The forced vibration of a partially delaminated structure such as an aircraft wing can result in catastrophic crack growth. In order to look at the underlying mechanism of the dynamics and failure of the material, a simplified model of a cantilever beam with a single delamination at its free end is considered. We investigate a number of aspects of this system, using mathematical models to gain insight into its behaviour.

The purpose of this work is to illuminate some of the non-smooth phenomena found in piecewise-smooth continuous and discrete dynamical systems, which do not occur in smooth systems. We will explain how such non-smooth phenomena arise in applications which experience impact, such as impact oscillators, and a type of rotating machine, called magnetic bearing systems. The study of their dynamics and sensitivity to parameter variation gives not just insights into the critical motion found in these applications, but also into the complexity and beauty in their own right. This work comprises two parts. The first part studies a general one-dimensional discontinuous power law map which can arise from impact oscillators with a repelling wall. Parameter variation and the influence of the exponent on the existence and stability of periodic orbits is presented. In the second part we analyse two coupled oscillators that model rotating machines colliding with a circular boundary under friction. The study of the dynamics of rigid bodies impacting with and without friction is approached in two ways. On the one hand existence and stability conditions for non-impacting and impacting invariant sets are derived using local and global methods. On the other hand the analysis of parameter variation reveals new non-smooth bifurcations. Extensive numerical studies confirm these results and reveal further phenomena not attainable otherwise.

Sound generated by impacting structures is of considerable importance in noise control. Sound is generated by a vibrating structure by inducing pressure fluctuations in the surrounding medium. Impact induced noise is the sound generated by a vibrating structure subjected to motion constraint. In such problems one has to study the vibration behavior of the oscillator, the impact mechanics, and the emanating acoustic field dynamics. A literature review carried out points to the fact that though there has been considerable work on vibration behavior of impact oscillators and the acoustics of impact of rigid masses, there is very little work reported on the sound generated due to vibration and impact. This thesis couples vibration analysis of oscillators undergoing impact with its acoustic behavior. The vibration behavior is nonlinear on account of the impact. Therefore the vibration analysis as well as the resulting acoustic field analysis has to be in the time-domain. This investigation is concerned with the effect of structural dynamics, impact dynamics, and acoustic field boundary conditions, on the sound pressure generated due to vibration and impact. We have considered a single degree of freedom as well as a flexible Euler-Bernoulli beam vibration model. The former is the simplest for studying vibro-acoustic response. The numerical model of the beam is derived using the finite element method resulting in a finite dimensional system with more than one degree of freedom. The dynamics of each degree of freedom are distinct in terms of amplitude and phase and are a function of the nature of linear dependence on other degrees of freedom and the nature of excitation. An impacting beam introduces interesting interactions between the dynamics of the degrees of freedom as a consequence of nonlinearity due to the motion constraint. The impact of the oscillator mass with a barrier is modeled using a simple coefficient of restitution model based on Hertzian contact theory. There is velocity reversal on contact with the barrier. The contact force is finite acting within a finite interval of time. The contact force is assumed to vary in time during the contact interval. This effectively models contact as linearly elastic. The pressure perturbation due to vibration of the oscillator mass is shown equivalent to the pressure perturbation due to an acoustic dipole. The acoustic dipole is placed at the equilibrium position of the vibrating mass. The dipole pressure is then a function of motion of the oscillator. In the case of a single degree of freedom oscillator the dipole axis is along the direction of motion. The sound pressure due to a vibrating beam is modeled as an array of acoustic dipoles placed at the finite element nodes of the beam and stationary at the beam's static equilibrium configuration. The dipole axis is once again aligned with the direction of vibration of the beam that is transverse to the beam neutral axis. Anechoic as well as perfectly reflecting acoustic boundary conditions are simulated in the time-domain. The resulting governing equation of motion of the single degree freedom oscillator as well as the beam are integrated numerically in time to compute its response. The acoustic pressure is shown to be critically dependent on the excitation frequency of the oscillator, dynamic properties of the oscillator, coefficient of restitution of impact and impact dynamics, and acoustic field boundary conditions.