Postcritical behavior of a discrete Nicolai column

Abstract

The Nicolai paradox, concerning the loss of stability of a column subjected to an evanescent follower torque, so far analyzed in the literature in the linear field, is here studied in the nonlinear regime. A paradigmatic, discrete, minimal mechanical system is introduced, roughly describing the column. The bifurcation equations ruling the dynamics around a semi-simple Hopf bifurcation are derived via the Multiple Scale Method, and the postcritical behavior of the system is investigated. A slight asymmetry of the model, which splits the critical eigenvalues, is also taken into account. It is shown that, while the trivial equilibrium configuration of the system is rendered unstable by vanishingly small circulatory forces, nonlinearities can, under certain circumstances, limit the amplitude of the oscillations. The existence of unsafe initial conditions is discussed, and the dangerous effects of the follower torque, also in nonlinear regime, are highlighted.