An Eigenvalue Inclusion Principle for Simple, Rotationally Periodic Structures

This paper describes an eigenvalue inclusion principle for a simple, rotationally periodic structure P whose i-th substructure S_i is connected to a neighboring substructure S_i+1 through a single-degree-of-freedom interface constraint I_i+1. The state vector v_i+1 at the interface I_i+1, consisting of the displacement and the force at the interface, is represented in terms of the state vector v_i at the interface I_i through transfer functions of the substructure S_i. The periodicity of the structure P then requires that a linear combination of the transfer functions of S_i be zero. As a consequence, a simple periodic structure P with period N will have exactly N eigenvalues lying between two consecutive eigenvalues of the substructure S_i. Finally, this eigenvalue inclusion property is illustrated on a periodic structure with known exact eigensolutions.