Non-linear control of buckled beams under step loading

The present paper studies a strategy for the active non-linear control of t he oscillations of simply supported buckled beams, in order to mitigate the effects of dynamic loading on the vibration amplitudes and prevent dangero us instability phenomena. First, an analysis of the symmetric non-linear be haviour of the structure without any control system is carried out. In orde r to control the non-linear vibrations of the beam, an active tendon contro l system is adopted. A control method based on non-linear optimal control u sing state feedback is developed and the solution of the non-linear optimal control problem is obtained by representing system non-linearities and per formance indices by power series with the help of algebraic tensor theory. In this work, general polynomial representations of the non-linear control law are obtained up to the fifth order. This solution procedure is employed to analyse the influence of the resulting non-linear control laws on the d ynamic behaviour of a buckled beam under a lateral step load. This arch-lik e structural system is highly non-linear and under compressive lateral load ing may suffer snap-through buckling. This may cause undesirable stresses a nd/or displacements, leading as a rule to a failure of the structural syste m. So, special attention is given to the determination of the potential of the present control methodology for efficiently limiting extreme state resp onses and preventing the snap-through buckling. Numerical results indicate that the control algorithm can effectively increase the load-carrying capac ity of the buckled beam without demanding large control forces. Also, this study can be used as a basis for the non-linear control of more complex str uctures and for the design of control systems. (C) 2000 Academic Press.