Autonomous Nonconservative Dynamical Systems

Linear beams on nonlinear elastic soil, solicited by compression follower forces, are analyzed. Formally, they are continuous dynamical systems, not explicitly dependent on time, and therefore said autonomous; moreover, they are nonconservative, due to the nature of the forces. At a critical value of the force, which acts as a bifurcation parameter, they manifest a dynamic (or Hopf) bifurcation. By using the Multiple Scale Method, the critical value of the force is determined, and the family of limit cycles, originating from the bifurcation point, is built-up. In particular, the amplitude and frequency of the cycle are determined as function of the bifurcation parameter, and stability investigated. The effect of a gravitational force added to the beam, coexisting with the follower force, and acting as a second bifurcation parameter, is successively investigated. Such nonconservative system also exhibits a static bifurcation, which depends on the combination between the two bifurcation parameters. The divergence geometrical locus at which the static bifurcation takes place on the parameter plane is determined, and the nontrivial equilibria states branching from the locus are evaluated. A stability analysis is finally carried out.