Longitudinal vibration of cracked nanobeams using nonlocal elasticity theory

The aim of this paper is to study the longitudinal frequency of a cracked nanobeam. The frequency equation of the nanobeam with clamped–clamped and clamped–free boundary conditions is derived based on the nonlocal elasticity theory. According to the equation, it can be found that the effects of the crack parameter, crack location, and nonlocal parameter on the longitudinal frequency of the cracked nanobeam are significant. The frequency decreases with an increase of the crack parameter. However, the increasing nonlocal parameter results in a decrease of the crack effect on the frequency. In addition, when the crack location is near the support, a larger decrease in the frequency can be observed.     

Size-dependent geometrically nonlinear free vibration analysis of fractional viscoelastic nanobeams based on the nonlocal elasticity theory

In recent decades, mathematical modeling and engineering applications of fractional-order calculus have been extensively utilized to provide efficient simulation tools in the field of solid mechanics. In this paper, a nonlinear fractional nonlocal Euler–Bernoulli beam model is established using the concept of fractional derivative and nonlocal elasticity theory to investigate the size-dependent geometrically nonlinear free vibration of fractional viscoelastic nanobeams. The non-classical fractional integro-differential Euler–Bernoulli beam model contains the nonlocal parameter, viscoelasticity coefficient and order of the fractional derivative to interpret the size effect, viscoelastic material and fractional behavior in the nanoscale fractional viscoelastic structures, respectively. In the solution procedure, the Galerkin method is employed to reduce the fractional integro-partial differential governing equation to a fractional ordinary differential equation in the time domain. Afterwards, the predictor–corrector method is used to solve the nonlinear fractional time-dependent equation. Finally, the influences of nonlocal parameter, order of fractional derivative and viscoelasticity coefficient on the nonlinear time response of fractional viscoelastic nanobeams are discussed in detail. Moreover, comparisons are made between the time responses of linear and nonlinear models.     

受迫振动非线性特性的教学拓展

受迫振动在大学物理和大学物理实验中均是重点教学内容,为了使学生更加深入理解受迫振动的非线性特性,本文基于波耳共振仪所涉及的非线性因素和实验数据,对受迫振动方程进行非线性修正,利用数值分析探讨其非线性特性.通过引入硬弹簧型杜芬方程,探讨系统由周期性运动进入混沌状态的演化,将受迫振动中相对稳定的平衡点与奇异吸引子进行类比,...     

Propagation of waves in nonlocal porous multi-phase nanocrystalline nanobeams under longitudinal magnetic field

ABSTRACT

This article investigates wave propagation behavior of a multi-phase nanocrystalline nanobeam subjected to a longitudinal magnetic field in the framework of nonlocal couple stress and surface elasticity theories. In this model, the essential measures to describe the real material structure of nanocrystalline nanobeams and the size effects were incorporated. This non-classical nanobeam model contains couple stress effect to capture grains micro-rotations. Moreover, the nonlocal elasticity theory is employed to study the nonlocal and long-range interactions between the particles. The present model can degenerate into the classical model if the nonlocal parameter, couple stress and surface effects are omitted. Hamilton’s principle is employed to derive the governing equations which are solved by applying an analytical method. The frequencies are compared with those of nonlocal and couple stress-based beams. It is showed that wave frequencies and phase velocities of a nanocrystalline nanobeam depend on the grain size, grain rotations, porosities, interface, magnetic field, surface effect and nonlocality.     

Surface solitons at the interface between semi-infinite linear and thermal nonlinear optical media

We investigate numerically surface-wave solitons occurring at the interface between semi-infinite linear and thermal nonlinear optical media, with the refractive index of the linear medium being greater than that of the nonlinear medium (in the absence of light). We find that the threshold energy flows of the existence of the surface solitons depend on the linear refractive index difference of the two media. Their fitting empirical formula has been obtained. Furthermore, we elucidate that the optical beams propagating in thermal nonlinear optical media, either as a single surface soliton or as a dipole surface soliton, can be attracted to the surface, even when launched from far away.     

Surface effect on the GSF energy of Al

The second-nearest-neighbor modified embedded atom method (2NN-MEAM) is used to calculate the generalized stacking fault (GSF) energy for (1 1 1) surface of Al crystal. It is found that the GSF energy curve is much lower for the fault in the first layer of the (1 1 1) surface than that in the bulk. When the fault exists in the second layer, the energy curve becomes considerably on the verge of that in the bulk. With a much lower unstable stacking fault energy γ_usf, the dislocation should be easier to set on at the outermost of the free surface. Expansion in relaxation always exists for the stacking fault either in bulk or near the surface and the GSF energy increases with the vertical expansion.     

Surface effect on buckling configuration of nanobeams containing internal flowing fluid: A nonlinear analysis

In this paper, the post-buckling behavior of supported nanobeams containing internal flowing fluid with two surface layers is studied based on a nonlinear theoretical model. The nonlinear governing equation, in which the surface effect and stretching-related nonlinearity are accounted for, is analytically solved for both clamped-clamped and pinned-pinned systems. The effects of nanobeam length, bulk thickness and several dimensionless parameters on the post-buckling behavior are analyzed. It is found that, the nanobeam with low flow velocity is stable at its original static equilibrium position and then undergoes a buckling instability at a critical flow velocity, which depends on the system parameters. When buckled, in all cases, the amplitude of the resultant buckling increases with the increasing flow velocity. Typically, the surface effect is explored by considering different nanobeam lengths and bulk thicknesses. The buckling amplitude is found to be length-dependent and thickness-dependent, showing that the effect of surface layers is considerably strong.     

Simulation of the nonlinear vibration of a string using the cellular automation method

In this study, the nonlinear dynamic responses of a string were simulated using the cellular automaton (CA) method. The local rules were set for the amplitude of vibration and the decay rate of amplitude. In the case of nonlinear systems, the velocity of wave propagation is not constant and depends on the amplitude. Thus, a new treatment of the time step was proposed, i.e., the time step in the CA method is adjusted to real time by considering the effect of the propagation velocity. As numerical examples, first, the dynamic responses of a string with linear characteristic were simulated and a typical resonance curve could be obtained. Secondly the dynamic responses of a string with nonlinear characteristic were simulated. Some characteristic types of vibration could be obtained. It was concluded that the linear and nonlinear dynamic responses of a string could be simulated using the CA method.     

Simulation of the nonlinear vibration of a string using the Cellular Automata based on the reflection rule

In this study, the nonlinear dynamic responses of a string are simulated using the Cellular Automata method based on the reflection rule. In the case of nonlinear systems, the velocity of wave propagation is not constant and depends on the amplitude. A new treatment of the dynamic time step is proposed for the Cellular Automata method considering the effect of the propagation velocity. As numerical examples, first, the dynamic responses of a string with linear characteristic are simulated using the Cellular Automata method. A typical resonance curve can be obtained. Second, the dynamic responses of a string with nonlinear characteristic are simulated using the proposed method. Some characteristic types of vibration can be obtained. It is concluded that the linear and nonlinear dynamic responses of a string may be obtained by simulation using the Cellular Automata method.     

关于受迫振动、共振的实验研究

简要分析了现行中学物理课本中关于受迫振动和共振的演示实验中存在的缺点,并针对其不足加以改进.改进后的实验装置演示效果更好,便于学生理解和掌握相应的物理规律.     

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.     

Surface effect on the necking of hyperelastic materials

Necking is a widely observed instability phenomenon. Surface tension often plays a significant role in the deformation of soft materials, especially at the micro scale, but its influence on the necking behavior of soft structures remains unclear. In this paper, we use the energy method to explore surface effects on the necking of soft plates and cylindrical bars under uniaxial tension. Analytical solutions are derived for the critical conditions of necking instability in the two types of structural components with a power-law constitutive relation. It is found that surface energy shows different impacts on their necking behaviors – it tends to postpone the occurrence of necking in soft plates but promote that in soft cylinders. This work not only deepens our understanding of the instability of soft materials but also helps design soft devices and robots.     

左手系材料界面上的非线性TE电磁波

针对电磁波在非线性左手系材料中的传播性质, 分别研究了左、右手系材料界面以及两左手系材料界面上非线性TE表面波的传播行为讨论了导波的频率特性、色散关系以及群速度随频率的变化规律分析表明,两种界面上的非线性TE表面波均存在频率通带和禁带,且带宽是传播功率的函数揭示了在一定条件下,左、右手系材料界面上既可以支持正向传播的非线性TE表面波,也可以支持反向传播的非线性TE表面波;两左手系材料界面上表面波的传播性质因材料参量的变化差异较大,一定参量条件下,该界面上仅支持反向传播的非线性TE表面波.     

光学系统受迫振动的激光测量方法

将激光信号与高帧频CCD结合,解决了光学系统中振动信号和激光信号之间的转换问题,不仅能够测量振动对系统光束指向稳定性的影响,而且能够得到振动信号本身的频率特性。利用该方法对振源为150 Hz和200 Hz两种条件下的光学系统受迫振动进行测量,得到了与输入信号相吻合的振动信号属性。通过实验与分析得知:时域振幅测量精度为6.25 m,频域分辨力为2 Hz,方法简便高效,测量结果准确,已应用于角多路准分子激光主振荡功率放大器系统打靶试验平台光束指向稳定性的研究中。     

Nonlinear vibration of viscoelastic embedded-DWCNTs integrated with piezoelectric layers-conveying viscous fluid considering surface effects

This article studies the nonlinear vibration of viscoelastic embedded nano-sandwich structures containing of a double walled carbon nanotube (DWCNT) integrated with two piezoelectric Zinc oxide (ZnO) layers. DWCNT and ZnO layers are subjected to magnetic and electric fields, respectively. This system is conveying viscous fluid and the related force is calculated by modified Navier–Stokes relation considering slip boundary condition and Knudsen number. Visco–Pasternak model with three parameters of the Winkler modulus, shear modulus, and damp coefficient is used for simulation of viscoelastic medium. The nano-structure is simulated as an orthotropic Timoshenko beam (TB) and the effects of small scale, structural damping and surface stress are considered based on Eringen's, Kelvin-voigt and Gurtin–Murdoch theories. Energy method and Hamilton's principle are employed to derive motion equations which are then solved using differential quadrature method (DQM). The detailed parametric study is conducted, focusing on the combined effects of small scale effect, fluid velocity, thickness of piezoelectric layer, boundary condition, surface effects, van der Waals (vdW) force on the frequency and critical velocity of nano-structure. Results indicate that the frequency and critical velocity increases with assume of surface effects.     

Surface profiling using fringe projection technique based on Lau effect

In this paper, the surface profile of objects is determined by using a grating projection system. The method is based on the concept of self-imaging, namely, the Lau effect. Periodic fringe patterns are generated from the projection of a grating illuminated by multiple mutually incoherent quasi-plane wavefront. The fringe patterns are then projected on to the object surface and the deformed grating image is captured by a CCD camera for subsequent analysis. Phase shifting techniques have been employed to determine the phase value. With suitable calibration of the system and the phase value obtained, the surface profile of the objects can be determined. The phase variation is achieved by using a linear translation stage incorporated to the grating. In this study, two specimens are tested to demonstrate the validity of the method. One is a spherical cap with a height of 4 mm, and the other is a coin. The experimental results are compared with results obtained by mechanical stylus method. In the case of the spherical cap the results are also compared with fringe projection method based on Talbot effect.     

计算机辅助受迫振动实验

在Matlab平台下编写了图形用户界面程序，该程序利用计算机的声卡采集音叉上的压电换能器信号，精确地计算出频率值和平均振幅，通过多组数据采集及插值计算，可自动绘制出不同阻尼下的共振曲线．     

Size effect of the elastic modulus of rectangular nanobeams:Surface elasticity effect

The size-dependent elastic property of rectangular nanobeams （nanowires or nanoplates） induced by the surface elas- ticity effect is investigated by using a developed modified core-shell model. The effect of surface elasticity on the elastic modulus of nanobeams can be characterized by two surface related parameters, i.e., inhomogeneous degree constant and surface layer thickness. The analytical results show that the elastic modulus of the rectangular nanobeam exhibits a distinct size effect when its characteristic size reduces below 1 O0 nm. It is also found that the theoretical results calculated by a mod- ified core-shell model have more obvious advantages than those by other models （core-shell model and core-surface model） by comparing them with relevant experimental measurements and computational results, especially when the dimensions of nanostructures reduce to a few tens of nanometers.     

Surface wave propagation at the interface between lossy metamaterials medium and nonlinear media with arbitrary nonlinearity

The dispersion relation in a system that consists of a lossy metamaterials (MTMs) film surrounded by a linear substrate and a nonlinear cladding with an arbitrary nonlinearity is derived. The surface plasmonic (SP) wave at the interfaces between metamaterials (MTMs) and the nonlinear cover is recovered by taking certain limits. Lossy MTMs have simultaneously complex-negative permeability μ and complex-negative permittivity ε. Results are presented by plotting the SP frequency as a function of the nonlinearity at chosen damping factors. Both the real and imaginary parts are studied. Results also display the wave frequency as a function of plasma frequency. For comparison, the imaginary part is set to zero and curves are reproduced.     

弹簧螺旋摆系统非线性振动的理论研究

运用牛顿第二运动定律对弹簧螺旋摆系统建立了模型方程,该方程为一组非线性微分方程,表明该系统具有复杂的非线性特征.理论分析表明,当系统固有的振动频率和摆动频率之比为2时,自由振动的弹簧螺旋摆系统存在内共振现象,数值求解结果证实了这一结论.并认为在一般情况下,弹簧螺旋摆系统的自由振动可能是准周期的.