Vibro-acoustic optimization using the moebius transformation

In complex variable analysis, the Moebius transformation maps straight lines or circles in one complex domain into straight lines or circles in another. In acoustic and dynamics, amplitude and phase information is frequently plotted in the complex plane. It is observed that the equations relating acoustic or mechanical impedance modifications to response under harmonic excitation at a certain frequency are in the form of Moebius transformation. Using the properties of the Moebius transformation, the optimal design that gives the desired response can be predicted. To prove the usefulness of this method in acoustic and mechanical system optimization, the approach is demonstrated for mufflers, enclosures and mounts. Impedance modifications are realized by changing duct dimensions for acoustic cases and adding mass for the mechanical case.