Mechanical wave velocity dispersion in rods and extension of the resonant bar method to 800 KHz: a study of the velocity dispersion of longitudinal waves in rods, and a feasibility study of extending the Resonant Bar method of internal friction measurement to 1 MHz

Abstract

The Resonant Bar method for measuring internal friction in solids has been investigated with the aim of extending this method to higher frequencies in the range 100 kHz to 1 MHz. This necessitated a study of the velocity dispersion of the fundamental longitudinal mode in rods of appropriate size in the region where the wavelength is of the same order of magnitude as the diameter. It is shown that the velocity dispersion of the fundamental longitudinal mode in a cylindrical rod obeys the exact solution of the Pochhammer - Chree equation calculated by Bancroft and Bradfield, over the range of diameter to wavelength ratio from zero to 2.0. The velocity dispersion of the lowest longitudinal mode of a rod of square cross-section has proved to be the same as that of a cylindrical rod of the same material, provided the equivalent diameter of the square cross-sectioned rod is taken to be 1.155 times its thickness. The theory of Nigro (1966, 1968) agreed only partially with the experimental velocity dispersion of a square rod (Booker 1969). As a result of this and a recent theory by Fraser (1969), Nigro has modified his theory, so that at present there is perfect agreement between theory and experiment. In cylindrical rods many resonant responses are excited that are not due to the fundamental mode (also called the Young's modulus mode) when the diameter to wavelength ratio becomes appreciable (usually about 0.7). These resonances are attributed to higher order longitudinal modes. Identification of the Young's modulus mode resonances becomes impossible when many of these higher order mode resonances occur. On several occasions one of the expected Young's modulus mode resonances is absent, its place being taken by two or more resonances extremely close together in frequency and of small vibration amplitude. Frequently some of the harmonics have a frequency above or below the expected value, so that they do not lie on a smooth graph of frequency versus harmonic number. It is shown that the above two phenomena occur when the value of Ω (= ωa/Ct) is close to one of the values ΩE, Ω1 to Ω2, or Ω3, where, at a value of: (1) Ω = ΩE, an end-resonance occurs; (2) Ω = Ω1, the first complex mode becomes real; (3) Ω = Ω2, the L(0,2) mode cuts the Ω axis on an Ω versus koa graph; (4) Ω = Ω3, the L(0,3) mode cuts the Ω axis. To extend the Resonant Bar method to frequencies approaching 1 MHz, it has been found essential to use rods approximately 31/2 inches long and 1/8 th of an inch in diameter. The design and construction of a bar support abstracting a negligible amount of energy from such slender rods proved almost impossible. Both a central three-pin support and a central three-wire support have proved to be unsuitable. A support consisting of two parallel horizontal tungsten wires of a quarter of a thousandth of an inch diameter has however been found to give damping values consistent to within 10%. Graphs are drawn of damping versus frequency for several aluminium and fused silica rods, showing that Q-1 varies but little with frequency and that on the whole it varies smoothly from one harmonic to the next. The Resonant Bar method is ideally suited for the determination of dynamic elastic moduli. A computer program has been developed and used to calculate Young's modulus and Poisson's ratio for any given rod from the resonance frequencies of longitudinal vibration, the dimensions and the mass.