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Nonlocal beam theory for nonlinear vibrations of a nanobeam resting on elastic foundation

Abstract

In the present study, nonlinear vibrations of an Euler-Bernoulli nanobeam resting on an elastic foundation is studied using nonlocal elasticity theory. Hamilton’s principle is employed to derive the governing equations and boundary conditions. The nonlinear equation of motion is obtained by including stretching of the neutral axis that introduces cubic nonlinearity into the equations. Forcing and damping effects are included in the equations of the motion. The multiple scale method, a perturbation technique for deriving the approximate solutions of the equations, is applied to the nonlinear systems. Natural frequencies and mode shapes for the linear problem are found and also nonlinear frequencies are found for a nonlocal Euler-Bernoulli nanobeam resting on an elastic foundation. In the numerical calculation, frequency-response curves are drawn for various parameters like nonlocal parameters, elastic foundation, and boundary conditions. The effects of the different nonlocal parameters (γ) and elastic foundation parameters (κ) as well as the effects of different boundary conditions on the vibrations are discussed.

Introduction

Due to the rapid improvement in the nano-mechanics, nanobeams have become one of the most important structures used extensively in nanotechnology, such as those in sensors and actuators. The nonlocal elasticity theory which was formally initiated by the papers of Eringen [1] on nonlocal elasticity can be used for nanotechnology applications due to the small length scale in nanoapplications of the beam. The main differences between continuum (local) elasticity theory and nonlocal elasticity theory come from stress definition. Continuum elasticity theory assumes that stress at a point is a function of strain at that point, whereas in the nonlocal elasticity theory stress at a point is a function of strains at all points in the continuum.

Numerical simulation and analysis of nanostructures have been presented extensively by the researchers. Researchers have searched these structures due to the difficulties in experimental specification at the nanoscale and due to their being time-consuming. Then researchers became interested in the theory applied to different mechanical analyses. The nonlocal elasticity theory has been used to examine the vibration, bending, and buckling of the beam depending on the beam model. Peddieson et al. [2] can be considered to be a pioneering work which first applied the nonlocal elasticity theory of Eringen [1] to the nanotechnology.

Vibration analyses are of first priority in the design of nanoelectromechanical systems (NEMS) and new nanodevices. Finding natural frequencies and mode shapes is of primary importance. The effect of the surrounding medium on the vibration response of nanobeams also has practical value. Beams resting on an elastic foundation are usually included in the design of aircraft structures. Niknam and Aghdam [3] studied natural frequency and buckling load of nonlocal functionally graded beams resting on a nonlinear elastic foundation using the Eringen’s nonlocal elasticity theory. Kiani [4] researched a single walled carbon nanotube (SWCNT) structure embedded in an elastic matrix by the nonlocal Euler-Bernoulli, Timoshenko, and higher order beams. Nonlocal elasticity and Timoshenko beam theory were implemented to investigate the stability response of SWCNT embedded in an elastic medium [5, 6]. Nonlocal Euler-Bernoulli theory was applied to investigate the influence of viscoelastic foundation on the nonconservative instability of cantilever CNTs under the action of a concentrated follower [7]. The critical buckling temperature of SWCNT, which is embedded in one parameter elastic medium (Winkler foundation) was estimated by using continuum mechanics theory [8]. Electro-thermal loadings on the transverse free vibration of double walled boron nitride nanotubes (DWBNNTs) embedded in an elastic medium were considered by Arani et al. [9], who investigated the influence of spring modulus, shear modulus, electric field, and temperature change on the natural frequency. Also this author and coauthors [10] carried out an electro-thermal vibration of the DWBNNTs which are coupled by a visco-Pasternak medium based on strain gradient theory. The thermal-mechanical vibration and buckling instability of a SWCNT conveying fluid and resting on an elastic medium were carried out to obtain the effects of temperature change, nonlocal parameter, and elastic medium on the vibration frequency [11]. The thermo-mechanical vibration of a SWCNT embedded in an elastic medium was studied by Murmu and Pradhan [12], who presented the effect of nonlocal small-scale effects, temperature change, Winkler constant, and vibration modes on the frequency. Also Murmu and Pradhan [13] applied a nonlocal beam model to the buckling analysis of SWCNT with the effect of temperature change and the surrounding elastic medium. An electro-thermo-mechanical vibration analysis of nonuniform and nonhomogeneous boron nitride nanorod embedded in elastic medium was presented [14]. Localized modes of free vibrations of SWCNTs embedded in a nonhomogeneous elastic medium were studied on the base of the nonlocal continuum shell theory [15]. The vibration of an axially loaded nonprismatic SWCNT embedded in a two parameter elastic medium [16], a viscous fluid conveying SWCNT embedded in an elastic medium [17], an elastically supported DWCNT embedded in an elastic foundation subject to axial load [18], SWCNT for delivering nanoparticles [19], a carbon nanotube resting on a linear viscoelastic Winkler foundation [20], SWCNT resting on elastic foundation [21], nonuniform SWCNT conveying fluid embedded in viscoelastic medium [22], carbon nanotubes embedded in an elastic medium [23], nanotubes embedded in an elastic matrix [24], and curved SWCNT on a Pasternak elastic foundation [25] were investigated based on the Euler-Bernoulli beam model and the Timoshenko beam model. Aydogdu and Arda [26] researched the torsional vibration behavior of DWCNTs based on nonlocal elasticity theory. Ahangar et al. [27] studied the size dependent vibration of a microbeam. The work of Marin is based on the thermoelasticity of initial stress bodies [28] and of dipolar bodies [29].

Differential equation solutions are much more complicated and time consuming. Further, in many cases, it is impossible to solve nonlinear equations exactly, and therefore there is a need for some approximate solution method. There are several numerical and analytical methods used for solution of nonlinear equations. Some of them are the homotopy perturbation method [30], the multiple scale method [3134], He’s variational method [3, 3538], the direct iterative method [39], the finite element method [11, 20, 22], and the differential quadrature method [5, 10, 21, 40, 41].

The above investigations clearly show that most of the studies presented in the literature are related to the linear vibration analysis of nanostructures, but studies on the nonlinear vibration are rather limited. Studies related to a nonlinear vibration analysis of nanotubes [32, 39, 40, 4247], functionally graded beams [37, 38, 48, 49], microbeams [36], nanobeams [35], and boron nitride nanotubes [41] have been reported. When searching the literature, most of the work is related to the frequency amplitude response of nanotubes/nanobeams. However, a nonlinear vibration of nanosystems with damping effect is very rare. The nonlinear free vibration of the nanotube with damping effect was studied by using nonlocal elasticity theory [32]. To the best of the knowledge of the author, there is no published work on a nonlinear free vibration of nanobeam resting on elastic foundation with the effect of damping and forcing terms. The nonlinearity of the problem is obtained by including stretching of the neutral axis that introduces a cubic nonlinearity into the equations. Nonlinear frequency-response curves are drawn for nanobeams with different end conditions.

Nonlocal elastic models

Recently, the nonlocal continuum mechanics method has been successfully applied to analyze the mechanical behaviors of nano-structures.

General theory

According to the Eringen’s nonlocal elasticity theory [1], the stress at a reference point x in an elastic continuum not only depends on the strain field at the same point but also on the strains at all other points of the body. So, the nonlocal stress tensor σ at a point x is given by

$$ \sigma_{ij} \bigl( x^{*} \bigr) = \int_{V} K \bigl( \bigl\vert x^{*} - x^{\prime *} \bigr\vert ,\gamma \bigr) C_{ijkl} \varepsilon_{kl} \bigl( x^{\prime *} \bigr)\,dV \bigl( x^{\prime *} \bigr) , $$
(1)

where $\sigma_{ij}$ and $\varepsilon_{ij}$ are the stress and strain tensors, respectively; ${C}_{ijkl}$ is the elastic modulus tensor of classical isotropic elasticity, and $K ( \vert x^{*} - x^{\prime *} \vert ,\gamma )$ is the kernel function. $\vert x^{*} - x^{\prime *} \vert $ is the Euclidean distance, and $\gamma = {e}_{0}a / {L}$, where ${e}_{0}$ is a constant appropriate to each material, a is an internal characteristic length (e.g., lattice parameter, granular distance), and L is an external characteristic length (e.g., the crack length, the wavelength). ${e}_{0}a$ used in this study is usually taken as a small-scale parameter. It is very hard to solve the elasticity problems by using the integral constitutive relation in equation (1). Therefore, a simplified constitutive relation in a differential form is given by Eringen [1] as follows:

$$ \bigl( 1 - ( e_{0}a )^{2}\nabla^{2} \bigr)\sigma = T. $$
(2)

Here $\nabla^{2}$ is the Laplacian operator. The nonlocal constitutive relation for the present nanobeam can be written as

$$ \sigma \bigl( x^{*} \bigr) - ( e_{0}a )^{2}\frac{\partial^{2}\sigma ( x^{*} )}{\partial x^{*2}} = E\varepsilon \bigl( x^{*} \bigr) , $$
(3)

where E is the elasticity modulus.

Governing equations of the nanobeam resting on elastic foundation

The considered nanobeam resting on elastic foundation as shown schematically in Figure 1. Simply supported and clamped-clamped supported boundary conditions are considered. The nonlocal Euler-Bernoulli beam model is used to model the nanobeam.

Figure 1
figure1

Boundary conditions for different beam supports.

Here $y^{*}$ the beam’s transverse displacement, $t^{*}$ the time variable, ρA the mass per unit length, k the elastic foundation stiffness, L the length of the beam, A the area of the cross-section of the beam, I the area moment of inertia, $e_{0}a$ the small-scale parameter, and N the axial force. The equations of motion are obtained by using Hamilton’s principle.

$$\begin{aligned} \mbox{\pounds} ={}& \frac{1}{2} \int_{0}^{L} \rho {A} \biggl( \frac{\partial y^{*}}{\partial t^{*}} \biggr)^{2}\,dx^{*} \\ &{} + \frac{1}{2} \int_{0}^{L} \biggl( ( e_{0}a )^{2}\rho A\frac{\partial^{2}y^{*}}{\partial t^{*2}} - ( e_{0}a )^{2}N \frac{\partial^{2}y^{*}}{\partial x^{*2}} - ( e_{0}a )^{2}k - EI\frac{\partial^{2}y^{*}}{\partial x^{*2}} \biggr) \frac{\partial^{2}y^{*}}{\partial x^{*2}}\,dx^{*} \\ &{} - \frac{1}{2} \int_{0}^{L} N \biggl( \frac{\partial y^{*}}{\partial x^{*}} \biggr)^{2}\,dx^{*} - \frac{1}{2} \int_{0}^{L} k y^{*2}\,dx^{*}. \end{aligned}$$
(4)

In equation (4), kinetic energy of the beam is shown in the first integral, the elastic energy induced by the bending is shown in the second integral, the elastic energy in extension due to stretching of the neutral axis is shown in the third integral, and the elastic energy due to elastic foundation is shown in the last integral. The dimensional form of nonlocal governing equation and the boundary conditions can be obtained by applying equation (4) as follows:

$$\begin{aligned} & EI\frac{\partial^{4}y^{*}}{\partial x^{*4}} + \rho {A}\frac{\partial^{2}}{\partial t^{*2}} \biggl( y^{*} - ( e_{0}a )^{2}\frac{\partial^{2}y^{*}}{\partial x^{*2}} \biggr) + k \biggl( y^{*} - ( e_{0}a )^{2}\frac{\partial^{2}y^{*}}{\partial x^{*2}} \biggr) \\ & \quad = \frac{EA}{2L} \biggl[ \int_{0}^{L} \biggl( \frac{\partial y^{*}}{\partial x^{*}} \biggr)^{2}\,dx^{*} \biggr] \biggl[ \frac{\partial^{2}y^{*}}{\partial x^{*2}} - ( e_{0}a )^{2}\frac{\partial^{4}y^{*}}{\partial x^{*4}} \biggr]. \end{aligned}$$
(5)

Furthermore, the following possible boundary conditions at the beam ends (at $x^{*} =0$ and $x ^{*} =L$) are obtained:

$$\begin{aligned} \textstyle\begin{array}{l@{\qquad}l} \text{Simple-Simple Case} &\text{Clamped-Clamped Case}\\ \textstyle\begin{array}{l@{\qquad }l} y^{*}(0) = 0, & y^{*}(L) = 0, \\ y^{\prime\prime *}(0) = 0, & y^{\prime\prime *}(L) = 0, \end{array}\displaystyle & \textstyle\begin{array}{l@{\qquad }l} y^{*}(0) = 0, & y^{*}(L) = 0, \\ y^{\prime *}(0) = 0, & y^{*\prime}(L) = 0. \end{array}\displaystyle \end{array}\displaystyle \end{aligned}$$

The following nondimensional parameters can be defined in order to obtain general results:

$$ x = \frac{x^{*}}{L},\qquad w = \frac{w^{*}}{L}, \qquad t = \frac{t^{*}}{L}\sqrt{\frac{EI}{\rho A}}, \qquad \gamma = \frac{e_{0}a}{L}, \qquad \kappa = \frac{kL^{4}}{EI}. $$
(6)

The equation of motion can be written in the dimensionless form by using equation (6) as follows:

$$ \frac{\partial^{4}y}{\partial x^{4}} + \frac{\partial^{2}y}{\partial t^{2}} - \gamma^{2}\frac{\partial^{4}y}{\partial x^{2}\,\partial t^{2}} + \kappa y - \kappa \gamma^{2}\frac{\partial^{2}y}{\partial x^{2}} = \frac{1}{2} \biggl[ \int_{0}^{L} \biggl( \frac{\partial y}{\partial x} \biggr)^{2}\,dx \biggr] \biggl[ \frac{\partial^{2}y}{\partial x^{2}} - \gamma^{2}\frac{\partial^{4}y}{\partial x^{4}} \biggr]. $$
(7)

Approximate solution

In order to obtain the approximate solution for the problem, the multiple scale method will be employed to the partial differential equation system and boundary conditions directly [50, 51]. Forcing and damping terms are included in equation (7):

$$\begin{aligned} \frac{\partial^{4}y}{\partial x^{4}} + \frac{\partial^{2}y}{\partial t^{2}} - \gamma^{2} \frac{\partial^{4}y}{\partial x^{2}\,\partial t^{2}} + \kappa y - \kappa \gamma^{2}\frac{\partial^{2}y}{\partial x^{2}} = {}&\frac{1}{2} \biggl[ \int_{0}^{L} \biggl( \frac{\partial y}{\partial x} \biggr)^{2}\,dx \biggr] \biggl[ \frac{\partial^{2}y}{\partial x^{2}} - \gamma^{2}\frac{\partial^{4}y}{\partial x^{4}} \biggr] \\ &{} + \overline{{F}} \cos \Omega {t} - 2\overline{\mu} \frac{\partial y}{\partial t}. \end{aligned}$$
(8)

Because of the absence of quadratic nonlinearities, a straightforward asymptotic expansion can be written in order to include stretching and damping effects at order $\varepsilon ^{3}$. Thus we assume that

$$ y(x,t;\varepsilon ) = \varepsilon y_{1}(x,T_{0};T_{2}) + \varepsilon^{3}y_{3}(x,T_{0};T_{2}) , $$
(9)

where ε is a small bookkeeping parameter to denote the deflections. Hence, a weakly nonlinear system can be investigated by this procedure. $T_{0} =t$ and $T_{2} = \varepsilon ^{2} t$ are the fast and slow time scales, respectively. These are used to characterize the modulation of the amplitude and phase due to damping, nonlinearity, and a possible resonance case. In this analysis, only the primary resonance case is considered. Hence, the damping and forcing terms are ordered as defined below so that they are a counter effect of nonlinearity:

$$\overline{\mu} = \varepsilon^{2}\mu,\qquad \overline{F} = \varepsilon^{3} F. $$

Using the chain rule, the derivatives with respect to time are transformed according to

$$ \frac{\partial}{\partial t} = D_{0} + \varepsilon^{2} D_{2},\qquad \frac{\partial^{2}}{\partial t^{2}} = D_{0}^{2} + 2 \varepsilon^{2}D_{0}D_{2}, $$
(10)

where $D_{n} = \partial / \partial T_{n}$ ($n=0,2$). In order to obtain the equations of motion and boundary conditions at different orders, we apply expansions as follows:

Order (ε):

$$ {y}_{1}^{\mathrm{iv}} + D_{0}^{2}{y}_{1} - \gamma^{2}D_{0}^{2}{y}''_{1} + \kappa {y}_{1} - \kappa \gamma^{2}{y}''_{1} = 0. $$
(11)

Order ( $\varepsilon ^{3}$ ):

$$\begin{aligned} & y_{3}^{\mathrm{iv}} + D_{0}^{2}y_{3} - \gamma^{2}D_{0}^{2}y''_{3} + \kappa {y}_{3} - \kappa \gamma^{2}{y}''_{3} \\ &\quad = - 2D_{0}D_{2}y_{1} + 2 \gamma^{2}D_{0}D_{2}y''_{1} \\ &\qquad {}+ \frac{1}{2} \biggl[ \int_{0}^{1} y_{1}^{\prime2}\,dx \biggr]y''_{1} - \frac{1}{2} \gamma^{2} \biggl[ \int_{0}^{1} y_{1}^{\prime2}\,dx \biggr]y_{1}^{\mathrm{iv}} + F \cos \Omega {t} - 2\mu D_{0}{y}_{1}. \end{aligned}$$
(12)

By using the first order of expansion, we obtain linear natural frequency values and by using the second order of expansion, we obtain a solvability condition.

Linear problem

The first terms in the expansions lead to the linear problem. The solution of the form is assumed as follows:

$$ y_{1} ( x,T_{0},T_{2} ) = \bigl[ A ( T_{2} )e^{i\omega {T}_{0}} + cc \bigr] Y ( x ) , $$
(13)

where cc represents the complex conjugate of the preceding terms. Equation (13) is substituted into equation (11) as follows:

$$\begin{aligned}& Y^{\mathrm{iv}}(x) + \bigl( \omega^{2} + \kappa \bigr) \gamma^{2}Y''(x) - \bigl( \omega^{2} - \kappa \bigr)Y(x) = 0, \end{aligned}$$
(14)
$$\begin{aligned}& \begin{aligned} &\mbox{S-F Case:} \quad Y(0) = 0,\qquad Y''(0) = 0,\qquad Y(1) = 0,\qquad Y''(1) = 0, \\ &\mbox{F-F Case:} \quad Y(0) = 0,\qquad Y'(0) = 0, \qquad Y(1) = 0, \qquad Y'(1) = 0. \end{aligned} \end{aligned}$$
(15)

The solutions to equation (15) are given by

$$ Y(x) = c_{1}e^{i\beta_{1}x} + c_{2}e^{i\beta_{2}x} + c_{3}e^{i\beta_{3}x} + c_{4}e^{i\beta_{4}x}. $$
(16)

The boundary conditions are applied; the frequency equations can be obtained. Mode shapes of the linear first three frequency are plotted in Figure 2 with the values $\kappa=1$ and $\gamma=1$ and simply supported and clamped-clamped supported boundary conditions, respectively.

Figure 2
figure2

First three vibration modes shape. (a) Simply supported for $\kappa=1$ and $\gamma=1$; (b) clamped-clamped supported for $\kappa=1$ and $\gamma=0.1$.

Non-linear problem

The solution of the nonlinear equation (12) gives corrections to the problem. They will have a solution only if a solvability condition is satisfied as explained in [51, 52]. The solvability condition is reached by separating the secular and nonsecular terms. Then the solution can be written by assuming the following expansions:

$$ y_{3} ( x,T_{0},T_{2} ) = \varphi ( x,T_{2} )e^{i\omega {T}_{0}} + cc + W ( x,T_{0},T_{2} ) . $$
(17)

Substituting equation (17) into equation (12), the terms that produce secular terms will be eliminated. Here, the solution related with nonsecular terms is represented by $W ( x,T_{0},T_{2} )$ and leads to

$$\begin{aligned} &\varphi^{\mathrm{iv}} - \omega^{2}\varphi - \gamma^{2}\omega^{2}\varphi '' + \kappa \varphi - \kappa \gamma^{2}\varphi '' \\ &\quad =- 2i\omega A'Y(x) + 2i\omega \gamma^{2}A'Y''(x) \\ &\qquad {} + \frac{3}{2}A^{2}\overline{A} \biggl( \int_{0}^{1} Y^{\prime2}\,dx \biggr)Y''(x) - \frac{3}{2}\gamma^{2}A^{2} \overline{A} \biggl( \int_{0}^{1} Y^{\prime2}(x)\,dx \biggr)Y^{\mathrm{iv}}(x) \\ &\qquad {} + \frac{1}{2}Fe^{i\sigma {T}_{2}} - 4i\mu \omega AY(x) + cc + NST, \end{aligned}$$
(18)

where NST stands for nonsecular terms. The nearness of the external excitation is represented by a detuning parameter of order 1σ defined by

$$ \Omega = \omega + \varepsilon^{2}\sigma. $$
(19)

Substituting equation (19) into equation (18) and after some algebraic manipulations, the solvability conditions are given by

$$ 2i\omega ( {D}_{2}A + 2\mu A ) + 2i\omega \gamma^{2}{D}_{2}Ab + \frac{3}{2}A^{2}\overline{A}\bigl( b^{2} + \gamma^{2}b\Lambda \bigr) - \frac{1}{2}{e}^{{i}\sigma {T}_{2}} f = 0. $$
(20)

Here the coefficients are defined as $\int_{0}^{1} Y^{2}\,dx = 1$, $\int_{0}^{1} Y^{\prime2}\,dx = b$, $\int_{0}^{1} Y^{\prime\prime2}\,dx = \Lambda$, $\int_{0}^{1} FY \,dx = f$. A is the complex amplitude in equation (20), which can be expressed as a real amplitude a and a phase θ,

$$ {A} = \frac{1}{2}{a}({T}_{2}){e}^{{i} \theta ({T}_{2})}, $$
(21)

where $a(T_{2})$ and $\theta(T_{2})$ represent the amplitude and phase angle of the response, respectively. Substituting equation (21) into equation (20), the modulation equation can be written as

$$ \begin{aligned} &\omega {aD}_{2}\Phi = \omega {a}\sigma + \omega \gamma^{2}{a}b\sigma - \omega \gamma^{2}{a}b{D}_{2} \Phi - \frac{3}{16}{a}^{3} \bigl( b^{2} + \gamma^{2}b\Lambda \bigr) + \frac{1}{2}f\cos \Phi, \\ &\omega {D}_{2}{a} \bigl( 1 + \gamma^{2} \bigr) + 2\mu \omega {a} = \frac{1}{2}f\sin \Phi, \end{aligned} $$
(22)

where $\Phi = \sigma T_{2} - \theta$. The nonlinear frequencies can be calculated from equation (22) by considering free undamped vibrations.

Numerical results

In this section, numerical results can be achieved using the multiple scale method explained in Section 3. Only γ (the nonlocal parameter) and κ (the dimensionless elastic foundation stiffness) values are required in the calculations because of the dimensionless equations obtained. In this study, two different boundary conditions are applied, i.e., simply supported and clamped-clamped supported. Nanobeams boundary conditions are represented by a two-letter symbol in order to simplify the notations. For example, the symbols S-S and C-C indicate that the nanobeam is simply supported and clamped-clamped supported, respectively. The linear natural frequencies for S-S and C-C boundary conditions are calculated for various γ and κ values. The nonlinear frequencies for free undamped vibrations were calculated. We took $\mu=f=\sigma=0$, and obtained

$$ {D}_{2}{a} = 0 \quad \mbox{and} \quad {a} = {a}_{0} \quad \mbox{(constant)} $$
(23)

from equation (17). ${a}_{0}$ is the steady-state real amplitude and hence the nonlinear frequency is

$$ \begin{aligned} &\omega_{{n}1} = \omega + {D}_{2}\theta = \omega + \frac{3}{16}\frac{{a}_{0}^{2} ( b^{2} + \gamma^{2}b\Lambda )}{\omega ( 1 + \gamma^{2}b )} , \\ &\omega_{{n}1} = \omega + {a}_{0}^{2}\lambda, \end{aligned} $$
(24)

where $\lambda = \frac{3}{16}\frac{ ( b^{2} + \gamma^{2}b\Lambda )}{\omega ( 1 + \gamma^{2}b )}$ is the nonlinear correction coefficient, a measure of the effect of stretching.

Table 1 and Table 2 show the linear frequencies and nonlinear correction terms for the first three frequencies of the S-S and C-C supported nanobeams and various γ and κ values, respectively. The effects of the support conditions and various nonlocal parameters γ and dimensionless elastic foundation parameters κ are given. The elastic foundation stiffness has an important influence on the vibration properties. Nondimensional elastic foundation parameters with $\kappa=1, 10, 50, 100, 200\mbox{ and }500$ are given in Table 1 and Table 2. An increase in the dimensionless elastic foundation stiffness κ increases the system stiffness and it becomes more stable than previously. It can be found in Table 1 that increasing the κ values generally increases the linear frequencies. It should be noted that the influence of elastic stiffness is more obvious for the smallest natural frequencies. This phenomenon is in agreement with that described by Ghavanloo et al. [20] and Rafiei et al. [22]. The same situation can be observed for the clamped-clamped boundary condition. The small-scale effects play an important role in the vibration analysis of the nanobeam. In this paper, the nonlocal parameter ($\gamma = \frac{e_{0}a}{L}$) is taken as $\gamma=0, 0.1, 0.2, 0.3, 0.4\mbox{ and }0.5$. As seen in Table 1 and Table 2, the linear frequencies obtained from the nonlocal theory for both boundary conditions are smaller than those obtained from the local (classical) theory. With the nonlocal parameter taken as $\gamma=0$, the nonlocal elasticity reduces to the classical beam theory. Furthermore, an increase in the nonlocal parameter (γ) decreases the linear frequencies for both boundary conditions, as expected, as shown in Table 1 and Table 2. It can be found in the same tables that the γ values increase, and the correction terms (λ) increase. The stretching effect is measured by λ. The C-C supported nanobeam has both linear and nonlinear frequencies higher than the S-S supported one, which can be seen in Table 1 and Table 2. The C-C supported nanobeam has the strongest end condition, while the S-S supported one has the weakest end condition.

Table 1 The first five frequencies and nonlinear correction term for various γ and κ values for simple-simple support condition
Table 2 The first five frequencies and nonlinear correction term for various γ and κ values for clamped-clamped support condition

We took ${D}_{2}{a} = 0$, $D_{2}\Phi = 0$ at the steady state. The detuning parameter frequency was defined by

$$\begin{aligned}& \sigma = \frac{3}{16}\frac{{a}^{2} ( b^{2} + \gamma^{2}b\Lambda )}{\omega ( 1 + \gamma^{2}b )} \mp \sqrt{ \frac{f^{2}}{4\omega^{2}{a}^{2} ( 1 + \gamma^{2}b )^{2}} - \mu^{2}}, \end{aligned}$$
(25)
$$\begin{aligned}& \sigma = {a}^{2}\lambda \mp \sqrt{\frac{f^{2}}{4\omega^{2}{a}^{2} ( 1 + \gamma^{2}b )^{2}} - \mu^{2}}. \end{aligned}$$
(26)

Vibrating system frequencies are sensitive to the selection of the nonlocal parameters (γ). Hence, the selection of the appropriate nonlocal parameter is very important for a particular system. Also, the elastic stiffness has a significant effect on the system frequencies. Figure 3 shows the nonlocal parameter effect on the nonlinear natural frequencies of the nanobeam with $\kappa=1$, the first mode and S-S and C-C boundary conditions. It can be observed that with the increase in nonlocal parameter ($\gamma =0, 0.1, 0.2, 0.3, 0.4\mbox{ and }0.5$), the nonlinear frequencies decrease. The dimensionless linear elastic foundation parameter effect on the nonlinear natural frequencies of the nanobeam with $\gamma=0.3$ and both boundary conditions are investigated in Figure 4 that plot the nonlinear natural frequency variation versus amplitude with $\kappa =1$, 10, 50, 100 and 200. It can be seen in the same figures that an increase in dimensionless linear elastic stiffness and a fixed in nonlocal parameter increase in nonlinear frequency value occur regardless of the type of boundary condition.

Figure 3
figure3

Nonlinear natural frequency versus amplitude for different nonlocal parameters with $\pmb{\kappa=1}$ and first mode. (a) S-S case; (b) C-C case.

Figure 4
figure4

Nonlinear natural frequency versus amplitude for different κ values with $\pmb{\gamma=0.3}$ . (a) S-S case; (b) C-C case.

The detuning parameter σ determines the instability region about the external excitation frequency when it is close to the natural frequency of system. The values ${f} =1$ and damping coefficient $\mu=0.1$ in equation (26) are assumed in drawing the figures. When $\sigma<0$, we have an increase in forcing term, an increase in amplitudes and when $\sigma>0$, an increase in forcing term, and a decrease in amplitudes at different values. The maximum amplitudes is reached when the detuning parameter is $\sigma>0$. Figures 5, 6, 7 plot the frequency response curves for various nonlocal parameters γ, mode numbers, and dimensionless linear elastic foundation parameter κ with two boundary conditions, i.e. S-S and C-C. From Figure 5, it is noted that the increase in γ values leads to a decrease in the amplitude of vibration in both types of boundary conditions. In the same figures we show that the maximum amplitude decreases by increasing the γ values. The same trend can also be seen in Figure 6 with an increase in the modes. Figure 7 shows that an increase in κ values leads to a decrease of the amplitude of the vibration. It can be noted that the stiffness of the elastic foundation has a significant effect on the frequency response curves of the nanobeam. The frequency response curve tends to be a straight line with an increase of the κ values. This indicates that the nonlinear vibration will return to the linear vibration when the stiffness is large enough. This phenomenon is in agreement with that described by Fu et al. [44].

Figure 5
figure5

Frequency-response curves for different nonlocal parameters with $\pmb{\kappa=1}$ and first mode. (a) S-S case; (b) C-C case.

Figure 6
figure6

Frequency-response curves for different modes with $\pmb{\kappa=1}$ and $\pmb{\gamma=0.1}$ . (a) S-S case; (b) C-C case.

Figure 7
figure7

Frequency-response curves for different κ values with first mode and $\pmb{\gamma=0.3}$ . (a) S-S case; (b) C-C case.

Conclusions

The nonlinear vibrations of a nanobeam resting on an elastic foundation are investigated for different end conditions. The nanobeam is described by the nonlocal Euler-Bernoulli beam model. The effect of stretching of the neutral axis is included in the nonlinear equations of motion. The multiple scale method, a perturbation technique, is used to obtain

approximate solutions. For the linear problem, exact solutions, and numerical values for natural frequencies are obtained. For the nonlinear problem, nonlinear correction terms are obtained. Nonlinear terms in the perturbation series appear as corrections to the linear problem. The effects of the nonlocal parameter (γ), dimensionless elastic foundation parameter (κ), and boundary conditions are discussed. For each of the end conditions the natural frequencies and mode shapes are tabulated and found. When nonlinear terms are added to the equations, corrections to the linear problem are introduced. The numerical result shows that the nonlinear frequency of the nanobeam decreases with increasing the nonlocal parameters. The present numerical results also reveal that an increase in the dimensionless elastic stiffness (κ) increases the nonlinear frequency value regardless of the type of boundary conditions.

References

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Keywords

  • vibratio
  • nanobeam
  • perturbation metho
  • nonlocal elasticity
  • elastic foundation