弹性直杆纵振时的混沌运动研究

建立了非线性材料弹性直杆在轴向扰力作用下的纵振方程。通过研究发现，非线性材料弹性直杆即使纵振也可能发生混沌运动。     

变截面直杆纵向自由振动的一种解法

本文根据均质变截面直杆的纵振与等截面直梁的弯曲问题的一些相似性质,把求变截面直杆纵振函数变为求梁弯曲时的截面上的弯矩问题。对应于杆的横截面面积的某些变化规律,得到了求杆的纵振函数及固有频率的计算公式。     

楔形杆轴纵、扭固有振动的修正摄动解

将含大参数的二阶线性方程的修正摄动解应用于楔型杆轴的固有纵振和扭振的研究,得到了其作固有纵振、扭振时的修正摄动解。并通过实例计算,比较了这种方法得出的解与Bessel方法一级近似解之间的误差。可以证明该修正摄动解具有相当高的精度。     

考虑桩体粘性的变阻抗桩受迫振动问题的解析解

建立了考虑桩体材料粘弹性、桩长有限、桩端为弹性支承时,考虑桩侧土作用、桩截面声阻抗发生突变状态下(有两个突变界面),桩顶受纵向激振时基桩受迫振动的定解问题,求得了稳态正弦及瞬态半正弦激振条件下的解析解,并对所得解进行了研究分析,结果显示,桩体材料粘性对变阻抗界面处的特征反射有着较大的影响。     

一类四阶非线性发展方程整体解的存在性与“blow up”现象

研究了描述粘弹性杆中纵向形变波传播的一类四阶非线性发展方程的第一初边值问题，用Ｇａｌｅｒｋｉｎ方法结合能量型先验估计证明了其整体解的存在性、惟一性及稳定性，并考虑了解的渐近性质及在一定条件下解的“ｂｌｏｗ ｕｐ”现象。     

粘弹性分析的复变边界积分方程法

本文应用弹性－粘弹性对应原理提出了基于复位势基本解的二维粘弹性分析的复变边界积分方程方法，给出了所有基本关系式，编制了相应的计算程序并给出计算实例，与已有工作相比，本方法具有公式统一，程序简洁通用，边界单元数少和效率高等特点。     

基于简谐激励的开槽圆盘粘弹性波导吸振器

在粘弹性变截面梁波导吸振器的基础上，研究简谐激励下的开槽圆盘粘弹性波导吸振器的耗能规律，定性地讨论了能量耗散率与粘弹性材料特性、开槽圆盘的几何参数及振动波频率的关系，并在家用冰柜压缩机上进行了减振试验。研究结果表明，开槽圆盘波导吸振器对低频振动具有良好的抑制作用。     

非线性弹性直杆纵振时的混沌行为

研究了S形本构关系的弹性直杆纵振时的混沌行为.用Galerkin原理将杆纵振时的动力控制方程转化为二阶三次非线性微分动力系统;给出了其产生同宿轨道和异宿轨道的条件,得到了同宿轨道的参数方程;借助Melnikov函数给出了系统发生混沌的临界条件;数值计算给出了混沌运动区域随β和γ的变化规律,用分岔图、位移时程曲线、相平面图和Poincaré映射判断了系统的运动行为即定常还是混沌.进一步的研究还表明本构关系中的二次非线性项对系统的动力响应具有很大的影响.     

一类四阶非线性发展方程整体解的存在性与“blowup”现象

研究了描述粘弹性杆中纵向形变波传播的一类四阶非线性发展方程的第一初边值问题 ,用 Galerkin方法结合能量型先验估计证明了其整体解的存在性、惟一性及稳定性 ,并考虑了解的渐近性质及在一定条件下解的“blow up”现象     

变截面梁式动力吸振器的宽带吸振机理

通过对变截面梁-板耦合结构的功率流研究，提出用变截面梁吸收弹性体的振动，论证了变截面梁式动力吸振器的宽带吸振性能，分析了梁的结构参数、梁在板上的连接位置等对吸振效果的影响，为设计适合于弹性主振系的新型梁式宽带动力吸振器奠定了基础。仿真结果表明：梁式动力吸振器吸振频带宽、吸振效果好，加之其结构简单．安装方便，有一定的工程应用前景。     

锥形纳米管纵向振动固有频率

研究变截面锥形纳米管/棒的纵向振动问题。由于其控制方程涉及变系数微分方程,提出适用范围广、简单精确的积分方程方法求解该问题,基于经典弹性杆理论与非局部弹性杆理论分别给出几种边界条件下的固有频率计算。对一端固定一端附加集中质量的碳纳米管的纵向振动,给出共振基频的近似简单表达式。与其它方法所得数值结果对比表明该方法有效。     

复杂变截面梁的轴向自由振动分析的近似方法

介绍了带有附加影响的变截面梁轴向自由振动问题的求解方法－模态摄动法，这一方法在由等截面均匀梁低阶主模态函数组成的模态子空间中，将复杂梁的变系数微分方程的求解化为线性代数方程组的求解，从而简化了计算过程，通过与其它方法的比较，说明了本文方法的优越性。     

Numerical models of steady rolling for non-linear viscoelastic structures in finite deformations

A Lagrangian formulation of constitutive laws for a viscoelastic material based on irreversible thermodynamics is first presented. These laws are expressed by a non-linear differential equation governing the evolution of an internal variable. Then equations describing the steady rolling of an axisymmetric viscoelastic structure are obtained from the conservation laws of continuum mechanics. A finite element approximation and a solution technique of the algebraic system is proposed. The eiimination of the internal variable allows one to keep an elastic-like algorithm with an independent solution of the viscoelastic equation. Numerical tests show the feasibility and the efficiency of the proposed methods in large three-dimensional situations.     

Free in-plane vibrations of curved nonuniform beams

Summary By introducing two physical parameters, the analysis of free in-plane vibrations of curved non-uniform beams is simplified. The explicit relations between the flexural displacement and the longitudinal displacement are derived. With these explicit relations, the two coupled governing differential equations are reduced to a complete sixth-order ordinary differential equation with variable coefficients in the longitudinal displacement, and a limiting study from the curved beam theory to the straight beam theory is successfully revealed. Finally, the influence of taper ratio, center angle and arc length on the first two natural frequencies of the beams is illustrated.     

Mode II Macrocrack Initiation in Orthotropic Composite Viscoelastic Plate

Safe loads and initiation time for a straight macrocrack in viscoelastic orthotropic material that is intended to model a fiber composite plate under shear loads is investigated. The composite material is modeled by viscoelastic orthotropic medium. Determination of expression for crack shear displacement as function of time is based on the corresponding elastic solution and the method of operator continued fractions. Initiation time is obtained as a solution of integral equation for the incubation period. Numerical calculations are given for mode II macrocrack initiation.     

Dissipation of energy of kinematically excited transverse oscillations in materials of elastic systems

We suggest a procedure for the calculation of kinematically excited transverse oscillations of rods with variable cross sections with regard for energy losses in cyclically strained materials. The procedure is based on the application of the asymptotic methods of nonlinear mechanics. We deduce analytic expressions both for the construction of amplitude-frequency characteristics of forced oscillations of the investigated dissipative elastic system and for the determination of the phase shift. These expressions make it possible to analyze the dynamic stressed states of rod-like engineering structures as functions of the parameters of applied loads in the stage of design.     

On extra boundary condition in the stagnation point flow of a second grade fluid

The two dimensional stagnation point flow of a second grade fluid is considered. The flow is governed by a boundary value problem in which the order of differential equations is one more than the number of available boundary conditions. It is shown that without augmenting the boundary conditions at infinity it is possible to obtain a numerical solution of the problem for all values of K, where K is the dimensionless viscoelastic fluid parameter. The numerical results using the algorithm foreshadow an asymptotic behavior for large K. The asymptotic solution is derived up to terms of O(K⁻¹). Perturbation solutions are also obtained up to the terms of O(K²). Finally an approximate solution is developed, based on stretching of the independent variable and minimizing the residual of the differential equation in the least square sense. All these solutions are compared with the exact numerical solution and the appropriate conclusions are drawn.     

Asymptotically approximate model equations for nonlinear dispersive waves in incompressible elastic rods

Summary In literature, nonlinear waves in elastic rods have been studied by many authors. Usually, the Navier-Bernoulli hypothesis (the assumption that plane cross-sections remain planar and normal to the rod axis) is used. Intuitively, one would expect that this would be a good approximation when one is mainly interested in longitudinal waves. However, there are no rigorous theoretical justifications available. Also, a defect of this assumption is that comparing with the exact three-dimensional theory the boundary conditions on the lateral surface can never be satisfied. Recently, three papers have been published to overcome this defect, but they contain some algebraic errors (which implies that the approach adopted there cannot be used to overcome this defect). So, this problem remains open. In this paper, we present our recent research results for this problem, and we have managed to establish asymptotically valid one-dimensional rod equations which are consistent with the lateral boundary conditions. Further, their dispersion relation can match with that of the exact three-dimensional field equations to any asymptotic order in the long-wave limit. For solitary waves in the far field, we derive to the leading order the KdV equation. Comparing its solitary-wave solution with that of the KdV equation obtained through the Navier-Bernoulli hypothesis, we find that the difference is very small. This provides some evidence to the validity of the assumption that plane cross sections remain planar and normal to the rod axis.     

Numerical modeling of micro- and macro-behavior of viscoelastic porous materials

This paper presents a numerical method for solving the two-dimensional problem of a polygonal linear viscoelastic domain containing an arbitrary number of non-overlapping circular holes of arbitrary sizes. The solution of the problem is based on the use of the correspondence principle. The governing equation for the problem in the Laplace domain is a complex hypersingular boundary integral equation written in terms of the unknown transformed displacements on the boundaries of the holes and the exterior boundaries of the finite body. No specific physical model is involved in the governing equation, which means that the method is capable of handling a variety of viscoelastic models. A truncated complex Fourier series with coefficients dependent on the transform parameter is used to approximate the unknown transformed displacements on the boundaries of the holes. A truncated complex series of Chebyshev polynomials with coefficients dependent on the transform parameter is used to approximate the unknown transformed displacements on the straight boundaries of the finite body. A system of linear algebraic equations is formed using the overspecification method. The viscoelastic stresses and displacements are calculated through the viscoelastic analogs of the Kolosov–Muskhelishvili potentials, and an analytical inverse Laplace transform is used to provide the time domain solution. Using the concept of representative volume, the effective viscoelastic properties of an equivalent homogeneous material are then found directly from the corresponding constitutive equations for the average field values. Several examples are given to demonstrate the accuracy of the method. The results for the stresses and displacements are compared with the numerical solutions obtained by commercial finite element software (ANSYS). The results for the effective properties are compared with those obtained with the self-consistent and Mori–Tanaka schemes.     

Asymptotic stress field in a cracked orthotropic strip

Christensen's theory of viscoelastic fracture allows the crack propagation velocity to be determined in terms of dissipation whose calculation requires the knowledge of the stress field in the vicinity of the crack tip: the simplest configuration leading to a constant velocity is that of a straight semi-infinite crack contained in an infinitely long strip whose clamped edges are displaced normal to the crack; although experimental data pertaining to this problem have been obtained for a number of materials, no analytical solution is available. When the material is highly anisotropic, an asymptotic solution involving a small parameter related to the ratio of shear modulus to the larger Young's modulus can be attempted. As the corresponding perturbation problem is singular, a matched asymptotic expansion has to be used: it is the sum of outer and inner approximations; both of these are solutions to simple boundary-value problems which can be solved in closed form. The so-constructed asymptotic solution is shown to agree with finite element results, even when the small parameter is as large as 0.2.