● TITLE: On non-smooth beam shapes for maximal transverse natural frequencies
● AUTHORS: Berger, M., Porat, I. (1934-2012)
● SOURCE: Journal of Sound and Vibration, v. 132, n. 3, pp. 423-432 (Aug. 8, 1989).
● LANGUAGE: English.
● ABSTRACT: The optimal shape of an Euler beam subject to space constraints (lower-upper bounds) is studied. Necessary conditions for a transverse natural frequency to be maximal are stated from which optimality conditions at shape 'corners' are derived. These conditions enable one to treat beam configurations comprising shapes of piecewise continuous derivatives, at the corners of which discontinuous slopes are permitted. For beams with a linear relationship between the cross-sectional moment of intertia and area optimal shapes of analytical form are derived. These previously unknown solutions are verified by a numerical method. For a free beam, the width of which is bounded between 0·02 and 0·1 relative to its length, the optimal shape attains a fundamental frequency 60% higher than that of a beam of uniform shape.

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