On the Response of Modulated Doublet Modes to Traveling Wave Excitation

The forced vibration of arotationally periodic structure when subjected to traveling wave excitation is discussed, with emphasis placed on the steady-state response of doublet modes having either repeated or split frequencies. Such vibration modes have spatially modulated shapes defined by (i) the numbers of nodal diameters present in the limiting case of axisymmetry, and (ii) certain additional, superposed, contaminating Fourier harmonics which distort their appearances. The natural frequency and mode structure of a model periodic structure is discussed in the context of an otherwise axisymmetric disk having evenly-spaced, sector-shaped, line distributions of stiffness and inertia. Through a perturbation analysis, the contamination wavenumbers present in a doublet having repeated frequency are shown to comprise two subsets, the members of which have sine and cosine coefficients of the same, or of differing, signs for each wavenumber present in the mode shape’s Fourier expansion. That structure for the wavenumber content is explored further with respect to the response of repeated and split doublets to a harmonic traveling wave excitation. The individual Fourier components comprising a modulated doublet mode shape can propagate in the same direction as the excitation, or opposite to it, depending on the wavenumber of the excitation and the subset to which the contamination wavenumber belongs. The response of the split frequency doublets and the circumstances under which traveling or standing wave responses, or a blend of the two, can occur in the structure’s reference frame are examined and discussed in the context of the model periodic structure. The qualitative character of the response, the forward or backward propagation direction of each mode’s constituent wavenumber components, and the phase speeds of those components are discussed in illustrative case studies.