On the similarity of the Coleman and Lyapunov–Floquet transformations for modal analysis of bladed rotor structures

Structures with isotropic bladed rotors can be modally analyzed by eigenvalue analysis of time-invariant Coleman transformed equations of motion related to the inertial frame or by Floquet analysis of the periodic equations of motion. The Coleman transformation is here shown to be a special case of the Lyapunov–Floquet (L–F) transformation which transforms system equations of structures with anisotropic bladed rotors into a time-invariant system using the transition matrix and Floquet eigenvectors as a basis. The L–F transformation is not unique, whereby eigensolutions of the time-invariant system are not directly related to the modal frequencies and mode shapes observed in the inertial frame. This modal frequency indeterminacy is resolved by requiring the periodic mode shapes from the L–F approach to be as similar as possible to the mode shapes from the Coleman approach. For an anisotropic rotor the Floquet analysis yields a periodic mode shape that contains harmonics of integer multiples of the rotor speed for inertial state variables. These harmonic components show up as resonance frequencies on the sides of the corresponding modal frequency in a computed frequency response function of a simple three-bladed turbine with an anisotropic rotor.