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.     

Nonlinear vibration of carbon nanotube embedded in viscous elastic matrix under parametric excitation by nonlocal continuum theory

In the present work, the nonlinear vibration of a carbon nanotube which is subjected to the external parametric excitation is studied. By the nonlocal continuum theory and nonlinear von Kármán beam theory, the governing equation of the carbon nanotube is derived with the consideration of the large deformation. The principle parametric resonance of the nanotube is discussed and the approximation explicit solution is presented by the multiple scale method. Numerical calculations are performed. It can be observed that when the mode number is 1, the stable region can be significantly changed by the parametric excitation, length-to-diameter ratio and matrix stiffness. This phenomenon becomes different to appear if the mode number increases. Moreover, the small scale effects have great influences on the positive bifurcation point for the short carbon nanotube, and the nonlocal continuum theory can present the proper model.     

Analysis of nonlinear steady state vibration of a multi-degree-of-freedom system using component mode synthesis method

In this paper, an analytical approach for nonlinear forced vibration of a multi-degree-of-freedom system is proposed using the component mode synthesis method. The whole system is divided into some components and a nonlinear modal equation of each component is derived using the free-interface vibration modes. The modal equations of all components and the conjunction conditions are solved simultaneously, and then the modal responses of components are derived. Finally, the dynamic responses of the whole system can be obtained. The degrees of freedom of modal equations can be reduced when the lower vibration modes are only adopted in each component. As a numerical example, a nine-degree-of-freedom system is considered, in which all spring have cubic type nonlinearity. As a result, it is shown that when there are no rigid modes in components, the compliance by the proposed method agrees very well with the exact one even if the lower vibration modes of components are only adopted. The other hand, in the case with rigid modes in components, the compliance has a little error compared with the exact result. It is recognized that the method proposed is very effective in the case without rigid modes in components for the actual application.     

Nonlinear vibration and stability analysis of a double-walled carbon nanotube under electrostatic actuation

In this paper, forced vibrations of a double-walled clamped–clamped carbon nanotube (DWNT) are studied. Two Euler–Bernoulli beams are used to model the inner and outer layers of the DWNT. An electrostatic actuation, which is comprised of DC and AC voltages is applied between the nanotubes and the electrode. In the system model, the nonlinear form of the interlayer van der Waals (vdW) force, and also, the mid-plane stretching are considered. The obtained equations are solved through Galerkin and multiple scales methods for primary and secondary resonances. The frequency response of the system is obtained as a function of some of the system parameters. A stability analysis of the response is conducted and bifurcation points are determined. The results demonstrate that the DWNT shows different behavior by changing the value of DC voltage. It is also observed that both layers of the DWNT vibrate with the same frequency under the primary and secondary resonance conditions.     

Nonlinear size-dependent longitudinal vibration of carbon nanotubes embedded in an elastic medium

In this paper, we study the longitudinal linear and nonlinear free vibration response of a single walled carbon nanotube (CNT) embedded in an elastic medium subjected to different boundary conditions. This formulation is based on a large deformation analysis in which the linear and nonlinear von Kármán strains and their gradient are included in the expression of the strain energy and the velocity and its gradient are taken into account in the expression of the kinetic energy. Therefore, static and kinetic length scales associated with both energies are introduced to model size effects. The governing motion equation along with the boundary conditions are derived using Hamilton's principle. Closed-form solutions for the linear free vibration problem of the embedded CNT rod are first obtained. Then, the nonlinear free vibration response is investigated for various values of length scales using the method of multiple scales.     

Nonlinear resonances of a single-wall carbon nanotube cantilever

The dynamics of an electrostatically actuated carbon nanotube (CNT) cantilever are discussed by theoretical and numerical approaches. Electrostatic and intermolecular forces between the single-walled CNT and a graphene electrode are considered. The CNT cantilever is analyzed by the Euler–Bernoulli beam theory, including its geometric and inertial nonlinearities, and a one-mode projection based on the Galerkin approximation and numerical integration. Static pull-in and pull-out behaviors are adequately represented by an asymmetric two-well potential with the total potential energy consisting of the CNT elastic energy, electrostatic energy, and the Lennard-Jones potential energy. Nonlinear dynamics of the cantilever are simulated under DC and AC voltage excitations and examined in the frequency and time domains. Under AC-only excitation, a superharmonic resonance of order 2 occurs near half of the primary frequency. Under both DC and AC loads, the cantilever exhibits linear and nonlinear primary and secondary resonances depending on the strength of the excitation voltages. In addition, the cantilever has dynamic instabilities such as periodic or chaotic tapping motions, with a variation of excitation frequency at the resonance branches. High electrostatic excitation leads to complex nonlinear responses such as softening, multiple stability changes at saddle nodes, or period-doubling bifurcation points in the primary and secondary resonance branches.     

Free vibration of an embedded single-walled carbon nanotube with various boundary conditions using the RMVT-based nonlocal Timoshenko beam theory and DQ method

The nonlocal Timoshenko beam theories (TBTs), based on the Reissner mixed variation theory (RMVT) and principle of virtual displacement (PVD), are derived for the free vibration analysis of a single-walled carbon nanotube (SWCNT) embedded in an elastic medium and with various boundary conditions. The strong formulations of the nonlocal TBTs are derived using Hamilton's principle, in which Eringen's nonlocal constitutive relations are used to account for the small-scale effect. The interaction between the SWCNT and its surrounding elastic medium is simulated using the Winkler and Pasternak foundation models. The frequency parameters of the embedded SWCNT are obtained using the differential quadrature (DQ) method. In the cases of the SWCNT without foundations, the results of RMVT- and PVD-based nonlocal TBTs converge rapidly, and their convergent solutions closely agree with the exact ones available in the literature. Because the highest order with regard to the derivatives of the field variables used in the RMVT-based nonlocal TBT is lower than that used in its PVD-based counterpart, the former is more efficient than the latter with regard to the execution time. The former is thus both faster and obtains more accurate solutions than the latter for the numerical analysis of the embedded SWCNT.     

Simplified formulae to investigate flexural vibration characteristics of piezoelectric tubes in ultrasonic micro-actuators

Based on the Rayleigh energy theory combining with Timoshenko beam model, the flexural vibration characteristics of piezoelectric tubes in ultrasonic micro-actuators are investigated. Additionally, the simplified formulae are derived to study the fundamental flexural resonance frequencies of the piezoelectric tubes with free-free ends and cantilevers. By changing the sizes of the tubes and the mass loads at the free ends, the variations of the flexural resonance frequencies of the piezoelectric tubes and cantilevers are calculated theoretically. To verify accuracy of the simplified formulae, by changing the lengths of the tubes and the mass loads the flexural resonance frequencies of the piezoelectric tube with free-free ends are measured experimentally. The theoretical results agree well with the experimental measurements, which demonstrate that the simplified formulae are accurate and effective for analyzing the flexural vibration characteristics of the piezoelectric tubes in the ultrasonic micro-actuators.     

Free vibration characteristics of double-walled carbon nanotubes embedded in an elastic medium

In this Letter, a theoretical analysis of the resonant vibration of double-walled carbon nanotubes (DWCNTs) and the DWCNTs embedded in an elastic medium is presented based on Euler-Bernoulli beam model and Winkler spring model. The vibration modes of DWCNTs are quite different from those of single-walled carbon nanotubes (SWCNTs). The resonant vibrations of DWCNTs are found to have in-phase and anti-phase modes, in which the deflections of the inner and outer nanotubes occur in the same and opposite directions, respectively. For the vibration of DWCNTs with the same harmonic numbers, the resonant frequencies of anti-phase mode are larger than the ones of in-phase mode. Moreover, influence of the surrounding medium on the resonant vibrations is investigated using the Winkler spring model. The results show that surrounding medium makes a strong impact on the vibration frequencies of in-phase mode, but little on those of anti-phase mode.     

Harmonic balancing approach to nonlinear oscillations of a punctual charge in the electric field of charged ring

The harmonic balance method is used to construct approximate frequency-amplitude relations and periodic solutions to an oscillating charge in the electric field of a ring. By combining linearization of the governing equation with the harmonic balance method, we construct analytical approximations to the oscillation frequencies and periodic solutions for the oscillator. To solve the nonlinear differential equation, firstly we make a change of variable and secondly the differential equation is rewritten in a form that does not contain the square-root expression. The approximate frequencies obtained are valid for the complete range of oscillation amplitudes and excellent agreement of the approximate frequencies and periodic solutions with the exact ones are demonstrated and discussed.     

Strongly localized states in a single carbon nanotube with polymer molecules

A single-walled carbon nanotube (SWCNT) with conjugated polymer molecules is analyzed via optical spectroscopy. The presence of strongly localized excitonic states in the SWCNT is confirmed using time-integrated photoluminescence (PL). The PL spectrum exhibits extremely narrow width (~0.8 meV) which is attributed to the strong confinement of the states by polymer molecules. In addition, I observed that the excited states are gradually filled as a function of the excitation power, which supports the localized excitonic behavior. Only the ground excitonic state is observed at low excitation powers, but three additional PL peaks appear as the excitation power is increased. Especially, the power-dependent PL spectrum shows a blueshift and increased width, which can be elucidated in terms of quantum confined stark effect and the screening of induced electric fields. Overall, I demonstrate that the presence of polymer molecules induces several localized states in a single SWCNT.     

Flexural vibration analyses of piezoelectric ceramic tubes with mass loads in ultrasonic actuators

Based on Timoshenko beam model, a theoretical model of radially polarized piezoelectric ceramic tubes is investigated. In the model, the piezoelectric effects are considered, and the shear correction factor is introduced which reveals effects of the size of the cross-section and Poisson’s ratio. Based on the model, the particular attentions are devoted to effects of the boundary conditions at two ends on flexural resonance frequencies of the piezoelectric ceramic tubes. Changing the sizes of the tubes and the mass loads at both free ends, the variations of the flexural resonance frequencies of free–free piezoelectric ceramic tubes are calculated theoretically. Besides, the flexural resonance frequencies of the piezoelectric ceramic tube cantilevers with mass loads at one free end are also investigated theoretically. To verify accuracy of the theoretical mode, the flexural resonance frequencies for different lengths of the piezoelectric ceramic tubes and different loaded masses are measured experimentally. The theoretical results agree well with the experimental measurement, which demonstrates that the model is accurate for analyzing the flexural resonance frequencies of the piezoelectric ceramic tubes with mass loads.     

Optical limiting properties of trimeric metallo-phthalocyanines/polymer composite films

The nonlinear absorption and optical limiting properties of two trimeric metallo-phthalocyanines namely, 2,4,6-tris[2-oxa-9,10,16,17,23,24-hexa(hexylthio) phthalocyaninato M(II)]-s-triazine (M=Zn for compound ZnPc and Cu for compound CuPc) doped polyvinyl chloride (PVC) thin film in the nanosecond regime were investigated by using the open-aperture Z-scan technique. The measurements were performed using 4 ns pulses generated from a frequency-doubled Nd:YAG laser at 532 nm wavelength. OL parameters of the ratio of the excited state to ground state absorption cross-sections κ, the effective nonlinear absorption coefficient β_eff, the linear absorption coefficient α₀ and the saturation density or energy density F_sat values were determined. The results show that MPc/PVC composite displays much larger nonlinear absorption coefficient and lower saturable fluence for optical limiting when compared to the same Pc molecules in solution. The results indicated that both compounds exhibited good OL performances. ZnPc shows slightly better OL parameters than that of CuPc.     

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.     

Multi-scale bending,buckling and vibration analyses of carbon fiber/carbon nanotube-reinforced polymer nanocomposite plates with various shapes

Using a finite element-based multi-scale modeling approach, the bending, buckling and free vibration of hybrid polymer matrix composites reinforced by carbon fibers and carbon nanotubes (CF/CNT-RP) are analyzed herein. Thick composite plates with rectangular, circular, annular and elliptical shapes are considered. First, the equivalent material properties of CF/CNT-RP are calculated for different volume fractions of CF and CNT. To accomplish this aim, a two-step procedure is presented through which the coupled effects of nano- and micro-scale are taken into account. In the first step, modeling of dispersion of CNTs into the polymer matrix is done with considering interphase formed by their chemical interaction with the matrix, and the equivalent properties of resulting composite material are determined accordingly. CFs are then dispersed into CNT-RP which is considered a homogenous material in this step. Both distributions of CNTs and CFs are assumed to be random. After computing the equivalent properties of CF/CNT-RP for different volume fractions of its constituents, the bending, buckling and free vibration analyses of plates with different shapes are performed. It is shown that the reinforcement of the polymer matrix with both CF and CNT significantly affects the bending, buckling and free vibration characteristics of plates.     

通量可控的双壁碳纳米管水分子泵

以双壁碳纳米管作为基本单元设计了一种新型纳米机械水泵, 其内管固定作为水分子通道, 外管做活塞式轴向运动. 分子动力学计算表明, 水分子净通量及管内水分子电偶极矩分布均与外管运动速率有强烈耦合效应. 该设计可以实现水分子的高效单向运输, 且输运效率可以通过外管活塞运动的速率进行调控. 这些发现可为未来实用纳米分子泵器件的设计提供新的思路.     

Femtosecond laser pulse generation with a fiber taper embedded in carbon nanotube/polymer composite

We propose and demonstrate a new saturable absorber based on a fiber taper embedded in a carbon nanotube/polymer composite. Greater than a 10% reduction in absorption (due to saturation) is directly measured for our saturable absorber. Using an embedded fiber-taper saturable absorber, we built an all-fiber mode-locked ring laser, which produces 594 fs/1.7 nJ pulses with a repetition rate of 13.3 MHz.     

Axisymmetric vibration of single-walled carbon nanotubes in water

A double shell-Stokes flow model is developed to study the axisymmetric vibration of single-walled carbon nanotubes (SWCNTs) immerged in water. In contrast to macroscopic solid-liquid system, a submerged SWCNT is coupled with surrounding water via the van der Waals interaction. It is shown that this unique feature substantially reduces viscous damping of the axisymmetric radial, longitudinal and torsional vibrations and significantly up-shifts the frequency of the radial vibration of an SWCNT. The study offers a theoretical explanation for the experimental observation and molecular dynamics simulations available in particular cases, and provides an efficient modelling tool and useful guidance for the study of the general dynamic behaviour of SWCNTs in a fluid.     

Nonlinear resistance of polymer composites with carbon nanotube additives in the percolation state

The electrical properties of a polymer composite with carbon nanotube additives have been analyzed. The state of the system near the percolation threshold, when charge is transferred along a single percolation path, has been considered. For this state, the current–voltage characteristics of a percolation chain made up of carbon nanotubes have been calculated under the assumption that the contact resistance between neighboring nanotubes is much higher than the intrinsic resistance of the nanotubes. According to recent data, the distance between neighboring (contacting) nanotubes has been assumed to be randomly distributed. It has been shown that, under the given conditions, the current–voltage characteristic is essentially nonlinear. This indicates the nonohmic conductivity of the composites. The dependence of the current–voltage characteristic on the spread of the contact distribution over distances has been discussed.