Showing papers on "Rotary inertia published in 2010"

Nonlinear free vibration of single-walled carbon nanotubes using nonlocal Timoshenko beam theory

Abstract: Nonlinear free vibration of single-walled carbon nanotubes (SWCNTs) is studied in this paper based on von Karman geometric nonlinearity and Eringen's nonlocal elasticity theory. The SWCNTs are modeled as nanobeams where the effects of transverse shear deformation and rotary inertia are considered within the framework of Timoshenko beam theory. The governing equations and boundary conditions are derived by using the Hamilton's principle. The differential quadrature (DQ) method is employed to discretize the nonlinear governing equations which are then solved by a direct iterative method to obtain the nonlinear vibration frequencies of SWCNTs with different boundary conditions. Zigzag (5, 0), (8, 0), (9, 0) and (11, 0) SWCNTs are considered in numerical calculations and the elastic modulus is obtained through molecular mechanics (MM) simulation. A detailed parametric study is conducted to study the influences of nonlocal parameter, length and radius of the SWCNTs and end supports on the nonlinear free vibration characteristics of SWCNTs.

Explicit analytical approximation to large-amplitude non-linear oscillations of a uniform cantilever beam carrying an intermediate lumped mass and rotary inertia

Abstract: This paper is concerned with analytical treatment of non-linear oscillations of planar, flexural large amplitude free vibrations of a slender, inextensible cantilever beam carrying a lumped mass with rotary inertia at an intermediate position along its span. An analytic approximate technique, namely Optimal Homotopy Asymptotic Method (OHAM) is employed for this purpose. It is proved that OHAM provide accurate solutions for large amplitudes and large modal constants in the considered nonlinear equations, when other classical methods fail. Our procedure provides us with a convenient way to optimally control the convergence of solution, such that the accuracy is always guaranteed. An excellent agreement of the approximate frequencies and periodic solutions with the numerical results and published results has been demonstrated. Two examples are given and the results reveal that this procedure is very effective, simple and accurate. This paper demonstrates the general validity and the great potential of the OHAM for solving strongly nonlinear problems.

Frequency analysis of nanotubes with consideration of surface effects

Abstract: Consideration of surface effects in microscaled and nanoscaled materials is important for accurate prediction of their dynamic behavior. In this study, the Timoshenko beam model is modified to include the surface effects and used to analyze the vibration of nanotubes as well as calculate their natural frequencies. The thin surface layers have been taken into account for rotary inertia computation. Through an example it is shown that dynamic behavior of nanoscaled tubes with consideration of surface effects considerably deviates from the results obtained by classical theories. Plots illustrating such deviations are given to support the conclusions.

Vibration analysis of a rotating tapered Timoshenko beam using DTM

Abstract: In this study, free vibration analysis of a rotating, tapered Timoshenko beam that undergoes flapwise bending vibration is performed. Derivation of the equations of motion of a rotating, uniform Timoshenko beam was made step by step in a previous work of the authors. Therefore, differential equations of motion are given directly without making any derivations in this paper. The parameters for the hub radius, rotational speed, taper ratio, rotary inertia, shear deformation and slenderness ratio are incorporated into the equations of motion. In the solution part, an efficient mathematical technique called the Differential Transform Method, DTM, is used. Finally, using the computer package Mathematica, the natural frequencies are calculated and the effects of the incorporated parameters are examined. Moreover, numerical examples are solved to make comparisons with the existing results in open literature and it is observed that the agreement between the results is very good.

Mindlin plate theory for damage detection: imaging of flexural inhomogeneities.

Abstract: The scattering of plate waves by localized damage or defects that can be modeled as flexural inhomogeneities is examined within the framework of Mindlin plate theory. These inhomogeneities are characterized by variations in one or more of the four plate-theory parameters: the bending stiffness, shear stiffness, rotary inertia, and transverse inertia. It is shown that the Born approximation for the scattered field leads to a plate-theory analog of the Fourier diffraction theorem, which relates the far-field scattering amplitude to the spatial Fourier transform of the inhomogeneity variations. The application of this result is illustrated by using synthetic data derived for an idealized model of a delamination as a flexural inhomogeneity, ignoring mode coupling effects. A computationally efficient implementation of the filtered back-propagation algorithm, based on the eigensystem of the scattering operator, is employed for image reconstruction. The implications for in-situ imaging of structural damage in plate-like structures are briefly discussed, and some directions for further work are indicated.

Dynamic stability of axially accelerating Timoshenko beam: Averaging method

Abstract: This study investigates dynamic stability in transverse parametric vibrations of an axially accelerating tensioned beam of Timoshenko model on simple supports. The axial speed is assumed as a harmonic fluctuation about the constant mean speed. The Galerkin method is applied to discretize the governing equation into a finite set of ordinary differential equations. The method of averaging is applied to analyze the instability phenomena caused by subharmonic and combination resonance. Numerical examples demonstrate the effects of the mean axial speed, bending stiffness, rotary inertia and shear modulus on the instability boundaries.

Nonlocal elasticity effect on vibration characteristics of protein microtubules

Abstract: For various cellular functions of microtubules (MTs), vibration of microtubules is one of the issues of major concern. In this paper, the vibration characteristics of protein microtubules (MTs) are examined based on a nonlocal Timoshenko beam model and using the wave propagation approach. The small scale effect on MTs wave propagation dispersion relation is explicitly revealed for different MTs wave numbers by theoretical analyses and numerical simulations. The research work reveals the significance of the effects of small scale, transverse shear deformation and rotary inertia on vibrations characteristics of protein microtubules. It is believed that the present model offers a simple and effective new approach to studying vibration characteristics of microtubules.

The Influence of shearing and rotary inertia on the resonant properties of gold nanowires

Abstract: In a previous publication [ P. A. T. Olsson, J. Appl. Phys. 108, 034318 (2010) ], molecular dynamics (MD) simulations have been performed to study the resonant properties of gold nanowires. It has been documented in the aforementioned publication that the eigenfrequencies of the fundamental mode follows the continuum mechanically predicted behavior when Bernoulli–Euler beam theory is used, whereas the higher order modes tend to be low in comparison to Bernoulli–Euler beam theory predictions. In this work, we have studied the resonant properties of unstressed and prestressed nanowires to explain why the eigenfrequencies of the fundamental mode follows the behavior predicted by Bernoulli–Euler beam theory while those of higher order modes are low in comparison. This is done by employing Timoshenko beam theory and studying the nanowire deformations for different modes. We find good agreement between the MD results and Timoshenko predictions due to the increasing importance of shearing and rotary inertia for higher order resonant modes. Furthermore, we argue that this type of behavior is merely a geometric effect stemming from low aspect ratio for the considered structures as a converging type of behavior is found when the aspect ratios fall between 15 and 20. Finally, we have found that classical Timoshenko beam theory that neglects nanoscale surface effects is able to, simply through utilization of the size dependent Young’s modulus, capture the dynamic properties of the gold nanowires as calculated through MD. (Less)

Out-of-plane free vibration analysis of functionally graded circular curved beams supported on elastic foundation

Abstract: As a first endeavor, the out-of-plane free vibration analysis of thin-to-moderately thick functionally graded (FG) circular curved beams supported on two-parameter elastic foundation is presented. The formulation is derived based on the first-order shear deformation theory (FSDT), which includes the effects of shear deformation and rotary inertia due to both torsional and flexural vibrations. The material properties are assumed to be graded in the direction normal to the plane of the beam curvature. The differential quadrature method (DQM), as an efficient and accurate method, is employed to discretize the equations of motion and the related boundary conditions. In order to assure the accuracy of the formulation and the method of solution, convergence behavior of the nondimensional natural frequencies is examined for FG circular curved beams and comparison studies with those of isotropic curved beams, available in the literature, are performed. The effects of the elastic foundation coefficients, boundary conditions, the material graded index and different geometrical parameters on the natural frequency parameters of the FG circular curved beams are investigated. The new results can be used as benchmark solutions for future research works.

Stochastic Free Vibration of Laminated Composite Plates in Thermal Environments

Abstract: This work deals with the stochastic free vibration of laminated composite plates subjected to a thermal loading with general boundary conditions by taking into account the randomness in lamina material properties and thermal expansion coefficients The system equations have been derived based on higher order shear deformation theory incorporating rotary inertia effects A C 0 finite element method is used for treating the random eigenvalue problem A mean centered first-order perturbation technique is adopted to examine the stochastic characteristics of thermal free vibration response The results have been compared with those available in the literature and independent Monte Carlo simulation

Application of differential transform in free vibration analysis of timoshenko beams resting on two-parameter elastic foundation

Abstract: Non-prismatic beams have received great attention from engineers due to their capability in optimizing the strength and weight of the structure. In recent years, many researchers have worked on engineering problems related to static and dynamic analysis of either Euler–Bernoulli [1–3] or Timoshenko [4,5] beams. For short thick beams and rotating machineries, the Timoshenko beam theory presents a more realistic model in comparison with the Euler–Bernoulli beam theory due to both the shear deformation and rotary inertia. When encountering problems such as beams on different types of elastic foundations, including buried pipelines, shallow foundations, and piles, understanding the static and dynamic response of Timoshenko beams on elastic foundations seems to be of great significance. To achieve this, a perception of the interaction between the soil and structural elements is required. Some researchers have investigated the effect of different soils on structural members. Mahmood and Ahmed [6] evaluated the sensitivity of concrete-reinforced beam structures to different behaviors of the soil and the interface layer when influenced by an earthquake excitement. Due to the complex behavior of different types of soils, it is difficult to obtain analytical solutions; therefore, simplified mechanical models have been proposed by several researchers, among which are one-parameter, two-parameter ,and three-parameter elastic foundations. First, Winkler [7] proposed a simple model with only one parameter, that is, the stiffness of linearly elastic and mutually independent vertical springs. Yankelevesky and Eisenberger [8] performed an exact analytical solution for a finite element beam-column resting on Winkler foundation leading to derivation of exact static stiffness matrix. Later, Yankelevsky

Energy Expressions and Free Vibration Analysis of a Rotating Uniform Timoshenko Beam Featuring Bending—torsion Coupling

Abstract: In this study; free vibration analysis of a uniform, rotating, cantilever Timoshenko beam featuring coupling between flapwise bending and torsional vibrations is performed. At the beginning of the study, kinetic and potential energy expressions of a rotating Timoshenko beam having single cross-sectional symmetry are derived by using several explanatory tables and figures. In the following section, Hamilton’s principle is applied to the derived energy expressions to obtain the governing differential equations of motion. The parameters for the hub radius, rotational speed, rotary inertia, shear deformation, slenderness ratio and bending—torsion coupling are incorporated into the equations of motion. In the solution part, an efficient mathematical technique, called the differential transform method, is used to solve the governing differential equations of motion. Using the computer package Mathematica, the mode shapes are plotted and the effects of the incorporated parameters on the natural frequencies are i...

Free vibration analysis of a embarked rotating composite shaft using the hp- version of the fem

Abstract: This paper presents the study of the vibratory behavior of rotating composite shafts. The composite shaft contains isotropic rigid disks and is supported by bearings that are modeled as springs and viscous dampers. An hp-version of the Finite Element Method (FEM) is used to model the structure. A hierarchical finite element of beam type with six degrees of freedom per node is developed. The assembly is made by the standard version of the finite element method for several elements. A theoretical study allows the establishment of the kinetic energy and the strain energy of the system (shaft, disk and bearings) necessary to the result of the equations of motion. In this study the transverse shear deformation, rotary inertia and gyroscopic effects, as well as the coupling effect due to the lamination of composite layers have been incorporated. A program is elaborate for the calculation of the eigen-frequencies and critical speeds of the system. The results obtained compared with those available in the literature show the speed of convergence, the exactitude and the effectiveness of the method used. Several examples are treated, and a discussion is established to determine the influence of the various parameters and boundary conditions.

Controller of electric motor having function of estimating inertia and friction simultaneously

Abstract: A controller estimates Coulomb friction itself together with inertia and viscous friction, and reduces the influence of the Coulomb friction on the accuracy of the estimated inertia. In addition, the controller estimates inertia, viscous friction and Coulomb friction simultaneously with sequential adaptation in which a Fourier transformer is not used but an inverse transfer function model is used in order to minimize the estimated error. Data sampled for a predetermined time need not be accumulated, as a result, a large amount of data memory is unnecessary.

Free vibration of a paraboloidal shell of revolution including shear deformation and rotary inertia

Abstract: The governing strain-displacement and curvature-displacement equations for paraboloidal shells including shear deformation and rotary inertia are solved for free vibration of closed shells. The finite element method is used to obtain three-dimensional frequency of vibration solutions for a variety of boundary conditions, free, fixed and simply supported. Assumptions concerning the circumferential vibrational behavior are incorporated that reduce the analysis to a single coordinate and the element shape function is formulated using the meridional coordinate. The results for frequency of vibration compare favorably with the available literature. Selected results for frequency of vibration are presented in tabular form for several shell parameters, including free, pinned and fixed boundary conditions. Representative mode shapes are plotted for a fixed boundary condition.

Geometrically nonlinear transient analysis of functionally graded shell panels using a higher-order finite element formulation

Abstract: Nonlinear transient analysis of functionally graded curved panels is carried out employing a higher-order C0 finite element formulation. The element consists of nine degrees-of-freedom per node with higher-order terms in the Taylor’s series expansion which represents the higher-order transverse cross sectional deformation modes. The formulation includes Sanders’ approximation for doubly curved shells considering the effects of rotary inertia, transverse shear and moderately large rotations in the von Karman sense. A realistic parabolic distribution of transverse shear strains through the shell thickness is assumed and the use of shear correction factor is avoided. The accuracy of the formulation is validated by comparing the results with those available in the literature. The transient dynamic responses of the functionally graded shell panels are investigated by varying the volume fraction index using a simple power law distribution. Material properties are assumed to be temperature-independent and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. Heat conduction between ceramic and metal constituents is neglected. Effects of different panel geometry parameters, boundary conditions and loadings are studied. Key words: Functionally graded materials, higher-order formulation, geometric nonlinearity, transient analysis.

Chatter problems in micro- and macrocutting operations, existing models, and influential parameters—a review

Abstract: Chatter presents one of the main problems in quality of machined surfaces limiting tool life, productivity, and tolerances. Chatter in milling and turning operations has been extensively analyzed; however, drilling operations have been neglected due to the complexity of drilling tools and problems that develop in the modeling of the tool. In this paper, an overview of chatter vibrations and chatter suppression in drilling has been presented. Models such as torsional–axial model, bending model, and the combination of axial and bending models have been presented showing the different effects each considers. Influence of parameters such as drill geometry, chisel edge, drill flank, pilot hole, margin engagement, stick–slip interaction, gyroscopic effect, and rotary inertia effects has been incorporated into the various models. In addition, the problem of chatter analysis and suppression in micromachining processes has been investigated. The differences in macro- and micro-scale problems have been addressed giving an overview of the current research directions and future work for both areas. The paper presents experimental as well as simulation data with the comparison of the two for clear understanding of the complex problems of chatter in drilling operations.