Vibration analysis of Nano-Rotor's Blade applying Eringen nonlocal elasticity and generalized differential quadrature method

This study investigates the small scale effect on the flapwise bending vibrations of a rotating nanoplate. The nanoplate is modeled with a classical plate theory and considering cantilever and propped cantilever boundary conditions. Due to the rotation, the axial forces are included in the model as true spatial variation. Hamilton's principle is used to derive the governing equation and boundary conditions of the classical plate theory based on Eringen's nonlocal elasticity theory. The generalized differential quadrature method is employed to solve the governing equation. The effect of small-scale parameter, non-dimensional angular velocity, non-dimensional hub radius, aspect ratio, and different boundary conditions in the first four non-dimensional frequencies is discussed. Due to considering rotating effects, results of this study are applicable in nano-machines such as nano-motors and nano-turbines and other nanostructures.     

Predicting nonlinear dynamic response of internal cantilever beam system on a steadily rotating ring

This paper is focused on nonlinear dynamic response of internal cantilever beam system on a steadily rotating ring via a nonlinear dynamic model. The analytical approximate solutions to the oscillation motion are obtained by combining Newton linearization with Galerkin's method. Numerical solutions could be obtained by using the shooting method on the exact governing equation. Compared with numerical solutions, the approximate analytical solutions here show excellent accuracy and rapid convergence. Two different kinds of oscillating internal cantilever beam system on a steadily rotating ring are investigated by using the analytical approximate solutions. These include symmetric vibration through three equilibrium points, and asymmetric vibration through the only trivial equilibrium point. The effects of geometric and physical parameters on dynamic response are useful and can be easily applied to design practical engineering structures. In particular, the ring angular velocity plays a significant role on the period and periodic solution of the beam oscillation. In conclusion, the analytical approximate solutions presented here are sufficiently precise for a wide range of oscillation amplitudes.     

Vibration of fluid-conveying nanotubes subjected to magnetic field based on the thin-walled Timoshenko beam theory

In this article, the flutter vibrations of fluid-conveying thin-walled nanotubes subjected to magnetic field is investigated. For modeling fluid structure interaction, the nonlocal strain gradient thin-walled Timoshenko beam model, Knudsen number and magnetic nanoflow are assumed. The Knudsen number is considered to analyze the slip boundary conditions between the fluid-flow and the nanotube's wall, and the average velocity correction parameter is utilized to earn the modified flow velocity of nano-flow. Based on the extended Hamilton's principle, the size-dependent governing equations and associated boundary conditions are derived. The coupled equations of motion are transformed to a general eigenvalue problem by applying extended Galerkin technique under the cantilever end conditions. The influences of nonlocal parameter, strain gradient length scale, magnetic nanoflow, longitudinal magnetic field, Knudsen number on the eigenvalues and critical flutter velocity of the nanotubes are studied.     

Dynamic analysis of non-uniform taper bars in post-elastic regime under body force loading

This paper presents a simulation study of the free flexural vibration behavior of non-uniform taper bars of circular and rectangular cross-section under body force loading due to gravity. The loading is controlled statically to take the bar to its post-elastic state so as to predict its dynamic behavior in the presence of plastic deformation. Hence the analysis is carried out in two parts; first the static problem under axial gravity loading is solved, then the dynamic problem is solved in this loaded condition. Appropriate variational method is employed to derive the set of governing equations for both the problems. The formulation is based on unknown displacement field which is approximated by finite linear combinations of orthogonal admissible functions. The present method is validated successfully with a well-known finite element package. Results are presented to investigate the effect of shape and size on the dynamic behavior of non-uniform taper bars. The study can be extended to study the post-elastic dynamic behavior of other related problems such as rotating beams and rotating disks.     

On wave propagation in two-dimensional functionally graded porous rotating nano-beams using a general nonlocal higher-order beam model

This paper studies the wave propagation of two-dimensional functionally graded (2D-FG) porous rotating nano-beams for the first time. The rotating nano-beams are made of two different materials, and the material properties of the nano-beams alter both in the thickness and length directions. The general nonlocal theory (GNT) in conjunction with Reddy's beam model are employed to formulate the size-dependent model. The GNT efficiently models the dispersions of acoustic waves when two independent nonlocal fields are modelled for the longitudinal and transverse acoustic waves. The governing equations of motion for the 2D-FG porous rotating nano-beams are established using Hamilton's principle as a function of the axial force due to centrifugal stiffening and displacement. The analytic solution is applied to obtain the results and solve the governing equations. The effect of the features of different parameters such as functionally graded power indexes, porosity, angular velocity, and material variation on the wave propagation characteristics of the rotating nano-beams are discussed in detail.     

Thermal-mechanical vibration and instability of a fluid-conveying single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity theory

Based on the theories of thermal elasticity mechanics and nonlocal elasticity, an elastic Bernoulli-Euler beam model is developed for thermal-mechanical vibration and buckling instability of a single-walled carbon nanotube (SWCNT) conveying fluid and resting on an elastic medium. The finite element method is adopted to obtain the numerical solutions to the model. The effects of temperature change, nonlocal parameter and elastic medium constant on the vibration frequency and buckling instability of SWCNT conveying fluid are investigated. It can be concluded that at low or room temperature, the fundamental natural frequency and critical flow velocity for the SWCNT increase as the temperature change increases, on the other hand, while at high temperature the fundamental natural frequency and critical flow velocity decrease as the temperature change increases. The fundamental natural frequency for the SWCNT decreases as the nonlocal parameter increases, both the fundamental natural frequency and critical flow velocity increase as elastic medium constant increases.     

An exact transfer matrix expression for bending vibration analysis of a rotating tapered beam

This paper presents a transfer matrix expression that can be used to determine the eigenpairs of a rotating beam with a cross section height that linearly decreases along the length of the beam element. The proposed method considers the effect of centrifugal force, including the effects of the axial force, hub radius, and taper ratio. Differential equations are solved for the in-plane bending vibration using the Frobenius method for a power series. The effect of the rotational speed on the eigenpairs of a rotating tapered beam is first investigated, followed by an examination of the contribution rates of the bending strain and additional strain energies generated by centrifugal forces for each mode by analyzing the variation of the energies computed from the strain and kinetic energies. To compute these contribution rates, we used a shape function that was defined by the displacement at both ends of the beam elements. The effect of tapering on the eigenfrequencies of the transverse vibration of rotating beams is analyzed by using various examples, and the contribution rates are examined by using taper ratios of 0 and 0.5.     

Reflection of plane waves in generalized thermoelasticity of type III with nonlocal effect

The generalized thermoelasticity theory based upon the Green and Naghdi model III of thermoelasticity as well as the Eringen's nonlocal elasticity model is used to study the propagation of harmonic plane waves in a nonlocal thermoelastic medium. We found two sets of coupled longitudinal waves, which are dispersive in nature and experience attenuation. In addition to the coupled waves, there also exists one independent vertically shear-type wave, which is dispersive but experiences no attenuation. All these waves are found to be influenced by the elastic nonlocality parameter. Furthermore, the shear-type wave is found to face a critical frequency, while the coupled longitudinal waves may face critical frequencies conditionally. The problem of reflection of the thermoelastic waves at the stress-free insulated and isothermal boundary of a homogeneous, isotropic nonlocal thermoelastic half-space has also been investigated. The formulae for various reflection coefficients and their respective energy ratios are determined in various cases. For a particular material, the effects of the angular frequency and the elastic nonlocal parameter have been shown on phase speeds and the attenuation coefficients of the propagating waves. The effect of the elastic nonlocality on the reflection coefficients and the energy ratios has been observed and depicted graphically. Finally, analysis of the various results has been interpreted.     

Stability analysis of a single-walled carbon nanotube conveying pulsating and viscous fluid with nonlocal effect

For a single-walled carbon nanotube (CNT) conveying fluid, the internal flow is considered to be pulsating and viscous, and the resulting instability and parametric resonance of the CNT are investigated by the method of averaging. The partial differential equation of motion based on the nonlocal elasticity theory is discretized by the Galerkin method and the averaging equations for the first two modes are derived. The stability regions in frequency–amplitude plane are obtained and the influences of nonlocal effect, viscosity and some system parameters on the stability of CNT are discussed in detail. The results show that decrease of nonlocal parameter and increase of viscous parameter both increase the fundamental frequency and critical flow velocity; the dynamic stability of CNT can be enhanced due to nonlocal effect; the contributions of the fluid viscosity on the stability of CNT depend on flow velocity; the axial tensile force can only influence natural frequencies of the system however the viscoelastic characteristic of materials can enhance the dynamic stability of CNT. The conclusions drawn in this paper are thought to be helpful for the vibration analysis and structural design of nanofluidic devices.     

The vibrations of a rotating thin piezoceramic cylindrical shell

G. H. Bryan [8] discovered and studied the dynamic phenomena that arise during the vibrations of thinwalled shells in natural modes while rotating uniformly about an axis of symmetry. In the present article a more detailed mathematical analysis has been conducted for the solution of the problem of the vibrations of a ring rotating with constant angular velocity under feasible conditions of perturbation of the vibrations and realistic mechanical properties of ceramic materials. It is established that two slowly precessing waves arise during the vibration of the ring, one of which (the one discovered by Bryan) is an inverse precession wave having an angular velocity of precession less than the angular velocity of rotation of the ring, while the second is a direct precession wave having angular velocity larger than the angular velocity of rotation of the ring. Under the actual working conditions for piezoceramic resonators the amplitudes of the direct and inverse wave become comparable. Consequently the vibrations present as a modulated standing wave, that is, the phenomenon of beats in the mode of steady-state normal vibration occurs. Translated fromMatematichni Metodi ta Fiziko-mekhanichni Polya, Vol. 39, No. 1, 1996, pp. 7–18.     

Small scale effect on vibrational response of single-walled carbon nanotubes with different boundary conditions based on nonlocal beam models

The free vibration response of single-walled carbon nanotubes (SWCNTs) is investigated in this work using various nonlocal beam theories. To this end, the nonlocal elasticity equations of Eringen are incorporated into the various classical beam theories namely as Euler-Bernoulli beam theory (EBT), Timoshenko beam theory (TBT), and Reddy beam theory (RBT) to consider the size-effects on the vibration analysis of SWCNTs. The generalized differential quadrature (GDQ) method is employed to discretize the governing differential equations of each nonlocal beam theory corresponding to four commonly used boundary conditions. Then molecular dynamics (MD) simulation is implemented to obtain fundamental frequencies of nanotubes with different chiralities and values of aspect ratio to compare them with the results obtained by the nonlocal beam models. Through the fitting of the two series of numerical results, appropriate values of nonlocal parameter are derived relevant to each type of chirality, nonlocal beam model, and boundary conditions. It is found that in contrast to the chirality, the type of nonlocal beam model and boundary conditions make difference between the calibrated values of nonlocal parameter corresponding to each one.     

Free vibration of centrifugally stiffened axially functionally graded tapered Timoshenko beams using differential transformation and quadrature methods

The free bending vibration of rotating axially functionally graded (FG) Timoshenko tapered beams (TTB) with different boundary conditions are studied using Differential Transformation method (DTM) and differential quadrature element method of lowest order (DQEL). These two methods are capable of modelling any beam whose cross sectional area, moment of inertia and material properties vary along the beam. In order to verify the competency of these two methods, natural frequencies are obtained for problems by considering the effect of material non-homogeneity, taper ratio, shear deformation parameter, rotating speed parameter, hub radius and tip mass. The results are tabulated and compared with the previous published results wherever available.     

Reflection of plane waves from the stress-free isothermal and insulated boundaries of a nonlocal thermoelastic solid

The generalized thermoelasticity theory based upon the Green and Naghdi model II of thermoelasticity as well as the Eringen’s nonlocal elasticity model is used to study the propagation of harmonic plane waves in a nonlocal thermoelastic medium. We found two sets of coupled longitudinal waves which are dispersive in nature and associated with attenuation. In addition to the coupled waves, there also exists one independent vertically shear type wave which is dispersive but without any attenuation. All these waves are found to be influenced by the elastic nonlocality parameter. Furthermore, the shear type wave is found to to be associated with a critical frequency, while the coupled longitudinal waves may have critical frequencies under constraints. The problem of reflection of the thermoelastic waves at the stress-free insulated and isothermal boundary of a homogeneous, isotropic nonlocal thermoelastic half-space has also been investigated. The formulae for various reflection coefficients and their respective energy ratios are determined in various cases. For a particular material, the effects of the angular frequency and the elastic nonlocal parameter have been shown on the phase speeds and the attenuation coefficients of the propagating waves. The effect of the elastic nonlocality on the reflection coefficients as well as the energy ratios has been observed and depicted graphically. Finally, analysis of the various results has been interpreted.     

Flow of a Reiner‐Rivlin fluid between two infinite coaxial rotating disks

Present study deals with the steady flow and heat transfer of a non‐Newtonian Reiner‐Rivlin fluid between two coaxially rotating infinite disks. Using similarity transformations, the governing equations are reduced to a set of nonlinear, highly coupled ordinary differential equations and by means of an effective analytical method called homotopy analysis method; analytical solutions are constructed in series form. Different cases, such as, when one disk is at rest and the other is rotating with constant angular velocity, two disks rotating with different angular velocities in same as well as opposite sense, two disks rotating with same angular velocities in opposite sense, are discussed. The effects of non‐Newtonian parameter, Reynolds number, are also discussed, and results are presented graphically.     

Stability analysis of cantilever carbon nanotubes subjected to partially distributed tangential force and viscoelastic foundation

Dynamic instability of cantilever carbon nanotubes conveying fluid embedded in viscoelastic foundation under a partially distributed tangential force is investigated based on nonlocal elasticity theory and Euler–Bernouli beam theory. The present study has incorporated the effects of nonlocal parameter, Knudsen number, surface effects and magnetic field. And two main parameters have also considered, namely partially distributed tangential force and foundation. It is assumed that viscoelastic foundation has modeled as Kelvin–Voigt, Maxwell and Standard linear solid types. The size-dependent governing equation of transverse vibration is derived using Hamilton’s variational principle and discretized by the Galerkin truncation method. A detailed parameter study is carried out, indicating the stability behavior of the nanotubes. In the light of numerical results, it is shown that variables considered in nondimensional equations have significant effects on natural frequencies and flutter velocities, especially for the foundation distribution length and model as well as the partially distributed tangential force.     

Flow-induced and mechanical stability of cantilever carbon nanotubes subjected to an axial compressive load

In the present study, a modified nonlocal elasticity theory is used for flutter and divergence analyses of the cantilever carbon nanotubes (CNTs) conveying fluid. The CNT is embedded in viscoelastic foundation and is subjected to an axial compressive load acting at the free end. An extreme high-order governing equation as well as higher-order boundary conditions is developed using Hamilton's principle for vibration and stability analysis of the CNT. The numerical solution for flutter and divergence velocities is computed using the extended Galerkin method. The validity of the present analysis is confirmed by comparing with molecular dynamics simulation (MDS) and numerical solutions available in the literature. In the numerical results, the effects of nonlocal parameter, surface effects, viscoelastic foundation and compressive axial load on the stability boundaries of the system are investigated. The results show that the stability boundaries of the CNT are strongly dependent on the small scale coefficient and surface effects.     

Vibration analysis of Euler–Bernoulli nanobeams by using finite element method

In the present study, an efficient finite element model for vibration analysis of a nonlocal Euler–Bernoulli beam has been reported. Nonlocal constitutive equation of Eringen is proposed. Equations of motion for a nonlocal Euler–Bernoulli are derived based on varitional statement. The finite element method is employed to discretize the model and obtain a numerical approximation of the motion equation. The model has been verified with the previously published works and found a good agreement with them. Vibration characteristics, such as fundamental frequencies, are illustrated in graphical and tabulated form. Numerical results are presented to figure out the effects of nonlocal parameter, slenderness ratios, rotator inertia, and boundary conditions on the dynamic characteristics of the beam. The above mention effects play very important role on the dynamic behavior of nanobeams.     

变截面二维功能梯度微梁的振动和屈曲特性北大核心CSCD

基于修正的偶应力理论和Timoshenko梁理论,应用变分原理建立了变截面二维功能梯度微梁的自由振动和屈曲力学模型.模型中包含金属组分和陶瓷组分的材料内禀特征尺度参数,可以预测微梁力学行为的尺度效应.采用Ritz法给出了任意边界条件下微梁振动频率和临界屈曲载荷的数值解.数值算例表明:微梁厚度减小时,无量纲一阶频率和无量纲临界屈曲载荷增大,尺度效应增强.锥度比对微梁一阶频率的影响与边界条件密切相关,同时,对应厚度和对应宽度锥度比的影响也有明显差异.变截面微尺度梁无量纲一阶频率随着陶瓷和金属的材料内禀特征尺度参数比的增加而增大,且不同边界条件时增大程度不同.厚度方向和轴向功能梯度指数对微梁的一阶频率和屈曲载荷也有显著的影响.     

Parametric resonance of a FG cylindrical thin shell with periodic rotating angular speeds in thermal environment

Parametric resonance of a functionally graded (FG) cylindrical thin shell with periodic rotating angular speeds subjected to thermal environment is studied in this paper. Taking account of the temperature-dependent properties of the shell, the dynamic equations of a rotating FG cylindrical thin shell based upon Love's thin shell theory are built by Hamilton's principle. The multiple scales method is utilized to obtain the instability boundaries of the problem with the consideration of time-varying rotating angular speeds. It is shown that only the combination instability regions exist for a rotating FG cylindrical thin shell. Moreover, some numerical examples are employed to systematically analyze the effects of constant rotating angular speed, material heterogeneity and thermal effects on vibration characteristics, instability regions and critical rotating speeds of the shell. Of great interest in the process is the combined effect of constant rotating angular speed and temperature on instability regions.     

Rotating response on the vibrations of functionally graded zigzag and chiral single walled carbon nanotubes

This study deals with the vibration analysis of zigzag and chiral rotating functionally graded carbon nanotubes (FG-CNT) invoking Love's shell theory using wave propagation approach. The frequency equation is formed in the eigenvalue form. It has been shown that with the increase of angular speed, frequencies of forwarding curve decrease and backward curve increase. The phenomena of frequency versus length-and height-to-radius ratios are noted as decreasing and increasing, respectively, for rotating CNTs. The backward and forward frequency curves of clamped-free are lower throughout the computation than the clamped-clamped zigzag and chiral carbon nanotube depending upon the rotating speed. MATLAB software is used to calculate the rotating (backward and forward) frequencies of SWCNTs and the frequency peaks in the present results show excellent stability across a wide range of parameters. Using geometrical and material parameters, the vibration results are given in tabular and graphical form. It is thus desirable to produce more precise estimations of the vibrational frequencies of CNTs. The present results are compared with earlier literature using simply supported boundary conditions and show a good coincidence.