70. Dynamic Equations for an Anisotropic Cylindrical Shell

Dynamic equations for an anisotropic cylindrical shell are derived using a series expansion technique. First the displacement components are expanded into power series in the thickness coordinate relative to the mid-surface of the shell. By using these expansions, the three-dimensional elastodynamic equations yield a set of recursion relations among the expansion functions that can be used to eliminate all but the six of lowest order and to express higher order expansion functions in terms of these of lowest orders. Applying the boundary conditions on the surfaces of the cylindrical shell and eliminating all but the six lowest order expansion functions give the shell equations as a power series in the shell thickness. These six differential equations can in principle be truncated to any order. The method is believed to be asymptotically correct to any order. For the special case of a ring, the eigenfrequencies are compared with exact two-dimensional theory, generally with a good correspondence.