Large Rotatory Oscillations of Transversely Isotropic Rods: Spatio-Temporal Symmetry-Breaking Bifurcation

Time-periodic motions of an initially straight, transversely isotropic, cantilever rod, modeled as a general nonlinearly elastic Cosserat curve, are considered. By inspection, the problem possesses $O(2)$ spatial symmetry and, due to the requirement of time periodicity, $O(2)$ temporal symmetry (invariance under time shifts, modulo the period, and time reversal). First, the mathematical realization of these symmetries is investigated in terms of transformation groups acting on the governing partial differential equations (PDEs), which ultimately simplifies the task of finding (special classes of) solutions. The main point of this paper is twofold: The process just described (i) is difficult to implement in this problem (we provide an unconventional reformulation of the Cosserat model that mitigates the difficulty), and (ii) uncovers new solutions. More specifically, the existence of solutions, invariant under a certain “twisted” subgroup of the spatio-temporal symmetry group is established. These solutions correspond physically to free, large-amplitude oscillations of the cantilever, in which the centroidal curve precesses steadily about the undeformed centroidal axis, while each cross section “wobbles” periodically with a distinct angular velocity.