Nonlinear vibration of embedded SWBNNTs based on nonlocal Timoshenko beam theory using DQ method

In the present work, effect of von Kàrmàn geometric nonlinearity on the vibration behavior of a single-walled boron nitride nanotube (SWBNNT) is investigated based on nonlocal piezoelasticity theory. The SWBNNT is considered as a nanobeam within the framework of Timoshenko beam (TB). Loading is composed of a temperature change and an imposed axially electric potential throughout the SWBNNT. The interactions between the SWBNNT and its surrounding elastic medium are simulated by Winkler and Pasternak foundation models. The higher order governing equations of motion are derived using Hamilton's principle and the numerical solution of equations is obtained using Differential Quadrature (DQ) method. The effects of geometric nonlinearity, elastic foundation modulus, electric potential field, temperature change and nonlocal parameter on the frequency of the SWBNNT are studied in detail.     

Assessment of nanotube structures under a moving nanoparticle using nonlocal beam theories

Dynamic analysis of nanotube structures under excitation of a moving nanoparticle is carried out using nonlocal continuum theory of Eringen. To this end, the nanotube structure is modeled by an equivalent continuum structure (ECS) according to the nonlocal Euler-Bernoulli, Timoshenko and higher order beam theories. The nondimensional equations of motion of the nonlocal beams acted upon by a moving nanoparticle are then established. Analytical solutions of the problem are presented for simply supported boundary conditions. The explicit expressions of the critical velocities of the nonlocal beams are derived. Furthermore, the capabilities of various nonlocal beam models in predicting the dynamic deflection of the ECS are examined through various numerical simulations. The role of the scale effect parameter, the slenderness ratio of the ECS and velocity of the moving nanoparticle on the time history of deflection as well as the dynamic amplitude factor of the nonlocal beams are scrutinized in some detail. The results show the importance of using nonlocal shear deformable beam theories, particularly for very stocky nanotube structures acted upon by a moving nanoparticle with low velocity.     

Small-scale effect on the vibration of thin nanoplates subjected to a moving nanoparticle via nonlocal continuum theory

The vibration of elastic thin nanoplates traversed by a moving nanoparticle involving Coulomb friction is investigated using the nonlocal continuum theory of Eringen. The eigen function technique and the Laplace transform method are employed to solve the governing equations of the nanoplate. The explicit expressions of the in-plane and transverse displacements are obtained when the moving nanoparticle traverses the nanoplate on an arbitrary straight line. In a special case, the obtained results are also compared with those of other researchers and a reasonably good agreement is achieved. The effects of small-scale parameters and velocity of the moving nanoparticle on the dynamic response as well as the dynamic amplitude factors (DAFs) of the in-plane and transverse displacements are then explored in some detail. The results indicate that the magnitude of DAF of the transverse displacement of the nanoplate (i.e., DAF_w) increases with the first small-scale effect parameter, irrespective of the values of the second small-scale effect parameter and the velocity of the moving nanoparticle. As the first small-scale effect parameter grows, the maximum values of DAF_w as a function of the moving nanoparticle velocity increase and generally occur in the lower levels of the moving nanoparticle velocity.     

Buckling analysis and smart control of SLGS using elastically coupled PVDF nanoplate based on the nonlocal Mindlin plate theory

This study presents an analytical approach for buckling analysis and smart control of a single layer graphene sheet (SLGS) using a coupled polyvinylidene fluoride (PVDF) nanoplate. The SLGS and PVDF nanoplate are considered to be coupled by an enclosing elastic medium which is simulated by the Pasternak foundation. The PVDF nanoplate is subjected to an applied voltage in the thickness direction which operates in control of critical load of the SLGS. In order to satisfy the Maxwell equation, electric potential distribution is assumed as a combination of a half-cosine and linear variation. The exact analysis is performed for the case when all four ends are simply supported and free electrical boundary condition. Adopting the nonlocal Mindlin plate theory, the governing equations are derived based on the energy method and Hamilton's principle. A detailed parametric study is conducted to elucidate the influences of the small scale coefficient, stiffness of the internal elastic medium, graphene length, mode number and external electric voltage on the buckling smart control of the SLGS. The results depict that the imposed external voltage is an effective controlling parameter for buckling of the SLGS. This study might be useful for the design and smart control of nano-devices.     

Direct assembly of nanoparticles for large-scale fabrication of nanodevices and structures

Non-uniform electric fields are utilized to direct the large scale assembly of colloidal nanoparticles in nanoscale structures over large areas. Using micro- and nanoscale templates, various nanoparticles can be directly assembled into parallel wires, cross-wires, and many other complex structures. The assembly process is controlled by electric field, time, and geometric design of templates. The results show that single nanoparticle wires as small as 10 nm wide and 100,000 nm long as well as other nanoparticle structures can be fabricated using electrophoresis over a large area. In addition, the directed assembly of polymeric and conductive nanoparticle nanowires and networks has been demonstrated using dielectrophoresis. The nanoparticle wires can be further oriented along the direction of an externally introduced hydrodynamic flow. The presented technique is a promising approach for large scale manufacturing of nanoscale devices for many applications including biosensors and nanoelectronics.     

Modeling and active vibration suppression of a single-walled carbon nanotube subjected to a moving harmonic load based on a nonlocal elasticity theory

This paper investigates active vibration suppression of a single-walled carbon nanotube (SWCNT) under the action of a moving harmonic load using Eringen’s nonlocal elasticity theory. The SWCNT is modeled according to the nonlocal Euler–Bernoulli beam theory. A Dirac-delta function is used to describe the position of the moving load along the SWCNT. Next, a linear classical optimal control algorithm with displacement-velocity feedback is used to suppress vibration in the SWCNT with control forces acting as actuators. The effects of a small-scale parameter, slenderness ratio, moving load velocity, and the excitation frequency of a moving load on the dynamic deflection of the SWCNT are examined. Finally, the ability of the control algorithm to suppress the response of the SWCNT under the effects of a moving load with a number of controlled modes and control forces is surveyed.     

Radial breathing-mode frequency of elastically confined spherical nanoparticles subjected to circumferential magnetic field

Knowledge of the vibrational properties of nanoparticles is of fundamental interest since it is a signature of their morphology, and it can be utilized to characterize their physical properties. In addition, the vibration characteristics of the nanoparticles coupled with surrounding media and subjected to magnetic field are of recent interest. This paper develops an analytical approach to study the radial breathing-mode frequency of elastically confined spherical nanoparticles subjected to magnetic field. Based on Maxwell's equations, the nonlocal differential equation of radial motion is derived in terms of radial displacement and Lorentz's force. Bessel functions are used to obtain a frequency equation. The model is justified by a good agreement between the results given by the present model and available experimental and atomic simulation data. Furthermore, the model is used to elucidate the effect of nanoparticle size, the magnetic field and the stiffness of the elastic medium on the radial breathing-mode frequencies of several nanoparticles. Our results reveal that the effects of the magnetic field and the elastic medium are significant for nanoparticle with small size.     

Nonlocal continuum-based modeling of a nanoplate subjected to a moving nanoparticle. Part II: Parametric studies

The possible usage of nanoplates in transporting of nanovehicles encouraged the author to propose some nonlocal plate models in the companion paper where the nanovehicle (i.e., moving nanoparticle) was modeled by a moving point load by considering its friction with the upper surface of the nanoplate. In this paper, a comprehensive parametric study is carried out to study the effects of length to thickness ratio of the nanoplate, small-scale parameter, and velocity (or angular velocity) of the moving nanoparticle on dynamic response of nonlocal Kirchhoff, Mindlin, and higher-order plates subjected to a moving nanoparticle. Herein, dynamic response of the nanoplate covers both time histories and dynamic amplitude factors of the in- and out-of-plane displacements. The capabilities of various nonlocal plate models in predicting the displacement field caused by friction and mass weight of the moving nanoparticle are then explored through various numerical analyses for two cases: (i) the moving nanoparticle moves along a diagonal of the nanoplate; (ii) the moving nanoparticle orbits on an ellipse path whose center is coincident with the nanoplate's center. The obtained results indicate that due to the incorporation of small-scale effect into shear strain energy of the nanoplate, an appropriate nonlocal plate model should be used. The results show that the choice of the nanoplate model to use relies on the small-scale parameter, geometrical properties of the nanoplate, and velocity of the moving nanoparticle.     

基态非简并聚合物中的极化子和双极化子的动力学(Ⅰ)--弱电场下研究

文中模拟了在基态非简并聚合物中的极化子和双极化子在弱电场中的运动,研究了在不同简并参数的系统中极化子和双极化子的动力学稳定性,发现在同一个系统中,双极化子比极化子的运动速度慢,晶格振荡小; 在简并参数大的系统中,极化子和双极化子的运动速度都变慢.极化子和双极化子在弱电场下都存在饱和速度,达到饱和速度后, 电场的能量发生了转换.     

Vibration of Timoshenko beam on hysteretically damped elastic foundation subjected to moving load

The vibration of beams on foundations under moving loads has many applications in several fields, such as pavements in highways or rails in railways. However, most of the current studies only consider the energy dissipation mechanism of the foundation through viscous behavior; this assumption is unrealistic for soils. The shear rigidity and radius of gyration of the beam are also usually excluded. Therefore, this study investigates the vibration of an infinite Timoshenko beam resting on a hysteretically damped elastic foundation under a moving load with constant or harmonic amplitude. The governing differential equations of motion are formulated on the basis of the Hamilton principle and Timoshenko beam theory, and are then transformed into two algebraic equations through a double Fourier transform with respect to moving space and time. Beam deflection is obtained by inverse fast Fourier transform. The solution is verified through comparison with the closed-form solution of an Euler-Bernoulli beam on a Winkler foundation. Numerical examples are used to investigate:(a) the effect of the spatial distribution of the load, and(b) the effects of the beam properties on the deflected shape, maximum displacement, critical frequency, and critical velocity. These findings can serve as references for the performance and safety assessment of railway and highway structures.     

Simulation and optimization of dynamic characteristics of piezoelectric smart structures

The paper deals with optimization of dynamic characteristics of smart structures based on piezoelectric materials with external electric circuits comprising resistance, capacitance and inductance. The dynamic parameters to be optimized are resonance frequencies and damping properties. For numerical estimation of the dynamic characteristics of the model system, a natural vibration problem of an electroviscoelastic solid with differing external electric circuits is proposed. Model examples are given to demonstrate the efficiency of the natural vibration problem in finding dynamically optimum piezoelectric smart structures with external electric circuits.     

Nonlocal continuum-based modeling of a nanoplate subjected to a moving nanoparticle. Part I: Theoretical formulations

The potential applications of nanoplates in energy storage, chemical and biological sensors, solar cells, field emission, and transporting of nanocars have been attracted the attentions of the nanotechnology community to them during recent years. Herein, the later application of nanoplates from nonlocal elastodynamic point of view is of interest. To this end, dynamic response of a nanoplate subjected to a moving nanoparticle is examined within the context of nonlocal continuum theory of Eringen. The fully simply supported nanoplate is modeled based on the nonlocal Kirchhoff, Mindlin, and higher-order plate theories. The non-dimensional equations of motion of the nonlocal plate models are established. The effects of moving nanoparticle's weight and existing friction between the surfaces of the moving nanoparticle and nanoplate on the in-plane and out-of-plane vibrations of the nanoplate are incorporated into the formulations of the proposed models. The eigen function expansion and the Laplace transform methods are employed for discretization of the governing equations in the spatial and the time domains, respectively. The analytical expressions of the dynamic deformation field associated with each nonlocal plate theory are obtained when the moving nanoparticle traverses the nanoplate on an arbitrary straight path (an opened path) as well as an ellipse path (a closed path). The dynamic in-plane forces and moments of each nonlocal plate model are also derived. Furthermore, the critical velocity and the critical angular velocity of the moving nanoparticle for the proposed models are expressed analytically for the aforementioned paths. Part II of this work consists in a comprehensive parametric study where the effects of influential parameters on dynamic response of the proposed nonlocal plate models are scrutinized in some detail.     

Reply to the comments of M.E. Golmakani and J. Rezatalab,Comment on “Nonlocal third-order shear deformation plate theory with application to bending and vibration of plates” (by R. Aghababaei and J.N. Reddy,Journal of Sound and Vibration 326 (2009) 277–289), Journal of Sound Vibration, 333 (2014) 3831–3835

Golmakani and Rezatalab [1] suggested in their paper that the deflection of a simply supported nonlocal elastic plate under uniform load is not affected by the small length scale terms. They based their proof on the use of Navier?s method using a sinusoidal-based deflection solution. This insensitivity of the deflection solution of a simply supported nonlocal elastic plate with respect to the small length terms of Eringen?s model is not correct, as already detailed in the literature (for example, see [2] for beam problems). In fact, the deflection of the nonlocal plate (in the Eringen sense) is larger than the one of the local case, as shown in many papers available in the literature. We prove in this reply to the authors that the Navier?s method has to be correctly applied for highlighting the specific sensitivity phenomenon of the deflection solution, as compared to exact analytical solution.     

Effect of metallic surface microroughness on the probability and spectrum of plasmon excitation by a fast charged particle moving parallel to the surface

The Green’s function of the electric field of plasmons is determined in a semi-infinite medium with an abrupt plasma boundary where nonequilibrium conduction electrons either undergo elastic reflection from the boundary or “stick” to it and give rise to a stationary surface charge. The angular reflection of elastically scattered electrons can be either specular or diffuse. The Green’s function is used to find the singleevent spectrum of energy loss by a fast electron moving parallel to the boundary. The effect of electronboundary scattering parameters on the structure of bulk and surface plasmon resonances is analyzed. The probability of transition radiation of bulk plasmon by an electron moving in vacuum is examined. A new type of surface resonance is found under conditions of perfectly elastic scattering of conduction electrons from the plasma boundary, similar in structure to a tangential surface plasmon.     

Vehicle-passenger-structure interaction of uniform bridges traversed by moving vehicles

An investigation into the dynamics of vehicle-occupant-structure-induced vibration of bridges traversed by moving vehicles is presented. The vehicle including the driver and passengers is modelled as a half-car planar model with six degrees-of-freedom, and the bridge is assumed to obey the Euler-Bernoulli beam theory with arbitrary conventional boundary conditions. Due to the continuously moving location of the variable loads on the bridge, the governing differential equations become rather complicated. The numerical simulations presented here are for the case of vehicle travelling at a constant speed on a uniform bridge with simply supported end conditions. The relationship between the bridge vibration characteristics and the vehicle speed is rendered, which yields into a search for a particular speed that determines the maximum values of the dynamic deflection and the bending moment of the bridge. Results at different vehicle speeds demonstrate that the maximum dynamic deflection occurs at the vicinity of the bridge mid-span, while the maximum bending moment occurs at ±20% of the mid-span point. It is shown that one can find a critical speed at which the maximum values of the bridge dynamic deflection and the bending moment attain their global maxima.     

Evaluation of the contact resonance frequencies in atomic force microscopy as a method for surface characterisation (invited)

The combination of ultrasound with atomic force microscopy (AFM) opens the high lateral resolution of scanning probe techniques in the nanometer range to ultrasonics. One possible method is to observe the resonance frequencies of the AFM sensors under different tip-sample interaction conditions. AFM sensors can be regarded as small flexible beams. Their lowest flexural and torsional resonance frequencies are usually found to be in a range between several kHz and several MHz depending on their exact geometrical shape. When the sensor tip is in a repulsive elastic contact with a sample surface, the local indentation modulus can be determined by the contact resonance technique. Contact resonances in the ultrasonic frequency range can also be used to improve the image contrast in other dynamic techniques as, for example, in the so-called piezo-mode. Here, an alternating electric field is applied between a conducting cantilever and a piezoelectric sample. Via the inverse piezoelectric effect, the sample surface is set into vibration. This excitation is localised around the contact area formed by the sensor tip and the sample surface. We show applications of the contact resonance technique to piezoelectric ceramics.     

电子束曝光机偏转系统及可动物镜分析

电子束大扫描场偏转系统设计中 ,像差的确定是一个必须解决的实际问题。以SDS 3电子束曝光机的磁复合偏转系统为基础 ,以双通道扫描原理进行扫描 ,分析了像差与电子束主轨迹的关系。给出了可动物镜的条件 ,并利用这一条件分析电子束最佳轨迹的结构和方法。并使用计算机辅助设计研究电子束曝光机聚焦偏转系统的结构。由该电子束曝光机试验结果表明 ,复合系统结构简单紧凑 ,像差小而可以不用动态校正。采用矢量描写电子轨迹 ,以积分式表示像差。以 0 .0 0 5rad半张角 ,5× 10 -5的高压纹波 ,5 0mm的像距 ,10mm× 10mm扫描场的边脚处 ,使动态校正前的总像差为 0 .0 3μm。     

细电子束像差与畸变参量非劣解中的最优解

以电子束偏转磁场的像差和畸变为目标,对参量进行非劣解中找出贴近理想的最优解。以SDS-3电子束曝光机的磁复合偏转系统为基础,分析了像差与电子束主轨迹的关系。并给出了磁聚焦和静电的偏转场相复合情况下竖轴的3级几何像差系数和1级色差系数公式。应用中表明,复合系统结构简单紧凑,像差小而可以不用动态校正。     

Longitudinal waves in carbon nanotubes in the presence of transverse magnetic field and elastic medium

In the present paper, the coupling effect of transverse magnetic field and elastic medium on the longitudinal wave propagation along a carbon nanotube (CNT) is studied. Based on the nonlocal elasticity theory and Hamilton's principle, a unified nonlocal rod theory which takes into account the effects of small size scale, lateral inertia and radial deformation is proposed. The existing rod theories including the classic rod theory, the Rayleigh-Love theory and Rayleigh-Bishop theory for macro solids can be treated as the special cases of the present model. A two-parameter foundation model (Pasternak-type model) is used to represent the elastic medium. The influence of transverse magnetic field, Pasternak-type elastic medium and small size scale on the longitudinal wave propagation behavior of the CNT is investigated in detail. It is shown that the influences of lateral inertia and radial deformation cannot be neglected in analyzing the longitudinal wave propagation characteristics of the CNT. The results also show that the elastic medium and the transverse magnetic field will also affect the longitudinal wave dispersion behavior of the CNT significantly. The results obtained in this paper are helpful for understanding the mechanical behaviors of nanostructures embedded in an elastic medium.     

Free torsional vibration of nanotubes based on nonlocal stress theory

A new elastic nonlocal stress model and analytical solutions are developed for torsional dynamic behaviors of circular nanorods/nanotubes. Unlike the previous approaches which directly substitute the nonlocal stress into the equations of motion, this new model begins with the derivation of strain energy using the nonlocal stress and by considering the nonlinear history of straining. The variational principle is applied to derive an infinite-order differential nonlocal equation of motion and the corresponding higher-order boundary conditions which contain a nonlocal nanoscale parameter. Subsequently, free torsional vibration of nanorods/nanotubes and axially moving nanorods/nanotubes are investigated in detail. Unlike the previous conclusions of reduced vibration frequency, the solutions indicate that natural frequency for free torsional vibration increases with increasing nonlocal nanoscale. Furthermore, the critical speed for torsional vibration of axially moving nanorods/nanotubes is derived and it is concluded that this critical speed is significantly influenced by the nonlocal nanoscale.