Nonstationary probability densities of system response of strongly nonlinear single-degree-of-freedom system subject to modulated white noise excitation

The nonstationary probability densities of system response of a single-degree-of -freedom system with lightly nonlinear damping and strongly nonlinear stiffness subject to modulated white noise excitation are studied.Using the stochastic averaging method based on the generalized harmonic functions,the averaged Fokker-Planck-Kolmogorov equation governing the nonstationary probability density of the amplitude is derived. The solution of the equation is approximated by the series expansion in terms of a set...     

TRIAL FUNCTION METHOD FOR THE LONGITUDINAL VIBRATION OF RODS WITH VARIABLE CROSS-SECTIONS

给出了一种试探函数法,并研究了变截面杆的纵振动问题. 先给出振动控制方程的特殊函数形式的试探解,然后要求此解满足控制方程,反过来确定了控制方程各种可能的系数函数（即截面变化函数）并得到了控制方程的精确解. 作为例子,给出了一种变截面杆在3 种边界条件下的频率方程,计算出了固有频率. 研究表明,试探函数法简单、直接,适合于研究变截面杆的纵振动问题. 对于杆扭转振动、薄膜振动以及管中波传播等问题,该方法同样有推广应用价值.     

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.     

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.     

非线性粘弹性大挠度梁的动力学模型及其简化

本文建立了描述几何非线性均匀梁动力学行为的偏微分－－积分方程,梁的材料满足Ｌｅａｄｅｍａｎ非线性本构关系,对于两端简支的情形用Ｇａｌｅｒｋｉｎ方法进行了截断简化为常微分－－积分方程,然后引进附加变量的方法进一步简化为常微分方程组。     

The meshless analog equation method: I. Solution of elliptic partial differential equations

A new purely meshless method for solving elliptic partial differential equations (PDEs) is presented. The method is based on the principle of the analog equation of Katsikadelis, hence its name meshless analog equation method (MAEM), which converts the original equation into a simple solvable substitute one of the same order under a fictitious source. The fictitious source is represented by multiquadric radial basis functions (MQ-RBFs). The integration of the analog equation yields new RBFs, which are used to approximate the sought solution. Then inserting the approximate solution into the PDE and boundary conditions (BCs) and collocating at the mesh-free nodal points results in a system of linear equations, which permit the evaluation of the expansion coefficients of the RBFs series. The method exhibits key advantages over other RBF collocation methods as it is highly accurate and the coefficient matrix of the resulting linear equations is always invertible. The accuracy is achieved using optimal values for the shape parameters and the centers of the multiquadrics as well as of the integration constants of the analog equation, which are obtained by minimizing the functional that produces the PDE. Without restricting its generality, the method is illustrated by applying it to the general second order 2D and 3D elliptic PDEs. The studied examples demonstrate the efficiency and high accuracy of the developed method.     

Exact analytic method for solving variable coefficient differential equation

Many engineering problems can be reduced to the solution of a variable coefficient differential equation. In this paper, the exact analytic method is suggested to solve variable coefficient differential equations under arbitrary boundary condition. By this method, the general computation format is obtained. Its convergence is proved. We can get analytic expressions which converge to exact solution and its higher order derivatives unifornuy. Four numerical examples are given, which indicate that satisfactory results can be obtained by this method.     

Difference schemes basing on coefficient approximation

Introduction AdiscreteprocedureofthevariablecoefficientODEisdividedintotwosteps:Firstthe coefficientisfrozenasaconstantoneverydiscretesubinterval,soaseriesofapproximate differentialequationsareobtainedonglobalsolutioninterval.Secondly,thesetofalgebraic eq…     

Simplified lattice Boltzmann method for non-Newtonian power-law fluid flows

In this paper, we present a simplified lattice Boltzmann method for non-Newtonian power-law fluid flows. The new method adopts the predictor-corrector scheme and reconstructs solutions to the macroscopic equations recovered from the lattice Boltzmann equation through Chapman-Enskog expansion analysis. The truncated power-law model is incorporated into this method to locally adjust the physical viscosity and the associated relaxation parameter, which recovers the non-Newtonian behaviors. Compared with existing non-Newtonian lattice Boltzmann models, the proposed method directly evolves the macroscopic variables instead of the distribution functions, which eliminates the intrinsic drawbacks like high cost in virtual memory and inconvenient implementation of physical boundary conditions. The validity of the method is demonstrated by benchmark tests and comparisons with analytical solution or numerical results in the literature. Benchmark solutions to the three-dimensional lid-driven cavity flow of non-Newtonian power-law fluid are also provided for future reference.     

STRESS CONCENTRATIONS IN CYLINDRICAL SHELLS WITH LARGE OPENINGS

Based on Donnell's shallow shell equation, a new method is given in this paper to ana-lyze theoretical solutions of stress concentrations about cylindrical shells with large openings. With themethod of complex variable function, a series of conformal mapping functions are obtained from dif-ferent cutouts' boundary curves in the developed plane of a cylindrical shell to the unit circle. And,the general expressions for the equations of a cylindrical shell, including the solutions of stress concen-trations meeting the boundary conditions of the large openings' edges, are given in the mapping plane.Furthermore, by applying the orthogonal function expansion technique, the problem can be summa-rized into the solution of infinite algebraic equation series. Finally, numerical results are obtained forstress concentration factors at the cutout's edge with various opening's ratios and different loadingconditions. This new method, at the same time, gives a possibility to the research of cylindrical shellswith large non-circular openings or with nozzles.     

Solutions of the generaln-th order variable coefficients linear difference equation

In this paper.variable operator and its product with shifting operator are studied.The product of power series of shifting operator with variable coefficient is defined andits convergence is proved under Mikusinski’s sequence convergence.After turning ageneral variable coefficient linear difference equation of order n into a set of operatorequations.we can obtain the solutions of the general n-th order variable coefficientlinear difference equation.     

Non-linear waves in a fluid-filled inhomogeneous elastic tube with variable radius

In the present work, by employing the non-linear equations of motion of an incompressible, inhomogeneous, isotropic and prestressed thin elastic tube with variable radius and the approximate equations of an inviscid fluid, which is assumed to be a model for blood, we studied the propagation of non-linear waves in such a medium, in the longwave approximation. Utilizing the reductive perturbation method we obtained the variable coefficient Korteweg–de Vries (KdV) equation as the evolution equation. By seeking a progressive wave type of solution to this evolution equation, we observed that the wave speed decreases for increasing radius and shear modulus, while it increases for decreasing inner radius and the shear modulus.     

Localized coherent structures based on variable separation solution of the (2+1)-dimensional Boiti–Leon–Pempinelli equation

A new mapping equation (coupled Riccati equations) method is used to obtain three kinds of variable separation solutions with two arbitrary functions of the (2+1)-dimensional Boiti?CLeon?CPempinelli equation. Based on the variable separation solution and by selecting appropriate functions, two type of multidromion excitations, that is, dromion lattice and multidromion solitoffs, are investigated. Moreover, we can discuss head-on collision and ??chase and collision?? phenomena between two multidromions.     

SOLITARY WAVES IN FINITE DEFORMATION ELASTIC CIRCULAR ROD

A new nonlinear wave equation of a finite deformation elastic circular rod simultaneously introducing transverse inertia and shearing strain was derived by means of Hamilton principle. The nonlinear equation includes two nonlinear terms caused by finite deformation and double geometric dispersion effects caused by transverse inertia and transverse shearing strain. Nonlinear wave equation and corresponding truncated nonlinear wave equation were solved by the hyperbolic secant function finite expansion method. The solitary wave solutions of these nonlinear equations were obtained. The necessary condition of these solutions existence was given also.     

A computational method for simulating transient motions of an incompressible inviscid fluid with a free surface

A new numerical method has been developed for the analysis of unsteady free surface flow problems. The problem under consideration is formulated mathematically as a two-dimensional non-linear initial boundary value problem with unknown quantities of a velocity potential and a free surface profile. The basic equations are discretized spacewise with a boundary element method and timewise with a truncated forward-time Taylor series. The key feature of the present paper lies in the method used to compute the time derivatives of the unknown quantities in the Taylor series. The use of the Taylor series expansion has enabled us to employ a variable time-stepping method. The size of time increment is determined at each time step so that the remainders of the truncated Taylor series should be equal to a given small error limit. Such a variable time-stepping technique has made a great contribution to numerically stable computations. A wave-making problem in a two-dimensional rectangular water tank has been analysed. The computational accuracy has been verified by comparing the present numerical results with available experimental data. Good agreement is obtained.     

Nonstationary probability densities of strongly nonlinear single-degree-of-freedom oscillators with time delay

The approximate nonstationary probability density of a nonlinear single-degree-of-freedom (SDOF) oscillator with time delay subject to Gaussian white noises is studied. First, the time-delayed terms are approximated by those without time delay and the original system can be rewritten as a nonlinear stochastic system without time delay. Then, the stochastic averaging method based on generalized harmonic functions is used to obtain the averaged Itô equation for amplitude of the system response and the associated Fokker–Planck–Kolmogorov (FPK) equation governing the nonstationary probability density of amplitude is deduced. Finally, the approximate solution of the nonstationary probability density of amplitude is obtained by applying the Galerkin method. The approximate solution is expressed as a series expansion in terms of a set of properly selected basis functions with time-dependent coefficients. The proposed method is applied to predict the responses of a Van der Pol oscillator and a Duffing oscillator with time delay subject to Gaussian white noise. It is shown that the results obtained by the proposed procedure agree well with those obtained from Monte Carlo simulation of the original systems.     

Sound wave propagation through incompressible flows

This paper derives a variable coefficient, multidimensional Burgers equation which models the propagation of a nonlinear sound wave through an incompressible background flow. The equation is derived from the compressible Euler equations by the combination of a weakly nonlinear acoustics expansion for the sound wave and an incompressible expansion for the background flow. The main effect of the incompressible flow on the sound wave is the advection of the sound wave by the transverse velocity component of the flow.     

Propagation of longitudinal deformation wave along a lifting rope of variable length

The problem of motion of the rope of variable length consists of solving the boundary value problem with a variable boundary for the one-dimensional wave equation. A change of the rope length is caused by the force acting at the upper cross section of the rope. Studying the wave propagation process along the rope is based on the known integro-differential relation. The problem is reduced to solving two ordinary differential equations with a retarded argument that describe the variable length of the rope and the position of its lower end. The value of argument for functions involved in the right-hand side of the equations lag behind the argument value in the left-hand side of the equations by a time interval it takes for a propagation of the deformation wave throughout the current rope length. The problem is solved by using a technique of the sequential continuation of solution in the cyclic mode for each of the equation alternately. A computer realization of this technique presents no problem. A pilot computer program has been developed for solving the problem. Results of the numerical solution are presented in the case that the active force varies with time according to a piecewise linear relation.     

Weakly non-linear waves in a fluid-filled elastic tube with variable prestretch

In the present work, by utilizing the non-linear equations of motion of an incompressible, isotropic thin elastic tube subjected to a variable prestretch both in the axial and the radial directions and the approximate equations of motion of an incompressible inviscid fluid, which is assumed to be a model for blood, we studied the propagation of weakly non-linear waves in such a medium, in the long wave approximation. Employing the reductive perturbation method we obtained the variable coefficient KdV equation as the evolution equation. By seeking a travelling wave solution to this evolution equation, we observed that the wave speed is variable in the axial coordinate and it decreases for increasing circumferential stretch (or radius). Such a result seems to be plausible from physical considerations.