Stability of elastic and viscoelastic systems under a random perturbations of their parameters

Summary The present paper is concerned with the investigation of the almost sure stability of elastic and viscoelastic systems, when their parameters assume a random wide-band stationary process. The parameters are parametric loads, characteristics of external damping and material viscosity. With the help of Liapunov's direct method, the sufficient condition of the almost sure asymptotic stability for distributed parameter systems with respect to perturbations of initial conditions of an arbitrary form is obtained. It is shown that, in some cases, this condition coincides with a similar condition derived from the assumption that the form of sure and required perturbations coincides with the first eigenfunction of system oscillations. However, an example is given for the stability of a viscoelastic rod, when the perturbations of initial conditions are more dangerous, if their form differs from the first eigenfunction.This research was sponsored by the Russian Foundation of Fundamental Research of the Russian Academy of Sciences under Grant 94-01-01522.     

Nonlinear random stability of viscoelastic cable with small curvature

The nonlinear planar mean square response and the random stability of a viscoelastic cable that has a small curvature and subjects to planar narrow band random excitation is studied. The Kelvin viscoelastic constitutive model is chosen to describe the viscoelastic property of the cable material. A mathematical model that describes the nonlinear planar response of a viscoelastic cable with small equilibrium curvature is presented first. And then a method of investigating the mean square response and the almost sure asymptotic stability of the response solution is presented and regions of instability are charted. Finally , the almost sure asymptotic stability condition of a viscoelastic cable with small curvature under narrow band excitation is obtained.     

Influence of rotatory inertia on dynamic stability of the viscoelastic symmetric cross-ply laminated plates

The dynamic stability problem of the symmetrically laminated cross-ply plates made of the viscoelastic Voigt–Kelvin material, compressed by time-dependent stochastic membrane forces, is investigated. The effect of rotatory inertia is included in the present formulation. It is assumed that all elastic moduli have the same retardation times. By using the direct Liapunov method, bounds of the almost sure stability of cross-ply plates as a function of viscous damping coefficient, retardation time, variances of the stochastic forces, ratio of the principal lamina stiffnesses, number of layers, plate aspect ratio, cross-ply ratio and intensity of the deterministic components of axial loading are obtained. Numerical calculations are performed for the Gaussian process with a zero mean, as well as an harmonic process with random phase.     

A variational principle of perturbed motion on viscoelastic thin plates with applications

In this paper, in the light of the Boltzmann superposition principle in linear viscoelasticity, a mathematical model of perturbed motion on viscoelastic thin plates is established. The corresponding variational principle is obtained in a convolution bilinear form. For application the problems of free vibration, forced vibration and stability of a viscoelastic simply-supported rectangular thin plate are considered. The results show that numerical solutions agree well with analytical solutions. The project was supported by the National Natural Science Foundation of China (No. 19772027) and the Shanghai Municipal Development Foundation of Science and Technology (No. 98JC14032).     

Creep buckling and post-buckling analysis of the laminated piezoelectric viscoelastic functionally graded plates

The creep buckling and post-buckling of the laminated piezoelectric viscoelastic functionally graded material (FGM) plates are studied in this research. Considering the transverse shear deformation and geometric nonlinearity, the Von Karman geometric relation of the laminated piezoelectric viscoelastic FGM plates with initial deflection is established. And then nonlinear creep governing equations of the laminated piezoelectric viscoelastic FGM plates subjected to an in-plane compressive load are derived on the basis of the elastic piezoelectric theory and Boltzmann superposition principle. Applying the finite difference method and the Newmark scheme, the whole problem is solved by the iterative method. In numerical examples, the effects of geometric nonlinearity, transverse shear deformation, the applied electric load, the volume fraction and the geometric parameters on the creep buckling and post-buckling of laminated piezoelectric viscoelastic FGM plates with initial deflection are investigated.     

Response and stability of strongly non-linear oscillators under wide-band random excitation

A new stochastic averaging procedure for single-degree-of-freedom strongly non-linear oscillators with lightly linear and (or) non-linear dampings subject to weakly external and (or) parametric excitations of wide-band random processes is developed by using the so-called generalized harmonic functions. The procedure is applied to predict the response of Duffing–van der Pol oscillator under both external and parametric excitations of wide-band stationary random processes. The analytical stationary probability density is verified by digital simulation and the factors affecting the accuracy of the procedure are analyzed. The proposed procedure is also applied to study the asymptotic stability in probability and stochastic Hopf bifurcation of Duffing–van der Pol oscillator under parametric excitations of wide-band stationary random processes in both stiffness and damping terms. The stability conditions and bifurcation parameter are simply determined by examining the asymptotic behaviors of averaged square-root of total energy and averaged total energy, respectively, at its boundaries. It is shown that the stability analysis using linearized equation is correct only if the linear stiffness term does not vanish.     

Almost sure stochastic stability of a viscoelastic double-beam system

This paper investigates the dynamic stability of a viscoelastic double-beam system under parametric excitations. It is assumed that the two beams, made from Voigt–Kelvin material, are simply supported and continuously joined by a Winkler elastic layer. Each pair of axial forces consists of a constant part and a time-dependent stochastic function. In the case of “non-white” excitations, by using the direct Liapunov method, bounds of the almost sure stability of the double-beam system as a function of retardation time, bending stiffness, stiffness modulus of the Winkler layer, variances of the stochastic forces and the intensity of the deterministic components of axial loading are obtained. Numerical calculations are performed for the Gaussian process with a zero mean, as well as a harmonic process with a random phase. When the excitations are wideband noises, almost sure stability is obtained within the concept of the Liapunov exponent. White noise and Ornstein–Uhlenbeck processes are considered as models of wideband noises.     

Nonlinear vibrations and stability of elastic and viscoelastic systems under random stationary loads

The paper deals with numerical analysis of nonlinear vibrations of viscoelastic systems under a stochastic action in the form of a Gaussian stationary process with rational spectral density. The analysis is based on numerical simulation of the original stationary process, numerical solution of the differential equations describing the motion of the system, and computation of the maximum Lyapunov exponent if the stability of this motion is studied. An example of a plate subjected to a random stationary load applied in its plane is used to consider specific issues concerning the application of the proposed method and the peculiarities of the behavior of geometrically nonlinear elastic and viscoelastic stochastic systems. Special attention is paid to the interaction of a deterministic periodic action and a stochastic action from the viewpoint of stability of the system motion. It is shown that in some cases imposing a “colored” noise may stabilize an unstable system subjected to a periodic load.     

移动的线源平稳随机荷载激励下梁的随机响应

利用广义Ｄｕｈａｍｅｌ积分和积分变换，研究了粘弹性Ｋｅｌｖｉｎ地基上无限长梁在运动的线源平稳随机荷载作用下的随机响应．发现此时梁的挠度响应为非平稳随机过程．通过引入随动坐标系，建立了有明确物理意义的随动谱分析方法，使随机位移响应在随动坐标系下成为平稳随机过程     

Influence of corner stresses on the stability characteristics of composite skew plates

Stability characteristics of composite skew plates subjected to in-plane compressive load are investigated here using the shear deformable finite element approach. The influences of high prebuckling stresses at the corner regions of isotropic and composite skew plates on their stability characteristics are emphasized for different load direction, boundary condition and laminate stacking sequence. The non-linear governing equations based on von Kármán's assumptions are solved by Newton-Raphson technique to get the hitherto unreported postbuckling equilibrium paths of composite skew plates loaded between two rigid flat platens. The variation of out-of-plane deformation and end-shortening with compressive in-plane load are examined for simply supported and clamped skew plates made of isotropic, symmetric and unsymmetric laminates. Marguerre's shallow shell theory is employed to study the effect of sinusoidal imperfection on the non-linear behavior of composite skew plates.     

On a probabilistic stability theory for imperfection sensitive structures

The concept of almost sure sample stability and sample stability in probability are formulated for elastic systems. Using a Koiter type approach these concepts are used in the analysis of imperfection sensitive structures. The applied load and the initial geometric imperfections are introduced into the analysis as random quantities. A compressed beam of finite length on a nonlinear elastic foundation is used in an example calculation.     

The transient fracture behavior for a functionally graded layered structure subjected to an in-plane impact load

The transient fracture behavior of a functionally graded layered structure subjected to an in-plane impact load is investigated. The studied structure is composed of two homogeneous layers and a functionally graded interlayer with a crack perpendicular to the boundaries. The impact load is applied on the face of the crack. Fourier transform and Laplace transform methods are used to formulate the present problem in terms of a singular integral equation in Laplace transform domain. Considering variations of parameters such as the nonhomogeneity constant, the thickness ratio and the crack length, the dynamic stress intensity factors (DSIFs) in time domain are studied and some meaningful conclusions are obtained.The project supported by the National Science Foundation for Excellent Young Investigators (10325208), the National Natural Science Foundation of China (10432030) and the China Postdoctoral Science Foundation (2004036018)The English text was polished by Ron Marshall.     

复合材料层合板蠕变屈曲与变形的优化问题

将瞬时弹性失稳荷载、持久临界荷载和后屈曲持久变形刚度作为复合材料层合板蠕变屈曲与变形的优化指标,对几组纤维铺设方式进行讨论,构造多目标优化模型,就纤维铺设角进行优化分析.考虑横向剪切变形效应,分别利用Lap lace变换和准弹性方法,导出了这三个优化指标的计算公式.     

FORCED WAVE PROPAGATION IN VISCOELASTIC CABLE WITH SMALL SAG

Based on the linear viscoelastic differential constitutive law and cable structural model, the coupled longitudinal-transverse waves that propagate along a viscoelastic cable with small curvature is investigated. A mathematical model is presented that describes the three-dimensional nonlinear response of a viscoelastic cable. An asymptotic form of this model is obtained for the linear response of cables having small equilibrium curvature. The spectral relation governing the propagating waves is derived using transform methods. The spectral relation is employed in deriving a Green's function that is then used to construct solutions for in-plane response under distributed harmonic excitation. Analysis of forced response reveals the existence of two types of periodic waves that propagate through the cable, one characterizing extension-comprehensive deformation and the other characteristic transverse deformation. Project supported by the National Natural Science Foundation of China (No. 59635140).     

Dynamical Behavior of Viscoelastic Cylindrical Shells Under Axial Pressures

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｏｒｙｏｆｖｉｓｃｏｅｌａｓｔｉｃｉｔｙａｎｄｉｔｓａｐｐｌｉｃａｔｉｏｎｓｈａｖｅｂｅｅｎｂｅｃｏｍｉｎｇａｖｅｒｙａｔｔｒａｃｔｉｖｅｒｅｓｅａｒｃｈｆｉｅｌｄｓｉｎｃｅｖｉｓｃｏｅｌａｓｔｉｃｓｔｒｕｃｔｕｒｅｓｗｅｒｅｗｉｄｅｌｙａｐｐｌｉｅｄｉｎｔｏｖａｒｉｏｕｓｅｎｇｉｎｅｅｒｉｎｇ .Ｂｕｔｔｏｔｈｅａｕｔｈｏｒｓ’ｋｎｏｗｌｅｄｇｅ,ｔｈｅｒｅａｒｅｒａｒｅｆｏｒｐａｐｅｒｓｃｏｎｃｅｒｎｉｎｇｗｉｔｈｄｙｎａｍｉｃａｌｂｅｈａｖｉｏｒｏｆｖｉｓｃｏｅ…     

Analysis of microbuckling for monomolecular layers adhering to a substrate

A two-dimensional linear spring model is established to study the microbuckling of a plane monomolecular layer adhering to a substrate. The model is for the layer subjected to a compressive load having an arbitrary angle with the chemical bond of the layer. The effects of the load angle, the strength of adhesion and the bending stiffness and shearing stiffness (the capability of resisting transverse bending and in-plane shearing) of the layer on the minimal buckling force and the critical buckling mode are discussed. It is found that the minimal buckling force increases with increasing load angle and, for a given bending stiffness, increases with increasing strength of adhesion and decreasing shearing stiffness. Furthermore, a critical condition under which the buckling of the layer can just occur is obtained, which is helpful to avoid buckling in an engineering application. The project supported by the National Distinguished Young Scientist Fund, Cheung Kong Scholars Programme, the National Natural Science Foundation of China (10272082, 10172068) and Shanghai Postdoctoral Science Foundation     

Almost sure stability condition of weakly coupled linear nonautonomous random systems

In this study, the sufficient condition of almost sure stability of twodimensional oscillating systems under parametric excitations is investigated. The systems considered are assumed to be composed of two weakly coupled subsystems. The driving actions are considered to be stationary stochastic processes satisfying ergodic properties. The properties of quadratic forms are used in conjunction with the bounds for the eigenvalues to obtain, in a closed form, the sufficient condition for the almost sure stability of the systems.     

A new approach to almost-sure asymptotic stability of stochastic systems of higher dimension

The almost sure asymptotic stability of higher-dimensional linear stochastic systems and of a special class of nonlinear stochastic systems with homogeneous drift and diffusion coefficients of order one is studied. Based on the well-known Khasminskii's theorem, a new approach for obtaining the regions of almost sure asymptotic stability and instability without evaluating the stationary probability density of the diffusion process defined on unit hypersphere is proposed. Two examples of two and three degree-of-freedom linear stochastic systems are given to illustrate the application and effectiveness of the proposed approach combined with stochastic averaging.     

Two-mode Galerkin approach in dynamic stability analysis of viscoelastic plates

ＩｎｔｒｏｄｕｃｔｉｏｎＤｙｎａｍｉｃｓｔａｂｉｌｉｔｙａｎａｌｙｓｉｓｏｆｖｉｓｃｏｅｌａｓｔｉｃｓｔｒｕｃｔｕｒｅｓｉｓｍｕｃｈｍｏｒｅｃｏｍｐｌｉｃａｔｅｄｓｉｎｃｅｔｈｅｍａｔｈｅｍａｔｉｃａｌｍｏｄｅｌｔｕｒｎｓｏｕｔｔｏｂｅａｓｙｓｔｅｍｏｆｉｎｔｅｇｒｏ＿ｐａｒｔｉａｌ＿ｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｓ,ｒａｔｈｅｒｔｈａｎａｓｙｓｔｅｍｏｆｐａｒｔｉａｌｄｉｆｆｅｒｅｎｔｉａｌｏｎｅｓａｓｉｎｔｈｅｅｌａｓｔｉｃｓｔｒｕｃｔｕｒｅ[1].Ｍａｎｙｗｏｒｋｓｈａｖｅｂｅｅｎｐ…     

Some Global Stability Results for Shear Flows of Viscoelastic Fluids

We consider plane shear flows of viscoelastic fluids. For a number of constitutive models, we prove stability of the rest state for perturbations of arbitrary size. We also consider stability of plane Poiseuille flow in a few special cases. This research was supported by the National Science Foundation under Grant DMS-0405810.