B.S. Wu

Large amplitude non-linear oscillations of a general conservative system

Abstract This paper presents new, approximate analytical solutions to large-amplitude oscillations of a general, inclusive of odd and non-odd non-linearity, conservative single-degree-of-freedom system. Based on the original general non-linear oscillating system, two new systems with odd non-linearity are to be addressed. Building on the approximate analytical solutions of odd non-linear systems developed by the authors earlier, we construct the new approximate analytical solutions to the original general non-linear system by combinatory piecing of the approximate solutions corresponding to, respectively, the two new systems introduced. These approximate solutions are valid for small as well as large amplitudes of oscillation for which the perturbation method either provides inaccurate solutions or is inapplicable. Two examples with excellent approximate analytical solutions are presented to illustrate the great accuracy and simplicity of the new formulation.

Analytical approximation to large-amplitude oscillation of a non-linear conservative system

This paper deals with non-linear oscillation of a conservative system having inertia and static non-linearities. By combining the linearization of the governing equation with the method of harmonic balance, we establish analytical approximate solutions for the non-linear oscillations of the system. Unlike the classical harmonic balance method, linearization is performed prior to proceeding with harmonic balancing, thus resulting in a set of linear algebraic equations instead of one of non-linear algebraic equations. Hence, we are able to establish analytical approximate formulas for the exact frequency and periodic solution. These analytical approximate formulas show excellent agreement with the exact solutions, and are valid for small as well as large amplitudes of oscillation.

Application of vector-valued rational approximations to a class of non-linear oscillations

Abstract By combining a perturbation technique with a rational approximation of vector-valued function, we propose a new approach to non-linear oscillations of conservative single-degree-of-freedom systems with odd non-linearity. The equation of motion does not require to contain a small parameter. First, the Lindstedt–Poincare perturbation method is used to obtain an asymptotic analytical solution. Then the range of validity of the analytical representation is extended by using the vector-valued rational approximation of functions. For constructing the rational approximations, all that is needed is the coefficients of the perturbation expansion being considered. General approximate formulas for period and the corresponding periodic solution of a non-linear system are established. Two examples are used to illustrate the effectiveness of the proposed method.

Constructive analysis of buckling mode interactions with single Z2-symmetry

Abstract This paper is concerned with the buckling mode interactions in a conservative system with single Z 2 -symmetry. The system depends on a load parameter and an auxiliary parameter (e.g., aspect ratio of structures, stiffness of supports, etc.). Attention is focused on those auxiliary parameters near their critical values, at which a two-mode compound buckling occurs. By using the Liapunov–Schmidt–Koiter approach and utilizing the potential energy derivatives, we give the conditions of secondary bifurcation, construct the asymptotical expansions of primary and secondary postbuckling states, and identify the stability of each postbuckling state by calculating eigenvalues of a simple matrix of order 2. Both of the non-symmetric primary and secondary postbuckling states are shown to be unstable. A cylindrical panel under axial compression is used as a typical example of the systems of the type described. The secondary bifurcation and complex stability behavior of the system are revealed.

A note on the critical points of equilibrium paths in imperfect structures

In a recent work (Int. J. Solids Struct. 37 (2000) 1561) by one of the authors, an extended system for calculating critical points of equilibrium paths in imperfect structures was presented. However, the extremum nature of these points was not analyzed explicitly in that paper. In this note, we will fill in the gap and establish a sufficient condition for determining the buckling strength of imperfect structures.

Efficient computation for lower bound dynamic buckling loads of imperfect systems under impact loading

This paper is concerned with the computation of lower bound dynamic buckling loads of discrete imperfect systems under impact loading. An extended system that is based on the energy criterion for establishing the lower bound dynamic buckling loads without solving the highly non-linear initial-value problems is proposed. For the extended system, the newly introduced parameters are regular solutions and thus standard methods can be used to compute them. From the solutions, one can obtain directly the lower bound dynamic buckling loads without tracing the postbuckling equilibrium paths. Two numerical examples are given to show the characteristics and effectiveness of this algorithm.