Vibration Of Carbon Nanotubes Research Articles

Investigation of vibration of carbon nanotube and quality factor with confined and submerged fluid under hammer impact Test: A molecular dynamics study

Because of their unique dynamic properties and high-quality factors, one and two-dimensional (1D and 2D) materials have been increasingly in favorin recent years for the construction of a new generation of nano sensors. While numerous research has explored these qualities in vacuum, there is a need to analyze the influence of van der Waals interaction on their vibrational properties when they are exposed to fluids such as water which is significant as it plays a vital role in defining the stability of CNT-based nano-fluidic systems. Using molecular dynamics approach, the behavior of a single walled carbon nanotube (SWCNT) affected by water environment inside or around the structure is investigated. Moreover, the influence of nanotube dimensions and water density are examined using the newly presented method called hammer impact test in nanoscale, which is a conventional modal analysis method in macroscale. The findings show thatthe presented hammer impact test can predict the frequencies and quality factor with high precision and at a far reduced computational cost. The results show that the presence of fluid reduces the natural frequency of the nanotube, and that the natural frequency of the nanotube drops as fluid density increases. In addition, increasing the diameter of fluid carrier nanotubes leads to a rise in the natural frequency of the nanotubes in varied chiralities. At a constant density of water, comparing the results of interior fluid CNT with those of submerged CNT, the results show a meaningful reduction in natural frequency, quality factor, and vibration amplitudes, which makes the interior fluid CNT an ideal case for future investigation on CNT-based nano-fluidic systems. Our resultsreveal a novel aspect offluid-CNTinteraction and examines its impact to design high performance nano sensors.

Combined Effects of Surface Energy, Initial Stress and Nonlocality on Vibration of Carbon Nanotubes Conveying Fluid Resting on Elastic Foundations in a Thermo-Magnetic Environment

This paper scrutinizes the simultaneous impacts of surface elasticity, initial stress, residual surface stress and nonlocality on the nonlinear vibration of carbon nanotube conveying fluid resting while resting on linear and nonlinear elastic foundations and operating in a thermo-magnetic environment. The derived partial differential equation is decomposed into spatial and temporal equations using Galerkin method of decomposition. Thereafter, the temporal differential equation is solved with the aid of method of homotopy perturbation. Studies of the significance of the model parameters reveal that the negative value of the surface stress enhances the frequency ratio while the positive value of the surface stress abates the ratio. At any given value of nonlocal parameters, the surface effect is lessened for enhancing value of the length of the nanotube. The frequency ratio is abated as strength of the magnetic field, nonlocal parameter and the length of the nanotube are increased. The nonlocality lessens the surface effects and ratio of the frequencies. At high values of nonlocal parameter and nanotube length, the natural frequency of the structure gradually approaches nonlinear Euler–Bernoulli beam limit. The ratio of the frequencies is heightened when the temperature change is reduced at high temperature while at room/low temperature, such ratio is enhanced as the temperature change is augmented. Also, the frequency ratio at low temperatures is lower than at high temperatures. The present work will be very useful in the design and control of carbon nanotubes in thermo-magnetic environment while resting on elastic foundations.

Nonlinear Vibrations of Carbon Nanotubes with Thermal-Electro-Mechanical Coupling

Carbon nanotubes (CNTs) have wide-ranging applications due to their excellent mechanical and electrical properties. However, there is little research on the nonlinear mechanical properties of thermal-electro-mechanical coupling. In this paper, we study the nonlinear vibrations of CNTs by a thermal-electro-mechanical coupling beam theory. The Galerkin method is used to discretize the partial differential equation and obtain two nonlinear ordinary differential equations that describe the first- and second-order mode vibrations. Then, we obtain the approximate analytical solutions of the two equations for the primary resonance and the subharmonic resonance using the multi-scale method. The results indicate the following three points. Firstly, the temperature and electric fields have a significant influence on the first-mode vibration, while they have little influence on the second-mode vibration. Under the primary resonance, when the load amplitude of the second mode is 20 times that of the first mode, the maximal vibrational amplitude of the second is only one-fifth of the first. Under the subharmonic resonance, it is more difficult to excite the subharmonic vibration at the second-order mode than that of the first mode for the same parameters. Secondly, because the coefficient of electrical expansion (CEE) is much bigger than the coefficient of thermal expansion (CTE), CNTs are more sensitive to changes in the electric field than the temperature field. Finally, under the primary resonance, there are two bifurcation points in the frequency response curves and the load-amplitude curves. As a result, they will induce the jump phenomenon of vibrational amplitude. When the subharmonic vibration is excited, the free vibration term does not disappear, and the steady-state vibration includes two compositions.

Application of the modified Fourier series method and the genetic algorithm for calibration of small‐scale parameters in the nonlocal strain gradient nanobeams

In recent years, nonlocal strain gradient theory (NSGT) has been widely applied by researchers for the analysis of nanostructures. In this theory, the appropriate selection of small‐scale parameters and the type of high‐order boundary conditions is of great importance. In the current paper, free vibrations of carbon nanotubes are studied using a nonlocal strain gradient Timoshenko beam model, and the small‐scale parameters are calibrated based on the molecular dynamics (MD) results. The calibration process is conducted by defining an optimization problem, and the genetic algorithm is applied for obtaining the best values of the small‐scale parameters. The governing equations are derived based on Hamilton's principle and solved analytically using the modified Fourier series method. The variational consistent high‐order boundary conditions are obtained applying the weighted residual approach. The small‐scale parameters are calibrated for the first three natural frequencies considering two different boundary conditions. Moreover, the type of high‐order boundary conditions that is more consistent with the MD results is determined. The outcomes of this paper provide an appropriate reference for researchers who are using the NSGT for the analysis of micro/nanobeams.

Robust active vibration suppression of single-walled carbon nanotube using adaptive sliding-mode control and electrostatic actuators

In this paper, the electrostatic actuator-based active vibration control of a single-wall carbon nanotube conveying fluid is studied. A double electrostatic actuator scheme is developed and its decoupling method also is proposed. Then, a novel adaptive sliding-mode control (SMC) scheme is developed. The newly proposed scheme future improves existing results from two aspects: (1) the newly proposed adaptive SMC can work well under high uncertain case in which all parameters are uncertain and the uncertainties cannot be separated from control force and (2) a double electrostatic actuators scheme which can suppress the vibration of carbon nanotube in multiple directions is designed, and a novel decoupling scheme for the two actuators is developed. The stability of the closed-loop system is analyzed by using Lyapunov stability theory. Finally, the simulation results illustrate the effectiveness of the proposed scheme.

Review of research solid mechanics laboratory for 2020–2022

The article presents an overview of the research of the Laboratory of Solid Mechanics IMech UFRS RAS for 2020-2022. All studies were published as articles (there are two exceptions in the list of references, registered in the Register of Computer Programs) in well-known domestic and (or) foreign scientific journals on mechanics. During the three years under consideration, a number of new problems of aerohydroelasticity were solved and important results were obtained on the dynamic behavior of thin-walled structural elements interacting with external and internal continuous media. In particular, the linear bending of a cantilever rod loaded with all-round pressure and longitudinal force is considered in static and dynamic formulations. The areas of attraction of the deflection to the upper and lower equilibrium positions of a two-support pipe during its spatial bending-rotational vibrations are determined. The interaction of forced and parametric vibrations of the pipeline has been studied. The influence of the internal and external added masses of continuous media on the frequency of natural vibrations of a pipe moving with acceleration in the transverse direction is studied. The eigenfrequencies of bending vibrations of a micro- and nanometer-sized rod clamped at the ends are calculated. From the solution of the inverse problem for the changed values of natural frequencies, the coordinate and magnitude of the added mass are found. Linear oscillations of micro- and nanostrings are also considered when the pressure in a gaseous medium changes, taking into account surface effects. Using a model of molecular dynamics with a reduced number of degrees of freedom, the eigenfrequencies of bending vibrations of carbon nanotubes (CNTs) of different diameters are calculated under conditions of a plane-deformed state. In addition, part of the work was devoted to the study of the phenomenon of the ascent of an underwater gas pipeline and the determination of its buoyancy parameters. The scientific articles presented in this review are arranged in chronological order. For all articles, their summary is given and the main conclusions are formulated.

RESEARCH ON THE NONLINEAR VIBRATION OF CARBON NANOTUBE EMBEDDED IN FRACTAL MEDIUM

The vibration of carbon nanotubes embedded in elastic medium is a hot topic and many research findings are available. But when the medium is a fractal medium such as chips, ceramics, etc., its vibration characteristics are difficult to explain with the existing theories. So in this paper, we mainly study the nonlinear vibration of carbon nanotube embedded in a fractal medium. The fractal variational principle of the problem is successfully established, and is proved to have a strong minimum condition according to the He–Weierstrass Theorem. Then the Hamiltonian theory combined with the residual equation is implemented to find the solution of the fractal nonlinear vibration. Finally, the impacts of different fractal orders on the vibration characteristics are also presented.

Chaotic vibrations of carbon nanotubes subjected to a traversing force considering nonlocal elasticity theory

The existence of chaos in the lateral vibration of the carbon nanotube (CNT) can contribute to source of instability and inaccuracy within the nano mechanical systems. So, chaotic vibrations of a simply supported CNT which is subjected to a traversing harmonic force are studied in this paper. The model of the system is formulated by using nonlocal Euler–Bernoulli beam theory. The equation of motion is solved using the Rung–Kutta method. The effects of the nonlocal parameter, velocity and amplitude of the traversing harmonic force on the nonlinear dynamic response of the system are analyzed by the bifurcation diagrams, phase plane portrait, power spectra analysis, Poincaré map and the maximum Lyapunov exponent. The results indicate that the nonlocal parameter, velocity and amplitude of the traversing harmonic force have considerable effects on the bifurcation behavior and can be used as effective control parameters for avoiding chaos.

Three-Dimensional Thermal Stress Effects on Nonlinear Torsional Vibration of Carbon Nanotubes Embedded in an Elastic Medium

ABSTRACT Nonlinear torsional vibration of nanorods embedded in an elastic medium under three-dimensional thermal stresses is investigated in this study. The scale effect is introduced to the equation of motion using the nonlocal theory. The nanorods are under the effect of a three-dimensional thermal environment. The elastic medium is modeled by infinite rotational springs around the nanorod. Galerkin’s and He’s variational methods are used to solve the differential equation of motion and obtain torsional frequencies. An uncertainty analysis is done to show the effect of the uncertain parameters on the frequencies. The frequency sensitivities are obtained to demonstrate the frequency sensitivities to uncertain parameters. Effect of temperature changes, elastic medium stiffness, vibration amplitude, nonlocal scale coefficient, and nanorod length and diameter on the nonlinear torsional frequencies are investigated. The effect of temperature on the frequencies is dependent on the values of elastic medium stiffness, vibration amplitude, and nanorod length and diameter.

Nonlinear free and forced vibration of carbon nanotubes conveying magnetic nanoflow and subjected to a longitudinal magnetic field using stress-driven nonlocal integral model

In this paper primary and secondary resonance of carbon nanotube conveying magnetic nanofluid and subjected to a longitudinal magnetic field resting on viscoelastic foundation with different boundary conditions is investigated. To investigate the small scale effects, stress driven nonlocal integral model has been used and to show the more correctness of stress driven nonlocal integral model response, in studying the behavior of carbon nanotube with different boundary conditions, its results are compared with strain gradient model. The governing partial differential equations are derived from the Bernoulli–Euler beam theory utilizing the von Kármán strain–displacement relations. Using the Galerkin method, the governing equations are reduced to a nonlinear ordinary differential equation. The nonlinear natural frequencies are obtained from the perturbation method and the divergence and flutter instability due to the increase in nanofluid velocity is investigated. Then the frequency response for primary, subharmonic and superharmonic resonance is obtained using the method of multiple scales.Finally, the effects of length small scale parameters, longitudinal magnetic field, magnetic nanofluid and boundary conditions on nonlinear free and forced vibration of carbon nanotube are investigated. As the most important results, as the intensity of the magnetic field increases, the critical flow velocity increases and divergence and flutter occur later. But the critical flow velocity decreases with increasing the intensity of the magnetic field for a carbon nanotube conveying magnetic nanofluid. In forced vibration, increasing the intensity of the magnetic field increases the amplitude of the response for all boundary conditions in primary and secondary resonance.

Nonlinear Vibration of Multi-Walled Carbon Nanotubes with Initial Curvature Resting on Elastic Foundations in a Nonlinear Thermo-Magnetic Environment

The present work focuses on nonlinear dynamics models of multi-walled carbon nanotubes with initial curvature resting on Winkler-Pasternak elastic foundations in a nonlinear thermomagnetic environment using nonlocal elasticity theory. The derived systems of nonlinear vibration models are solved with the aid of the Galerkin decomposition and the homotopy perturbation method. Effects of temperature, magnetic field, multi-layer, and other thermomechanical parameters on the dynamic responses of the slightly curved multi-walled carbon nanotubes are investigated and discussed. As the temperature increases, the frequency ratio decreases as the linear natural frequency of the system increases. The results reveal that the frequency ratios decrease as the number of nanotube walls, temperature, spring constants, magnetic field strength, and the ratio of the radius of curvature to the length of the slightly curved nanotubes increase. These trends are the same for all the boundary conditions considered. However, clamped-simple and clamped-clamped supported multi-walled nanotube have the highest and lowest frequency ratio, respectively. Also, from the parametric studies to control nonlinear vibration of the carbon nanotubes, it is shown that quadruple-walled carbon nanotubes can be taken as pure linear vibration even at any value of linear Winkler and Pasternak constants. Therefore, this can be used for the restraining of the chaos vibration in the objective structure. These research findings will assist the designers and manufacturers in developing multi-walled carbon nanotubes for various structural, electrical, mechanical, and biological applications, especially in the areas of designing nanoelectronics, nanodevices, nanomechanical systems, nanobiological devices, and nanocomposites, and particularly when they are subjected to thermal loads, magnetic fields and elastic foundations.

A mathematical model of torsional vibrations of SWCNTs incorporating surface irregularity effects

The aim of the present work is to investigate the surface irregularity effects on torsional vibrations of single-walled carbon nanotubes (SWCNTs). Equation of motion and corresponding closed form solutions were derived based Hamilton’s model. The equations of motion are solved analytically and the influence of surface irregularity on the natural frequency of torsional vibrations of SWCNTs is studied in detail. Numerical caculations were performed for chiral graphene SWCNT of (12, 6) and the results of torsional vibrations were discussed and presented graphically. The obtained numerical results reveal that, the surface irregularity has notable effects on the natural frequency of torsional vibrations of SWCNTs. The impacts of surface irregularities on the natural frequency of nano materials, especially for the natural frequancies of torsional vibration of SWCNTs, have not been studied and most of previous studied were carried out for regular carbon nanotubes. In this sense, the present study is novel, and it is expected that the results obtained will be useful in the design and analysis of the torsional vibration of carbon nanotubes (CNTs) and nanostructures.

Coupled effects of magnetic field, number of walls, geometric imperfection, temperature change, and boundary conditions on nonlocal nonlinear vibration of carbon nanotubes resting on elastic foundations

The present study focusses on the investigations of the combined effects of magnetic field, temperature, nonlocal parameter, number of walls, initial geometric imperfection and end supports on the nonlinear transverse vibrations of carbon nanotubes resting on linear and nonlinear elastic foundations. With the aids of Erigen's nonlocal elasticity, Euler-Bernoulli beam theories and van der Waal forces equation, systems of nonlinear partial differential equations governing the dynamics responses of slightly curved multi-walled carbon nanotubes resting on Winkler and Pasternak foundations in a thermal-magnetic environment are developed. The developed equations are decomposed into spatial and temporal equations using Galerkin separation approach. The time-dependent nonlinear differential equations are solved by homotopy perturbation method. Detailed studies are conducted to investigate the effects of the models parameters on the nonlinear vibrations of the structures. The parametric studies reveal that the frequency ratio decreases as the number of magnetic field strength, number of nanotube walls, temperature, spring constants and the ratio of the radius of curvature to the length of the slightly curved nanotubes increase. It is established that these trends are the same for all the boundary conditions considered. The study also reveals that the clamped-simple supported multi-walled nanotubes have the highest frequency ratio while the clamped-clamped supported multi-walled nanotubes have the lowest frequency ratio. Further investigations show that quadruple-walled carbon nanotubes can be taken as pure linear vibration even at any value of linear Winkler and Pasternak foundations constants. Such result can be used to control the nonlinear vibration of the carbon nanotubes and also restrains the chaos vibration in the objective structure. The findings in this work will help in the design of multi-walled carbon nanotubes for various structural, electrical, mechanical and biological applications in a thermal and magnetic environment.

Собственные частоты изгибных колебаний углеродных нанотрубок

Using a molecular dynamics model with a reduced number of degrees of freedom, the natural frequencies of bending vibrations of carbon nanotubes (CNTs) of various diameters are calculated under plane strain conditions. It is shown that the theory of thin cylindrical shells provides high accuracy in estimating the frequencies of low-amplitude natural vibrations even for relatively small CNT diameters. It is shown that with an increase in the amplitude, the frequency of natural vibrations decreases, which is consistent with the data available in the literature. The results obtained are necessary for the design of terahertz resonators based on CNTs and high-precision mass and force nanosensors based on the effect of electromechanical coupling that CNTs exhibit.

Thermal effects on the free vibration of joined FG-CNTRC conical-conical shells

In this paper, the effects of temperature on free vibration of carbon nanotube reinforced composite (CNTRC) joined conical-conical shells are investigated. Carbon nanotubes (CNTs) are embedded across the thickness uniformly or functionally. Material properties of the shell are assumed to be functions of temperature. To study the free vibration of shells settled in thermal environments, it requires solving the static equations. Therefore, in order to derive these equations, first order shear deformation theory (FSDT) accompanied by von Kármán types of geometrical nonlinearity are employed in Hamilton's principle. Then, initial stress resultants due to equilibrium state, are extracted via linear membrane solution. Afterward, equations of motion are derived with regard to initial deformations and stresses. With considering the equations of motion for two cones together with boundary and continuity conditions, a set of governing equations is provided. To discretize the governing equations meridionally, generalized differential quadrature (GDQ) method is proposed. The results are primarily validated via comparing them with those of relevant researches in this field. Thereafter, some studies are carried out to exhibit the significant role of temperature variations, CNT distribution patterns, CNT volume fractions, cones' angles and boundary conditions on the free vibrational behaviors of joined conical shells. For the sake of preventing buckling occurrence during the present vibration analysis, critical buckling temperature Tcr is obtained by the procedure that the natural frequency takes zero value for the lowest temperature.

Free vibration analysis of nanotube based sensors including rotary inertia based on the Rayleigh beam and modified couple stress theories

In this study, a finite element formulation is presented to analyze the free vibration of carbon nanotube based sensor in conjunction with modified couple stress and Rayleigh beam theories. Rotary inertia effect and size dependency are considered for vibration problem of the cantilever single walled carbon nanotube. The aim of this paper is to examine the vibrational frequencies of single-walled carbon nanotube with these effects. Therefore, for the finite element solution, stiffness and mass matrices have been obtained that include these effects in the calculations. Numerical results are presented to show the variation of the frequencies with a variety of parameters such as the material length scale parameter, number of the finite elements, length of the nanotube and mode number.

Real-Time Video Imaging of Mechanical Motions of a Single Molecular Shuttle with Sub-Millisecond Sub-Angstrom Precision

Abstract Miniaturized machines have open up a new dimension of chemistry, studied usually as an average over numerous molecules or for a single molecule bound on a robust substrate. Mechanical motions at a single molecule level, however, are under quantum control, strongly coupled with fluctuations of its environment — a system rarely addressed because an efficient way of observing the nanomechanical motions in real time is lacking. Here, we report sub-millisecond sub-Å precision in situ video imaging of a single fullerene molecule shuttling, rotating, and interacting with a vibrating carbon nanotube at 0.625 milliseconds(ms)/frame or 1600 fps, using an electron microscope, a fast camera, and a denoising algorithm. We have achieved in situ observation of the mechanical motions of a molecule coupled with vibration of a carbon nanotube with standard error as small as 0.9 millisecond in time and 0.01 nm in space. We have revealed rich molecular dynamics, where motions are non-linear, stochastic and often non-repeatable, and a work and energy relationship at a molecular level previously undetected by time-averaged measurements or microscopy. The molecular video recording at a 1600-fps rate exceeds by 100 times the previous records of continuous recording of molecular motions.

Quantum capacitance mediated carbon nanotube optomechanics

Cavity optomechanics allows the characterization of a vibration mode, its cooling and quantum manipulation using electromagnetic fields. Regarding nanomechanical as well as electronic properties, single wall carbon nanotubes are a prototypical experimental system. At cryogenic temperatures, as high quality factor vibrational resonators, they display strong interaction between motion and single-electron tunneling. Here, we demonstrate large optomechanical coupling of a suspended carbon nanotube quantum dot and a microwave cavity, amplified by several orders of magnitude via the nonlinearity of Coulomb blockade. From an optomechanically induced transparency (OMIT) experiment, we obtain a single photon coupling of up to g0 = 2π ⋅ 95 Hz. This indicates that normal mode splitting and full optomechanical control of the carbon nanotube vibration in the quantum limit is reachable in the near future. Mechanical manipulation and characterization via the microwave field can be complemented by the manifold physics of quantum-confined single electron devices.

تحلیل غیرموضعی ارتعاشات آشوبناک، رزونانس اولیه و سوپرهارمونیک نانولولههای کربنی تکلایه در محیط حرارتی

In this article, a nonlinear elastic Bernoulli–Euler beam model is presented to investigate the chaotic behavior and primary and superharmonic resonance of single walled carbon nanotubes embedded in a visco-elastic medium at an elevated temperature. Using the Galerkin method and fourthorder Runge-Kutta method the governing equation is solved. The bifurcation diagram and largest Lyapunov exponent are employed to detect the critical amplitude of external force of periodic and chaotic response of single walled carbon. Having known the critical values, phase portrait and Poincare maps are presented to observe the periodic and chaotic behavior of the system. Moreover, the amplitude– frequency response for the primary superharmonic resonance of system is derived with the multiple scale method to investigate the feasibility of jump phenomenon. The sensitivity of jump phenomenon are studied for the selected viscoelastic foundation parameters, detuning parameter and external amplitude load. The results show that the amplitude of external force, viscoelastic foundation parameters, detuning parameter and temperature change in the cases of high and low temperature have a significant effect on the frequency response with jump phenomenon of system. In addition, the chaotic vibration of carbon nanotube can be controlled by changing of amplitude of external force.

Exponential and harmonic forced torsional vibration of single-walled carbon nanotube in an elastic medium

In this study, free torsional vibration and forced torsional vibration analysis under the time-dependent exponential and harmonic torsional loadings in single-walled carbon nanotube are investigated. The SWCNT is embedded in an elastic medium. Eringen's theory among the small-scale theories is selected. The nonlocal differential constitutive relation and corresponding boundary condition are derived via Hamilton's principle. Clamped–clamped boundary condition is utilized. The assumed modes method is employed for the dynamic torsional vibration in order to discretize the derived governing equations. The novelty of this work is devoted to the analysis of forced torsional vibration of a carbon nanotube embedded in an elastic medium under the various loadings. The angular displacement for the resonance frequency neglecting the elastic medium is illustrated. For the free analysis, the first three nondimensional natural frequencies with various small-scale parameters and stiffness of the elastic medium are calculated. The results are compared with another study for the first 10 mode numbers. The effects of the nonlocal parameter, length of carbon nanotube, stiffness of the elastic medium, thickness, time constant, and excitation frequency on the nondimensional and dimensional angular displacements are investigated, dynamically. For the greater values of the stiffness of the medium, the nonlocal parameter becomes negligible. When a time-dependent exponential torque is applied to the model, the angular displacement becomes greater and then lower by an increase in the value of the length, but the nondimensional angular displacement decreases continuously by increasing the value of the length under the time-dependent harmonic loading. Moreover, the angular displacement for a determined time becomes lower first and then becomes greater by increasing the time constant.