有限变形弹性杆中的非线性波及周期解

利用Ham ilton变分原理,导出了计及有限变形和横向Possion效应的弹性杆中非线性纵向波动方程.利用Jacob i椭圆正弦函数展开和第三类Jacob i椭圆函数展开法,对该方程和截断的非线性方程进行求解,得到了非线性波动方程的两类准确周期解及相应的孤波解和冲击波解,讨论了这些解存在的必要条件.     

NONLINEAR WAVES AND PERIODIC SOLUTION IN FINITE DEFORMATION ELASTIC ROD

A nonlinear wave equation of elastic rod taking account of finite deformation, transverse inertia and shearing strain is derived by means of the Hamilton principle in this paper. Nonlinear wave equation and truncated nonlinear wave equation are solved by the Jacobi elliptic sine function expansion and the third kind of Jacobi elliptic function expansion method. The exact periodic solutions of these nonlinear equations are obtained, including the shock wave solution and the solitary wave solution. The necessary condition of exact periodic solutions, shock solution and solitary solution existence is discussed.     

On the reduction of some second-order nonlinear ODEs in physics and mechanics to first-order nonlinear integrodifferential and Abel’s classes of equations

Second-order ordinary differential equations (ODEs) with strong nonlinear stiffness terms (cubic nonlinearities) governing wave motions, dynamic crack propagations, nonlinear oscillations etc. in physics and nonlinear mechanics are analyzed. Selecting as guide line a second-order nonlinear ODE of the form of the forced Duffing equation and using admissible functional transformations it is possible to reduce it to an equivalent first-order nonlinear integrodifferential equation. The reduced equation is exact. In the limits of small or large values of the parameter characterizing this nonlinear problem, it is shown that further reductions lead to a nonlinear ODE of the Abel classes. Taking into account the known exact analytic solutions of this equivalent equation it is proved that there does not exist an exact analytic solution of this type of equations. However, in cases when convenient functional relations connecting all parameters of the corresponding null equation and the characteristics of the driving force exist, approximate analytic solutions to the problem under consideration are provided.     

梁中非线性弯曲波传播特性的研究

研究了梁中的非线性弯曲波的传播特性,同时考虑了梁的大挠度引起的几何非线性效应和 梁的转动惯性导致的弥散效应,利用Hamilton变分法建立了梁中非线性弯曲波的波动方程. 对该方程进行了定性分析,在不同的条件下,该方程在相平面上存在同宿轨道或异宿轨道, 分别对应于方程的孤波解或冲击波解. 利用Jacobi椭圆函数展开法,对该非线性方程进行 求解,得到了非线性波动方程的准确周期解及相对应的孤波解和冲击波解,讨论了这些解存 在的必要条件,这与定性分析的结果完全相同. 利用约化摄动法从非线性弯曲波动方程中导 出了非线性Schr\"{o}dinger方程,从理论上证明了考虑梁的大挠度和转动惯性时梁中存在 包络孤立波.     

Nonlinear oscillation of a two-dimensional lift body

It is very difficult to obtain an exact analytical solution to a nonlinear ordinary differential equation, so till now analytical solutions are rare in this area. The author has obtained the exact analytical solutions of this type of nonlinear oscillations. In this paper as an example, the exact analytical solution of nonlinear oscillation of a two-dimen-sional lift body, which has attracted the attention of research workers for a long time, is given.     

Localized solutions to the nonlinear wave equation for flexural motions of an elastic beam traveling in an air-filled tube

Steady-progressive-wave solutions are sought to the nonlinear wave equation derived previously [J. Fluids Struct. 16 (2002) 597] for flexural motions of an elastic beam traveling in an air-filled tube along its center axis at a subsonic speed. Fluid-structure interactions are taken into account through aerodynamic loading on the lateral surface of the beam subjected to small but finite deflection but end effects and viscous effects are neglected. Linear dispersion characteristics are first examined by exploiting the small ratio of the induced mass to the mass of the beam per unit length. Centered around the traveling speed of the beam, there exists such a narrow range of propagation velocity that the linear steady propagation is prohibited. In this range, it is revealed that some interesting nonlinear solutions exist. The periodic wavetrain is found to exist as the exact solution. Asymptotic analysis is then made by applying the method of multiple scales and the stationary nonlinear Schrödinger equation is derived for a complex amplitude. A monochromatic solution to this equation corresponds to the exact periodic solution. Imposing undisturbed boundary conditions at infinity, it is revealed that the localized solution exists as a result of balance between the linear instability and the nonlinearity. This solution is checked by solving the nonlinear equation numerically. It is further revealed that the amplitude-modulated wavetrain exists not only in the range of the velocity mentioned above but also outside of it.     

基于FPK方程的减摆器的非线性阻尼测试方法

受高斯白噪声外激的一阶非线性动力学方程能通过求解对应的FPK方程得到精确稳态解．本文基于这一结果导出减摆器非线性阻尼力与系统速度输出的概率结构的关系,将动力学系统中非线性阻尼力参数的测试问题转化测量系统的概率结构,并通过仿真进行了验证．     

To obtain approximate probability density functions for a class of axially moving viscoelastic plates under external and parametric white noise excitation

The probability density function plays an essential role to investigate the behaviors of stochastic linear or nonlinear systems. This function can be evaluated by several approaches but due to its analytical theme, the Fokker–Planck–Kolmlgorov (FPK) approach is preferable. FPK equation is a nonlinear PDE gives the probability density function for a stochastic linear or nonlinear system. Many researches have been done in literature tried to specify the conditions, in which the FPK equation gives an exact solution. Although, the exact probability density function can be achieved by solving the FPK equation even for some nonlinear systems, many types of systems cannot satisfy the conditions for exact solution. In this article, the axially moving viscoelastic plates under both external and parametric white noise excitation as one of the newest and applicable research areas are studied. Due to strong nonlinearities recognized in the governing equation of the system, the exact probability density function cannot be obtained, however, via an approximate method; some precise approximate solutions for different but comprehensive case studies are evaluated, validated, and discussed.     

On the response of purely nonlinear oscillators: An Ateb-type solution for motion and an Ateb-type external excitation

This study is concerned with free and forced undamped purely nonlinear oscillators. First, the exact closed-form solution for free vibrations given in terms of the Ateb function is discussed. An insight is provided with respect to the period of vibrations and the harmonic content of the response. Then, forced purely nonlinear oscillators with an Ateb-type external excitation are considered. The exact solution for the forced response is obtained, the amplitude-frequency equation derived and frequency-response curves investigated. It is also shown how one can adjust the system parameters to cause a constant frequency/period of the forced response.     

非圆截面杆中非线性扭转波方程的解析求解

研究了非圆截面杆中非线性扭转波动方程的精确求解问题. 利用直接积分与微分变换相结合的方法,得到了该方程的隐式通解. 通过对积分常数和方程系数的不同情形的讨论, 给出了该方程的三角函数、双曲函数、椭圆函数、指数函数以及它们的组合形式的解,分别对应于的非线性扭转波的孤立波、周期波以及冲击波等多种传播形式.     

Exact solution of the Boltzmann equation

We build up immediate connection between the nonlinear Boltzmann transport equation and the linear AKNS equation, and classify the Boltzmann equation as the Dirac equation by a new method for solving the Boltzmann equation out of keeping with the Chapman, Enskog and Grad’s way in this paper. Without the effect of other external fields, the exact solution of the Boltzmann equation can be obtained by the inverse scattering method.     

经典梁热过屈曲问题的解析解

基于非线性经典梁理论,建立了控制轴向和横向变形的基本方程,将两个非线性方程化简为一个关于横向挠度的四阶非线性积分-微分方程。对于本文所考虑的三类边界条件,该方程与相应的边界条件构成了微分特征值问题;直接求解该问题,得到热过屈曲构形的解析解,该解是外加热载荷的函数。为考察热载荷以及边界条件的影响,根据得到的解析解给出了一些数值算例,讨论了梁过屈曲行为的性质。本文得到的解析解可用于验证或改进各类近似理论和数值方法。     

INTEGRABLE TYPES OF NONLINEAR ORDINARY DIFFERENTIAL EQUATION SETS OF HIGHER ORDERS

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Nonlinear asymptotic analysis in elastica of straight bars—analytical parametric solutions

It is shown that by a series of admissible functional transformations the already derived (third-order) strongly nonlinear ordinary differential equation (ODE), describing the elastica buckling analysis of a straight bar under its own weight [Int.J.Solids Struct.24(12), 1179–1192, 1988, The Theory of Elastic Stability, McGraw-Hill, New York, 1961], is reduced to a first-order nonlinear integrodifferential equation. The absence of exact analytic solutions of the reduced equation leads to the conclusion that there are no exact analytic solutions in terms of known (tabulated) functions of this elastica buckling problem. In the limits of large or small values of the slope of the deflected elastica, we expand asymptotically the above integrodifferential equation to nonlinear ODEs of the Emden–Fowler or Abel nonlinear type. In these cases, using the solution methodology recently developed in Panayotounakos [Appl. Math. Lett. 18:155–162, 2005] and Panayotounakos and Kravvaritis [Nonlin. Anal. Real World Appl., 7(2):634–650, 2006], we construct exact implicit analytic solutions in parametric form of these types of equations and thus approximate implicit analytic solutions of the original elastica buckling nonlinear ODE.     

Solution of the first boundary value problem of nonlinear theory of steady-state creep for a half-space in antiplane deformation

The first boundary value problem of nonlinear theory of steady-state creep with powerlaw relationship between the stresses and the strain rates is considered for a half-space under conditions of antiplane (out-of-plane) deformation when tangential distributed forces are given on the half-space boundary. By using the introduced harmonic pseudostress function, we reduce solving this problem to solving a nonlinear singular integral equation admitting an exact solution.     

Explicit solution to the exact Riemann problem and application in nonlinear shallow‐water equations

The Riemann solver is the fundamental building block in the Godunov‐type formulation of many nonlinear fluid‐flow problems involving discontinuities. While existing solvers are obtained either iteratively or through approximations of the Riemann problem, this paper reports an explicit analytical solution to the exact Riemann problem. The present approach uses the homotopy analysis method to solve the nonlinear algebraic equations resulting from the Riemann problem. A deformation equation defines a continuous variation from an initial approximation to the exact solution through an embedding parameter. A Taylor series expansion of the exact solution about the embedding parameter provides a series solution in recursive form with the initial approximation as the zeroth‐order term. For the nonlinear shallow‐water equations, a sensitivity analysis shows fast convergence of the series solution and the first three terms provide highly accurate results. The proposed Riemann solver is implemented in an existing finite‐volume model with a Godunov‐type scheme. The model correctly describes the formation of shocks and rarefaction fans for both one and two‐dimensional dam‐break problems, thereby verifying the proposed Riemann solver for general implementation. Copyright © 2007 John Wiley & Sons, Ltd.     

Exact solution for finite deformation problems of cantilever beam with variable section under the action of arbitrary transverse loads

This paper deals with finite deformation problems of cantilever beam with variable section under the action of arbitrary transverse loads. By the use of a method of variable replacement, the nonlinear differential equation with varied coefficient for the problem can be transformed into an equation with variable separable. The exact solution can be obtained by the integration method. Some examples are given in the paper, and the results of these examples show that this exact solution includes the existing solutions in references as special cases. Projects Supported by the Science Foundation of the Chinese Academy of Sciences.     

Propagation of acoustic waves and shock waves radiated by a sinusoidal pulsation of a sphere during a single period

A spherical sound wave is emitted by a sphere which executes a small sinusoidal pulsation of a single period at high frequency in an inviscid fluid. Nonlinear propagation of the waves is formulated as an initial boundary value problem and is analysed in detail. The governing equation is linear near the sphere, while it is a nonlinear hyperbolic equation in a far field. The nonlinearity has a significant effect there, leading to the formation of two shocks. The exact solution to match the near field solution can easily be obtained for the far field equation. The nonlinear distortion of waveform and the shock formation distance are evaluated from the representation of the solution with strained coordinates. The evolution and nonlinear attenuation of the two shock discontinuities are also examined by making use of the equal-areas rule. In its asymptotic form the entire profile is an N wave with a long tail.     

Stochastic averaging of energy harvesting systems

A stochastic averaging method is proposed for nonlinear energy harvesters subjected to external white Gaussian noise and parametric excitations. The Fokker–Planck–Kolmogorov equation of the coupled electromechanical system of energy harvesting is a three variables nonlinear parabolic partial differential equation whose exact stationary solutions are generally hard to find. In order to overcome difficulties in solving higher dimensional nonlinear partial differential equations, a transformation scheme is applied to decouple the electromechanical equations. The averaged Itô equations are derived via the standard stochastic averaging method, then the FPK equations of the decoupled system are obtained. The exact stationary solution of the averaged FPK equation is used to determine the probability densities of the displacement, the velocity, the amplitude, the joint probability densities of the displacement and velocity, and the power of the stationary response. The effects of the system parameters on the output power are examined. The approximate analytical outcomes are qualitatively and quantitatively supported by the Monte Carlo simulations.     

On the structure of characteristic surfaces related to partial differential equations of first and higher orders. II

The generalized method of characteristics is developed within the framework of the geometric Monge picture. Hopf-Lax-type extremality solutions are obtained for a broad class of Cauchy problems for nonlinear partial differential equations of the first and higher orders. A special Hamilton-Jacobi-type case is analyzed separately. An exact extremality Hopf-Lax-type solution of the Cauchy problem for the nonlinear Burgers equation is obtained, and its linearization to the Hopf-Cole expression and to the corresponding Airy-type linear partial differential equation is found and discussed. Published in Neliniini Kolyvannya, Vol. 8, No. 4, pp. 529–543, October–December, 2005.