Dynamic stability of harmonically excited nanobeams including axial inertia

2018 ◽  
Vol 25 (4) ◽  
pp. 820-833 ◽  
Author(s):  
Mustafa Arda ◽  
Metin Aydogdu
Keyword(s):  
Dynamic Stability ◽  
Longitudinal Vibration ◽  
Perturbation Expansion ◽  
Axial Loading ◽  
Arbitrary Point ◽  
Expansion Method ◽  
Small Scale ◽  
Beam Model ◽  
Critical Dynamic ◽  
Euler Bernoulli Beam

This article is concerned with the dynamic stability problem of a nanobeam under a time-varying axial loading. The nonlocal Euler–Bernoulli beam model has been used for the continuum modeling of the nanobeam structure. This problem leads to a time-dependent Mathieu-Hill equation and has been solved by using the Lindstedt–Poincaré perturbation expansion method. The effect of a small-scale parameter on the dynamic displacement and critical dynamic buckling load of nanobeams has been investigated. Stability regions have been obtained from the local and nonlocal elasticity theories. The effect of the longitudinal vibration of nanobeams on instability regions has been included in the present analysis. Amplitudes of an arbitrary point of a nanobeam due to harmonic loads have been determined. Nonlocal and longitudinal vibration effects reduce the area of the instability region and increase amplitudes.

On the size-dependent nonlinear dynamics of viscoelastic/flexoelectric nanobeams

2020 ◽  
pp. 107754632095222
Author(s):  
Rasoul Bagheri ◽  
Yaghoub Tadi Beni
Keyword(s):  
Boundary Conditions ◽  
Viscoelastic Medium ◽  
Medium Effect ◽  
Small Scale ◽  
Beam Model ◽  
Governing Equations ◽  
Size Dependent ◽  
Flexoelectric Effect ◽  
Nonlinear Forced Vibration ◽  
Euler Bernoulli Beam

In this study, size-dependent nonlinear forced vibration of viscoelastic/flexoelectric nanobeams has been investigated. By calculating enthalpy and kinetic energy and using Hamilton’s principle, the coupled governing equations of viscoelastic/flexoelectric nanobeams are derived along with dependent electrical and mechanical boundary conditions. Furthermore, to take the effects of the small scale into account, the nonclassical theory of continuous medium has been used and the Euler–Bernoulli beam model has been adopted to model the nanobeams. Finally, the governing equations are solved using numerical methods for distributed loaded and clamped–clamped boundary conditions. By comparing the results, it is determined that the parameters of the size effect and the viscoelastic medium effect can increase the vibrational frequency of the nanobeams. Also, the results show that the frequency of nanobeams outside of the viscoelastic medium strongly depends on the size-dependent parameters, and the increase in the length and thickness of the nanobeam decreases the frequency. The results also show that with the increasing flexoelectric effect, the amplitude of the nonlinear oscillation increases.

Buckling Analysis of Nanobeams Based on Nonlocal Timoshenko Beam Model by the Method of Initial Values

2019 ◽  
Vol 19 (04) ◽  
pp. 1950036 ◽  
Author(s):  
Erol Demirkan ◽  
Reha Artan
Keyword(s):  
Timoshenko Beam ◽  
Beam Theory ◽  
Small Scale ◽  
Beam Model ◽  
Timoshenko Beam Model ◽  
Initial Values ◽  
Nonlocal Timoshenko Beam Theory ◽  
Small Scale Effect ◽  
Carry Over ◽  
Euler Bernoulli Beam

Investigated herein is the buckling of nanobeams based on a nonlocal Timoshenko beam model by the method of initial values within the framework of nonlocal elasticity. Since the nonlocal Timoshenko beam theory is of higher order than the nonlocal Euler–Bernoulli beam theory, it is known to be superior in predicting the small-scale effect. The buckling determinants and critical loads for bars with various kinds of supports are presented. The Carry-Over matrix (Transverse Matrix) is presented and the priorities of the method of initial values are depicted. To the best of the researchers’ knowledge, this is the first work that investigates the buckling of nonlocal Timoshenko beam with the method of initial values.

Variational Principle of Carbon Nanotubes with Temperature Changes Based on Nonlocal Euler-Bernoulli Beam Model

2010 ◽  
Vol 452-453 ◽  
pp. 785-788
Author(s):  
Tao Fan
Keyword(s):  
Variational Principle ◽  
Small Scale ◽  
Beam Model ◽  
Bernoulli Beam ◽  
Variational Integral ◽  
Temperature Changes ◽  
Mechanical Coupling ◽  
Thermal Expansion Coefficients ◽  
Euler Bernoulli Beam ◽  
Bernoulli Beam Model

In this paper, the CNTS are considered as the Euler-Bernoulli beams which have been used in many references about the CNTS. Taken the thermal-mechanical coupling and small scale effect into account, the variational principle for the CNTS is presented by the variational integral method. With the derivation of the varitional principle, the vibration governing equation is illustrated, which will be benefit for the numerical simulation with finite element method in further investigations. From the stationary value conditions deduced by the variational principle, the influences of the temperature changes and the thermal expansion coefficients based on nonlocal Euler-Bernoulli beam model are presented.

scholarly journals Dynamics of Space-Fractional Euler–Bernoulli and Timoshenko Beams

Materials ◽  
2021 ◽  
Vol 14 (8) ◽  
pp. 1817
Author(s):  
Paulina Stempin ◽  
Wojciech Sumelka
Keyword(s):  
Small Scale ◽  
Rotational Inertia ◽  
Beam Model ◽  
Static Case ◽  
Applicability Limit ◽  
Beam Geometry ◽  
Slender Beams ◽  
Euler Bernoulli Beam ◽  
Good Agreement ◽  
Non Locality

This paper investigates the dynamics of the beam-like structures whose response manifests a strong scale effect. The space-Fractional Euler–Bernoulli beam (s-FEBB) and space-Fractional Timoshenko beam (s-FTB) models, which are suitable for small-scale slender beams and small-scale thick beams, respectively, have been extended to a dynamic case. The study provides appropriate governing equations, numerical approximation, detailed analysis of free vibration, and experimental validation. The parametric study presents the influence of non-locality parameters on the frequencies and shape of modes delivering a depth insight into a dynamic response of small scale beams. The comparison of the s-FEBB and s-FTB models determines the applicability limit of s-FEBB and indicates that the model (also the classical one) without shear effect and rotational inertia can only be applied to beams significantly slender than in a static case. Furthermore, the validation has confirmed that the fractional beam model exhibits very good agreement with the experimental results existing in the literature—for both the static and the dynamic cases. Moreover, it has been proven that for fractional beams it is possible to establish constant parameters of non-locality related to the material and its microstructure, independent of beam geometry, the boundary conditions, and the type of analysis (with or without inertial forces).

Vibration and Stability Analysis of DWCNT-Based Spinning Nanobearings

2017 ◽  
Vol 17 (09) ◽  
pp. 1750102 ◽  
Author(s):  
Rohollah Dehghani Firouz-Abadi ◽  
Hassan Mohammad-Khani ◽  
Mohammad Rahmanian
Keyword(s):  
Nonlocal Theory ◽  
Elastic Stiffness ◽  
Free Vibrations ◽  
Theory Of Elasticity ◽  
Small Scale ◽  
Beam Model ◽  
Stability Margins ◽  
The Stability ◽  
Spinning Speed ◽  
Euler Bernoulli Beam

This paper aims at investigating free vibrations and stability of double-walled carbon nanotube (DWCNT)-based spinning nanobearings. The so-called nanobearing consists of two coaxial carbon nanotubes (CNTs) where either of the two CNTs can be a rotor while the other takes the role of stator. Euler–Bernoulli beam model along with the Eringen’s nonlocal theory of elasticity are employed to obtain governing equations of transverse vibrations for the CNTs. The coupling of the two CNTs originates from the van-der-Waals (vdW) forcing present in the interface of the two CNTs. The coupling is taken into account as distributed spring foundation with an equivalent elastic stiffness. Based on the obtained model, effects of small-scale parameter, vdW interaction between CNTs and the diameter ratio of CNTs on the critical spinning speed are investigated. Finally, the stability margins of the nanobearings are determined and some general conclusions are drawn.

scholarly journals Numerical Solution of Bending of the Beam with Given Friction

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 898
Author(s):  
Michaela Bobková ◽  
Lukáš Pospíšil
Keyword(s):  
Sliding Friction ◽  
Quadratic Function ◽  
Building Construction ◽  
Beam Model ◽  
Bernoulli Beam ◽  
Sustainable Building ◽  
Construction Design ◽  
Euler Bernoulli Beam ◽  
Bernoulli Beam Model ◽  
Fixed Beam

We are interested in a contact problem for a thin fixed beam with an internal point obstacle with possible rotation and shift depending on a given swivel and sliding friction. This problem belongs to the most basic practical problems in, for instance, the contact mechanics in the sustainable building construction design. The analysis and the practical solution plays a crucial role in the process and cannot be ignored. In this paper, we consider the classical Euler–Bernoulli beam model, which we formulate, analyze, and numerically solve. The objective function of the corresponding optimization problem for finding the coefficients in the finite element basis combines a quadratic function and an additional non-differentiable part with absolute values representing the influence of considered friction. We present two basic algorithms for the solution: the regularized primal solution, where the non-differentiable part is approximated, and the dual formulation. We discuss the disadvantages of the methods on the solution of the academic benchmarks.

Study on Dynamic Stability of Single Floating Pier Under Waves

2021 ◽  
Author(s):  
Yikuan He ◽  
Bing Han ◽  
Wenyu Ji
Keyword(s):  
Dynamic Stability ◽  
Expansion Method ◽  
Diffraction Effect ◽  
Exterior Region ◽  
Factors Affecting ◽  
Eigenfunction Expansion Method ◽  
Floating Bridge ◽  
Response Equation ◽  
Floating Pier ◽  
Wave Excitation Force

Abstract Considering the upper structure restraint effect of the floating bridge, the diffraction effect and radiation effect of linear monochromatic waves, the dynamic response equation of floating pier is derived and the factors affecting the dynamic stability of the floating pier are analyzed in this paper. Based on the theory of potential flow, the calculation domain is divided into the interior region and the exterior region. The wave diffraction and radiation problems are solved by the matched eigenfunction expansion method (MEEM). After obtaining the wave excitation force, additional mass and radiation damping coefficient, considering the restraint effect of the upper structure of the floating bridge, the motion differential equation of the floating pier is established, and the response amplitude operator (RAOs) of the floating pier is obtained. The effects of span, mass and stiffness of upper structure, as well as the draft depth, size and net height of floating pier on dynamic stability of floating pier under wave are analyzed. The results show that the increase in the span of upper structure will significantly increase the peak RAOs of sway and heave, and the increase in stiffness is helpful to reduce the peak RAOs of sway and heave. The increase of the floating pier radius can reduce the heave RAO, and the net height on the water surface of the floating pier increases the heave and roll.

Asymmetric Bifurcation of Initially Curved Nanobeam

2015 ◽  
Vol 82 (9) ◽  
Author(s):  
X. Chen ◽  
S. A. Meguid
Keyword(s):  
Symmetry Breaking ◽  
Degrees Of Freedom ◽  
Surface Elasticity ◽  
Beam Theory ◽  
Surface Effects ◽  
Beam Model ◽  
Bernoulli Beam ◽  
Reduced Order ◽  
Euler Bernoulli Beam ◽  
Bifurcation Behavior

In this paper, we investigate the asymmetric bifurcation behavior of an initially curved nanobeam accounting for Lorentz and electrostatic forces. The beam model was developed in the framework of Euler–Bernoulli beam theory, and the surface effects at the nanoscale were taken into account in the model by including the surface elasticity and the residual surface tension. Based on the Galerkin decomposition method, the model was simplified as two degrees of freedom reduced order model, from which the symmetry breaking criterion was derived. The results of our work reveal the significant surface effects on the symmetry breaking criterion for the considered nanobeam.

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