Dynamic Instabilities in Structures with Nonholonomic Constraints

This thesis delves into the dynamics of structures subject to velocity dependent restrictions, which are known as nonholonomic constraints. These constraints are applied to elastic structures, and their nonlinear dynamics are analyzed within the framework of bifurcation and stability theory. Initially, the stability of Ziegler's double pendulum is analyzed to show its interesting dynamic behavior and capability of flutter via a Hopf bifurcation. The main contribution of the thesis is the development of models based on a double pendulum equipped with a non-holonomic constraint, which is a variant of Ziegler's double pendulum. The first investigation that has been carried out, is to consider this device to have a charge concentrated at its tip, where the nonholonomic constraint is located. As such, the new device is named the 'charged Ziegler's double pendulum'. This device is then placed within an ideal solenoid so that there is an additional interaction due to Lorentz force coming from the magnetic induction field within the solenoid. After determining the equilibrium of the system, a stability analysis is preformed, and it is shown that the device still undergoes a Hopf bifurcation. This then allows for a post critical study, whereby focus is brought onto the interaction between the Lorentz force and the other forces prescribed to the device. The study is not limited to the case where the device lies inside the solenoid; the case where it is outside is also examined. Within this setting, the Lorentz force induced is solely a property of the electric field as the magnetic induction field outside the solenoid vanishes. This property is known in electromagnetism as the 'Maxwell--Lodge effect'. The second investigation revisits Ziegler's double pendulum under a nonholonomic constraint and examines its interaction with an oscillator, modeled as a plate attached to an axial spring. A linear stability analysis is then carried out on this system, revealing again the possibility of the system to undergo a Hopf bifurcation. However, when in the dynamic regime, the fluttering which is brought on by the Hopf bifurcation, interacts with the movement of the plate leading to a self-induced resonance phenomenon. When the frequency of the self-induced vibrations arising from a Hopf bifurcation approaches the natural frequency of the plate oscillator, both the limit-cycle amplitude and the mean structural velocity diverge. Moreover, instability regions emerge in which the Hopf bifurcation becomes subcritical; within these regions, the structure may exhibit multiple coexisting limit cycles or transition to chaotic dynamics. These instability mechanisms are of practical importance: resonance between Hopf-induced oscillations and structural modes can severely reduce stability margins and promote fatigue. Conversely, the same mechanisms may be harnessed for controlled amplification and nonlinear sensing applications.