A system with asymmetric stability analyzed with optimization techniques

Abstract

We consider a discrete model composed of two horizontal rigid articulated bars of length L, massless, initially collinear, with a linear spring of stiffness K inclined connected to the bars at their central pinned joint. We consider a hinged fixed support at the left end of the system and a mobile articulated support at the right end. A horizontal compressive axial load P is applied to the mobile support in the direction of their original primary stable horizontal configuration. As this force grows, a critical buckling load may be reached so that this primary configuration becomes instable and the system assumes a new secondary inclined position of the bars that perform a rotation angle theta with respect to their original positions. In this model, for positive valued theta angles the secondary configuration is stable, but for negative valued angles it is instable, a potentially dangerous asymmetric stability problem that may arise in aerospace applications. In our analysis, we compute the Total Potential Energy (TPE) of the system. According to Lagrange’s theorem, equilibrium points correspond to stationary TPE, being stable if it is a point of minimum and instable if a point of maximum. We use optimization algorithms to find stable and instable equilibrium points for very close axial load values near the critical one.