Two-bar planar truss with non linear precritical behavior: Cusp bifurcation

The paper analyzes the critical and post critical behavior of a two-bar planar truss subjected to a vertical static load at its top joint. Such system is the representative mathematical model of a wide variety of engineering applications such as large-span cable roofs, tensegrity structures and rapidly deployable structural systems. The main goal is the study of fixed points bifurcation, in order to model dramatic phenomena (snap through instability) and their dependence on parameters. By varying parameters, the state of the system changes and we obtain a cusp catastrophe surface. The catastrophe manifold representation allows to get qualitative features of system behavior: hysteresis and jump. The mechanical system is simplified as Duffing's oscillator and its mathematical model is formulated by using Lagrange equation. Since the integration of the obtained non-linear system isn't straightforward, we have investigated the system stability in a qualitative way by linearizing the problem about fixed point. The system equilibrium stability and the codimension-two bifurcation are investigated using the Jacobian eigenvalues and a qualitative geometrical analysis.