Divergence, Hopf and double-zero bifurcations of a nonlinear planar beam

The multiple scale method is directly applied to a one-dimensional continuous model to derive the equations governing the system asymptotic dynamic around a bifurcation point. The theory is illustrated with reference to a specific example, namely an internally constrained planar beam, equipped with a lumped visco-elastic device and loaded by a follower force. Nonlinear, integro-differential equations of motion are derived and expanded upto cubic terms in the transversal displacements and velocities of the beam. The linear stability of the trivial equilibrium is first studied. It reveals the existence of divergence and Hopf and double-zero bifurcations. The spectral properties of the linear operator are studied at the bifurcation points by obtaining closed-form expressions. Notably, the system posses an incomplete system of eigenvectors at the double-zero point (i.e. it is defective or nilpotent), thus entailing the need to find a generalized eigenvector. A multiple-scale analysis is then performed for the three bifurcations and the relevant bifurcation equations are derived directly in their normal forms. Finally, they are numerically studied to furnish a comprehensive scenario of the postcritical behaviour around the bifurcations.