scholarly journals Free Vibrations of a Cantilevered SWCNT with Distributed Mass in the Presence of Nonlocal Effect

2015 ◽  
Vol 2015 ◽  
pp. 1-12 ◽  
Author(s):  
M. A. De Rosa ◽  
M. Lippiello ◽  
H. D. Martin
Keyword(s):  
Nonlocal Elasticity ◽  
Equations Of Motion ◽  
Nonlocal Elasticity Theory ◽  
Added Mass ◽  
Free Vibrations ◽  
Point Of View ◽  
Vibration Modes ◽  
Nonlocal Effects ◽  
Single Walled Carbon Nanotube ◽  
Parametric Relationships

The Hamilton principle is applied to deduce the free vibration frequencies of a cantilever single-walled carbon nanotube (SWCNT) in the presence of an added mass, which can be distributed along an arbitrary part of the span. The nonlocal elasticity theory by Eringen has been employed, in order to take into account the nanoscale effects. An exact formulation leads to the equations of motion, which can be solved to give the frequencies and the corresponding vibration modes. Moreover, two approximate semianalytical methods are also illustrated, which can provide quick parametric relationships. From a more practical point of view, the problem of detecting the mass of the attached particle has been solved by calculating the relative frequency shift due to the presence of the added mass: from it, the mass value can be easily deduced. The paper ends with some numerical examples, in which the nonlocal effects are thoroughly investigated.

scholarly journals Parametric vibrations of graphene sheets based on the double mode model and the nonlocal elasticity theory

2021 ◽  
Author(s):  
Jan Awrejcewicz ◽  
Grzegorz Kudra ◽  
Olga Mazur
Keyword(s):  
Elasticity Theory ◽  
Nonlocal Elasticity ◽  
Nonlinear Oscillations ◽  
Scale Effects ◽  
Equations Of Motion ◽  
Nonlocal Elasticity Theory ◽  
Small Scale ◽  
Largest Lyapunov Exponent ◽  
Graphene Sheets ◽  
Parametric Vibrations

AbstractParametric vibrations of the single-layered graphene sheet (SLGS) are studied in the presented work. The equations of motion govern geometrically nonlinear oscillations. The appearance of small effects is analysed due to the application of the nonlocal elasticity theory. The approach is developed for rectangular simply supported small-scale plate and it employs the Bubnov–Galerkin method with a double mode model, which reduces the problem to investigation of the system of the second-order ordinary differential equations (ODEs). The dynamic behaviour of the micro/nanoplate with varying excitation parameter is analysed to determine the chaotic regimes. As well the influence of small-scale effects to change the nature of vibrations is studied. The bifurcation diagrams, phase plots, Poincaré sections and the largest Lyapunov exponent are constructed and analysed. It is established that the use of nonlocal equations in the dynamic analysis of graphene sheets leads to a significant alteration in the character of oscillations, including the appearance of chaotic attractors.

Delayed-Acceleration Feedback for Active-Multimode Vibration Control of Cantilever Beams

2007 ◽  
Author(s):  
Khaled A. Alhazza ◽  
Ali H. Nayfeh ◽  
Mohammed F. Daqaq
Keyword(s):  
Cantilever Beam ◽  
Controller Design ◽  
Equations Of Motion ◽  
Time Integration ◽  
Free Vibrations ◽  
Delayed Feedback Control ◽  
Vibration Modes ◽  
Stability Boundaries ◽  
Single Input Single Output ◽  
The Stability

We present a single-input single-output multimode delayed-feedback control methodology to mitigate the free vibrations of a flexible cantilever beam. For the purpose of controller design and stability analysis, we consider a reduced-order model consisting of the first n vibration modes. The temporal variation of these modes is represented by a set of nonlinearly-coupled ordinary-differential equations that capture the evolving dynamics of the beam. Considering a linearized version of these equations, we derive a set of analytical conditions that are solved numerically to assess the stability of the closed-loop system. To verify these conditions, we characterize the stability boundaries using the first two vibration modes and compare them to damping contours obtained by long-time integration of the full nonlinear equations of motion. Simulations show excellent agreement between both approaches. We analyze the effect of the size and location of the piezoelectric patch and the location of the sensor on the stability of the response. We show that the stability boundaries are highly dependent on these parameters. Finally, we implement the controller on a cantilever beam for different controller gain-delay combinations and assess the performance using time histories of the beam response. Numerical simulations clearly demonstrate the controller ability to mitigate vibrations emanating from multiple modes simultaneously.

scholarly journals Electromechanical Analysis of Flexoelectric Nanosensors Based on Nonlocal Elasticity Theory

Micromachines ◽  
2020 ◽  
Vol 11 (12) ◽  
pp. 1077
Author(s):  
Yaxuan Su ◽  
Zhidong Zhou
Keyword(s):  
Elasticity Theory ◽  
Electric Potential ◽  
Nonlocal Elasticity ◽  
Classical Model ◽  
Nonlocal Elasticity Theory ◽  
Vital Role ◽  
Generalized Variational Principle ◽  
Scaling Effects ◽  
Nonlocal Effects ◽  
External Loads

Flexoelectric materials have played an increasingly vital role in nanoscale sensors, actuators, and energy harvesters due to their scaling effects. In this paper, the nonlocal effects on flexoelectric nanosensors are considered in order to investigate the coupling responses of beam structures. This nonlocal elasticity theory involves the nonlocal stress, which captures the effects of nonlocal and long-range interactions, as well as the strain gradient stress. Based on the electric Gibbs free energy, the governing equations and related boundary conditions are deduced via the generalized variational principle for flexoelectric nanobeams subjected to several typical external loads. The closed-form expressions of the deflection and induced electric potential (voltage) values of flexoelectric sensors are obtained. The numerical results show that the nonlocal effects have a considerable influence on the induced electric potential of flexoelectric sensors subjected to general transverse forces. Moreover, the induced electric potential values of flexoelectric sensors calculated by the nonlocal model may be smaller or larger than those calculated by the classical model, depending on the category of applied loads. The present research indicates that nonlocal effects should be considered in order to understand or design basic nano-electromechanical components subjected to various external loads.

ON THE DYNAMICS OF A BEAM PARTIALLY SUPPORTED BY AN ELASTIC FOUNDATION: AN EXACT SOLUTION-SET

2013 ◽  
Vol 13 (08) ◽  
pp. 1350045 ◽  
Author(s):  
ANTONIO CAZZANI
Keyword(s):  
Elastic Foundation ◽  
Elastic Stiffness ◽  
Free Vibrations ◽  
Point Of View ◽  
Beam Model ◽  
Vibration Modes ◽  
Rigid Body Modes ◽  
Transcendental Functions ◽  
First Time ◽  
Euler Bernoulli Beam

Free vibrations of straight beams which are partially supported by an elastic foundation are analyzed. For the sake of simplicity, only the Euler–Bernoulli beam model coupled with a Winkler-type elastic foundation is considered. This structural system can be used to study, in a rather accurate way, the dynamic response of partially embedded piles (like those used for telecommunications) when dealing with the problem of identifying their mechanical properties during operative conditions. The study makes clear that different kinds of vibration modes may occur in the part of the beam which is supported by the continuous elastic foundation: indeed apart from the classical modes, corresponding to the dynamics of a free beam, it is possible to have vibration modes which are similar to the static deflection of a beam on an elastic support or even corresponding to rigid-body modes. For the same beam it is shown that transition between these vibration modes can appear when switching from the fundamental natural frequency to subsequent ones. This effect is the focus of the presented numerical examples. In particular, the analytic expression of the transcendental functions governing the vibration modes, and of the coefficients of the eigenfunctions for all occurring cases, are given here — to the best of the author's knowledge — for the first time. From the practical point of view, the reported results allow to define a suitable range of the elastic stiffness parameter such that the behavior of a partially supported beam can be conveniently approximated with that of a single-span beam, having one built-in end and the other free.

Nonlinear Vibration Analysis of Single-Walled Carbon Nanotube With Shell Model Based on the Nonlocal Elasticity Theory

2016 ◽  
Vol 11 (1) ◽  
Author(s):  
P. Soltani ◽  
J. Saberian ◽  
R. Bahramian
Keyword(s):  
Carbon Nanotube ◽  
Differential Equations ◽  
Nonlinear Vibration ◽  
Nonlocal Elasticity ◽  
Geometrical Nonlinearity ◽  
Nonlocal Parameter ◽  
Nonlocal Elasticity Theory ◽  
Single Walled Carbon Nanotube ◽  
Single Walled Carbon ◽  
Aspect Ratios

In this paper, nonlinear vibration of a single-walled carbon nanotube (SWCNT) with simply supported ends is investigated based on von Karman's geometric nonlinearity and nonlocal shell theory. The SWCNT is designated as an individual shell, and the Donnell's formulations of a cylindrical shell are used to obtain the governing equations. The Galerkin's procedure is used to discretized partial differential equations (PDEs) into the ordinary differential equations (ODEs) of motion, and the method of averaging is applied to obtain an analytical solution of the nonlinear vibration of (10,0), (20,0), and (30,0) zigzag SWCNTs. The effects of the nonlocal parameters, nonlinear parameters, different aspect ratios, and different circumferential wave numbers are investigated. The results of the classical and the nonlocal models are compared with different nonlocal elasticity constants (e0a). It is shown that the nonlocal parameter predicts different resonant frequencies in comparison to the local models. The softening and/or hardening nonlinear behaviors of the CNTs may change against the nonlocal parameters. Hence, considering the geometrical nonlinearity and the nonlocal elasticity effects, the dynamical models of the SWCNTs predict their vibration behaviors accurately and should not be ignored during theoretical modeling.

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