Postbuckling Analysis of Nonconservative Elastic Systems

Abstract

Previous work on the postbuckling and imperfection-sensitivity of elastic structures has concentrated on conservative systems. The results of Koiterand others have led to a general theory of nonlinear stability behavior for these systems. The theory must be modified when nonconservative forces are present, and this is the aim of the present paper.

Discrete, nonconservative, elastic systems which exhibit static (divergence) instability are considered. The nonlinear behavior in the neighborhood of a critical point is analyzed by means of a perturbation procedure. When the critical point is simple, the results are similar to those for conservative systems. When a coincident critical point exists, however, different types of behavior occur. In many cases there is no bifurcation at all, with only the fundamental (trivial) equilibrium path passing through the critical point. Imperfection-sensitivity is more severe than for the typical bifurcation points and can even occur when the perfect system has no bifurcation. The results are illustrated with the use of a nonlinear double pendulum model subjected to a partial follower load.     

Global Attractors: Topology and Finite-Dimensional Dynamics

Many dissipative evolution equations possess a global attractor with finite Hausdorff dimension d. In this paper it is shown that there is an embedding X of into , with N=[2d+2], such that X is the global attractor of some finite-dimensional system on with trivial dynamics on X. This allows the construction of a discrete dynamical system on which reproduces the dynamics of the time T map on and has an attractor within an arbitrarily small neighborhood of X. If the Hausdorff dimension is replaced by the fractal dimension, a similar construction can be shown to hold good even if one restricts to orthogonal projections rather than arbitrary embeddings.     

Nonlinear Dynamics of Mechanical Systems Under Complicated Forcing

This paper considers the problem of simple linear mechanical oscillators forced by complicated forcing functions, such as those induced by fluids or humans. Such problems arise in civil engineering in the design of bridges, grandstands and towers. The paper questions whether low dimensional models have much to contribute to the understanding of such problems, given the complicated possibilities of high-dimensional forcing.     

Stability of Thin-Walled Open Beams Under Nonconservative Loads

ABSTRACT

A finite element formulation for free-vibration analysis of straight prismatic beams of general thin-walled open cross-section, under conservative and nonconservative loads, is presented. The formulation is used to calculate the flutter load for a number of beam problems and is verified by comparison with pre-existing numerical solutions.     

Generalization of an Energetic Optimality Condition for Nonconservative Systems

ABSTRACT

An optimality condition based on strain energy densities, derived by earlier investigators, has been generalized. The generalized condition for an optimum structure is in bilinear form. The concept of the adjoint system is utilized. The resulting condition is applied to the minimization of the mass of a column loaded by a tangential follower force. Some results are presented.     

Boundedness of Solutions of Certain Volterra Systems in a Finite-Dimensional Space

We give conditions for the boundedness and periodicity of solutions of Volterra systems with matrix coefficients. The conditions are formulated in terms of the coefficients of these systems.     

非完整非保守系统在相空间的Noether定理及其逆定理

本文根据动力学系统在相空间的作用积分在无穷小群变换下的变换性质，得到了非完整非保守系统在相空间的Ｎｏｅｔｈｅｒ定理和逆定理。并举例说明。     

The Dynamics of Thin-Walled Structures Under Nonstationary Loads

Studies are reviewed in which nonstationary loads determining the dynamics of thin-walled structures are found and the initial concepts are formulated for the theory and methods of stress–strain analysis of shells with holes and different discrete reinforcements, as well as for coupled problems of hydro- and aeroelasticity. Special attention is drawn to the design and strength and dynamic analyses of thin-walled blasting chambers. The basic investigations conducted at the S. P. Timoshenko Institute of Mechanics of the National Academy of Sciences of Ukraine are outlined     

Dynamics of Holonomic Rigid-Body Systems

The theory behind analytical algorithms for computer-algebra systems is discussed. A computer system is described which is capable of optimizing input data, constructing equations of equilibrium and motion, formulating and solving the basic mechanics problems for a broad class of holonomic systems with elastic and dissipative constraints on the basis of the Lagrange-D'Alembert principle __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 7, pp. 130–138, July 2005.     

迟滞系统受随机激励的响应计算

首先为随机激励设计了具有较高精度的滤波器，把迟滞系统受有色噪声激励的随机振动问题转化为由迟滞系统与滤波器组成的扩阶系统受白噪声激励的随机振动问题，然后详细论述了等效线性化法对此问题的具体求解，着重推导了高斯随机变量联合矩的递推关系和李亚普诺夫方程解法的实现，最后根据数字模拟结果分析了等效线性化法的精度和误差来源，给出了迟滞减振系统参数设计的有关建议。     

The Dynamics of Discrete Mechanical Systems with Perfect Unilateral Constraints

The dynamics of discrete mechanical systems with perfect unilateral constraints is formulated in a very general setting. The well-posedness of the resulting evolution problem is studied. It is proved that existence and uniqueness of a maximal solution is ensured provided strong assumptions are made on the regularity of the data: they are supposed to be analytic. Simple examples show that this regularity assumption may not be relaxed. Sufficient conditions to ensure that the maximal solution is defined for all time are supplied. The continuous dependence of the solution on initial conditions is also studied and the numerical computation of the solution is discussed. Accepted April 18, 2000?Published online September 18, 2000     

Configurational Forces and the Dynamics of Planar Cracks in Three-Dimensional Bodies

This paper develops a three-dimensional framework for the evolution of planar cracks, concentrating on the derivation of balances and constitutive equations that describe the motion of the crack tip. The theory is based on the notion of configurational forces in conjunction with a mechanical version of the second law. This revised version was published online in August 2006 with corrections to the Cover Date.     

Theory of Complex Systems and Economic Dynamics

The purpose of this paper is to discuss the significance of the theory of complex systems for analyzing economic evolution. We emphasize the necessity of the structural dynamic approach and discuss possible implications of the theory of complex systems for studying economic processes with different speeds of change. We illustrate a way to construct a general economic theory that includes the main economic theories of competition with government intervention as special cases in the structural sense.     

Dynamics and Control of Initialized Fractional-Order Systems

Due to the importance of historical effects in fractional-order systems,this paper presents a general fractional-order system and control theorythat includes the time-varying initialization response. Previous studieshave not properly accounted for these historical effects. Theinitialization response, along with the forced response, forfractional-order systems is determined. The scalar fractional-orderimpulse response is determined, and is a generalization of theexponential function. Stability properties of fractional-order systemsare presented in the complex w-plane, which is a transformation of thes-plane. Time responses are discussed with respect to pole positions inthe complex w-plane and frequency response behavior is included. Afractional-order vector space representation, which is a generalizationof the state space concept, is presented including the initializationresponse. Control methods for vector representations of initializedfractional-order systems are shown. Finally, the fractional-orderdifferintegral is generalized to continuous order-distributions whichhave the possibility of including all fractional orders in a transferfunction.     

Response of Systems Under Non-Gaussian Random Excitations

The approach of nonlinear filter is applied to model non-Gaussian stochastic processes defined in an infinite space, a semi-infinite space or a bounded space with one-peak or multiple peaks in their spectral densities. Exact statistical moments of any order are obtained for responses of linear systems jected to such non-Gaussian excitations. For nonlinear systems, an improved linearization procedure is proposed by using the exact statistical moments obtained for the responses of the equivalent linear systems, thus, avoiding the Gaussian assumption used in the conventional linearization. Numerical examples show that the proposed procedure has much higher accuracy than the conventional linearization in cases of strong system nonlinearity and/or high excitation non-Gaussianity. An erratum to this article is available at .     

The Dynamics of Systems of Rigid and Elastic Bodies as Applied to Spacecraft

The basic results of the scientific research conducted at the S. P. Timoshenko Institute of Mechanics of the National Academy of Sciences of Ukraine (NASU) in a creative cooperation with the M. K. Yangel' SDO "Yuzhnoe" and the Institute of Technical Mechanics of NASU and the National Space Agency of Ukraine (NSAU) are generalized and systematized. The research addressed mathematical models and the dynamics of objects of space-rocket engineering such as controlled systems of rigid and elastic bodies, systems of rigid bodies of variable configuration, and systems of bodies with unilateral connections. The following information is detailed here: methods for creating spatial program motions of elastic space structures about the center of mass, methods and algorithms for mathematical simulation of the dynamics of reconfigurable spacecraft, unconventional concepts of simulating the state of weightlessness of a reconfiguring spacecraft under earth conditions on a special stand using suspension cables, different aspects of the dynamics of space cable systems, and other problems resolved within the framework of the subjects indicated