Energy transfer between interconnected mechanical systems is important in many real world applications. When a finite dimensional system is coupled to an infinite dimensional system, energy can be radiated from the finite dimensional system and absorbed by the infinite dimensional system, which we call radiation damping. For example, satellites and space stations have a central rigid body which can radiate energy through flexible components such as solar panels and antennae. Radiation damping can also describe dissipation in a conservative context, where energy of one form (such as motion of a mechanical system) is transformed into energy of another form (such as heat) of a larger conservative system. An early physical model of radiation damping, investigated by Horace Lamb, describes an oscillator coupled to a string modeling free vibrations of a nucleus in an external medium. This system exhibits radiation damping via energy radiating from the oscillator into waves of the string that are carried off to infinity. We extend the Lamb model to a linear mechanical system with gyroscopic forces coupled to a wave equation. As energy radiates from this system into wave modes, the system can become unstable. Adding dispersion in the wave equation eliminates the mechanical system's access to low wave modes and allows a band of stability for small coupling. We focus on three types of coupling: the gyroscopic Lamb model, wave field coupling, and heat bath coupling. In the gyroscopic Lamb model, a system has a boundary constraint through which energy is transferred into the wave. In the wave field coupling, the mechanical system excites a wave field. In the heat bath coupling, the mechanical system is coupled to a collection of independent oscillators. We also study a class of mechanical systems intermittently coupled to a surface. Such intermittent coupling arises naturally when describing animal or robotic locomotion which have a stance phase and a aerial phase as part of their gait. Just as radiation damping destabilizing a relative equilibrium, intermittent contact can destabilize periodic motion of mechanical systems. As a example of intermittent coupling, we analyze the aerial and the stance phases of an inhomogeneous cylinder impacting a surface. We show the existence of periodic orbits with no energy loss and investigate their stability.Ph.D.MathematicsPure SciencesUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/125423/2/3016860.pd
