Resonance frequency of chiral single-walled carbon nanotubes using Timoshenko beam theory

This Letter develops a model that analyzes the resonant frequency of the chiral single-walled carbon nanotubes (SWCNTs) subjected to a thermal vibration by using Timoshenko beam model, including the effect of rotary inertia and shear deformation. The analytical solution is derived and the frequency equation is obtained. The results based on the beam model show that the frequency increases with decreasing the nanotube aspect ratio of length to diameter. In addition, the frequency obtained by Timoshenko beam model is lower than that calculated by Euler beam model. As the nanotube aspect ratio of length to diameter decreased, the discrepancy is more obvious. Furthermore, as the effect of thermal vibration increases, the frequency for chiral SWCNTs decreases.     

Vibration analysis of scanning thermal microscope probe nanomachining using Timoshenko beam theory

In this paper, the effect of thermal vibration on the resonant frequency of transverse vibration of scanning thermal microscope (SThM) cantilever probe is analyzed using the Timoshenko beam theory, including the effects of rotary inertia and shear deformation. The thermal vibration effect can be considered as an axial force and is dependent of temperature distribution of the probe. In this analysis, the temperature is assumed to be distributed in accordance with the constant, linear, and quadratic models along the probe length. The Rayleigh–Ritz method is used to solve the vibration problem of the probe. The numerical results show that the frequency obtained with the constant model is the highest, while it is the lowest for the quadratic model. The frequency of vibration modes of the probe increases with increasing the temperature of the probe. As the ratio of probe length to its thickness increases, the frequency of vibration modes decreases. In addition, the effects of rotary inertia and shear deformation on the frequency are significant, especially in higher order modes and smaller values of the ratio of the probe length to its thickness.     

Scale effects on thermal buckling properties of carbon nanotube

In this Letter, the thermal buckling properties of carbon nanotube with small scale effects are studied. Based on the nonlocal continuum theory and the Timoshenko beam model, the governing equation is derived and the nondimensional critical buckling temperature is presented. The influences of the scale coefficients, the ratio of the length to the diameter, the transverse shear deformation and rotary inertia are discussed. It can be observed that the small scale effects are significant and should be considered for thermal analysis of carbon nanotube. The nondimensional critical buckling temperature becomes higher with the ratio of length to diameter increasing. Furthermore, for smaller ratios of the length to the diameter and higher mode numbers, the transverse shear deformation and rotary inertia have remarkable influences on the thermal buckling behaviors.     

Vibration of scanning near-field optical microscope probe with laser-induced thermal effect using Timoshenko beam theory

The Timoshenko beam theory, including the effects of rotary inertia and shear deformation, is used to analyze the resonant frequency of lateral vibration of scanning near-field optical microscope (SNOM) tapered probe with a laser-induced thermal effect. In the analysis, the thermal effect can be considered as an axial force and is dependent of temperature distribution of the probe. The Rayleigh–Ritz method is used to solve the vibration problem of the probe. According to the analysis, the frequencies of the first three vibration modes increase when the thermal effect is taken into account. The effects of shear deformation and rotary inertia on the frequency ratio of a Timoshenko beam to an Euler beam increase when the mode number increases and the contact stiffness decreases. In addition, the frequency of mode 1 increases with increasing taper angle and coating thickness of the probe. Comparison of the frequency of a SNOM probe coated with Al, Ag, or Au, the highest is with Al coating, and the lowest is with Au coating.     

Free vibration analysis of Mindlin plates with linearly varying thickness

A method based on the variational principles in conjunction with the finite difference technique is applied to examine the free vibration characteristics of isotropic rectangular plates of linearly varying thickness by including the effects of transverse shear deformation and rotary inertia. The validity of the present approach is demonstrated by comparing the results with other solutions proposed for plates with uniform and linearly varying thickness. Natural frequencies and mode shapes of Mindlin plates with simply supported and clamped edges are determined for various values of relative thickness ratio and the taper thickness constant.     

Vibration and instability of a single-walled carbon nanotube in a three-dimensional magnetic field

The vibration and instability of a single-walled carbon nanotube (SWCNT) under a general magnetic field are of particular interest to researchers. Using nonlocal Rayleigh beam theory and Maxwell’s equations, the dimensionless governing equations pertinent to the free vibration of a SWCNT due to a general magnetic field were derived. The effects of the longitudinal and transverse magnetic fields on the longitudinal and flexural frequencies as well as their corresponding phase velocities were addressed and are discussed below. The critical transverse magnetic field (CTMF) associated with the lateral buckling of the SWCNT was obtained. The obtained results reveal that the CTMF increases with the longitudinally induced magnetic field. Further, its value decreases as the effect of the small-scale parameter increases.     

Free vibration analysis of a cracked shear deformable beam on a two-parameter elastic foundation using a lattice spring model

The free vibration of a shear deformable beam with multiple open edge cracks is studied using a lattice spring model (LSM). The beam is supported by a so-called two-parameter elastic foundation, where normal and shear foundation stiffnesses are considered. Through application of Timoshenko beam theory, the effects of transverse shear deformation and rotary inertia are taken into account. In the LSM, the beam is discretised into a one-dimensional assembly of segments interacting via rotational and shear springs. These springs represent the flexural and shear stiffnesses of the beam. The supporting action of the elastic foundation is described also by means of normal and shear springs acting on the centres of the segments. The relationship between stiffnesses of the springs and the elastic properties of the one-dimensional structure are identified by comparing the homogenised equations of motion of the discrete system and Timoshenko beam theory.     

Coupled transverse and axial vibratory behaviour of cracked beam with end mass and rotary inertia

In this paper the vibrational behaviour of a cracked cantilever beam carrying end mass and rotary inertia is investigated. The transverse and axial vibrations of the beam are coupled through the crack model. The values of the ratio between the cracked and uncracked beam natural frequencies, the frequency ratio, are examined and are shown to follow well-defined trends with respect to the crack parameters and end mass and rotary inertia. However, the coupling between the transverse and axial vibrations is shown to be weak for the first two modes for moderate values of crack depth ratio. High crack depth ratios appear to increase the coupling effects. Low aspect ratios are expected to show strong coupling effects and further investigation is recommended using Timoshenko beam theory.     

Free vibration of a rotating non-uniform beam with arbitrary pretwist, an elastically restrained root and a tip mass

The governing differential equations for the coupled bending-bending vibration of a rotating beam with a tip mass, arbitrary pretwist, an elastically restrained root, and rotating at a constant angular velocity, are derived by using Hamilton's principle. The frequency equation of the system is derived and expressed in terms of the transition matrix of the transformed vector characteristic governing equation. The influence of the tip mass, the rotary inertia of the tip mass, the rotating speed, the geometric parameter of the cross-section of the beam, the setting angle, and the pretwist parameters on the natural frequencies are investigated. The difference between the effects of the setting angle on the natural frequencies of pretwisted and unpretwisted beams is revealed.     

Rayleigh-Ritz vibration analysis of Mindlin plates

The Rayleigh-Ritz method is applied to the prediction of the natural frequencies of flexural vibration of square plates having general boundary conditions. The analysis is based on the use of Mindlin plate theory so that the effects of shear deformation and rotary inertia are included. The spatial variations of the plate deflection and the two rotations over the plate middle surface are assumed to be series of products of appropriate Timoshenko beam functions. Results are presented for a number of types of plate and these demonstrate the manner of convergence of the method as the number of terms in the assumed series increases.     

Stability and vibration of isotropic,orthotropic and laminated plates according to a higher-order shear deformation theory

A higher-order shear deformation theory is used to determine the natural frequencies and buckling loads of elastic plates. The theory accounts for parabolic distribution of the transverse shear strains through the thickness of the plate and rotary inertia. Exact solutions of simply supported plates are obtained and the results are compared with the exact solutions of three-dimensional elasticity theory, the first-order shear deformation theory, and the classical plate theory. The present theory predicts the frequencies and buckling loads more accurately when compared to the first-order and classical plate theories.     

Instability of single-walled carbon nanotubes conveying Jeffrey fluid

We report instability of the single-walled carbon nanotubes(SWCNT) filled with non-Newtonian Jeffrey fluid.Our objective is to get the influences of relaxation time and retardation time of the Jeffrey fluid on the vibration frequency and the decaying rate of the amplitude of carbon nanotubes.An elastic Euler-Bernoulli beam model is used to describe vibrations and structural instability of the carbon nanotubes.A new vibration equation of an SWCNT conveying Jeffrey fluid is first derived by employing Euler-Bernoulli beam equation and Cauchy momentum equation taking constitutive relation of Jeffrey fluid into account.The complex vibrating frequencies of the SWCNT are computed by solving a cubic eigenvalue problem based upon differential quadrature method(DQM).It is interesting to find from computational results that retardation time has significant influences on the vibration frequency and the decaying rate of the amplitude.Especially,the vibration frequency decreases and critical velocity increases with the retardation time.That is to say,longer retardation time makes the SWCNT more stable.     

Vibration characteristics of a rotating flexible arm with ACLD treatment

In this paper, the vibration behavior and control of a clamped–free rotating flexible cantilever arm with fully covered active constrained layer damping (ACLD) treatment are investigated. The arm is rotating in a horizontal plane in which the gravitational effect and rotary inertia are neglected. The stress–strain relationship for the viscoelastic material (VEM) is described by a complex shear modulus while the shear deformations in the two piezoelectric layers are neglected. Hamilton's principle in conjunction with finite element method (FEM) is used to derive the non-linear coupled differential equations of motion and the associated boundary conditions that describe the rigid hub angle rotation, the arm transverse displacement and the axial deformations of the three-layer composite. This refined model takes into account the effects of centrifugal stiffening due to the rotation of the beam and the potential energies of the VEM due to extension and bending. Active controllers are designed with PD for the piezosensor and actuator. The vibration frequencies and damping factors of the closed-loop beam/ACLD system are obtained after solving the characteristic complex eigenvalue problem numerically. The effects of different rotating speed, thickness ratio and loss factor of the VEM as well as different controller gain on the damped frequency and damping ratio are presented. The results of this study will be useful in the design of adaptive and smart structures for vibration suppression and control in rotating structures such as rotorcraft blades or robotic arms.     

Large amplitude vibration of an initially stressed cross ply laminated plates

In this paper, nonlinear equations of large amplitude vibration for a laminated plate in a general state of nonuniform initial stress are derived. The equations include the effects of transverse shear and rotary inertia. Using these derived governing equations, the large amplitude vibration behaviour of an initially stressed cross-ply laminated plate is studied. The initial stress is taken to be a combination of pure bending stress plus an extensional stress in the plane of the plate. The Galerkin method is used to reduce the governing nonlinear partial differential equations to ordinary nonlinear differential equations and the Runge-Kutta method is used to obtain the nonlinear to linear frequencies. The frequency responses of nonlinear vibration are sensitive of the vibration amplitude, aspect ratio, thickness ratio, modulus ratio, stack sequence, layer number and state of initial stresses. The effects of various parameters on the large amplitude free vibrations are presented.     

Application of elastically supported single-walled carbon nanotubes for sensing arbitrarily attached nano-objects

The potential application of SWCNTs as mass nanosensors is examined for a wide range of boundary conditions. The SWCNT is modeled via nonlocal Rayleigh, Timoshenko, and higher-order beam theories. The added nano-objects are considered as rigid solids, which are attached to the SWCNT. The mass weight and rotary inertial effects of such nanoparticles are appropriately incorporated into the nonlocal equations of motion of each model. The discrete governing equation pertinent to each model is obtained using an effective meshless technique. The key factor in design of a mass nanosensor is to determine the amount of frequency shift due to the added nanoparticles. Through an inclusive parametric study, the roles of slenderness ratio of the SWCNT, small-scale parameter, mass weight, number of the attached nanoparticles, and the boundary conditions of the SWCNT on the frequency shift ratio of the first flexural vibration mode of the SWCNT as a mass sensor are also discussed.     

Wave propagation in fluid-conveying viscoelastic single-walled carbon nanotubes with surface and nonlocal effects

In this paper, the transverse wave propagation in fluid-conveying viscoelastic single-walled carbon nanotubes is investigated based on nonlocal elasticity theory with consideration of surface effect. The governing equation is formulated utilizing nonlocal Euler-Bernoulli beam theory and Kelvin-Voigt model. Explicit wave dispersion relation is developed and wave phase velocities and frequencies are obtained. The effect of the fluid flow velocity, structural damping, surface effect, small scale effects and tube diameter on the wave propagation properties are discussed with different wave numbers. The wave frequency increases with the increase of fluid flow velocity, but decreases with the increases of tube diameter and wave number. The effect of surface elasticity and residual surface tension is more significant for small wave number and tube diameter. For larger values of wave number and nonlocal parameters, the real part of frequency ratio raises.     

The vibrations of generally orthotropic beams,a finite element approach

This paper presents two finite element models for the prediction of free vibrational natural frequencies of fixed-free beams of general orthotropy. The discrete models include the transverse shear deformation effect and the rotary inertia effect. Numerical studies show that the convergence rate of the approximations calculated from the finite element analysis is dependent on the fibre orientation.     

Analytical formulation and evaluation for free vibration of naturally curved and twisted beams

An analytical study for free vibration of naturally curved and twisted beams with uniform cross-sectional shapes is carried out using spatial curved beam theory based on the Washizu's static model. In the governing equations of motion of the beams, all displacement functions and the generalized warping coordinate are defined at the centroid axis and also the effects of rotary inertia, transverse shear deformations and torsion-related warping are included in the proposed model. Explicit analytical expressions are derived for the vibrating mode shapes of a curved, bending-torsional-shearing coupled beam under clamped-clamped boundary condition with the help of symbolic computing package Mathematica, and a process of searching is used to determine the natural frequencies. Comparisons of the present results with the FEM results using beam elements in ANSYS code show good accuracy in computation and validity of the model. Further, the present model is used for cylindrical helical springs with circular cross-section fixed at both ends, and the results indicate that the natural frequencies agree well with the theoretical and experimental results available.     

The effects of rotation,tip mass,and hub radius on the stability of beck's column

The stability of a cantilever beam subjected to a follower force at its free end and rotating at a uniform angular velocity is investigated. The beam is assumed to be offset from the axis of rotation, carries a tip mass at its free end, and undergoes deflection in a direction perpendicular to the plane of rotation. The equations of motion are formulated within the Euler-Bernoulli and Timoshenko beam theories for the case of a Kelvin model viscoelastic beam. The associated adjoint boundary value problems are derived and appropriate adjoint variational principles are introduced. These variational principles are used for the purpose of determining approximately the values of the critical flutter load of the system as it depends upon its damping parameters, tip mass and its rotary inertia, hub radius, and speed of rotation. The variation of the critical flutter load with these parameters is revealed in a series of several graphs. The numerical results show that the critical load can be reduced significantly due to (a) the transverse and rotary inertia of the tip mass and (b) increasing values of the internal damping parameter associated with the transverse shear deformation of the rotating beam.     

Free vibration analysis of planar curved beams by wave propagation

In this paper, a systematic approach for the free vibration analysis of a planar circular curved beam system is presented. The system considered includes multiple point discontinuities such as elastic supports, attached masses, and curvature changes. Neglecting transverse shear and rotary inertia, harmonic wave solutions are found for both extensional and inextensional curved beam models. Dispersion equations are obtained and cut-off frequencies are determined. Wave reflection and transmission matrices are formulated, accounting for general support conditions. These matrices are combined, with the aid of field transfer matrices, to provide a concise and efficient method for the free vibration problem of multi-span planar circular curved beams with general boundary conditions and supports. The solutions are exact since the effects of attenuating wave components are included in the formulation. Several examples are presented and compared with other methods.