Spectrum of the Torsional and Longitudinal Natural Frequencies of Cantilever with an Arbitrary Number of Layers with the Piece wise Constant Mechanical Properties, Densities and Thicknesses

Here, we derived analytically a transcendental equation, which positive roots are the spectrum of torsional and longitudinal natural frequencies of cantilever composed of an arbitrary number of layers with the piecewise constant mechanical properties, densities and thicknesses. This transcendental equation has been obtained in the Laplace image space and the one connects the mechanical properties, density and layer thickness. The present solution can be particularly useful for either a non-destructive material testing or design of the cantilever structures.