On the hard loss of stability of the Beck’s beam

This paper investigates the complex nonlinear dynamics of non-conservative mechanical systems under different sources of nonlinear damping, including internal damping from material behavior and external damping due to fluid–structure interactions. A Beck’s beam, namely, a viscoelastic cantilever beam subjected to a follower force at its free end, is taken as a paradigmatic example. The governing equations of motion are derived using a variational principle, then reformulated into an integro-differential form and discretized through the Galerkin method. Starting from the Hopf bifurcation, identified via the linear stability analysis, the nonlinear post-critical behavior of the discretized system close to it is analyzed using the Multiple Scales Method. This perturbation technique yields bifurcation equations whose analysis reveals new aspects of damping-induced destabilization, including the dual nature of nonlinear damping, which can either promote stability or induce instability in the bifurcated response of the system, as well as the emergence of the so-called Hard Loss of Stability phenomenon, analytically predicted by a second-order amplitude modulation equation. Numerical analyses are finally performed and corroborate the analytical findings of the study.