Modal and Impedance Modeling of a Conical Bore for Control Applications
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The research presented in this thesis focuses on the use of feedback control for lowering acoustic levels within launch vehicle payload fairings. Due to the predominance of conical geometries within payload fairings, our work focused on the analytical modeling of conical shrouds using modal and impedance based models. Incorporating an actuating boundary condition within a sealed enclosure, resonant frequencies and mode shapes were developed as functions of geometric and mechanical parameters of the enclosure and the actuator. Using a set of modal approximations, a set of matrix equations have been developed describing the homogeneous form of the wave equation. Extending to impedance techniques, the resonant frequencies of the structure were again calculated, providing analytical validation of each model. Expanding this impedance model to first order form, the acoustic model has been coupled with actuator dynamics yielding a complete model of the system relating pressure to control voltage. Using this coupled state-space model, control design using Linear Quadratic Regulator and Positive Position Feedback techniques has also been presented. Using the properties of LQR analysis, an analytical study into the degree of coupling between actuator and cavity as a function of actuator resonance has been conducted. Constructing an experimetnal test-bed for model validation and control implementation, a small sealed enclosure was built and outfitted with sensors. Placing a control speaker at the small end of the cone the large opening was sealed with a rigid termination. An internal acoustic source was used to excite the system and pressure measurements were captured using an array of microphones located throughout the conic section. Using the parameters of this experimental test-bed, comparisons were made between LQR and PPF control designs. Using an impulse disturbance to excite the system, LQR simulations predicted reductions of 53.2% below those of the PPF design, while the control voltages corresponding to these reductions were 43.8% higher for LQR control. Actual application of these control designs showed that the ability to manually set PPF gains made this design technique much more convenient for actual implementation. Yielding overall attenuation of 38% with control voltages below 200 mV, single-channel low authority control was seen to be an effective solution for low frequency noise reduction. Control was then expanded to a larger geometry representative of Minotaur fairings. Designing strictly from experimental results, overall reductions of 38.5% were observed. Requiring slightly larger control voltages than those of the conical cavity, peak voltages were still found to be less than 306 mV. Extrapolating to higher excitation levels of 140 dB, overall power requirements for 38.5% pressure reductions were estimated to be less than 16 W.
- Masters Theses