Abstract
As the self-excited phenomenon of the acoustic pipe system, the self-excited sound of the Rijke tube, combustion vibration, thermoacoustic phenomenon, and musical sound of a leadless wind instrument such as a flute are known. These systems include a self-exciting mechanism and oscillate at the resonance frequency of the acoustic system to generate a loud sound. For these systems, research on the self-excited mechanism is being carried out for each subject. However, since heat and fluid are involved, a clear self-excited mechanism cannot be explained from the viewpoint of mechanical dynamics. In this study, firstly the wave equation of an acoustic pipe excited by a point is derived. Secondly, it is replaced with a vibration system with one degree of freedom by using modal analysis. Then, the self-excitation mechanism is investigated by positively feeding back the state quantities (particle velocity and sound pressure) of the acoustic system to the input of point excitation. As a result, positive feedback of particle velocity is equivalent to proportional control (P control) that assumes displacement output in a 1-DOF vibration system. And positive feedback of sound pressure is equivalent to the differentiated control (D control) that assumes displacement output in a 1-DOF vibration system. It is shown that the feedback path with delay time element causes negative damping and self-excited oscillation. It is also shown that the result of this calculation using modal analysis is completely in agreement with the result calculated by the cellular automaton method (CA method) which describes the solution of D'Alembert in the wave equation.
