Dynamical parametric instability of carbon nanotubes under axial harmonic excitation by nonlocal continuum theory

Structures under parametric load can be induced to the parametric instability in which the excitation frequency is located the instability region. In the present work, the parametric instability of double-walled carbon nanotubes is studied. The axial harmonic excitation is considered and the nonlocal continuum theory is applied. The critical equation is derived as the Mathieu form by the Galerkin's theory and the instability condition is presented with the Bolotin's method. Numerical calculations are performed and it can be seen that the van der Waals interaction can enhance the stability of double-walled nanotubes under the parametric excitation. The parametric instability becomes more obvious with the matrix stiffness decreasing and small scale coefficient increasing. The parametric instability is going to be more significant for higher mode numbers. For the nanosystem with the soft matrix and higher mode number, the small scale coefficient and the ratio of the length to the diameter have obvious influences on the starting point of the instability region.     

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.     

A nonlinear system under combined periodic and random excitation

The anharmonic oscillator under combined sinusoidal and white noise excitation is studied using the Gaussian closure approximation. The mean response and the steady-state variance of the system is obtained by the WKBJ approximation and also by the Fokker-Planck equation. The multiple steadystate solutions are obtained and their stability analysis is presented. Numerical results are obtained for a particular set of system parameters. The theoretical results are compared with a digital simulation study to bring out the usefulness of the present approximate theory.     

Phase equation with nonlinear excitation for nonlocally coupled oscillators

Some reaction-diffusion systems feature nonlocal interaction and, near the point of Hopf bifurcation, can be represented as a system of nonlocally coupled oscillators. Phase of oscillations satisfies an evolution pde which takes different forms depending on the values of parameters. In the simplest case the equation is effectively a diffusion equation which is excitation-free. However, more complex forms are possible such as the Nikolaevskii equation and the Kuramoto–Sivashinsky equation incorporating linear excitation. We analyse a situation when the phase equation is based on nonlinear excitation. We derive conditions on the values of the parameters leading to the situation and show that the values satisfying the conditions exist.     

Modulational instability and envelope solutions of nonlinear dispersive wave equations

The nonlinear Schrödinger equation describing the evolution of the plane wave solutions of the Hirota equation and of the Boussinesq equation are obtained. The conditions for modulational instability and the localised stationary solutions are derived.     

Chaos and chaotic control in a relative rotation nonlinear dynamical system under parametric excitation

This paper studies the chaotic behaviours of a relative rotation nonlinear dynamical system under parametric excitation and its control. The dynamical equation of relative rotation nonlinear dynamical system under parametric excitation is deduced by using the dissipation Lagrange equation. The criterion of existence of chaos under parametric excitation is given by using the Melnikov theory. The chaotic behaviours are detected by numerical simulations including bifurcation diagrams, Poincar map and maximal Lyapunov exponent. Furthermore, it implements chaotic control using non-feedback method. It obtains the parameter condition of chaotic control by the Melnikov theory. Numerical simulation results show the consistence with the theoretical analysis. The chaotic motions can be controlled to period-motions by adding an excitation term.     

Analysis of parametric oscillatory instability in Fabry-Perot cavity with Gauss and Laguerre-Gauss main mode profile

We calculate the parametric instabilities in Fabry-Perot cavities of advanced VIRGO and LIGO interferometers with different main mode profiles. All unstable combinations of elastic and Stokes modes both for the case with TEM₀₀ and LG₃₃ as a carriers are deduced.     

一类准周期参激非线性相对转动动力系统的稳定性与时滞反馈控制

建立了一类含准周期参数激励和时滞反馈的相对转动非线性系统的动力学方程. 采用多尺度法求解1/2亚谐波主参数共振下的分岔响应方程,并分析了系统的稳定性. 在求解非受控系统的定常解的基础上,通过讨论系统的动力学特性,研究了准周期参数激励对系统响应的影响. 采用时滞反馈控制的方法对系统分岔和极限环(域)进行控制,数值模拟的结果表明通过改变时滞参数可以实现对系统分岔的控制,并能有效地控制极限环(域)的幅值和稳定性. 关键词： 相对转动 准周期参激 时滞反馈 极限环     

Stability and instability of periodic travelling wave solutions for the critical Korteweg-de Vries and nonlinear Schrödinger equations

In this paper we establish new results about the existence, stability, and instability of periodic travelling wave solutions related to the critical Korteweg-de Vries equation

u_t+5u⁴u_x+u_xxx=0,     

随机参激和外激联合作用下非线性动力系统的路径积分解

将基于Gauss Legendre公式的路径积分法推广到随机参激和外激联合作用的非线性动力系统.研究了受高斯白噪声参激和外激联合作用的非线性振子，将所得路径积分解与其精确解（满足一定条件时该系统存在精确解）或Monte Carlo随机模拟结果相比较，充分验证了路径积分的准确性.并借助路径积分数值解，捕捉到该随机系统的一维P分岔. 关键词： 路径积分 P分岔 随机参激 随机外激     

Dynamics and stability of a new class of periodic solutions of the optical parametric oscillator

The stability and dynamics of a new class of periodic solutions is investigated when a degenerate optical parametric oscillator system is forced by an external pumping field with a periodic spatial profile modeled by Jacobi elliptic functions. Both sinusoidal behavior as well as localized hyperbolic (front and pulse) behavior can be considered in this model. The stability and bifurcation behaviors of these transverse electromagnetic structures are studied numerically. The periodic solutions are shown to be stabilized by the nonlinear parametric interaction between the pump and signal fields interacting with the cavity diffraction, attenuation, and periodic external pumping. Specifically, sinusoidal solutions result in robust and stable configurations while well-separated and more localized field structures often undergo bifurcation to new steady-state solutions having the same period as the external forcing. Extensive numerical simulations and studies of the solutions are provided.     

Studies in nonlinear stochastic processes. III. Approximate solutions of nonlinear stochastic differential equations excited by Gaussian noise and harmonic disturbances

A fusion of the highly successful methods of harmonic and statistical linearization is used as a first approximation in determining, either iteratively or via a nonlinear integral equation, the effects of higher harmonics and non-Gaussian distortion terms on the second-order statistics of a wide variety of nonlinear stochastic differential equations perturbed by some linear combination of Gaussian noise and a periodic deterministic/stochastic excitation. Physical a posteriori applicability criteria are presented which justify when these higher order effects may be neglected. A simple modification of this statistical-harmonic linearization procedure based upon the Fokker-Planck variance is proposed.This work was supported in part by the National Science Foundation under grant CHE75-20624.     

A new method for constructing soliton solutions and periodic solutions of nonlinear evolution equations

With the aid of the symbolic computation, we improve Xie's algorithm [F. Xie, Z.Y. Yan, H. Zhang, Phys. Lett. A 285 (2001) 76], and present a new extended method. Based on the new general ansatz (3), we successfully solve a compound KdV-MKdV equation, and obtain some special solutions which contain soliton solutions, and periodic solutions. The method can also be applied to other nonlinear partial differential equations.     

窄带随机激励双稳压电悬臂梁响应机制与能量采集研究

具有中心频率的窄带随机振动是一种典型的环境振动,其振动特征与环境的变化密切相关.本文以双稳压电悬臂梁能量采集系统为研究对象,分析系统在不同磁铁间距下的等效线性固有频率特性,以带通滤波器输出一定带宽的窄带随机激励模拟环境振动,研究系统的响应和能量采集特征.研究表明,对于一定带宽的窄带随机激励,一方面系统始终存在一个固定的磁铁间距使其输出达到峰值,另一方面当激励中心频率在一定范围内变化时,系统还分别存在另外两个或一个不同磁铁间距也能使系统输出达到峰值,而且该峰值特性是系统在其等效线性固有频率处诱导双稳或单稳“共振”形成的.研究结果可为具有窄带随机激励特征的振动能量采集提供一定的理论和技术支持.     

Nonlinear Rayleigh–Taylor instability of two superposed magnetic fluids under parallel rotation and a normal magnetic field

The Rayleigh–Taylor instability (RTI) of a ferrofluid has been the subject of recent research, because of its implications on the stability of stellar and planetary interiors. This paper analyzes the effects of rotation and magnetic field on nonlinear RTI of two superposed ferrofluids. It is considered that the system is subjected to uniform parallel rotation and normal magnetic field. Surface tension acts at the interface. The method of multiple scales is utilized to obtain the solutions and dispersion relations are obtained for the nonlinear problem of RTI of magnetic fluids. Finally the stability of the problem is discussed.     

Output-only identification and dynamic analysis of time-varying mechanical structures under random excitation: A comparative assessment of parametric methods

This article addresses the problem of parametric time-domain identification and dynamic analysis for time-varying (TV) mechanical structures under unobservable random excitation. The methods presented are based on time-dependent autoregressive moving average (TARMA) models, and are classified according to the mathematical structure imposed on the TV parameter evolution as unstructured parameter evolution, stochastic parameter evolution, and deterministic parameter evolution. The features and relative merits of each class are outlined. A representative method from each is then assessed through its application to the identification and dynamic analysis of a laboratory TV structure consisting of a beam with a mass moving on it. The results are mutually compared and contrasted to those obtained through “frozen-configuration” (multiple experiment) baseline identification.     

A general model for nano-cantilever switches with consideration of surface effects and nonlinear curvature

A general model for nano-cantilever switches with consideration of surface stress, nonlinear curvature, the location and length of the fixed electrode is developed. Some representative cantilever switch architectures are incorporated into this model. The governing equation is derived by using Hamilton principal and solved numerical. Results show that the influence of nonlinear curvature and surface effect on the pull-in instability and free vibration is significant for a switch with a large gap-length ratio and a short fixed electrode (the length of the fixed electrode is smaller than that of the cantilever nanobeam). The length and position of the fixed electrode have a significant effect on the pull-in parameters.     

Envelope self-similar solutions for the nonautonomous and inhomogeneous nonlinear Schrödinger equation

By means of the similarity transformation connecting with the solvable stationary equation, the self-similar combined Jacobian elliptic function solutions and fractional form solutions of the generalized nonlinear Schrödinger equation (NLSE) are obtained when the dispersion, nonlinearity, and gain or absorption are varied. The propagation dynamics in a periodic distributed amplification system is investigated. Self-similar cnoidal waves and corresponding localized waves including bright and dark similaritons (or solitons) for NLSE and arch and kink similaritons (or solitons) for cubic-quintic NLSE are analyzed. The results show that the intensity and the width of chirped cnoidal waves (or similaritons) change more distinctly than that of chirp-free counterparts (or solitons).     

SnSe分子外场下的基态性质和激发态性质

对Sn原子使用SDB-cc-pVTZ基组, Se原子采用6-311++G^**基组, 利用密度泛函中的B3LYP方法研究了电场强度为-0.04–0.04 a.u.的外电场对SnSe基态分子的几何结构、 电荷布居分布、 HOMO能级、 LUMO能级、 能隙、 费米能级、 谐振频率和红外光谱强度的影响. 继而使用含时密度泛函(TD-B3LYP) 方法研究了SnSe分子在外场下的激发特性. 结果表明, 外电场的大小和方向对SnSe分子基态的这些性质有明显影响. 在所加的电场范围内(-0.04 a.u.–0.04 a.u.), 随着正向电场的增大, 核间距先减小后增大, 在F=0.03 a .u.时取得最小值0.2317 nm; 分子电偶极矩μ近似线性地增大; E_L, E_H、 费米能级E_F和能隙E_g均减小. 随着正向电场逐渐增大, 分子总能量和谐振频率均先增大后减小; 红外谱强度则先减小后增大, 在F=0.03 a.u.时, 取得最小值 0.1138 km·mol^-1. 由基态到第1–10个单重激发态的波长均随着正向电场的增大而增大. 激发能均随着正向电场的增大而减小. 电场的引入可改变SnSe分子激发态出现的顺序并使得一些禁止的跃迁变得可能. 关键词： SnSe 外电场 能隙 激发特性     

Similarity reductions and new nonlinear exact solutions for the 2D incompressible Euler equations

For the 2D and 3D Euler equations, their existing exact solutions are often in linear form with respect to variables x, y, z. In this paper, the Clarkson–Kruskal reduction method is applied to reduce the 2D incompressible Euler equations to a system of completely solvable ordinary equations, from which several novel nonlinear exact solutions with respect to the variables x and y are found.