A modified nonlocal beam model for vibration and stability of nanotubes conveying fluid

In this paper, a new, modified nonlocal beam model is developed for analyzing the vibration and stability of nanotubes conveying fluid, in which one single nonlocal nanoscale parameter is included. Using Hamilton’s principle, a new higher-order differential equation of motion and the corresponding higher-order, non-classical boundary conditions are obtained for nanotubes conveying fluid. Based on this modified nonlocal model, effect of nonlocal nanoscale parameter on natural frequencies and critical flow velocities is presented and discussed through numerical calculations. It is found that this factor has great influence on the vibration and stability of nanotubes conveying fluid. In particular, the nonlocal effect tends to induce higher natural frequencies and higher critical flow velocities as compared to the results obtained from the classical and partial nonlocal beam models.     

Longitudinal,transverse, and torsional vibrations and stabilities of axially moving single-walled carbon nanotubes

Thanks to the brilliant mechanical properties of single-walled carbon nanotubes (SWCNTs), they are suggested as high speed nanoscale vehicles. To date, various aspects of vibrations of SWCNTs have been addressed; however, vibrations and instabilities of moving SWCNTs have not been thoroughly assessed. Herein, vibrational properties of an axially moving SWCNT with simply supported ends are studied using nonlocal Rayleigh beam theory. Employing assumed mode and Galerkin methods, the discrete governing equations pertinent to longitudinal, transverse, and torsional motions of the moving SWCNT are obtained. The resulting eigenvalue equations are then numerically solved. The speeds corresponding to the initiation of the instability within the moving nanostructure are calculated. The roles of the speed of the moving SWCNT, small-scale parameter, and aspect ratio on the characteristics of longitudinal, transverse, and torsional vibrations of axially moving SWCNTs are scrutinized. The obtained results show that the appearance of the small-scale parameter would result in the occurrence of both divergence and flutter instabilities at lower levels of the speed.     

轴向加速运动黏弹性梁受迫振动中的混沌动力学

研究了黏弹性轴向运动梁在外部激励和参数激励共同作用下横向振动的混沌非线性动力学行为. 引入有限支撑刚度, 并考虑黏弹性本构关系取物质导数, 同时计入由梁轴向加速度引起的沿径向变化的轴力, 建立轴向运动黏弹性梁横向非线性振动的偏微分-积分模型. 通过Galerkin截断方法研究了外部激励的频率和因速度简谐脉动引起的参数激励的频率在不可通约关系时轴向运动连续体的非线性动力学行为, 并对不同截断阶数的数值预测进行了对比. 基于对控制方程的Galerkin截断, 得到离散化的常微分方程组, 使用四阶Runge-Kutta方法求解. 基于此数值解, 运用非线性动力学时间序列分析方法, 通过Poincaré 映射, 观察到轴向运动梁随扰动速度幅值的倍周期分岔现象, 并比较了有无外部激励对倍周期分岔的影响. 分别在低速以及近临界高速运动状态下, 从相平面图、Poincaré 映射以及频谱分析的角度识别了系统中存在的准周期运动形态. 关键词： 轴向运动梁 非线性 混沌 分岔     

Frictional vibration transmission from a laterally moving surface to a traveling beam

As the density of information stored in automated magnetic tape libraries continues to increase, greater requirements are placed on the precision of mechanical positioning in order to successfully read and write data bits. The location of the read/write head in the direction across the tape's width (termed the lateral direction) is actively controlled in order to maintain alignment between the head and data tracks, even in the presence of the tape's lateral vibration. However, during repositioning, vibration is undesirably transmitted from the laterally moving head structure to the axially moving tape because of frictional contact between the two adjacent surfaces. As an analog of that interaction, a model is developed here to describe frictional vibration transmission from a surface having prescribed lateral motion to a tensioned beam that travels and slides over it. For a transport speed that is high when compared to the lateral vibration velocity, Coulomb friction between the surface and the beam can be well-approximated by an equivalent form of viscous damping. The beam is divided into contiguous regions corresponding to free spans and the beam's portion that contacts the surface. A critical engagement length between the beam and the surface exists for which vibration transmission at a particular natural frequency can be substantially reduced, and for a given mode, that length depends weakly on the surface's position along the beam's span. By contouring the surface to have portions of differing radii of curvature, the extent of vibration transmission can be reduced over a broad range of frequency.     

Free vibration and stability of a cantilever beam attached to an axially moving base immersed in fluid

Free vibration and stability are investigated for a cantilever beam attached to an axially moving base in fluid. The equations of motion of the slender cantilever beam affiliated to an axially moving base at a known rate while immersed in an incompressible fluid are derived first. An “axially added mass coefficient” is taken into account in the obtained equations. Then, a coordinate transformation is introduced to fix the boundaries. Based on the Galerkin approach, the natural frequencies of the beam system are numerically analyzed. The effects of moving speed of the base and several other system parameters on the dynamics and stability of the beam are discussed in detail. It is found that when the moving speed exceeds a certain value the beam becomes unstable and the instability type is sensitive to the system parameters. When the values of system parameters, such as mass ratio and axially added mass coefficient, are big enough, however, no instabilities are detected. The variations of the lowest unstable critical moving speed with respect to several key parameters are also investigated.     

Nonlocal-integro-differential modeling of vibration of elastically supported nanorods

In the previously established nonlocal continuum-based models, small characteristic length was commonly incorporated into the mass matrix and the driving force vector which is a bit in contradiction with our sense regarding these factors. Herein, a nonlocal-integro-differential version of the constitutive relations is employed for the bulk and the surface layer of the nanorod. By adopting Hamilton's principle, integro-partial differential equations of motion of elastically supported nanorods are established accounting for both nonlocality and surface energy effects. Then, these are solved by an efficient meshless methodology. For fixed–fixed and fixed–free nanorods, modal analysis of the problem is also performed and the explicit expressions of the mass and stiffness matrices are derived. For these special cases, the obtained results by the meshless technique are successfully verified with those of the modal solution. In the newly developed numerical model, the small-scale parameter is only incorporated into the stiffness matrix which gives us a more realistic sense about the nonlocality effect. Subsequently, the roles of the surface energy, small-scale parameter, elastic supports, and kernel function on natural frequencies of the nanostructure are discussed and explained. This work can be considered as a pivotal step towards a more reasonable nonlocal modeling of vibration of nanoscale structures.     

Molecular dynamics study of the torsional vibration characteristics of boron-nitride nanotubes

In recent years, synthesizing inorganic nanostructures such as boron nitride nanotubes (BNNTs) has led to extensive studies on their exceptional properties. In this study, the torsional vibration behavior of boron-nitride nanotubes (BNNTs) is explored on the basis of molecular dynamics (MD) simulation. The results show that the torsional frequency is sensitive to geometrical parameters such as length and boundary conditions. The axial vibration is found to be induced by torsional vibration of nanotubes which can cause instability in the nanostructure. It is also observed that the torsional frequency of BNNTs is higher than that of their carbon counterpart. Moreover, the shear modulus is predicted by incorporating MD simulation numerical results into torsional vibration frequency obtained through continuum-based model of tubes. Finally, it is seen that the torsional frequency of double-walled boron-nitride nanotubes (DWBNNTs) is between the frequencies of their constituent inner and outer tubes.     

Nonlocal discrete and continuous modeling of free vibration of stocky ensembles of vertically aligned single-walled carbon nanotubes

Free dynamic analysis of transverse motion of vertically aligned stocky ensembles of single-walled carbon nanotubes is of particular interest. A linear model is developed to take into account the van der Waals forces between adjacent SWCNTs because of their bidirectional transverse displacements. Using Hamilton's principle, the discrete equations of motion of free vibration of the nanostructure are obtained based on the nonlocal Rayleigh, Timoshenko, and higher-order beam theories. The application of such discrete models for frequency analysis of highly populated ensembles would be associated with so much computational effort. To overcome such a problem, some useful nonlocal continuous models are established. The obtained results reveal that the newly developed models can successfully capture the predicted fundamental frequencies of the discrete models. Through various numerical studies, the roles of slenderness ratio, radius of the SWCNT, small-scale parameter, population of the ensemble, and intertube distance on the fundamental flexural frequency of the nanostructure are examined and discussed. The capabilities of the proposed nonlocal continuous models in predicting flexural frequencies of the nanostructure are also addressed.     

Longitudinal vibration and stability analysis of carbon nanotubes conveying viscous fluid

Nowadays, carbon nanotubes (CNT) play an important role in practical applications in fluidic devices. To this end, researchers have studied various aspects of vibration analysis of a behavior of CNT conveying fluid. In this paper, based on nonlocal elasticity theory, single-walled carbon nanotube (SWCNT) is simulated. To investigate and analyze the effect of internal fluid flow on the longitudinal vibration and stability of SWCNT, the equation of motion for longitudinal vibration is obtained by using Navier-Stokes equations. In the governing equation of motion, the interaction of fluid-structure, dynamic and fluid flow velocity along the axial coordinate of the nanotube and the nano-scale effect of the structure are considered. To solve the nonlocal longitudinal vibration equation, the approximate Galerkin method is employed and appropriate simply supported boundary conditions are applied. The results show that the axial vibrations of the nanotubesstrongly depend on the small-size effect. In addition, the fluid flowing in nanotube causes a decrease in the natural frequency of the system. It is obvious that the system natural frequencies reach zero at lower critical flow velocities as the wave number increases. Moreover, the critical flow velocity decreases as the nonlocal parameter increases.     

Analysis of the coupled thermoelastic vibration for axially moving beam

The coupled thermoelstic vibration characteristics of the axially moving beam are investigated. The differential equation of motion of the axially moving beam under the thermoelastic coupling is established based to the equilibrium equation and the thermal conduction equation involving deformation term. The eigenequation is deduced and the dimensionless complex frequencies of the axially moving beam with different boundary conditions under the coupled thermoelastic case are calculated by the differential quadrature method. The curves of the real parts and imaginary parts of the first three-order dimensionless complex frequencies versus the dimensionless axially moving speed are obtained. The effects of the dimensionless coupled thermoelastic factor, the ratio of length to height, the dimensionless moving speed on the stability of the beam are analyzed.     

Theoretical and experimental study on the transverse vibration properties of an axially moving nested cantilever beam

An axially moving nested cantilever beam is a type of time-varying nonlinear system that can be regarded as a cantilever stepped beam. The transverse vibration equation for the axially moving nested cantilever beam with a tip mass is derived by D’Alembert?s principle, and the modified Galerkin?s method is used to solve the partial differential equation. The theoretical model is modified by adjusting the theoretical beam length with the measured results of its first-order vibration frequencies under various beam lengths. It is determined that the length correction value of the second segment of the nested beam increases as the structural length increases, but the corresponding increase in the amplitude becomes smaller. The first-order decay coefficients are identified by the logarithmic decrement method, and the decay coefficient of the beam decreases with an increase in the cantilever length. The calculated responses of the modified model agree well with the experimental results, which verifies the correctness of the proposed calculation model and indicates the effectiveness of the methods of length correction and damping determination. Further studies on non-damping free vibration properties of the axially moving nested cantilever beam during extension and retraction are investigated in the present paper. Furthermore, the extension movement of the beam leads the vibration displacement to increase gradually, and the instantaneous vibration frequency and the vibration speed decrease constantly. Moreover, as the total mechanical energy becomes smaller, the extension movement of the nested beam remains stable. The characteristics for the retraction movement of the beam are the reverse.     

The effect of calibrated nonlocal constant on the modal parameters and stability of axially compressed CNTs

Nowadays investigating the vibration behavior of carbon nanotubes (CNTs) has drawn considerable attention due to the superior mechanical properties of the CNTs. One of the powerful theoretical methods to study the vibration behavior of CNTs is implementing the nonlocal theory. Most of studies on the vibration behavior of CNTs have assumed a fixed value for small scale parameter for all vibration modes, however, this value is mode-dependent. Therefore, in this paper, the small scale parameter is calibrated for a single-walled carbon nanotube (SWCNT) with respect to each vibration mode. For this propose, the governing equation of motion based on the nonlocal beam theory is extracted by applying the Hamilton's principle. Then, by using the power series method, an eigenvalue problem is defined to derive the calibrated value of small scale constant and nonlocal mode shapes of the CNT. By using the expansion theory, the equation of motion is discretized, and the effect of nonlocality on the modal parameters and stability of the CNT under compressive force is investigated. Finally, the possibility of estimating nonlocal parameter based on simulated frequency domain response of the system by using modal analysis methods is studied. The results show that the calibration of small scale constant is important and the critical axial force is highly sensitive to this value.     

Galerkin method for steady-state response of nonlinear forced vibration of axially moving beams at supercritical speeds

The present paper investigates the steady-state periodic response of an axially moving viscoelastic beam in the supercritical speed range. The straight equilibrium configuration bifurcates in multiple equilibrium positions in the supercritical regime. It is assumed that the excitation of the forced vibration is spatially uniform and temporally harmonic. Under the quasi-static stretch assumption, a nonlinear integro-partial-differential equation governs the transverse motion of the axially moving beam. The equation is cast in the standard form of continuous gyroscopic systems via introducing a coordinate transform for non-trivial equilibrium configuration. For a beam constituted by the Kelvin model, the primary resonance is analyzed via the Galerkin method under the simply supported boundary conditions. Based on the Galerkin truncation, the finite difference schemes are developed to verify the results via the method of multiple scales. Numerical simulations demonstrate that the steady-state periodic responses exist in the transverse vibration and a resonance with a softening-type behavior occurs if the external load frequency approaches the linear natural frequency in the supercritical regime. The effects of the viscoelastic damping, external excitation amplitude, and nonlinearity on the steady-state response amplitude for the first mode are illustrated.     

Flow-thermoelastic vibration and instability analysis of viscoelastic carbon nanotubes embedded in viscous fluid

The flexural vibration of viscoelastic carbon nanotubes (CNTs) conveying fluid and embedded in viscous fluid is investigated by the nonlocal Timoshenko beam model. The governing equations are developed by Hamilton's principle, including the effects of structural damping of the CNT, internal moving fluid, external viscous fluid, temperature change and nonlocal parameter. Applying Galerkin’s approach, the resulting equations are transformed into a set of eigenvalue equations. The validity of the present analysis is confirmed by comparing the results with those obtained in literature. The effects of the main parameters on the vibration characteristics of the CNT are also elucidated. Most results presented in the present investigation have been absent from the literature for the vibration and instability of the CNT conveying fluid.     

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.     

A nonlocal Levinson beam model for free vibration analysis of zigzag single-walled carbon nanotubes including thermal effects

A nonlocal Levinson beam model is developed to study the free vibrations of a zigzag single-walled carbon nanotube (SWCNT) in thermal environments. The equivalent Young’s modulus and shear modulus for a zigzag SWCNT are derived using an energy-equivalent model. The present study illustrates that the vibration characteristics of an SWCNT are strongly dependent on the temperature change and on the chirality of a zigzag carbon nanotube. The investigation of the chirality and temperature effects on free vibration of carbon nanotubes may be used as a useful reference for the application and the design of nanoelectronic and nanodrive devices, nano-oscillators, and nanosensors, in which carbon nanotubes act as basic elements.     

Stability and bifurcation of an axially moving beam tuned to three-to-one internal resonances

This study analyzed the nonlinear vibration of an axially moving beam subject to periodic lateral force excitations. Attention is paid to the fundamental and subharmonic resonances, since the excitation frequency is close to the first two natural frequencies of the system. The incremental harmonic balance (IHB) method was used to evaluate the nonlinear dynamic behaviour of the axially moving beam. The stability and bifurcations of the periodic solutions for given parameters were determined by the multivariable Floquet theory using Hsu’s method. The solutions obtained from the IHB method agreed very well with those obtained from numerical integration. Furthermore, numerical examples are given to illustrate the effects of the three-to-one internal resonance on the response of the system.     

纵-扭复合模式夹心式功率超声压电换能器的研究

本文对纵-扭复合模式功率超声换能器进行了研究,该换能器由指数型前后金属喇叭盖板以及轴向极化和切向极化的两组压电陶瓷晶片组成。文中推出了换能器中纵向及扭转两种振动模式的共振频率方程。通过改变换能器前后指数型盖板的半径减编系数,得出了换能器中纵向与扭转两种振动模式同频共振的条件。实验表明,换能器的设计频率与测试频率基本一致,而且,换能器的纵向及扭转共振频率也比较接近,该换能器可望应用于超声加工及焊接等高振幅的功率超声技术中。     

Nonlinear free transverse vibration of an axially moving beam: Comparison of two models

Nonlinear free transverse vibration of an axially moving beam is investigated. A partial-differential equation governing the transverse vibration is derived from the Newton's second law. Under the assumption that the tension of beam can be replaced by the averaged tension over the beam, the partial-differential reduces to a widely used integro-partial-differential equation for nonlinear free transverse vibration. The method of multiple scales is applied directly to two equations to evaluate nonlinear natural frequencies. Numerical examples are presented to demonstrate the analytical results and to highlight the difference between two models. Two models yield the essentially same results for the weak nonlinearity, the small axial speed and the low mode, while the difference between two models increases with the nonlinear term, the axial speed, and the order of mode.     

On the vibrations of single-walled carbon nanotubes

In this paper, a detailed numerical study on the free and forced vibrations of single walled carbon nanotubes is presented. A simple and straightforward method developed such that the proximity of the mathematical model to the actual atomic structure of the nanotube is significantly retained, is used for this purpose. Both zigzag and armchair chiralities of the carbon nanotubes for clamped-free and clamped-clamped boundary conditions are analyzed and their natural frequencies and corresponding mode shapes are obtained. Results pertaining to axial, bending, and torsional modes of vibration are reported with discussions. These modes of vibration appear in the eigen-values and eigen-vectors without any distinction. The direct integration method by Newmark is used extensively along with the fast Fourier transform to identify different types of vibrational modes. In the case of zigzag nanotubes, the axial, bending, and torsional modes appear to be decoupled, whereas the armchair nanotubes show coupling between such modes.