Resolving Mechanical Resonances with Breit-Wigner Formula

Abstract

Vibrating mechanical systems are usually characterized by a number of resonances appearing in the Power Spectrum Density (PSD). Each resonance is featured by three quantities with physical meaning: the resonant frequency, the damping coefficient and the amplitude. Often, another parameter, the asymmetry factor, is required when neighbor resonances overlap. Resonances can be represented by a mathematical formula, so the four parameters can be obtained by fitting the PSD to that formula. The proper expression for fitting comes from the transfer function for resonances, using a pair of complex conjugate poles. Keeping in mind that the PSD stands only for positive frequencies, a Breit-Wigner formula is derived for fitting resonances. Breit-Wigner formula is widely used in Nuclear Physics but seldom in Mechanical Engineering and so the derivation of the formula is outlined. Applications are presented for resolved resonances and the method is tested in simple cases. There is a case study for unresolved resonances belonging to the vibration of a pressurized water reactor core barrel measured by ex-core neutron detectors. The four resonance parameters are changing along the time, so for preventive maintenance purposes, they must be supervised. A natural question arises: what is the percent change of the four parameters altogether? Such a question is replied by using quaternion numbers.