Viscoelastic effects on solid amplitude and phase transformers

Abstract

The longitudinal oscillations of nonuniform bars are used as displacement amplifiers in power tools and other devices. These often operate near resonance [E. Eisner, J. Acoust. Soc. Am. 35, 1367–1372 (1963)], so that the elastic model should be replaced by a viscoelastic one, which includes damping. In the present paper, the longitudinal vibrations of a tapered viscoelastic bar are discussed generally from the wave equations for the displacement and strain. Exact solutions are obtained for the exponential, catenoidal, sinusoidal, and inverse shapes, and also for the Gaussian and power‐law shapes. These solutions generalize earlier results for elastic bars; e.g., the elastic Gaussian bar [D. A. Bies, J. Acoust. Soc. Am. 34, 1567–1569 (1962)] is generalized to a viscoelastic one. Diagrams of wavenumber and damping ratio versus frequency and viscous relaxation time are presented for several shapes of tapered bar; they describe the propagation and dissipation of oscillations and their effects on amplitude and phase.