On the propagation of longitudinal stress waves in finite solid elastic horns

Martin, Gordon Eugene, 1925-
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This dissertation consists of an investigation of the propagation of longitudinal stress waves in finite solid elastic horns or tapered bars. A large number of publications during the last several decades have reported improved theories for other structures such as rods of uniform cross section. However, there appears to have been no publication wherein the effects of lateral inertia and/or shear have been included in a theory of longitudinal waves in horns. These effects are included in the results presented here. A new wave equation and appropriate boundary conditions are obtained, and the special case of the conical horn is discussed in detail including theoretical and experimental results. The historical progress in the study of fluid as well as solid horns is surveyed. A previously-unpublished duality principle is derived which shows that the results from fluid horn theory have a one-to-one correspondence with the one-dimensional theory of solid horns. The major advances in the theory of longitudinal waves in rods of uniform cross section are reviewed also. The engineering method for the derivation of improved theories of wave propagation is applied to horns. Namely, assumed forms for the components of particle displacement are used to form the Lagrangian and incorporate it into Hamilton's Principle. The solution of the resultant integral equation leads to an (Euler) wave equation and the corresponding boundary conditions. It is shown that the horn theory satisfies the reciprocity conditions for a physical system only if the appropriate boundary conditions are used. The conical horn is discussed as a special case of a horn with lateral inertia effects only. It is shown that the solution of the wave equation is a linear combination of the two Legendre functions divided by the axial coordinate. The theory of the finite conical horn with arbitrary end conditions is derived. The new theory is compared with the one-dimensional theory with the conclusion that small but significant lateral effects are present if the horn is long and thin. Comparison of theoretical results with experimental data published three decades ago indicated that the previous experimental results were not sufficiently accurate to test the theory. Therefore, an experimental study is reported wherein two identical horns are cemented with a thin ferroelectric disc between them to provide excitation. Measured values of critical frequencies of resonances are compared with results from theory incorporating the effects of the cement joint and the disc as a linear piezoelectric device. A separate experiment for the evaluation of the dynamic compliance of cement joints is described. The comparison of theoretical and experimental frequencies of resonances of conical horns shows that the relative error is reasonably small (.01 - 1.%). Possible causes for the error due to experimental aspects are given. It is concluded further that the theory should be extended with higher order representations of particle displacement used to include more correctly the effects of lateral inertia and shear.