Stochastic stability and global bifurcations in gyroscopic mechanical systems

The stability and bifurcation behavior of mechanical systems parametrically excited by small periodic or stochastic perturbations is studied. In the case of random excitation, the almost-sure and moment asymptotic stability of two- and four-dimensional dynamical systems subject to small intensity noise is investigated. The almost-sure stability is defined by the sign of the maximal Lyapunov exponent, the exponential growth rate of solutions to a linear stochastic system. Similarly, the moment stability of such a system is determined by the sign of the moment Lyapunov exponent which describes the exponential growth rate of the moments of solutions to a linear stochastic system.;A perturbative approach is employed to construct an asymptotic expansion for the maximal Lyapunov exponent of a four-dimensional gyroscopic dynamical system driven by a small intensity real noise. This method involves solving a series of partial differential equations, along with the corresponding solvability conditions, to obtain successive terms in the expansion for the top Lyapunov exponent. The perturbative technique developed is then applied to study the lateral vibration instability in rotating shafts subject to stochastic axial loads and stationary shafts in cross flow with randomly varying flow velocity. A second perturbative method is developed to compute the asymptotic expansion for the moment Lyapunov exponent.;The local and global bifurcation behavior of nonlinear deterministic gyroscopic systems subject to periodic parametric excitation is also examined. Throughout this work, it is assumed that the dissipation, imperfections and amplitudes of parametric excitations are small. In this way, it is possible to treat these problems as weakly Hamiltonian systems. Most of the analysis presented here is based on the recent work of perturbed Hamiltonian systems. Although the local and global results presented here have a wide range of applications, the motivating problem throughout this phase of the research is the investigation of the dynamics and stability of the rotating shaft subject to a periodic parametric excitation. This system is a fundamental component in many mechanical and power generating systems. The parametric excitations in this system arise due to the action of adjacent components on the rotating shaft.;The final phase of this research is the design and construction of laboratory facilities dedicated to the verification of the analytical techniques developed in this research. The theoretical results will serve as a guideline for locating stability boundaries and predicting post-critical behavior. The extent to which the theory and experimental results match will provide an insight into the accuracy of the mathematical models and theoretical approximations. The experiments will, in turn, guide the development and refinement of the theories developed to incorporate any new phenomena observed.