{v}_{\mathrm{3}}(z,t)}{\partial z^{\mathrm{3}}}\delta \left(v_{\mathrm{3}}(z,t)\right)\right]}_{{\gamma }_{\mathrm{2}}L}^L\right)\text{dt}=\mathrm{0}.\end{array}$$ (A.7)

Finally, from (A.6) the following equations can be deduced (cf. (4)):

$$\begin{array}{c}\text{EI}\frac{{\partial }^{\mathrm{4}}{v}_{\mathrm{1}}(z,t)}{\partial z^{\mathrm{4}}}-{\mu }^{\mathrm{2}}\rho A\frac{{\partial }^{\mathrm{4}}{v}_{\mathrm{1}}(z,t)}{\partial z^{\mathrm{2}}\partial t^{\mathrm{2}}}+\rho A\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{1}}(z,t)}{\partial t^{\mathrm{2}}}=\mathrm{0},\\ \phantom{\text{ggggggggggggggggggggggggggggggg}\mathrm{1}}\mathrm{0}<z<{\gamma }_{\mathrm{1}}L,\\ \text{EI}\frac{{\partial }^{\mathrm{4}}{v}_{\mathrm{2}}(z,t)}{\partial z^{\mathrm{4}}}-{\mu }^{\mathrm{2}}\rho A\frac{{\partial }^{\mathrm{4}}{v}_{\mathrm{2}}(z,t)}{\partial z^{\mathrm{2}}\partial t^{\mathrm{2}}}+\left(\rho A+\stackrel{-}{m}\right)\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{2}}(z,t)}{\partial t^{\mathrm{2}}}=\mathrm{0},\\ \phantom{\text{gggggggggggggggggggggggggggggggggggg}}{\gamma }_{\mathrm{1}}L<z<{\gamma }_{\mathrm{2}}L,\\ \text{EI}\frac{{\partial }^{\mathrm{4}}{v}_{\mathrm{3}}(z,t)}{\partial z^{\mathrm{4}}}-{\mu }^{\mathrm{2}}\rho A\frac{{\partial }^{\mathrm{4}}{v}_{\mathrm{3}}(z,t)}{\partial z^{\mathrm{2}}\partial t^{\mathrm{2}}}+\rho A\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{3}}(z,t)}{\partial t^{\mathrm{2}}}=\mathrm{0},\\ \phantom{ggggggggggggggggggggggggg1gg}{\gamma }_{\mathrm{2}}L<z<L,\end{array}$$ (A.8)

whereas the boundary conditions can be deduced from (A.7),

$$\begin{array}{c}{v}_{\mathrm{1}}\left(\mathrm{0},t\right)=\mathrm{0},\\ \frac{\partial v_{\mathrm{1}}\left(\mathrm{0},t\right)}{\partial z}=\mathrm{0},\end{array}$$ (A.9)

for z = 0 and for z = L,

$$\begin{array}{c}\text{EI}\frac{{\partial }^{\mathrm{3}}{v}_{\mathrm{3}}(L,t)}{\partial z^{\mathrm{3}}}-{\mu }^{\mathrm{2}}\rho A\frac{{\partial }^{\mathrm{3}}{v}_{\mathrm{3}}\left(L,t\right)}{\partial t^{\mathrm{2}}\partial z}=\mathrm{0},\\ -\text{EI}\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{3}}\left(L,t\right)}{\partial z^{\mathrm{2}}}+{\mu }^{\mathrm{2}}\rho A\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{3}}\left(L,t\right)}{\partial t^{\mathrm{2}}}=\mathrm{0}.\end{array}$$ (A.10)

The boundary conditions for z = γ ₁ L are

$$\begin{array}{c}{v}_{\mathrm{1}}\left({\gamma }_{\mathrm{1}}L,t\right)=v_{\mathrm{2}}\left({\gamma }_{\mathrm{1}}L,t\right),\\ \frac{\partial v_{\mathrm{1}}\left({\gamma }_{\mathrm{1}}L,t\right)}{\partial z}=\frac{\partial v_{\mathrm{2}}\left({\gamma }_{\mathrm{1}}L,t\right)}{\partial z},\\ {\mu }^2\rho A\frac{{\partial }^3{v}_1\left({\gamma }_{1}L,t\right)}{\partial t^2\partial z}-\text{EI}\frac{{\partial }^3{v}_1\left({\gamma }_{1}L,t\right)}{\partial z^3}-{\mu }^2\rho A\frac{{\partial }^3{v}_2\left({\gamma }_{1}L,t\right)}{\partial t^2\partial z}\\ \hspace{1em}\hspace{1em}+\text{EI}\frac{{\partial }^3{v}_2\left({\gamma }_{1}L,t\right)}{\partial z^3}=0,\\ {\mu }^{\mathrm{2}}\rho A\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{1}}\left({\gamma }_{\mathrm{1}}L,t\right)}{\partial t^{\mathrm{2}}}-\text{EI}\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{1}}\left({\gamma }_{\mathrm{1}}L,t\right)}{\partial z^{\mathrm{2}}}-{\mu }^{\mathrm{2}}\rho A\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{2}}\left({\gamma }_{\mathrm{1}}L,t\right)}{\partial t^{\mathrm{2}}}\\ \hspace{1em}\hspace{1em}+\text{EI}\frac{{\partial }^{\mathrm{2}}{v}_{\mathrm{2}}\left({\gamma }_{\mathrm{1}}L,t\right)}{\partial z^{\mathrm{2}}}=\mathrm{0}\end{array}$$ (A.11)

and those for z = γ ₂ L are

$$\begin{array}{c}{v}_{\mathrm{2}}\left({\gamma }_{\mathrm{2}}L,t\right)=v_{\mathrm{3}}\left({\gamma }_{\mathrm{2}}L,t\right),\\ \frac{\partial v_{\mathrm{2}}\left({\gamma }_{\mathrm{2}}L,t\right)}{\partial z}=\frac{\partial v_{\mathrm{3}}\left({\gamma }_{\mathrm{2}}L,t\right)}{\partial z},\\ {\mu }^2\rho A\frac{{\partial }^3{v}_2\left({\gamma }_{2}L,t\right)}{\partial t^2\partial z}-\text{EI}\frac{{\partial }^3{v}_2\left({\gamma }_{2}L,t\right)}{\partial z^3}-{\mu }^2\rho A\frac{{\partial }^3{v}_3\left({\gamma }_{2}L,t\right)}{\partial t^2\partial z}\\ \hspace{1em}\hspace{1em}+\text{EI}\frac{{\partial }^3{v}_3\left({\gamma }_{2}L,t\right)}{\partial z^3}=0,\\ {\mu }^2\rho A\frac{{\partial }^2{v}_2\left({\gamma }_{2}L,t\right)}{\partial t^2}-\text{EI}\frac{{\partial }^2{v}_2\left({\gamma }_{2}L,t\right)}{\partial z^2}-{\mu }^2\rho A\frac{{\partial }^2{v}_3\left({\gamma }_{2}L,t\right)}{\partial t^2}\\ \hspace{1em}\hspace{1em}+\text{EI}\frac{{\partial }^2{v}_3\left({\gamma }_{2}L,t\right)}{\partial z^2}=0.\end{array}$$ (A.12)

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.