The vincent circle

More than 30 years ago, A. H. Vincent of Westland Helicopters demonstrated that if a structure is excited harmonically at a fixed frequency, the response at another position will trace a circle in the complex plane as a result of a stiffness modification between two points. This can be more generally expressed in the following manner. The structural or acoustical response at any position will map a circle in the complex plane for any straight line impedance modification in the complex plane. This article reviews the basis for this little-known principle and illustrates its usefulness for vibro-acoustics problems. The principle is demonstrated for noise radiation from a cantilevered plate and operator equipment cab. The technique can also be applied to waveguides.