Wave propagation in single-walled carbon nanotube under longitudinal magnetic field using nonlocal Euler-Bernoulli beam theory

In the present work, the effect of longitudinal magnetic field on wave dispersion characteristics of equivalent continuum structure (ECS) of single-walled carbon nanotubes (SWCNT) embedded in elastic medium is studied. The ECS is modelled as an Euler-Bernoulli beam. The chemical bonds between a SWCNT and the elastic medium are assumed to be formed. The elastic matrix is described by Pasternak foundation model, which accounts for both normal pressure and the transverse shear deformation. The governing equations of motion for the ECS of SWCNT under a longitudinal magnetic field are derived by considering the Lorentz magnetic force obtained from Maxwell's relations within the frame work of nonlocal elasticity theory. The wave propagation analysis is performed using spectral analysis. The results obtained show that the velocity of flexural waves in SWCNTs increases with the increase of longitudinal magnetic field exerted on it in the frequency range: 0-20 THz. The present analysis also shows that the flexural wave dispersion in the ECS of SWCNT obtained by local and nonlocal elasticity theories differ. It is found that the nonlocality reduces the wave velocity irrespective of the presence of the magnetic field and does not influences it in the higher frequency region. Further it is found that the presence of elastic matrix introduces the frequency band gap in flexural wave mode. The band gap in the flexural wave is found to independent of strength of the longitudinal magnetic field. (C) 2011 Elsevier Inc. All rights reserved.

Coupled polynomial field approach for elimination of flexure and torsion locking phenomena in the Timoshenko and Euler-Bernoulli curved beam elements

The curvature related locking phenomena in the out-of-plane deformation of Timoshenko and Euler-Bernoulli curved beam elements are demonstrated and a novel approach is proposed to circumvent them. Both flexure and Torsion locking phenomena are noticed in Timoshenko beam and torsion locking phenomenon alone in Euler-Bernoulli beam. Two locking-free curved beam finite element models are developed using coupled polynomial displacement field interpolations to eliminate these locking effects. The coupled polynomial interpolation fields are derived independently for Timoshenko and Euler-Bernoulli beam elements using the governing equations. The presented of penalty terms in the couple displacement fields incorporates the flexure-torsion coupling and flexure-shear coupling effects in an approximate manner and produce no spurious constraints in the extreme geometric limits of flexure, torsion and shear stiffness. the proposed couple polynomial finite element models, as special cases, reduce to the conventional Timoshenko beam element and Euler-Bernoulli beam element, respectively. These models are shown to perform consistently over a wide range of flexure-to-shear (EI/GA) and flexure-to-torsion (EI/GJ) stiffness ratios and are inherently devoid of flexure, torsion and shear locking phenomena. The efficacy, accuracy and reliability of the proposed models to straight and curved beam applications are demonstrated through numerical examples. (C) 2012 Elsevier B.V. All rights reserved.

Analogy Between Rotating Euler-Bernoulli and Timoshenko Beams and Stiff Strings

The governing differential equation of a rotating beam becomes the stiff-string equation if we assume uniform tension. We find the tension in the stiff string which yields the same frequency as a rotating cantilever beam with a prescribed rotating speed and identical uniform mass and stiffness. This tension varies for different modes and are found by solving a transcendental equation using bisection method. We also find the location along the rotating beam where equivalent constant tension for the stiff string acts for a given mode. Both Euler-Bernoulli and Timoshenko beams are considered for numerical results. The results provide physical insight into relation between rotating beams and stiff string which are useful for creating basis functions for approximate methods in vibration analysis of rotating beams.

Closed-form solutions for non-uniform Euler-Bernoulli free-free beams

In this paper, the free vibration of a non-uniform free-free Euler-Bernoulli beam is studied using an inverse problem approach. It is found that the fourth-order governing differential equation for such beams possess a fundamental closed-form solution for certain polynomial variations of the mass and stiffness. An infinite number of non-uniform free-free beams exist, with different mass and stiffness variations, but sharing the same fundamental frequency. A detailed study is conducted for linear, quadratic and cubic variations of mass, and on how to pre-select the internal nodes such that the closed-form solutions exist for the three cases. A special case is also considered where, at the internal nodes, external elastic constraints are present. The derived results are provided as benchmark solutions for the validation of non-uniform free-free beam numerical codes. (C) 2013 Elsevier Ltd. All rights reserved.

Modal tailoring and closed-form solutions for rotating non-uniform Euler-Bernoulli beams

In this paper, the free vibration of a rotating Euler-Bernoulli beam is studied using an inverse problem approach. We assume a polynomial mode shape function for a particular mode, which satisfies all the four boundary conditions of a rotating beam, along with the internal nodes. Using this assumed mode shape function, we determine the linear mass and fifth order stiffness variations of the beam which are typical of helicopter blades. Thus, it is found that an infinite number of such beams exist whose fourth order governing differential equation possess a closed form solution for certain polynomial variations of the mass and stiffness, for both cantilever and pinned-free boundary conditions corresponding to hingeless and articulated rotors, respectively. A detailed study is conducted for the first, second and third modes of a rotating cantilever beam and the first and second elastic modes of a rotating pinned-free beam, and on how to pre-select the internal nodes such that the closed-form solutions exist for these cases. The derived results can be used as benchmark solutions for the validation of rotating beam numerical methods and may also guide nodal tailoring. (C) 2014 Elsevier Ltd. All rights reserved.

Tailoring the second mode of Euler-Bernoulli beams: an analytical approach

In this paper, we study the inverse mode shape problem for an Euler-Bernoulli beam, using an analytical approach. The mass and stiffness variations are determined for a beam, having various boundary conditions, which has a prescribed polynomial second mode shape with an internal node. It is found that physically feasible rectangular cross-section beams which satisfy the inverse problem exist for a variety of boundary conditions. The effect of the location of the internal node on the mass and stiffness variations and on the deflection of the beam is studied. The derived functions are used to verify the p-version finite element code, for the cantilever boundary condition. The paper also presents the bounds on the location of the internal node, for a valid mass and stiffness variation, for any given boundary condition. The derived property variations, corresponding to a given mode shape and boundary condition, also provides a simple closed-form solution for a class of non-uniform Euler-Bernoulli beams. These closed-form solutions can also be used to check optimization algorithms proposed for modal tailoring.

Meshless Local Petrov-Galerkin Method for Rotating Euler-Bernoulli Beam

Free vibration problem of a rotating Euler-Bernoulli beam is solved with a truly meshless local Petrov-Galerkin method. Radial basis function and summation of two radial basis functions are used for interpolation. Radial basis function satisfies the Kronecker delta property and makes it simpler to apply the essential boundary conditions. Interpolation with summation of two radial basis functions increases the node carrying capacity within the sub-domain of the trial function and higher natural frequencies can be computed by selecting the complete domain as a sub-domain of the trial function. The mass and stiffness matrices are derived and numerical results for frequencies are obtained for a fixed-free beam and hinged-free beam simulating hingeless and articulated helicopter blades. Stiffness and mass distribution suitable for wind turbine blades are also considered. Results show an accurate match with existing literature.

Free vibration and wave propagation analysis of uniform and tapered rotating beams using spectrally formulated finite elements

This paper presents a formulation of an approximate spectral element for uniform and tapered rotating Euler-Bernoulli beams. The formulation takes into account the varying centrifugal force, mass and bending stiffness. The dynamic stiffness matrix is constructed using the weak form of the governing differential equation in the frequency domain, where two different interpolating functions for the transverse displacement are used for the element formulation. Both free vibration and wave propagation analysis is performed using the formulated elements. The studies show that the formulated element predicts results, that compare well with the solution available in the literature, at a fraction of the computational effort. In addition, for wave propagation analysis, the element shows superior convergence. (C) 2007 Elsevier Ltd. All rights reserved.

Response time measurement in flow induced signal generation on semiconductors

Measurable electrical signal is generated when a gas flows over a variety of solids, including doped semiconductors, even at the modest speed of a few meters per second. The underlying mechanism is an interesting interplay of Bernoulli's principle and the Seebeck effect. The electrical signal depends on the square of Mach number (M) and is proportional to the Seebeck coefficient (S) of the solids. Here we present experimental estimate of the response time of the signal rise and fall process, i.e. how fast the semiconductor materials respond to a steady flow as soon as it is set on or off. A theoretical model is also presented to understand the process and the dependence of the response time on the nature and physical dimensions of the semiconductor material used and they are compared with the experimental observations. (c) 2007 Elsevier B.V. All rights reserved.

Non local scale effects on ultrasonic wave characteristics nanorods

In this paper, the nonlocal elasticity theory has been incorporated into classical Euler-Bernoulli rod model to capture unique features of the nanorods under the umbrella of continuum mechanics theory. The strong effect of the nonlocal scale has been obtained which leads to substantially different wave behaviors of nanorods from those of macroscopic rods. Nonlocal Euler-Bernoulli bar model is developed for nanorods. Explicit expressions are derived for wavenumbers and wave speeds of nanorods. The analysis shows that the wave characteristics are highly over estimated by the classical rod model, which ignores the effect of small-length scale. The studies also shows that the nonlocal scale parameter introduces certain band gap region in axial wave mode where no wave propagation occurs. This is manifested in the spectrum cures as the region where the wavenumber tends to infinite (or wave speed tends to zero). The results can provide useful guidance for the study and design of the next generation of nanodevices that make use of the wave propagation properties of single-walled carbon nanotubes. (C) 2010 Elsevier B.V. All rights reserved.

Theoretical Estimation of Length Dependent In-Plane Stiffness of Single Walled Carbon Nanotubes Using the Nonlocal Elasticity Theory

In the present paper, Eringen's nonlocal elasticity theory is employed to evaluate the length dependent in-plane stiffness of single-walled carbon nanotubes (SWCNTs). The SWCNT is modeled as an Euler-Bernoulli beam and is analyzed for various boundary conditions to evaluate the length dependent in-plane stiffness. It has been found that the nonlocal scaling parameter has a significant effect on the length dependent in-plane stiffness of SWCNTs. It has been observed that as the nonlocal scale parameter increases the stiffness ratio of SWCNT decreases. In nonlocality, the cantilever SWCNT has high in-plane stiffness as compared to the simply-supported and the clamped cases.

Supercritical age-dependent branching Markov processes and their scaling limits

This paper studies the long-time behavior of the empirical distribution of age and normalized position of an age-dependent supercritical branching Markov process. The motion of each individual during its life is a random function of its age. It is shown that the empirical distribution of the age and the normalized position of all individuals alive at time t converges as t -> infinity to a deterministic product measure.

Active Vibration Suppression of One-dimensional Nonlinear Structures Using Optimal Dynamic Inversion

A flexible robot arm can be modeled as an Euler-Bernoulli beam which are infinite degrees of freedom (DOF) system. Proper control is needed to track the desired motion of a robotic arm. The infinite number of DOF of beams are reduced to finite number for controller implementation, which brings in error (due to their distributed nature). Therefore, to represent reality better distributed parameter systems (DPS) should be controlled using the systems partial differential equation (PDE) directly. In this paper, we propose to use a recently developed optimal dynamic inversion technique to design a controller to suppress nonlinear vibration of a beam. The method used in this paper determines control forces directly from the PDE model of the system. The formulation has better practical significance, because it leads to a closed form solution of the controller (hence avoids computational issues).

Spectral Finite Element Formulation for Nanorods via Nonlocal Continuum Mechanics

In this article, the Eringen's nonlocal elasticity theory has been incorporated into classical/local Bernoulli-Euler rod model to capture unique properties of the nanorods under the umbrella of continuum mechanics theory. The spectral finite element (SFE) formulation of nanorods is performed. SFE formulation is carried out and the exact shape functions (frequency dependent) and dynamic stiffness matrix are obtained as function of nonlocal scale parameter. It has been found that the small scale affects the exact shape functions and the elements of the dynamic stiffness matrix. The results presented in this paper can provide useful guidance for the study and design of the next generation of nanodevices that make use of the wave dispersion properties of carbon nanotubes.

Dynamics of rotating composite beams: A comparative study between CNT reinforced polymer composite beams and laminated composite beams using spectral finite elements

This paper presents a spectral finite element formulation for uniform and tapered rotating CNT embedded polymer composite beams. The exact solution to the governing differential equation of a rotating Euler-Bernoulli beam with maximum centrifugal force is used as an interpolating function for the spectral element formulation. Free vibration and wave propagation analysis is carried out using the formulated spectral element. The present study shows the substantial effect of volume fraction and L/D ratio of CNTs in a beam on the natural frequency, impulse response and wave propagation characteristics of the rotating beam. It is found that the CNTs embedded in the matrix can make the rotating beam non-dispersive in nature at higher rotation speeds. Embedded CNTs can significantly alter the dynamics of polymer-nanocomposite beams. The results are also compared with those obtained for carbon fiber reinforced laminated composite rotating beams. It is observed that CNT reinforced rotating beams are superior in performance compared to laminated composite rotating beams. © 2012 Elsevier Ltd. All rights reserved.