Fundamental frequencies of mechanical systems with N-piecewise constant properties

This paper presents an analytical derivation of a set of transcendental equations for the normal modes of a one-dimensional mechanical system composed of an arbitrary number of components and provides experimental validation for the case of a two-component system consisting of a fiber partially submerged in liquid. The transcendental equations relate thickness of layers and their elastic moduli and allow possible experimental measurement of thin film thickness and elastic moduli. An effective model of the wave propagation in such systems gives an a-priori qualitative explanation of the natural frequencies change and allows one to predict them quantitatively.