Active non-linear control of buckling and vibrations of a flexible buckled beam

Buckled beams have been used in several engineering applications to support lateral loading. These arch-like structures are highly non-linear and may display several instability phenomena when subjected to dynamic and static loads, in particular snap-through buckling. In this paper the non-linear dynamic behaviour and snapping of a shallow, simply supported sinusoidal buckled beam subjected to a distributed step pressure load is studied over a range of arch geometries. The dynamic response of the buckled beam is shown to exhibit both symmetric and asymmetric solutions, depending on the value of the applied axial load. To control the non-linear oscillations of the arch, concentrated moments are applied at suitable points along the beam axis. A control method based on non-linear optimal control, using state feedback, is developed and the solution of the non-linear optimal control problem is obtained by representing system non-linearities and performance indices by power series of the state variables with the help of algebraic tensor theory. Using this procedure, control laws up to the third order are derived. The results show that the proposed control scheme cannot only mitigate the effects of dynamic loading on the vibration amplitudes of the beam but also prevent dangerous instability phenomena. In particular the numerical results indicate that, increasing the non-linearity of the control force, one cannot only decrease the peak and root mean square (rms) values of the controlled response but also increase significantly the dynamic buckling load, thus increasing the load-carrying capacity of the buckled beam.