A model for a two dimensional distributed resonator

Abstract In many distributed resonators the low order mode frequencies are sufficiently separated to permit satisfactory modelling in terms of only the single mode being used. It can be represented mechanically by mass, stiffness and energy loss by dissipation and coupling. The acoustically and electrically resonant frequency and the Q-factor are commonly used as parameters. A feature of high order modes not given by this model is the stepped nature of the transient response. It is also inadequate for the treatment of the interaction of adjacent frequency modes. The case of a low dissipation distributed resonator can be represented on the plane of the complex Laplace variable s, by a series of pairs of poles and zeros. Each pair corresponds to a mode of resonance and the positions in the plane are given by the frequency on the imaginary axis and the coupling on the real axis. A pair considered alone (the dominant pole approximation) is identical to the simple model. A comprehensive s-plane model would incorporate all the mode poles and zeros and the vibrational response could be found by performing a Bromwich contour integral round the points. In practice good results are obtained by using only a limited number of modes. The longitudinal vibrations of a rod driven at the end are well understood and this has been used as an example. A more difficult case, that of the in-plane vibrations of a thin disk, has been analyzed to obtain theoretical expressions for the coupling factors of the lower modes. The theoretical values have been checked experimentally and the resultant model used for material characterization experiments. When two adjacent resonances are excited by the drive signal a trial and error comparison between experimental and theoretical responses, by using computer graphics, enables the parameters of the two modes to be found. The s-plane mode can incorporate the effects of internal friction and by using it a convenient locus diagram representing the change in internal friction with temperature has been obtained.