Exploiting the Complex Plane: The Moebius Transformation and Vibro-Acoustic Optimization

If a mechanical or acoustical impedance modification is introduced between two positions, the effect of that modification can be plotted in the complex plane at a given frequency. It has been shown that the mechanical or acoustical response will trace a circle in the complex plane for straight-line modifications to impedance in the complex plane. In that case, the equations relating the response to an impedance modification are in a form consistent with the Moebius transformation, which maps straight lines or circles in one complex domain into straight lines or circles in another complex plane. It is demonstrated that the equations for muffler transmission loss and mount insertion loss are in a form consistent with the Moebius transformation for certain design changes. Accordingly, the usefulness of this linear system property will be illustrated for both muffler and mount design.