Free vibration of regular polygonal plates with simply supported edges

Abstract

The free vibration of regular polygonal plates with simply supported edges is studied by the dynamical analogy with membranes. A regular polygonal membrane is formed on a rectangular membrane by fixing several segments. With the reaction forces acting on all edges of an actual polygonal membrane regarded as unknown harmonic loads, the stationary response of the membrane to these loads is expressed by the eigenfunctions of the extended rectangular membrane without internal supports. The force distributions along the edges are expanded into Fourier sine series with unknown coefficients, and the homogeneous equations for the coefficients are derived by restraint conditions on the edges. The natural frequencies and the mode shapes of the actual membrane are determined by calculating the eigenvalues and eigenvectors of the equations. The method is applied to an equilateral triangular through a regular decagonal membrane, the natural frequencies and mode shapes are calculated numerically and the effect of the shape of membrane is discussed. The numerical values obtained for polygonal membranes are immediately converted into those of simply supported polygonal plates.