| We study the local and global bifurcation behavior of parametrically excited gyroscopic systems near a 0:1 resonance. A major goal of the analysis is to understand how energy may be transferred from the high frequency mode to the low frequency mode in these gyroscopic systems.; The first part of this research involves the derivation and simplification of the equations of motion for two parametrically excited gyroscopic systems: a pipe conveying fluid and a rotating shaft. The simplification of these equations involves deriving the normal form near a critical point at which the system possesses a non-semisimple double zero eigenvalue. To aid in this procedure, an algorithm is developed for calculating normal forms for non-autonomous Hamiltonian systems. The forms of the equations derived for the two gyroscopic systems are identical, allowing us to study the two systems simultaneously.; The second part of this research involves examination of the local dynamics of the two gyroscopic systems. First, using a previously obtained model for a rotating shaft with small symmetry-breaking, it is shown that the presence of combination resonance forcing can extend the stability boundaries of the system from those of the unforced case. Next, we study the local bifurcations of the gyroscopic models derived in this work, focusing on the subharmonic resonance case. We calculate the stability of the trivial solution, the bifurcating single mode branches and their stability, and the existence of multi-mode or periodic solutions. Regions where energy transfer may occur from high to low frequency modes are identified. The numerical bifurcation analysis software AUTO is used to support the analytical results.; The final part of this research involves study of the global bifurcations of the two gyroscopic systems. Using recently developed bifurcation methods, we detect the presence of multi-pulse orbits homoclinic to a slow manifold. In certain parameter regions, we can prove that multi-pulse orbits exist which are homoclinic to fixed points on the slow manifold, leading to chaotic dynamics in the system. These multi-pulse orbits provide the mechanism by which energy transfer between modes may occur. |
