Vibration of a Cracked Cylindrical Shell of Rectangular Planform

Harmonic vibrations of a circular, cylindrical shell of rectangular planform and with an arbitrarily located crack, are investigated. The problem is described by Donnell’s equations and solved using triple finite Fourier transformation of discontinuous functions. The unknowns of the problem are the discontinuities of the slope and of three displacement components across the crack. These last quantities are replaced, using constitutive equations, by curvatures and strain in order to improve convergence and to represent explicitly the singularities at the tips. The formulas for differentiation of discontinuous functions are derived using Green-Gauss theorem. Application of the boundary conditions at the crack leads to a homogeneous system of linear algebraic equations. The frequencies are obtained from the characteristic equation resulting from this system. Numerical results for special cases are provided.