Energy Trapped Resonances in Solid Structures

Abstract

The end resonant modes of vibration in solid and hollow cylinders, strips and plates are considered. Experimental results presented for solid cylinders, made of various materials, establish not only the position of cylindrical end modes, within the frequency spectrum, but also their dependance upon Poisson's Ratio. Additional experimental results, obtained for hollow cylinders of various wall thicknesses, indicate how the antisymmetric end resonant modes of solid cylinders are influenced by the presence of a concentric hole. Numerical solutions have been found, for the end resonant modes in plates and strips, by the method of summation of stresses associated with real and complex modes of propagation. In all, results have been obtained for eleven different values of Poisson's Ratio in the range 0.1 to 0.5. The modulus and phase of the amplitude coefficients for real and complex modes are presented in graphical form. A general method of obtaining expressions for the impedance of mechanical resonators is discussed. The technique, based on the principal of virtual work, makes use of normal mode expansions. Three examples are considered in detail, with special attention being given to the application, of the obtained impedance expressions, to an existing acoustic transmission line theory, as a means of analysing the echo technique used for the experimental work. An empirical equation is derived, which enables the calculation of the natural frequencies of a pair of dynamically clamped rectangular plates to be carried out. The equation, obtained by comparing experimental results with theoretical solutions for the corresponding statically clamped plate, relates the resonant frequency to plate dimensions and the known frequency factors for the static case.