Saint-Venant’s method to solve the Poisson equation is applied to the Helmholtz equation which governs membrane vibration. A family of non-circular membrane shapes with N-fold rotational symmetry is presented. Membrane shapes, properties, and fundamental frequencies are tabulated. The solutions are exact and can serve as accuracy standards for approximate methods.
Issue Section:Technical Brief
Keywords:membrane, vibration, exact solution, closed-form
C. Y., and
Structural Vibration, Exact Solutions for Strings, Membranes, Beams and Plates,
Boca Raton, FL.
S. P., and
Theory of Elasticity, 3rd ed.,
Fundamental Frequencies of Rounded Polygonal Membranes— A Class of Homotopy Shapes,”
ASME J. Vib. Acoust.,
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