The topic of this dissertation, modeling and identification of nonlinear oscillation, represents an area of mathematical systems theory that has received little attention in the past. Primarily, the types of oscillation of interest are those found in biological systems where theoretical foundations for mathematical models are insufficient. These oscillations are also observed in other systems including electrical, mechanical, and chemical. The contributions of this dissertation are a generalized class of autonomous differential equations that are found to exhibit stable limit cycles, and an investigation of a method of system identification that can be used to estimate the model parameters. Here the observed signal is modeled as the response of a nonlinear system that can be described by differential equations. Modeling the signal in this way shifts the emphasis from signal characteristics, such as spectral content, to system characteristics, such as parameter values and system structure. This shift in emphasis may provide a better method for monitoring complex systems that exhibit periodic behavior such as patients under anesthesia. A class of autonomous differential equations, called the generalized oscillator models, are presented as one nᵗʰ-order differential equations with nonlinear coefficients. The coefficients are chosen to change sign depending on the magnitude of the phase variables. The coefficients are negative near the origin and positive away from the origin. Motivated by the generalized Routh-Hurwitz criterion, this coefficient sign changing produces the desired oscillation. Properties of the generalized oscillator model are investigated using the describing function method of analysis and numerical simulation. Several descriptive examples are presented. Based on the generalized oscillator model as a set of candidate models, the system identification problem is formed as a mathematical programming problem. The method of quasilinearization is investigated as method of solving the identification problem. Two examples are presented that demonstrate the method. It is shown that in general, the method of quasilinearization as a solution to the system identification problem will not converge regardless of the initial starting point. This result indicates that although the quasilinearization method is useful for solving two-point boundary value problems, it is not useful (in its present form) for solving the system identification problem.

published_or_final_version / Civil and Structural Engineering / Doctoral / Doctor of Philosophy

The main objective of this thesis is to study and investigate the dynamics and stability of cantilevered structures subjected to internal, external, or simultaneous internal and external axial flows. This was accomplished, in some cases, by deriving the linear equations of motion using a Newtonian approach and, in other cases, by making the necessary modifications to existing theoretical models. The continuous cantilevered systems were then discretized using the Galerkin method in order to determine their complex eigenfrequencies. Moreover, numerous experiments were performed to compare and validate, or otherwise, the theoretical models proposed. More specifically, the four cantilevered systems studied were the following: (i) a pipe conveying fluid that is fitted with a stabilizing end-piece, which suppresses flutter by blocking the straight-through exit of flow at the downstream end; (ii) a pipe aspirating fluid, which flutters at low flow velocities in its first mode; (iii) a free-clamped cylinder (i.e. with the upstream end free and the downstream end clamped) in confined axial flow, which also flutters at low flow velocities in its first mode and eventually develops a buckling instability; and (iv) a pipe subjected to internal flow, which after exiting the pipe is transformed to a confined counter-current annular flow, that becomes unstable by flutter too.

No description available.

In this Thesis the theoretical understanding of oscillations in coronal structures is developed. In particular, coronal loops are modelled as magnetic slabs of plasma. The effect of introducing inhomogeneities on the frequency of oscillation is studied. Current observations indicate the existence of magnetohydrodynamic (MHD) modes in the corona, so there is room for improved modelling of these modes to understand the physical processes more completely. One application of the oscillations, on which this Thesis concentrates, is coronal seismology. Here, the improved theoretical models are applied to observed instances of coronal MHD waves with the aim of determining information regarding the medium in which these waves propagate. In Chapter two, the effect of gravity on the frequency of the longitudinal slow MHD mode is considered. A thin, vertical coronal slab of magnetised plasma, with gravity acting along the longitudinal axis of the slab is studied, and the effect on the frequency of oscillation for the uniform, stratified and structured cases is addressed. In particular, an isothermal plasma, a two-layer plasma and a plasma with a linear temperature profile are studied. Here, a thin coronal loop, with its footpoints embedded in the chromosphere-photosphere is modelled, and the effects introduced by both gravity and the structuring of density at the footpoint layers are studied. In this case, gravity increases the frequency of oscillation and causes amplification of the eigenfunctions by stratification. Furthermore, density enhancements at the footpoints cause a decrease in the oscillating frequency, and can inhibit wave propagation, depending on the parameter regime. In Chapter three, the effects introduced to the transverse fast MHD mode when gravity acts across a thin coronal slab of magnetised plasma are considered. This study concentrates on the modification of the frequency due to the dynamical effect of gravity in the equation of motion, neglecting the effect of stratification. Here, gravity causes a reduction of the oscillating frequency of the fundamental fast mode, and increases the lower cutoff frequency. In effect, for this configuration, gravity allows the transition between body and surface modes, in a slab geometry. It is found, in these two studies, that each harmonic is affected in a unique manner due to structuring or stratification of density. With this knowledge, in Chapter four, a new parameter is derived; P1/2P2, the ratio of the period of the fundamental harmonic of oscillation to twice the period of its first harmonic. This parameter is shown to be a measure of the longitudinal structuring of density along a coronal loop, and the departure of this ratio from unity can yield information regarding the lengthscales of the structure. This process is highlighted using the known observations, indicating that P1/2P2 may prove to be a useful diagnostic tool for coronal seismology. Finally, in Chapter five, outwardly propagating coronal slow MHD modes are observed and are used to infer coronal parameters. The possibility of using these oscillations to infer near-resolution lengthscales in coronal loops -- fine-scale strands -- is also discussed. TRACE observations are used to determine the average period, phase speed, detection length, amplitude and energy flux for the propagating slow MHD mode. The indication is that the source of these oscillations appears very localised in space, and the driver only acts for a few periods, suggesting the perturbations are driven by leaky p-modes (solar surface modes).

No description available.