VIBRATION OF ROTATIONALLY PERIODIC STRUCTURES

The purpose of this paper is to determine eigensolutions of a rotationally periodic structure P of period N in terms of those of the ith substructure S(i) of P. Let y(i) be the generalized displacement vector and f(i) be the generalized force vector at interface I(i), where substructures S(i) and S(i-1) are connected. Transfer function formulation of S(i) implies that there exists a linear operator G mapping y(i) and f(i) at interface I(i) to y(i + 1) and f(i+1) at interface I(i+1). In addition, the operator G depends on transfer functions (and therefore eigensolutions) of S(i). The periodicity of P then requires that GN be an identity map. This results in N self-adjoint Fredholm integral equations, the non-trivial solutions of which predict eigensolutions of P. As a consequence, a periodic structure P with period N will have exactly N eigenvalues lying between two consecutive eigenvalues of the substructure S(i) if the interface I(i) contains only one degree of freedom. Finally, three examples are illustrated. The first example considers anti-plane strain vibration of a linear elastic solid containing four slots of infinitesimal width. The second example derives exact eigensolutions of a string-mass periodic structure. The third example illustrates how eigensolutions of an axisymmetric structure can be recovered under the formulation of rotationally periodic structures. © 1994 Academic Press Limited.