On Multiple Hopf Bifurcations of Airflow Excited Vibration of a Translating String

Abstract

This paper presents the stability and bifurcation of transverse motion of translating strings excited by a steady wind flowfield. The stability of the equilibrium configuration is presented for loss of stability and generation of limit cycles via the Hopf bifurcation. It is demonstrated that there are single, double and quadruple Hopf bifurcations in the parametric space that lead to the limit cycle motion. The method of Incremental Harmonic Balance is used to solve the limit cycle response of which the stability is determined by computation of the Floquet multipliers. For the forced vibration, it is pointed out that the periodic and quasi-periodic motions exist as parameters are changed. The quench frequency and the Neimark-Sacker (NS) bifurcation and flip bifurcation are obtained. The continuity software MATCONT is adopted and the Resonance 1:1, 1:3 and 1:4 as well as NS to NS bifurcations are presented. The bifurcation behavior reveals the complexity of the string’s motion response induce by aerodynamic excitations.