Chebyshev polynomials in the study of transverse vibrations of nonuniform rectangular orthotropic plates

An analysis is presented for free transverse vibrations of thin, rectangular orthotropic plates with thickness varying exponentially in one direction only and resting on an elastic foundation of Winkler type. Following the Levy approach, that is, two parallel edges being simply supported, the fourth order differential equation with variable coefficients governing the motion of such plates has been solved numerically by using the Chebyshev polynomials for three different combinations of clamped, simply supported, and free boundary conditions at the other two edges. The effect of the elastic foundation together with the orthotropy, aspect ratio, and thickness variation on the natural frequencies of vibration is illustrated for the first three modes of vibration. Mode shapes have been presented for two different values of the taper parameter, keeping other plate parameters fixed, for all three boundary conditions. A comparison of the results with those available in literature has been presented.