Instability and chaos in the elastica type problem of parametrically excited columns

The analysis of a parametrically excited simply supported column, made of non-linear elastic material, including the effects of the (elastica type) large deflections and the changes in its length, is presented. The equation of motion derived is in the form of the non-linear Mathieu equation ÿ + (ω² + a cos θt)y + b cos θty³ + cy³ = 0, which was analytically investigated in a previous paper by the present authors and their colleagues [1]. The main results as obtained there are described. The influence of the various parameters involved and the initial conditions on the instability regions, as well as on the nature of the system’s response, are investigated. The possibility of chaotic behaviour is examined within the concept of Lyapunov exponents.