• Advances in nano research
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• v.16 no.3
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• pp.265-272
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• 2024

In this paper, the problem of static bending of axially functionally graded (AFG) nanobeam is formulated with the local stress(Lσ)- and strain-driven(εD) two-phase local/nonlocal integral models (TPNIMs). The novelty of the present study aims to compare the size-effects of nonlocal integral models on bending deflections of AFG Euler-Bernoulli nano-beams. The integral relation between strain and nonlocal stress components based on two types nonlocal integral models is transformed unitedly and equivalently into differential form with constitutive boundary conditions. Purely LσD- and εD-NIMs would lead to ill-posed mathematical formulation, and Purely εD- and LσD-nonlocal differential models (NDM) may result in inconsistent size-dependent bending responses. The general differential quadrature method is applied to obtain the numerical results for bending deflection and moment of AFG nanobeam subjected to different boundary and loading conditions. The influence of AFG index, nonlocal models, and nonlocal parameters on the bending deflections of AFG Euler-Bernoulli nanobeams is investigated numerically. A consistent softening effects can be obtained for both LσD- and εD-TPNIMs. The results from current work may provide useful guidelines for designing and optimizing AFG Euler-Bernoulli beam based nano instruments.

• Structural Engineering and Mechanics
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• v.59 no.6
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• pp.1139-1153
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• 2016

Elastic constraints are usually simplified as "spring forces" exerted on beam ends without considering the "spring deformation". The partial differential equation governing the free vibrations of a cantilever Bernoulli-Euler beam considering the deformation of elastic constraints is firstly established, and is nondimensionalized to obtain two dimensionless factors, $k_v$ and $k_r$, describing the effects of elastically vertical and rotational end constraints, respectively. Then the frequency equation for the above Bernoulli-Euler beam model is derived using the method of separation of variables. A numerical analysis method is proposed to solve the transcendental frequency equation for the continuous change of the frequency with $k_v$ and $k_r$. Then the mode shape functions are given. Finally, effects of $k_v$ and $k_r$ on free vibration characteristics of the beam with different slenderness ratios are calculated and analyzed. The results indicate that the effects of $k_v$ are larger on higher-order free vibration characteristics than on lower-order ones, and the impact strength decreases with slenderness ratio. Under a relatively larger slenderness ratio, the effects of $k_v$ can be neglected for the fundamental frequency characteristics, while cannot for higher-order ones. However, the effects of $k_r$ are large on both higher- and lower-order free vibration characteristics, and cannot be neglected no matter the slenderness ratio is large or small.

• Journal of the Computational Structural Engineering Institute of Korea
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• v.28 no.5
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• pp.477-483
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• 2015

In this paper, Euler-Bernoulli beam theories(EB-beam) are used, and Fast Fourier Transformation(FFT) analysis is then employed to extract their natural frequencies using both analytical approach and Co-rotational plane beam(CR-beam) EDISON program. EB-beam is used to analyze a spring-mass system with a single degree of freedom. Sinusoidal force with various frequencies and constant magnitude are applied to tip of each beam. After the oscillatory tip response is observed in EB-beam, it decreases and finally converges to the so-called 'steady-state.' The decreasing rate of the tip deflection with respect to time is reduced when the forcing frequency is increased. Although the tip deflection is found to be independent of the excitation frequency, it turns out that time to reach the steady state response is dependent on the forcing frequency.

• Journal of Institute of Control, Robotics and Systems
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• v.10 no.11
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• pp.1006-1011
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• 2004

Carbon nanotubes have outstanding properties which make them useful for a number of high-technology applications. Especially, single-walled carbon nanotube (SWNT), working under physical conditions (in aqueous solution) and converting electrical energy into mechanical energy directly, can be a good substitute for artificial muscle. The carbon nanotube structure simulated in this paper is an isotropic cantilever type with an adhesive tape which is sandwiched between two SWNTs. For predicting the geometrical and physical parameters such as deflection, slope, bending moment and induced force with various applied voltages, the analytical model for a 3 layer bimorph nanotube actuator is developed by applying Euler-Bernoulli beam theory. The governing equation and boundary conditions are derived from energy Principles. Also, the brief history of carbon nonotube is overviewed and its properties are compared with other functional materials. Moreover, an electro-mechanical coupling coefficient of the carbon nanotube actuator is discussed to identify the electro-mechanical energy efficiency.

• Structural Engineering and Mechanics
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• v.63 no.2
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• pp.137-146
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• 2017

The size-dependent behavior of single walled carbon nanotubes (SWCNT) embedded in the elastic medium and subjected to the initial axial force is investigated using the mixed finite element method. The SWCNT is assumed to be Euler-Bernoulli beam incorporating nonlocal theory developed by Eringen. The mixed finite element model shows its great advantage of dealing with nonlocal behavior of SWCNT subjected to a concentrated load owing to the existence of two coefficients ${\alpha}_1$ and ${\alpha}_2$. This is the first numerical approach to deal with a puzzling fact of nonlocal theory with concentrated load. Numerical examples are performed to show the accuracy and efficiency of the present method. In addition, parametric study is carefully carried out to point out the influences of nonlocal effect, the elastic medium, and the initial axial force on the behavior of the carbon nanotubes.

• Structural Engineering and Mechanics
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• v.65 no.3
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• pp.335-342
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• 2018

The transverse free vibration of chiral double-walled carbon nanotube (DWCNTs) embedded in elastic medium is modeled by the non-local elasticity theory and Euler Bernoulli beam model. The governing equations are derived and the solutions of frequency are obtained. According to this study, the vibrational mode number, the small-scale coefficient, the Winkler parameter and chirality of double-walled carbon nanotube on the frequency ratio (xN) of the (DWCNTs) are studied and discussed. The new features of the vibration behavior of (DWCNTs) embedded in an elastic medium and the present solutions can be used for the static and dynamic analyses of double-walled carbon nanotubes.

• Advances in nano research
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• v.6 no.4
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• pp.339-356
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• 2018

In this article, the critical buckling of a single-walled carbon nanotube (SWCNT) embedded in Kerr's medium is studied. Based on the nonlocal continuum theory and the Euler-Bernoulli beam model. The governing equilibrium equations are acquired and solved for CNTs subjected to mechanical loads and embedded in Kerr's medium. Kerr-type model is employed to simulate the interaction of the (SWNT) with a surrounding elastic medium. A first time, a comparison with the available results is made, and another comparison between various models Winkler-type, Pasternak-type and Kerr-type is studied. Effects of nonlocal parameter and aspect ratio of length to diameter of nanobeam, as well as the foundation parameters on buckling of CNT are investigated. These results are important in the mechanical design considerations of nanocomposites based on carbon nanotubes.

• Proceedings of the Korean Society for Noise and Vibration Engineering Conference
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• 2006.11a
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• pp.396-400
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• 2006

We studied the influences of open cracks in free vibrating beam with rectangular section using a numerical model. The crack was assumed to be single and always open during the free vibration and equivalent bending stiffness of a cracked beam was calculated based on the strain energy balance. By Galerkin's method, the frequencies of cantilever beam could he obtained with respect to various crack depths and locations. Also, the experiments on the cracked beams were carried out to find natural frequencies. The cracks were initiated at five locations and the crack depths were increased by five steps at each location. The experimental results were compared with the numerical results and the comparison results were discussed.

• Structural Engineering and Mechanics
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• v.45 no.1
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• pp.69-93
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• 2013

The present study deals with the dynamics of the flapwise (out-of-plane) vibrations of a rotating, internally damped (Kelvin-Voigt model) tapered Bernoulli-Euler beam carrying a heavy tip mass. The centroid of the tip mass is offset from the free end of the beam and is located along its extended axis. The equation of motion and the corresponding boundary conditions are derived via the Hamilton's Principle, leading to a differential eigenvalue problem. Afterwards, this eigenvalue problem is solved by using Frobenius Method of solution in power series. The resulting characteristic equation is then solved numerically. The numerical results are tabulated for a variety of nondimensional rotational speed, tip mass, tip mass offset, mass moment of inertia, internal damping parameter, hub radius and taper ratio. These are compared with the results of a conventional finite element modeling as well, and excellent agreement is obtained.

• Structural Engineering and Mechanics
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• v.47 no.3
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• pp.307-329
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• 2013

The bending problem of Euler-Bernoulli discontinuous beams is dealt with, in which the discontinuities are due to the loads and eventually to essential constrains applied along the beam axis. In particular, the loads are modelled as random delta-correlated processes acting along the beam axis, while the ulterior eventual discontinuities are produced by the presence of external rollers applied along the beam axis. This kind of structural model can be considered in the static study of bridge beams. In the present work the exact expression of the response quantities are given in terms of means and variances, thanks to the use of the stochastic analysis rules and to the use of the generalized functions. The knowledge of the means and the variances of the internal forces implies the possibility of applying the reliability ${\beta}$-method for verifying the beam.