在有限变形条件下损伤粘弹性梁的动力学行为

本文在有限变形条件下，根据损伤粘弹性材料的一种卷积型本构关系和温克列假设，建立了粘弹性基础上损伤粘弹性Timoshenko梁的控制方程。这是一组非线性积分——偏微分方程。为了便于分析，首先利用Galerkin方法对该方程组进行简化，得到一组非线性积分一常微分方程。然后应用非线性动力学中的数值方法，分析了粘弹性地基上损伤粘弹性Timoshenko梁的非线性动力学行为，得到了简化系统的相平面图、Poincare截面和分叉图等。考察了材料参数和载荷参数等对梁的动力学行为的影响。特别，考察了基础和损伤对粘弹性梁的动力学行为的影响。     

Burgers方程的小波精细积分算法

求解偏微分方程的常用方法包括有限差分法、有限元法等。近年来,小波分析在偏微分方程数值求解中的应用已引起很多学者的关注,例如采用Daubechies小波或shannon小波构造的小波配置方法已经取得较好的结果。钟万勰院士提出的偏微分方程的子域精细积分方法是一种半解析方法,方法简单,精度高。将小波方法和精细积分方法相结合应用于偏微分方程的数值求解中将有利于提高算法的精度和稳定性,为此本文以Burgers方程为例,提出了一种求解一维非线性抛物型偏微分方程的小波精积分方法。该方法用拟小波配点法对空间域进行离散,建立起对时间的常微分方程组,然后采用精细时程积分方法对该方程组求解。数值计算结果表明,该方法同其它方法相比,具有计算格式简单,数值稳定性和精度较高的优点。     

层间接触对沥青路面粘弹性动力响应影响研究

为分析沥青面层材料粘弹特性及路面各层间接触条件对沥青路面动力响应的影响,基于解析的方法,开展了层间非完全连续沥青路面粘弹性动力响应的求解工作．采用修正的Burgers模型定义沥青面层材料的粘弹性本构关系,考虑沥青路面层间接触条件,在车辆荷载作用下,建立沥青路面的理论计算模型;通过Laplace-Hankel积分变换将偏微分方程组转化为常微分方程组并对问题进行求解;采用转换矩阵表征层间接触条件,求得层间非完全连续沥青路面粘弹性动力响应的解析表达式．从沥青路面实例计算结果发现：修正的Burgers模型中的瞬时弹性模量参数是对弯沉计算结果影响最大的因素之一,路表弯沉随的增大呈下降趋势,特别是当较小时,这种趋势尤为明显;沥青混合料本构模型中的粘弹性修正系数B和黏性参数是影响路面路表弯沉计算结果的另两个重要因素,并且B和对弯沉峰值出现时间的影响具有相反的趋势;层间的非完全连续条件对沥青路面动态弯沉计算结果影响较大,并以面层与基层间的非完全连续对弯沉计算结果影响最为显著．     

粘弹性桩的混沌运动分析

研究了在轴向载荷和周期性横向载荷共同作用下非线性粘弹性嵌岩桩的混沌运动情况。假定桩和土体分别满足Leaderman非线性粘弹性和线性粘弹性本构关系，得到的运动方程为非线性偏微分．积分方程；利用Galerkin方法将方程简化为非线性常微分一积分方程，同时利用非线性动力系统中的数值方法，进行了数值计算，得到了不同载荷参数、几何参数、材料参数时粘弹性桩发生周期运动、多周期运动及混沌运动的时程曲线、相图、功率谱、Poincare截面图，同时得到了挠度-载荷、挠度-几何参数、挠度-材料参数等分叉图，考察了各种参数的影响。数值结果表明非线性粘弹性桩在一定的条件下可以通过倍周期分叉的方式进入混沌运动状态，且桩的载荷参数、几何参数、材料参数对其运动状态有较大的影响。     

粘弹性Timoshenko梁的变分原理和静动力学行为分析

从线性，各向同性，均匀粘弹性材料的Boltzmann本构定律出发，通过Laplace变换及其反变换，由三维积分型本构关系给出了Timoshenko梁的本构关系，并由此建立了小挠度情况下粘弹性Timoshenko梁的静动力学行为分析的数学模型，一个积分－偏微分方程组的初边值问题。同时，采用卷积，建立了相应的简化Gurtin型变分原理。给出了两个算例，考查了梁的厚度h与梁的长度l之比β对梁的力学行为的影响。     

结构粘弹性分析的线法

本文将以现代常微分方程求解器为支撑软件的线法推广应用于含时间变量的粘弹性力学问题。基基本思想是先用拦普拉斯积分变换把问题的控制偏微分方程组和定解条件进行交换,得到拉氏变换空间中的定常问题,然后用传统的线法借助于常微分方程求解器得到该变换空间中的数值解,再应用拉氏数值逆变换求得原时空域中的数值解。文中还给出方法的基本原理和实施过程。算例表明,采用拉普拉斯积分变换于线法分析结构粘弹性力学等含时间变量的     

升阶试函数族在矩形板大挠度问题中的应用

为解决加权残值法求近似解的计算精度问题，将摄动法与加权残值法相结合，首先以板中心挠度为摄动参数进行摄动，将矩形板大挠度非线性偏微分方程组分解为线性偏微分方程组，然后用最小二乘法求解．求解中构造并应用了可以由控制参数，调节的升阶试函数族，计算结果与实验结果基本一致，与以前的研究比较，计算精度明显提高．该方法对于寻求最佳试函数和最佳近似值是一种有效的方法．     

弹性-蠕变体理论的广义变分原理

在应力和应变成线性关系时,应用弹性-蠕变体理论于具体工程问题一般归结为求解第二类伏尔泰勒积分方程或一组积分-微分方程,或化为求解一组变系数微分方程.当应力与应变为非线性关系时,常归结为求解非线性积分方程,或化为求解一组非线性变系数微分方程.这在数学上都将遇到很大困难,因而往往都用数值解法.这样,寻求有足够精度的近似解成为十分必要.另一方面,或许是更重要的,这就是用有限元法来求解蠕变问题.与弹性理论有限元问题相似,在有限元中引入广义变分原理将大大地促进有限元法的发展,本文将为这两方面提供条件.     

粘弹性矩形板的混沌和超混沌行为

从薄板Karman理论的基本假设出发；利用线性粘弹性理论中的Boltzman叠加原理，建立了粘弹性薄板非线性动力学分析的初边值问题，其运动方程是一组非线性积分──微分方程．在空间域上利用Galerkin平均化法之后，得到了变型的非线性积分──微分型的Duffing方程．综合利用动力系统中的多种方法，揭示了粘弹性矩形板在横向周期激励下的丰富的动力学行为，如不动点、极限环、混沌、奇怪吸引子、超混沌等，其中，混沌和超混沌是交替出现的．     

Duhamel项的精细积分方法在非线性微分方程数值求解中的应用

基于Duhamel项的精细积分方法,构造了几种求解非线性微分方程的数值算法。首先将非线性微分方程在形式上划分为线性部分和非线性部分,对非线性部分进行多项式近似,利用Duhamel积分矩阵,导出了非线性方程求解的一般格式。然后结合传统的数值积分技术,例如Adams线性多步法等,构造了基于精细积分方法的相应算法。本文算法利用了精细积分方法对线性部分求解高度精确的优点,大大提高了传统算法的数值精度和稳定性,尤其是对于刚性问题。本文构造的算法不需要对线性系统矩阵求逆,可以方便的考察不同的线性系统矩阵对算法性能的影响。数值算例验证了本文算法的有效性,并表明非线性系统的线性化矩阵作为线性部分是比较合理的选择。     

Nonlinear forced vibration analysis of postbuckled beams

A numerical solution methodology is proposed herein to investigate the nonlinear forced vibrations of Euler–Bernoulli beams with different boundary conditions around the buckled configurations. By introducing a set of differential and integral matrix operators, the nonlinear integro-differential equation that governs the buckling of beams is discretized and then solved using the pseudo-arc-length method. The discretized governing equation of free vibration around the buckled configurations is also solved as an eigenvalue problem after imposing the boundary conditions and some complicated matrix manipulations. To study forced and nonlinear vibrations that take place around a buckled configuration, a Galerkin-based numerical method is applied to reduce the partial integro-differential equation into a time-varying ordinary differential equation of Duffing type. The Duffing equation is then discretized using time differential matrix operators, which are defined based on the derivatives of a periodic base function. Finally, for any given magnitude of axial load, the pseudo -arc-length method is used to obtain the nonlinear frequencies of buckled beams. The effects of axial load on the free vibration, nonlinear, and forced vibrations of beams in both prebuckling and postbuckling domains for the lowest three vibration modes are analyzed. This study shows that the nonlinear response of beams subjected to periodic excitation is complex in the postbuckling domain. For example, the type of boundary conditions significantly affects the nonlinear response of the postbuckled beams.     

求解非线性振动问题的一种新方法

首先把描述非线性振动的微分方程归结为一非线性积分微分方程，然后把此积分微分方程的求解转化为一无穷阶的非线性代数方程组的求解。从理论上讲，可得到满足任何精度要求的周期解。本文用此方法对Ｄｕｆｉｎｇ系统进行了分析。     

A method for studying waves with spatially localized envelopes in a class of nonlinear partial differential equations

A methodology for investigating stationary and travelling waves with spatially localized envelopes is presented. The nonlinear governing partial differential equations considered possess a constant first integral of motion, and are separable in space and time when the small parameter of the problem is set to zero. To study stationary waves, a coordinate transformation on the governing nonlinear partial differential equation is imposed which eliminates the time dependence from the problem. An amplitude modulation function is then introduced to express the response of the system at an arbitrary point as a nonlinear function of a reference response. Analytic approximations to the amplitude modulation function are developed by expressing it in power series, and asymptotically solving sets of singular functional equations at the various orders of approximation. Travelling solutions may be computed from stationary ones, by imposing appropriate Lorentz transformations. As an application of the methodology, stationary and travelling breathers of a nonlinear partial differential equation are analytically computed.     

Application of perturbation idea to well-known Hencky problem: A perturbation solution without small-rotation-angle assumption

In existing studies, the well-known Hencky problem, i.e. the large deflection problem of axisymmetric deformation of a circular membrane subjected to uniformly distributed loads, has been analyzed generally on small-rotation-angle assumption and solved by using the common power series method. In fact, the problem studied and the method adopted may be effectively expanded to meet the needs of larger deformation. In this study, the classical Hencky problem was extended to the problem without small-rotation-angle assumption and resolved by using the perturbation idea combining with power series method. First, the governing differential equations used for the solution of stress and deflection in the perturbed system were established. Taking the load as a perturbation parameter, the stress and deflection were expanded with respect to the parameter. By substituting the expansions into the governing equations and corresponding boundary conditions, the perturbation solution of all levels were obtained, in which the zero-order perturbation solution exactly corresponds to the small-rotation-angle solution, i.e. the solution of the unperturbed system. The results indicate that if the perturbed and unperturbed systems as well as the corresponding differential equations may be distinguished, the perturbation method proposed in this study can be extended to solve other nonlinear differential equations, as long as the differential equation of unperturbed system may be obtained by letting a certain parameter be zero in the corresponding equation of perturbed system.     

The Gelfand-Levitan, the Marchenko, and the Gopinath-Sondhi integral equations of inverse scattering theory, regarded in the context of inverse impulse-response problems

In this paper, we are concerned with interpreting some classical inverse-scattering theories so that they are relevant to the one-dimensional inverse impulse-response problem which arises in reflection seismology

First the Gelfand-Levitan integral equations (which arise in the inverse scattering theory for the Schrödinger equation) are derived strictly in the time domain. Originally these celebrated equations were derived as a means of solving an inverse spectral problem, which is naturally posed in the frequency domain.

We show not only that these equations have a time-dependent formulation, but that their derivation is actually simpler here than in its original context. Next we give a similar derivation for the Marchenko integral equation.

We then obtain, by an independent method, the integral equation of Gopinath and Sondhi, which is not unlike the Gelfand-Levitan linear equation. It was used by them as a means of solving a time-dependent inverse problem arising in speech synthesis. A new integral equation, similar to the Marchenko equation is also derived.

Finally the integral equations are related to each other by direct transformation independent of the corresponding differential equations.

The present paper opens with a section in which the equation of one-dimensional elastic waves and the corresponding seismic inverse impulse-response problem are transformed into a form to which the Gelfand-Levitan theory applies, and then into the equations which arose in Gopinath and Sondhi's work.

We regard the integral equation of Gopinath and Sondhi as being directly applicable to the interpretation of seismic reflection data. The Gelfand-Levitan theory is also applicable but only after considerable transformation.

Our calculations are entirely in the time domain. The resulting equations have the merit that in order to recover the unknown coefficient on a finite interval (0, L) we need to use the boundary data also only on a finite segment of the reflection time series. The length of the record is just the two-way travel time associated with length L. By contrast, frequency-domain approaches require that long records be Fourier transformed. We distinguish results which are true only when the unknown coefficient is continuous from those true when it is bounded but only piecewise continuous.

In this work we have not addressed the important problem of deconvolution.     

Chaotic Oscillation of Saddle Form Cable-Suspended Roofs under Vertical Excitation Action

The purpose of this paper is to investigate the dynamic behaviour of saddle form cable-suspended roofs under vertical excitation action. The governing equations of this problem are system of nonlinear partial differential and integral equations. We first establish a spectral equation, and then consider a model with one coefficient, i.e., a perturbed Duffing equation. The analytical solution is derived for the Duffing equation. Successive approximation solutions can be obtained in likely way for each time to only one new unknown function of time. Numerical results are given for our analytical solution. By using the Melnikov method, it is shown that the spectral system has chaotic solutions and subharmonic solutions under determined parametric conditions.     

空间杆系结构非线性动力稳定性分析的一种实用算法

本文对任意空间杆系结构进行非线性动力稳定性分析。     

强非线性动力系统的频率增量法

提出一类强非线性动力系统的暧时频率增量法,将描述动力系统的二阶常微分方程,化为以相位为自变量、瞬廛频率为未知函数的积分方程;用谐波平衡原理,将求解瞬时频率的积分问题,归结为求解以频率增量的Fourier系数为独立变量的线性代数方程组;给出了若干例子。     

Chaotic Analysis of Nonlinear Viscoelastic Panel Flutter in Supersonic Flow

In this paper chaotic behavior of nonlinear viscoelastic panels in asupersonic flow is investigated. The governing equations, based on vonKàarmàn's large deflection theory of isotropic flat plates, areconsidered with viscoelastic structural damping of Kelvin's modelincluded. Quasi-steady aerodynamic panel loadings are determined usingpiston theory. The effect of constant axial loading in the panel middlesurface and static pressure differential have also been included in thegoverning equation. The panel nonlinear partial differential equation istransformed into a set of nonlinear ordinary differential equationsthrough a Galerkin approach. The resulting system of equations is solvedthrough the fourth and fifth-order Runge–Kutta–Fehlberg (RKF-45)integration method. Static (divergence) and Hopf (flutter) bifurcationboundaries are presented for various levels of viscoelastic structuraldamping. Despite the deterministic nature of the system of equations,the dynamic panel response can become random-like. Chaotic analysis isperformed using several conventional criteria. Results are indicative ofthe important influence of structural damping on the domain of chaoticregion.