Stability, modal couplings and local bifurcations in bending -torsion forced vibrations of a three-dimensional elastic rod

Numerical and experimental studies of the coupled bending-torsion forced vibration in a 3-D thin elastica are conducted. The elastic rod is under the clamped-free boundary condition, with its long axis aligned in the vertical direction, and is subjected to a sinusoidal base excitation. Particular interest is focused on the phenomena associated at the transition of a planar motion to a nonplanar one.;The numerical study focuses on the stability of the planar motion and the related local bifurcations. The model developed by Joseph P. Cusumano from Cornell University is adopted. The structure of the instability band is observed in the stability diagram. A similar phenomenon shown by the Mathieu equation with a resonating coefficient is demonstrated to explain this unusual feature. Under resonance excitation, the modal convergence test at and right after the planar instability suggests a dominant modal coupling among the first bending mode, the resonance bending mode and the first torsional mode.;The local bifurcation at the planar instability is studied by the continuation method and the underlying symmetry structure of the model. A subcritical pitchfork bifurcation is identified at the loss of planar stability. Other bifurcation scenarios for a family of period one solutions are also given; these are the saddle-node, the pitchfork and the secondary Hopf bifurcations. Low-dimensional chaos is discovered by numerical simulations. The chaotic orbit is evolved from a set of 2-tori undergoing what is known as the torus doubling bifurcations. From the Poincare map, Fast Fourier Transform, Lyapunov exponents and the correlation dimension calculations, detailed descriptions of the phenomena are presented.;The experiment incorporating the constrained layer damping shows two similar features found from the numerical work. These features are, therefore, believed intrinsic to the vibration of beams with thin cross section. The first feature is suggested by the stability transition curves for beams under different damping treatments. A dominant modal coupling involving only lower bending modes and the torsional motion is observed at the loss of planar stability. The second feature is the low-dimensional chaotic motion right after the planar instability. The active degree of freedom of the chaotic motion is estimated from the delay-embedding map.