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).

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.

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.

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.

Exact solution for nonlocal axial buckling of linear carbon nanotube hetero-junctions

2013 ◽  
Vol 228 (2) ◽  
pp. 366-377 ◽  
Author(s):  
A Ghorbanpour Arani ◽  
R Kolahchi
Keyword(s):  
Boundary Conditions ◽  
Buckling Load ◽  
Small Scale ◽  
Bernoulli Beam ◽  
Electromechanical Systems ◽  
Various Boundary Conditions ◽  
Axial Buckling ◽  
Euler Bernoulli Beam ◽  
Good Agreement ◽  
Scale Coefficient

CNT hetero-junctions offer the possibility of being used in micro-electromechanical systems and nano-electromechanical systems. However, in the present work, nonlocal axial buckling analysis of linear CNT hetero-junctions based on Euler–Bernoulli beam (EBB) model is investigated. In order to mathematically model a linear CNT hetero-junction, a CNT with two segments is considered. The constitutive equations are derived based on the Eringen's theory for various boundary conditions, namely clamped–clamped (C–C), clamped–pinned (C–P) and pinned–pinned (P–P). An analytical approach is applied to obtain the dimensionless buckling load of the CNT hetero-junctions. A detailed parametric study is conducted to elucidate the influences of the small-scale coefficient, homogeneity parameter, boundary conditions and CNT length of each segment on the axial buckling of the linear CNT hetero-junctions. The results indicate that the length of each segment and homogeneity parameter have a significant effect on the buckling load of the CNT hetero-junctions and should therefore be considered in its optimum design. Furthermore, the results are in good agreement with the previous researches.

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.

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 Experimental and Numerical Investigation about Small Clearance Journal Bearings under Static Load Conditions

2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Carlo Alberto Niccolini Marmont Du Haut Champ ◽  
Fabrizio Stefani ◽  
Paolo Silvestri
Keyword(s):  
Three Dimensional ◽  
Journal Bearings ◽  
Radial Clearance ◽  
Small Scale ◽  
Test Rig ◽  
Small Clearance ◽  
Dimensional Simulation ◽  
Dynamic Loading Conditions ◽  
Static And Dynamic Loading ◽  
Good Agreement

The aim of the present research is to characterize both experimentally and numerically journal bearings with low radial clearances for rotors in small-scale applications (e.g., microgas turbines); their diameter is in the order of ten millimetres, leading to very small dimensional clearances when the typical relative ones (order of 1/1000) are employed; investigating this particular class of journal bearings under static and dynamic loading conditions represents something unexplored. To this goal, a suitable test rig was designed and the performance of its bearings was investigated under steady load. For the sake of comparison, numerical simulations of the lubrication were also performed by means of a simplified model. The original test rig adopted is a commercial rotor kit (RK), but substantial modifications were carried out in order to allow significant measurements. Indeed, the relative radial clearance of RK4 RK bearings is about 2/100, while it is around 1/1000 in industrial bearings. Therefore, the same original RK bearings are employed in this new test rig, but a new shaft was designed to reduce their original clearance. The new custom shaft allows to study bearing behaviour for different clearances, since it is equipped with interchangeable journals. Experimental data obtained by this test rig are then compared with further results of more sophisticated simulations. They were carried out by means of an in-house developed finite element (FEM) code, suitable for thermoelasto-hydrodynamic (TEHD) analysis of journal bearings both in static and dynamic conditions. In this paper, bearing static performances are studied to assess the reliability of the experimental journal location predictions by comparing them with the ones coming from already validated numerical codes. Such comparisons are presented both for large and small clearance bearings of original and modified RKs, respectively. Good agreement is found only for the modified RK equipped with small clearance bearings (relative radial clearance 8/1000), as expected. In comparison with two-dimensional lubrication analysis, three-dimensional simulation improves prediction of journal location and correlation with experimental results.

Heat-flow experiments in liquid 4He with a variable cylindrical geometry

1987 ◽  
Vol 174 ◽  
pp. 209-231 ◽  
Author(s):  
H. Gao ◽  
G. Metcalfe ◽  
T. Jung ◽  
R. P. Behringer
Keyword(s):  
Cylindrical Geometry ◽  
Small Scale ◽  
Onset Of Convection ◽  
Aspect Ratios ◽  
A Value ◽  
Prandtl Numbers ◽  
Convection Rolls ◽  
Flow Experiments ◽  
Increasing Function ◽  
Good Agreement

This paper first describes an apparatus for measuring the Nusselt number N versus the Rayleigh number R of convecting normal liquid 4He layers. The most important feature of the apparatus is its ability to provide layers of different heights d, and hence different aspect ratios [Gcy ]. The horizontal cross-section of each layer is circular, and [Gcy ] is defined by [Gcy ] = D/2d where D is the diameter of the layer. We report results for 2.4 [les ] [Gcy ] [les ] 16 and for Prandtl numbers Pr spanning 0.5 [lsim ] Pr [lsim ] 0.9 These results are presented in terms of the slope N1 = RcdN/dR evaluated just above the onset of convection at Rc. We find that N1 is only a slowly increasing function of [Gcy ] in the range 6 [lsim ] [Gcy ] [lsim ] 16, and that it has a value there which is quite close to 0.72. This value of N1 is in good agreement with variational calcuations by Ahlers et al. (1981) pertinent to parallel convection rolls in cylindrical geometry. Particularly for [Gcy ] [lsim ] 6, we find additional small-scale structure in N1 associated with changes in the number of convection rolls with changing [Gcy ]. An additional test of the linearzied hydrodynamics is given by measurements of Rc. We find good agreement between theory and our data for Rc.

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.

Sign in / Sign up

Export Citation Format

Share Document