RESPONSE OF MODULATED DOUBLET MODES TO TRAVELLING WAVE EXCITATION

The forced vibration of a rotationally periodic structure when subjected to travelling 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 (1) the number of nodal diameters present in the limiting case of axisymmetry, and (2) certain additional Fourier harmonics which contaminate and 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 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 in the mode shapeʹs Fourier expansion. The structure of the wavenumber content is explored further with respect to the response of repeated and split doublets to a harmonic travelling wave excitation. The individual Fourier components comprising a modulated doublet can respond and 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 travelling or standing wave responses, or a blend of the two, can occur in the structureʹs reference frame are also examined in the context of the model periodic structure. The qualitative character of the response, the forward or backward propagation direction of each modeʹs constituent wavenumber components, and the phase speeds of those components are discussed in illustrative case studies.