Coupled vibrations in hollow cylindrical shells of arbitrary aspect ratio

Abstract

Vibrations of hollow cylinders have been the subject of considerable interest for many years. Piezoelectric cylinders offer a convenient system to study the vibration mode shapes, resonance frequencies and their mode coupling do to the ability to strongly and symmetrically excite extensional circumferential and axial vibration modes as well as flexural bending axial modes. While the mode repulsion of coupled circumferential and axial modes is generally widely known, their interaction gives rise to tubular flexural resonances in cylinders of finite thickness. Junger et al. [JASA 26, 709–713 (1954)] appears to have been first to discredit the notion of a forbidden zone, a frequency band free of resonant modes, as being an artifact of treating thin cylinders in the membrane limit. Aronov [JASA 125(2), 803–818 (2009)] showed experimental and theoretical proof of the presence of resonant modes throughout the spectrum as a result of the extensional mode coupling induced symmetric tubular bending modes in cylinders and their relationships as a function of different piezoelectric polarizations. That analysis used the energy method and the Euler-Lagrange equations based on the coupling of assumed modes of vibration and the synthesis of results using equivalent electromechanical circuits. This paper aims to both summarize and generalize those results for the applicability of passive cylindrical shells.