Analytical solution for modulation sidebands associated with a class of mechanical oscillators

Abstract

Sideband structures are a commonly observed phenomenon in measured vibro-acoustic signatures of many types of mechanical systems, and especially in rotating machinery. Such spectral information is often used for fault diagnostic applications as well as noise and vibration control studies. A new theory is developed that examines in a critical manner the commonly held belief that simple amplitude and frequency or phase modulation processes are responsible for generating such sidebands, and instead provides a more logical explanation. A class of viscosity damped mechanical oscillators is examined having spatially periodic stiffness and displacement excitation functions that are exponentially modulated by the instantaneous vibratory displacement of the inertial element. To this end, a new family of dual domain periodic differential equations is introduced which is shown to be more pertinent to the study of force modulation inherent to many mechanical systems, such as a gear pair. Analytical expressions are then derived to predict sideband amplitudes in terms of key system parameters. The harmonic balance method and direct time domain integration techniques are employed to study various subsets of the problem.