Nonlinear vibration and instability of fluid-conveying DWBNNT embedded in a visco-Pasternak medium using modified couple stress theory

Nonlinear free vibration and instability of fluid-conveying double-walled boron nitride nanotubes (DWBNNTs) embedded in viscoelastic medium are studied in this paper. The effects of the transverse shear deformation and rotary inertia are considered by utilizing the Timoshenko beam theory. The size effect is applied by the modified couple stress theory and considering a material length scale parameter for beam model. The nonlinear effect is considered by the Von Kármán type geometric nonlinearity. The electromechanical coupling and charge equation are employed to consider the piezoelectric effect. The surrounding viscoelastic medium is described as the linear visco-Pasternak foundation model characterized by the spring and damper. Hamilton’s principle is used to derive the governing equations and boundary conditions. The differential quadrature method (DQM) is employed to discretize the nonlinear higher-order governing equations, which are then solved by a direct iterative method to obtain the nonlinear vibration frequency and critical fluid velocity of fluid-conveying DWBNNTs with clamped-clamped (C-C) boundary conditions. A detailed parametric study is conducted to elucidate the influences of the small scale coefficient, spring and damping constants of surrounding viscoelastic medium and fluid velocity on the nonlinear free vibration, instability and electric potential distribution of DWBNNTs. This study might be useful for the design and smart control of nano devices.     

A meshless approach for free transverse vibration of embedded single-walled nanotubes with arbitrary boundary conditions accounting for nonlocal effect

A single-walled nanotube structure embedded in an elastic matrix is simulated by the nonlocal Euler-Bernoulli, Timoshenko, and higher order beams. The beams are assumed to be elastically supported and attached to continuous lateral and rotational springs to take into account the effects of the surrounding matrix. The discrete equations of motion associated with free transverse vibration of each model are established in the context of the nonlocal continuum mechanics of Eringen using Hamilton's principle and an efficient meshless method. The effects of slenderness ratio of the nanotube, small scale effect parameter, initial axial force and the stiffness of the surrounding matrix on the natural frequencies of various beam models are investigated for different boundary conditions. The capabilities of the proposed nonlocal beam models in capturing the natural frequencies of the nanotube are also addressed.     

Calibration of the analytical nonlocal shell model for vibrations of double-walled carbon nanotubes with arbitrary boundary conditions using molecular dynamics

In the present study, the free vibration response of double-walled carbon nanotubes (DWCNTs) is investigated. Eringen's nonlocal elasticity equations are incorporated into the classical Donnell shell theory accounting for small scale effects. The Rayleigh-Ritz technique is applied to consider different sets of boundary conditions. The displacements are represented as functions of polynomial series to implement the Rayleigh-Ritz method to the governing differential equations of nonlocal shell model and obtain the natural frequencies of DWCNTs relevant to different values of nonlocal parameter and aspect ratio. To extract the proper values of nonlocal parameter, molecular dynamics (MD) simulations are employed for various armchair and zigzag DWCNTs, the results of which are matched with those of nonlocal continuum model through a nonlinear least square fitting procedure. It is found that the present nonlocal elastic shell model with its appropriate values of nonlocal parameter has the capability to predict the free vibration behavior of DWCNTs, which is comparable with the results of MD simulations.     

Vibration of multi-walled carbon nanotubes by generalized shear deformation theory

In this study, free vibration of simply supported multi-walled carbon nanotubes (CNTs) was investigated by using the generalized shear deformation-beam theory (GSDBT). Parabolic shear deformation theory (PSDT) is used in the specific solutions. Unlike Timoshenko beam theory present theory satisfies zero traction boundary conditions on the upper and lower surface of the structures so there is no need to use a shear correction factor. Free vibration frequencies and amplitude ratios were obtained and results are compared with previous studies. Results showed that significant difference exist between PSDT and Euler beam theory. Present results are slightly higher than the results of Timoshenko beam theory. Shear deformation effects are important especially for higher modes. It is obtained that van der Waals (vdW) forces should be considered for small inner radius.     

Free vibration of Timoshenko beam with finite mass rigid tip load and flexural–torsional coupling

In this paper, the free vibration of a cantilever Timoshenko beam with a rigid tip mass is analyzed. The mass center of the attached mass need not be coincident with its attachment point to the beam. As a result, the beam can be exposed to both torsional and planar elastic bending deformations. The analysis begins with deriving the governing equations of motion of the system and the corresponding boundary conditions using Hamilton's principle. Next, the derived formulation is transformed into an equivalent dimensionless form. Then, the separation of variables method is utilized to provide the frequency equation of the system. This equation is solved numerically, and the dependency of natural frequencies on various parameters of the tip mass is discussed. Explicit expressions for mode shapes and orthogonality condition are also obtained. Finally, the results obtained by the application of the Timoshenko beam model are compared with those of three other beam models, i.e. Euler–Bernoulli, shear and Rayleigh beam models. In this way, the effects of shear deformation and rotary inertia in the response of the beam are evaluated.     

Electro-thermo nonlocal nonlinear vibration in an embedded polymeric piezoelectric micro plate reinforced by DWBNNTs using DQM

In the present paper, electro-thermo nonlinear vibration of a piezo-polymeric rectangular micro plate made from polyvinylidene fluoride (PVDF) reinforced by zigzag double walled boron nitride nanotubes (DWBNNTs) is studied. This plate is embedded in an elastic medium which is simulated by Winkler and Pasternak foundation models. Using nonlinear strain-displacement relations and nonlocal elasticity plate theory as well as considering charge equation for coupling between electrical and mechanical fields, the motion equations are derived based on energy method and Hamilton??s principle. The differential quadrature method (DQM) is employed to computation of nonlinear frequency for different mechanical and free-free electrical boundary conditions. The results indicate that smart composite and consequently the generated G4 improved sensor and actuator applications in several process industries, because it increases the nonlinear vibration frequency. Furthermore, it can be also found that the nonlinear frequency increases as the values of the elastic medium constants, the geometrical aspect ratios and DWBNNTs volume fraction increase but it decreases as nonlocal parameter increases.     

Flow-induced vibration and stability analysis of multi-wall carbon nanotubes

The free vibration and flow-induced flutter instability of cantilever multi-wall carbon nanotubes conveying fluid are investigated and the nanotubes are modeled as thin-walled beams. The non-classical effects of the transverse shear, rotary inertia, warping inhibition, and van der Waals forces between two walls are incorporated into the structural model. The governing equations and associated boundary conditions are derived using Hamilton’s principle. A numerical analysis is carried out by using the extended Galerkin method, which enables us to obtain more accurate solutions compared to the conventional Galerkin method. Cantilevered carbon nanotubes are damped with decaying amplitude for a flow velocity below a certain critical value. However, beyond this critical flow velocity, flutter instability may occur. The variations in the critical flow velocity with respect to both the radius ratio and length of the carbon nanotubes are investigated and pertinent conclusions are outlined. The differences in the vibration and instability characteristics between the Timoshenko beam theory and Euler beam theory are revealed. A comparative analysis of the natural frequencies and flutter characteristics of MWCNTs and SWCNTs is also performed.     

Closed-form solutions for crack detection problem of Timoshenko beams with various boundary conditions

An analytical approach for crack identification procedure in uniform beams with an open edge crack, based on bending vibration measurements, is developed in this research. The cracked beam is modeled as two segments connected by a rotational mass-less linear elastic spring with sectional flexibility, and each segment of the continuous beam is assumed to obey Timoshenko beam theory. The method is based on the assumption that the equivalent spring stiffness does not depend on the frequency of vibration, and may be obtained from fracture mechanics. Six various boundary conditions (i.e., simply supported, simple–clamped, clamped–clamped, simple–free shear, clamped–free shear, and cantilever beam) are considered in this research. Considering appropriate compatibility requirements at the cracked section and the corresponding boundary conditions, closed-form expressions for the characteristic equation of each of the six cracked beams are reached. The results provide simple expressions for the characteristic equations, which are functions of circular natural frequencies, crack location, and crack depth. Methods for solving forward solutions (i.e., determination of natural frequencies of beams knowing the crack parameters) are discussed and verified through a large number of finite-element analyses. By knowing the natural frequencies in bending vibrations, it is possible to study the inverse problem in which the crack location and the sectional flexibility may be determined using the characteristic equation. The crack depth is then computed using the relationship between the sectional flexibility and the crack depth. The proposed analytical method is also validated using numerical studies on cracked beam examples with different boundary conditions. There is quite encouraging agreement between the results of the present study and those numerically obtained by the finite-element method.     

A novel approach for in-plane/out-of-plane frequency analysis of functionally graded circular/annular plates

An exact closed-form frequency equation is presented for free vibration analysis of circular and annular moderately thick FG plates based on the Mindlin's first-order shear deformation plate theory. The edges of plate may be restrained by different combinations of free, soft simply supported, hard simply supported or clamped boundary conditions. The material properties change continuously through the thickness of the plate, which can vary according to a power-law distribution of the volume fraction of the constituents, whereas Poisson's ratio is set to be constant. The equilibrium equations which govern the dynamic stability of plate and its natural boundary conditions are derived by the Hamilton's principle. Several comparison studies with analytical and numerical techniques reported in literature and the finite element analysis are carried out to establish the high accuracy and superiority of the presented method. Also, these comparisons prove the numerical accuracy of solutions to calculate the in-plane and out-of-plane modes. The influences of the material property, graded index, thickness to outer radius ratios and boundary conditions on the in-plane and out-of-plane frequency parameters are also studied for different functionally graded circular and annular plates.     

弹性地基上转动功能梯度材料Timoshenko梁自由振动的微分变换法求解

基于Timoshenko梁理论研究弹性地基上转动功能梯度材料（FGM）梁的自由振动。首先确定功能梯度材料Timoshenko梁的物理中面,利用广义Hamilton原理推导出该梁在弹性地基上转动时横向自由振动的两个控制微分方程。其次采用微分变换法（DTM）对控制微分方程及其边界条件进行变换,计算了弹性地基上转动功能梯度材料Timoshenko梁在夹紧-夹紧、夹紧-简支和夹紧-自由三种不同边界条件下横向自由振动的量纲一固有频率,与已有文献的计算结果进行比较,退化后结果一致。最后讨论了不同边界条件、转速、弹性地基模量和梯度指数对功能梯度材料Timoshenko梁自振频率的影响。结果表明：功能梯度材料Timoshenko梁的量纲一固有频率随量纲一转速和量纲一弹性地基模量的增大而增大;在量纲一转速和量纲一弹性地基模量一定的情况下,梁的量纲一固有频率随着功能梯度材料梯度指数的增大而减小。     

Free vibration analysis of arbitrary shaped thick plates by differential cubature method

In this paper, a new numerical solution technique, the differential cubature method, is applied to solve the free vibration problems of arbitrary shaped thick plates. The basic idea of the differential cubature method is to express a linear differential operation such as a continuous function or any order of partial derivative of a multivariable function, as a weighted linear sum of discrete function values chosen within the overall domain of a problem. By using the differential cubature procedure, the governing differential equations and boundary conditions are transformed into sets of linear homogeneous algebraic equations. This is an eigenvalue problem, of which the eigenvalues can be calculated numerically. The subspace iterative method is employed in search of the free vibration frequency parameters. Detailed formulations are presented, and the method is examined here for its suitability for solving the vibration problems of moderately thick plates governed by Mindlin shear deformation theory. The applicability, efficiency and simplicity of the method are demonstrated through solving some example plate vibration problems of different shapes. The numerical accuracy of the method is ascertained by comparing the vibration frequency solutions with those of existing literatures.     

Exact dynamic stiffness matrix of a Timoshenko three-beam system

An exact dynamic stiffness matrix is established for an elastically connected three-beam system, which is composed of three parallel beams of uniform properties with uniformly distributed-connecting springs among them. The formulation includes the effects of shear deformation and rotary inertia of the beams. The dynamic stiffness matrix is derived by rigorous use of the analytical solutions of the governing differential equations of motion of the three-beam system in free vibration. The use of the dynamic stiffness matrix to study the free vibration characteristics of the three-beam system is demonstrated by applying the Muller root search algorithm. Numerical results for the natural frequencies and mode shapes of the illustrative examples are discussed for 10 interesting boundary conditions and three different stiffness constants of springs.     

Natural frequencies,modes and critical speeds of axially moving Timoshenko beams with different boundary conditions

In this paper, natural frequencies, modes and critical speeds of axially moving beams on different supports are analyzed based on Timoshenko model. The governing differential equation of motion is derived from Newton's second law. The expressions for various boundary conditions are established based on the balance of forces. The complex mode approach is performed. The transverse vibration modes and the natural frequencies are investigated for the beams on different supports. The effects of some parameters, such as axially moving speed, the moment of inertia, and the shear deformation, are examined, respectively, as other parameters are fixed. Some numerical examples are presented to demonstrate the comparisons of natural frequencies for four beam models, namely, Timoshenko model, Rayleigh model, Shear model and Euler–Bernoulli model. Finally, the critical speeds for different boundary conditions are determined and numerically investigated.     

Free periodic vibrations of beams with large displacements and initial plastic strains

This paper intends to analyse free vibrations of beams in the geometrically non-linear regime and with plastic strains. The specific goal is to find how plastic strains combined with large displacements influence the non-linear modes of vibration, by analysing the influence of the former two factors in mode shapes and natural frequencies of vibration. The geometrical non-linearity is represented by the Von Kármán type strain-displacement relations. A stress-strain relation of the bilinear type, with isotropic strain hardening, is assumed, the Von Mises yield criterion is employed and the flow theory of plasticity applied. To obtain the time domain ordinary differential equations of motion the principle of virtual work is used and a Timoshenko p-version finite element model with hierarchical basis functions is adopted. The equations of motion are naturally different from the usual large displacement equations, due to the appearance of matrices and vectors related with plastic terms. In the cases studied, plastic strains are imposed on the beam by equally distributed static forces; the forces are then removed and a study on the free vibrations is carried out. It is assumed that, once defined, the plastic strain field does not change. The time domain equations are transformed to the frequency domain by the harmonic balance method and these frequency domain equations are solved by an arc-length continuation method. The variations of mode shapes of vibration and of natural frequencies with vibration amplitude are investigated. It is found that the plastic strain distribution defines if and how much softening spring effect occurs. Hardening spring effect always appears, but with some plastic strain fields hardening spring takes place only at large vibration amplitudes. Plastic deformations also have an important effect in the vibration shapes.     

Static analysis of nanobeams using nonlocal FEM

A very efficiently finite element model is developed for static analysis of nanobeams. Nonlocal differential equation of Eringen is exploited to reveal a scale effect of nanobeams through nonlocal Euler-Bernoulli beam theory. The equilibrium equation of nonlocal beam is derived based on the variational statement. The element stiffness matrix and force vector are presented. The novelty and accuracy of this model is presented and verified. It is found that, this model is more accurate than others and can consider as a benchmark. The effects of nonlocality, boundary conditions, and slenderness ratio are figured out. The deflection of multi-span nanobeam is also illustrated. The present model can be used for static analysis of single-walled carbon nanotubes. Complex geometry and nonlinear boundary conditions can also be included.     

Study of the sensitivity and resonant frequency of the torsional modes of an AFM cantilever with a sidewall probe based on a nonlocal elasticity theory

A relationship based on a nonlocal elasticity theory is developed to investigate the torsional sensitivity and resonant frequency of an atomic force microscope (AFM) with assembled cantilever probe (ACP). This ACP comprises a horizontal cantilever and a vertical extension, and a tip located at the free end of the extension, which makes the AFM capable of topography at sidewalls of microstructures. First, the governing differential equations of motion and boundary conditions for dynamic analysis are obtained by a combination of the basic equations of nonlocal elasticity theory and Hamilton's principle. Afterward, a closed‐form expression for the sensitivity of vibration modes has been obtained using the relationship between the resonant frequency and contact stiffness of cantilever and sample. These analysis accounts for a better representation of the torsional behavior of an AFM with sidewall probe where the small‐scale effect are significant. The results of the proposed model are compared with those of classical beam theory. The results show that the sensitivities and resonant frequencies of ACP predicted by the nonlocal elasticity theory are smaller than those obtained by the classical beam theory. Microsc. Res. Tech. 78:408–415, 2015. © 2015 Wiley Periodicals, Inc.     

Closed-form vibration analysis of thick annular functionally graded plates with integrated piezoelectric layers

This paper employs an analytical method to analyze vibration of piezoelectric coupled thick annular functionally graded plates (FGPs) subjected to different combinations of soft simply supported, hard simply supported and clamped boundary conditions at the inner and outer edges of the annular plate on the basis of the Reddy's third-order shear deformation theory (TSDT). The properties of host plate are graded in the thickness direction according to a volume fraction power-law distribution. The distribution of electric potential along the thickness direction in the piezoelectric layer is assumed as a sinusoidal function so that the Maxwell static electricity equation is approximately satisfied. The differential equations of motion are solved analytically for various boundary conditions of the plate. In this study closed-form expressions for characteristic equations, displacement components of the plate and electric potential are derived for the first time in the literature. The present analysis is validated by comparing results with those in the literature and then natural frequencies of the piezoelectric coupled annular FG plate are presented in tabular and graphical forms for different thickness-radius ratios, inner-outer radius ratios, thickness of piezoelectric, material of piezoelectric, power index and boundary conditions.     

Free vibration analysis of beams with non-ideal clamped boundary conditions

A non-ideal boundary condition is modeled as a linear combination of the ideal simply supported and the ideal clamped boundary conditions with the weighting factors k and 1-k, respectively. The proposed non-ideal boundary model is applied to the free vibration analyses of Euler-Bernoulli beam and Timoshenko beam. The free vibration analysis of the Euler-Bernoulli beam is carried out analytically, and the pseudospectral method is employed to accommodate the non-ideal boundary conditions in the analysis of the free vibration of Timoshenko beam. For the free vibration with the non-ideal boundary condition at one end and the free boundary condition at the other end, the natural frequencies of the beam decrease as k increases. The free vibration where both the ends of a beam are restrained by the non-ideal boundary conditions is also considered. It is found that when the non-ideal boundary conditions are close to the ideal clamped boundary conditions the natural frequencies are reduced noticeably as k increases. When the non-ideal boundary conditions are close to the ideal simply supported boundary conditions, however, the natural frequencies hardly change as k varies, which indicate that the proposed boundary condition model is more suitable to the non-ideal boundary condition close to the ideal clamped boundary condition.     

Vibration analysis of orthotropic circular and elliptical nano-plates embedded in elastic medium based on nonlocal Mindlin plate theory and using Galerkin method

In the present study a continuum model based on the nonlocal elasticity theory is developed for free vibration analysis of embedded orthotropic thick circular and elliptical nano-plates rested on an elastic foundation. The elastic foundation is considered to behave like a Pasternak type of foundations. Governing equations for vibrating nano-plate are derived according to the Mindlin plate theory in which the effects of shear deformations of nano-plate are also included. The Galerkin method is then employed to obtain the size dependent natural frequencies of nano-plate. The solution procedure considers the entire nano-plate as a single super-continuum element. Effect of nonlocal parameter, lengths of nano-plate, aspect ratio, mode number, material properties, thickness and foundation on circular frequencies are investigated. It is seen that the nonlocal frequencies of the nano-plate are smaller in comparison to those from the classical theory and this is more pronounced for small lengths and higher vibration modes. It is also found that as the aspect ratio increases or the nanoplate becomes more elliptical, the small scale effect on natural frequencies increases. Further, it is observed that the elastic foundation decreases the influence of nonlocal parameter on the results. Since the effect of shear deformations plays an important role in vibration analysis and design of nano-plates, by predicting smaller values for fundamental frequencies, the study of these nano-structures using thick plate theories such as Mindlin plate theory is essential.     

Natural frequencies and buckling of pressurized nanotubes using shear deformable nonlocal shell model

The small-scale effect on the natural frequencies and buckling of pressurized nanotubes is investigated in this study. Based on the firstorder shear deformable shell theory, the nonlocal theory of elasticity is used to account for the small-scale effect and the governing equations of motion are obtained. Applying modal analysis technique and based on Galerkin’s method a procedure is proposed to obtain natural frequencies of vibrations. For the case of nanotubes with simply supported boundary conditions, explicit expressions are obtained which establish the dependency of the natural frequencies and buckling loads of the nanotube on the small-scale parameter and natural frequencies obtained by local continuum mechanics. The obtained solutions generalize the results of nano-bar and -beam models and are verified by the literature. Based on several numerical studies some conclusions are drawn about the small-scale effect on the natural frequencies and buckling pressure of the nanotubes.