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A new analytical method is developed for transient solutions for bars in longitudinal vibration, shafts in torsional vibration and strings in transverse vibration. These vibrating continua have nonuniform distributions of geometric and material properties, and are governed by second-order partial differential equations (wave equations). Based on a transfer function formulation and a residue formula, the proposed method delivers exact solutions for several types of nonuniform systems subject to external, boundary and initial disturbances. The proposed method does not need system eigenfunctions, and is numerically efficient as it only involves simple operations of two-by-two matrices.

A discussion is given of early literature pertaining to the theory of vibration from the time of Pythagoras up through 1750. The paper attempts to give an analytical interpretation to early anecdotal works concerning Pythagoras and to publications of Galileo, Huygens, Hooke, Taylor, John Bernoulli, Leibniz and Euler. To bridge the “culture gap,” mathematical developments by the latter cited authors are, whenever appropriate, rephrased in modern notation, using, for the most part, only those techniques that should have been well known to the authors at the time. The emphasis is on what might be loosely called the physics (or the mathematical physics) of vibration.