An Experimental and Theoretical Investigation into the Acoustic Spectra of Solids Having Cylindrical Symmetry

Abstract

A highly sensitive acoustic line pulse-echo system was developed to measure and classify the spectra of a range of isotropic solids having cylindrical geometry. In a number of cases, these have been compared with exact theoretical solutions and in others they have been used to test the extent to which approximate solutions are valid. A theory for contour vibrations in thick disks was developed and numerical solutions obtained for a wide range of disk thickness-to-diameter ratios. These numerical eigenvalues and the resonant spectra of the specimen, which is in the form of a disk resonator(with no restriction on its thickness), have been used to determine the elastic constants of the specimen material. The in plane vibrations of annular rings, which is the general case of the disk at one extreme and the narrow ring at the other, have been analysed. Six series of resonances were identified, in which every disk mode progresses to a particular ring mode as the hole diameter is increased. The thick disk theory has been extended to the thick annular disk and numerical eigenvalues evaluated for a full range of hole sizes, Poisson's ratio and disk thicknesses up to one radius. The correspondence of the resonances in thick disks to those in cylindrical shells was examined. The six series of resonances observed in annular rings were present and in addition, there is one series each of pure radial (compressional), finite frequency and asymptotic (compressional) modes. Several limiting cases obtained by simplifying the thick annular disk frequency equation, using various approximations, were investigated. Of these, the most important one is the frequency equation which yields the cut-off frequencies (plane strain) of an infinite hollow cylinder. These features agree exactly with those of other researchers. It is shown that a material modulus transformation yields the plane stress (annular ring) results from the plane strain results. The interaction of a thin layer of gas on a vibrating body was investigated by constructing a series of tuning forks with various narrow gaps between the tines. An electronic system was constructed for this investigation which enabled the fork to bedriven into resonance and the subsequent decrement received from the fork was analysed to give the decrement parameters. A reactive component changing the frequency and a lossy component changing the logarithmic decrement were found by changing the gas pressure. Finally, the solution of non—axisymmetric vibrations in a disk-shaft combination has been considered. Resonant spectra were obtained for disk-shafts of various disk and shaft dimensions and the way in which their various modes of vibration were coupled together, was examined.