Nonlocal beam theory for nonlinear vibrations of a nanobeam resting on elastic foundation
- Necla Togun^{1}Email author
Received: 7 January 2016
Accepted: 14 February 2016
Published: 29 February 2016
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.
Keywords
1 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 [31–34], He’s variational method [3, 35–38], 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, 42–47], 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.
2 Nonlocal elastic models
Recently, the nonlocal continuum mechanics method has been successfully applied to analyze the mechanical behaviors of nano-structures.
2.1 General theory
2.2 Governing equations of the nanobeam resting on elastic foundation
3 Approximate solution
3.1 Linear problem
3.2 Non-linear problem
4 Numerical results
The first five frequencies and nonlinear correction term for various γ and κ values for simple-simple support condition
κ | γ | \(\boldsymbol{\omega_{1}}\) | \(\boldsymbol{\omega_{2}}\) | \(\boldsymbol{\omega_{3}}\) | \(\boldsymbol{\omega_{4}}\) | \(\boldsymbol{\omega_{5}}\) | λ | |
---|---|---|---|---|---|---|---|---|
Simple-Simple | 1 | 0 | 9.9201 | 39.4911 | 88.8321 | 157.917 | 246.742 | 1.84113 |
0.1 | 9.46883 | 33.4426 | 64.6491 | 98.3343 | 132.51 | 1.93244 | ||
0.2 | 8.41654 | 24.6026 | 41.6405 | 58.3889 | 74.8465 | 2.23004 | ||
0.3 | 7.25166 | 18.5286 | 29.6349 | 40.5001 | 51.2291 | 2.77290 | ||
0.4 | 6.2264 | 14.6293 | 22.7963 | 30.8283 | 38.7947 | 3.52508 | ||
0.5 | 5.39378 | 12.0161 | 18.4661 | 24.8405 | 31.1804 | 4.40404 | ||
10 | 0 | 10.3638 | 39.6049 | 88.8827 | 157.945 | 246.76 | 1.76231 | |
0.1 | 9.93271 | 33.5769 | 64.7187 | 98.38 | 132.544 | 1.84219 | ||
0.2 | 8.93522 | 24.7849 | 41.7484 | 58.4659 | 74.9066 | 2.10059 | ||
0.3 | 7.84771 | 18.7699 | 29.7864 | 40.611 | 51.3169 | 2.56230 | ||
0.4 | 6.91145 | 14.9337 | 22.9928 | 30.9739 | 38.9105 | 3.17568 | ||
0.5 | 6.17194 | 12.3849 | 18.7082 | 25.021 | 31.3244 | 3.84878 | ||
50 | 0 | 12.1412 | 40.1067 | 89.1074 | 158.072 | 246.841 | 1.50432 | |
0.1 | 11.7753 | 34.1674 | 65.027 | 98.5831 | 132.695 | 1.55392 | ||
0.2 | 10.9471 | 25.5791 | 42.2248 | 58.807 | 75.1732 | 1.71454 | ||
0.3 | 10.079 | 19.8068 | 30.4504 | 41.1005 | 51.7051 | 1.99506 | ||
0.4 | 9.36846 | 16.2178 | 23.8468 | 31.6131 | 39.4212 | 2.34281 | ||
0.5 | 8.83701 | 13.9063 | 19.7483 | 25.8079 | 31.9565 | 2.68806 | ||
100 | 0 | 14.0502 | 40.7252 | 89.3876 | 158.23 | 246.943 | 1.29992 | |
0.1 | 13.7353 | 34.8914 | 65.4103 | 98.8364 | 132.883 | 1.33218 | ||
0.2 | 13.0322 | 26.5385 | 42.8127 | 59.2306 | 75.505 | 1.44022 | ||
0.3 | 12.312 | 21.0311 | 31.2607 | 41.7044 | 52.1864 | 1.63322 | ||
0.4 | 11.7375 | 17.6923 | 24.8731 | 32.3942 | 40.0503 | 1.86995 | ||
0.5 | 11.3178 | 15.6008 | 20.9761 | 26.7591 | 32.7294 | 2.09886 | ||
200 | 0 | 17.2456 | 41.935 | 89.9452 | 158.546 | 247.145 | 1.05906 | |
0.1 | 16.99 | 36.2961 | 66.1703 | 99.341 | 133.259 | 1.07698 | ||
0.2 | 16.4267 | 28.36 | 43.9651 | 60.0688 | 76.1643 | 1.14261 | ||
0.3 | 15.8615 | 23.2875 | 32.8212 | 42.8865 | 53.1359 | 1.26773 | ||
0.4 | 15.4197 | 20.3228 | 26.808 | 33.9026 | 41.2799 | 1.42342 | ||
0.5 | 15.1027 | 18.5307 | 23.2378 | 28.5666 | 34.223 | 1.57287 | ||
500 | 0 | 24.442 | 45.3712 | 91.5977 | 159.489 | 247.751 | 0.74725 | |
0.1 | 24.2623 | 40.217 | 68.3997 | 100.84 | 134.38 | 0.75417 | ||
0.2 | 23.8713 | 33.2309 | 47.2539 | 62.5161 | 78.1089 | 0.78627 | ||
0.3 | 23.4859 | 29.0225 | 37.111 | 46.2521 | 55.8876 | 0.85618 | ||
0.4 | 23.1898 | 26.7024 | 31.9166 | 38.0708 | 51.7907 | 0.94648 | ||
0.5 | 22.9803 | 25.365 | 28.9827 | 33.4073 | 38.3564 | 1.03368 |
The first five frequencies and nonlinear correction term for various γ and κ values for clamped-clamped support condition
κ | γ | \(\boldsymbol{\omega_{1}}\) | \(\boldsymbol{\omega_{2}}\) | \(\boldsymbol{\omega_{3}}\) | \(\boldsymbol{\omega_{4}}\) | \(\boldsymbol{\omega_{5}}\) | λ | |
---|---|---|---|---|---|---|---|---|
Clamped-Clamped | 1 | 0 | 22.3956 | 61.6809 | 120.908 | 199.862 | 298.557 | 1.26716 |
0.1 | 21.1327 | 50.993 | 85.7223 | 121.352 | 156.744 | 1.45493 | ||
0.2 | 18.3167 | 36.4377 | 54.5331 | 71.6196 | 88.4926 | 1.73402 | ||
0.3 | 15.3861 | 27.0202 | 38.8468 | 49.687 | 60.7786 | 2.30212 | ||
0.4 | 12.9434 | 21.1635 | 29.9791 | 37.841 | 46.1611 | 3.33512 | ||
0.5 | 11.0368 | 17.3019 | 24.3529 | 30.501 | 37.1779 | 4.68740 | ||
10 | 0 | 22.5957 | 61.7538 | 120.945 | 199.884 | 298.572 | 1.25595 | |
0.1 | 21.3446 | 51.0811 | 85.7747 | 121.389 | 156.772 | 1.44049 | ||
0.2 | 18.5608 | 36.5609 | 54.6156 | 71.6824 | 88.5434 | 1.71122 | ||
0.3 | 15.6759 | 27.1862 | 38.9625 | 49.7775 | 60.8526 | 2.25956 | ||
0.4 | 13.2865 | 21.375 | 30.1288 | 37.9597 | 46.2584 | 3.24900 | ||
0.5 | 11.4372 | 17.5601 | 24.537 | 30.6482 | 37.2987 | 4.52332 | ||
50 | 0 | 23.4641 | 62.0769 | 121.11 | 199.984 | 298.639 | 1.20946 | |
0.1 | 22.2619 | 51.4712 | 86.0076 | 121.554 | 156.9 | 1.38114 | ||
0.2 | 19.6087 | 37.1039 | 54.9806 | 71.9608 | 88.769 | 1.61976 | ||
0.3 | 16.9036 | 27.9122 | 39.4725 | 50.1777 | 61.1804 | 2.09545 | ||
0.4 | 14.715 | 22.2911 | 30.7855 | 38.483 | 46.6888 | 2.93359 | ||
0.5 | 13.0694 | 18.6643 | 25.339 | 31.2939 | 37.8312 | 3.95841 | ||
100 | 0 | 24.5064 | 62.4783 | 121.316 | 200.109 | 298.723 | 1.15802 | |
0.1 | 23.3579 | 51.9546 | 86.2978 | 121.759 | 157.059 | 1.31633 | ||
0.2 | 20.8447 | 37.7717 | 55.4334 | 72.3074 | 89.0502 | 1.52372 | ||
0.3 | 18.323 | 28.7939 | 40.1008 | 50.6734 | 61.5876 | 1.93312 | ||
0.4 | 16.3258 | 23.3857 | 31.5871 | 39.1272 | 47.2212 | 2.64414 | ||
0.5 | 14.8597 | 19.9588 | 26.3071 | 32.0829 | 38.4863 | 3.48149 | ||
200 | 0 | 26.4682 | 63.2735 | 121.728 | 200.359 | 298.89 | 1.07219 | |
0.1 | 25.4085 | 52.9083 | 86.8752 | 122.169 | 157.377 | 1.21010 | ||
0.2 | 23.1193 | 39.073 | 56.3282 | 72.9956 | 89.6099 | 1.37381 | ||
0.3 | 20.8742 | 30.481 | 41.3289 | 51.6507 | 62.3942 | 1.69686 | ||
0.4 | 19.145 | 25.4341 | 33.1323 | 40.3849 | 48.2685 | 2.25478 | ||
0.5 | 17.9112 | 22.3239 | 28.1437 | 33.6052 | 39.7643 | 2.88835 | ||
500 | 0 | 31.6317 | 65.6013 | 122.954 | 201.106 | 299.392 | 0.89717 | |
0.1 | 30.7505 | 55.6712 | 88.585 | 123.391 | 158.327 | 0.99988 | ||
0.2 | 28.8878 | 42.7399 | 58.931 | 75.0224 | 91.2685 | 1.09948 | ||
0.3 | 27.1244 | 35.0584 | 44.8116 | 54.4775 | 64.7537 | 1.30586 | ||
0.4 | 25.8173 | 30.7716 | 37.3865 | 43.9424 | 51.282 | 1.67204 | ||
0.5 | 24.9161 | 28.2552 | 33.0464 | 37.8062 | 43.3728 | 2.07632 |
5 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.
Declarations
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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