Wave Propagation in Circularly Curved Beams and Rings,
NEW YORK INST OF TECH OLD WESTBURY
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The present theory which deals with the problem in its general form was developed from variational considerations using Hamiltons principle to derive the exact equations of motion for thin circularly curved beams and rings, together with consistent boundary, discontinuity and initial conditions in terms of the radial and tangential midsurface displacements, and the rotation of the normal. The theory accounts for the effects of extensional, flexural and sharing deformations, and rotatory inertia. The effects of distributed elastic foundtions in the directions of the radial and tangential displacements and the rotation are also incorporated into the equations of motion. The vibration and wave propagation analyses on which the present investigation is based properly begin with the resolution of the foregoing equations of motion into three-uncoupled sixth order homogeneous differential equations in terms of the radial and tangential midsurface displacements, and the rotation. Using the classical form for the traveling wave solution, the frequency equation is derived in closed-form, in terms of the flexural, transverse shearing and extensional stiffness as well as the three spring constants of the elastic foundations as precisely identifiable parameters.