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The steady-state response of spatially modulated doublet modes that occur in low count flexible bladed disks is investigated for the case in which the structure is driven by a harmonic traveling wave excitation source. Finite element simulation and modal testing of prototypical bladed disk structures demonstrate the presence of particular wavenumbers, beyond the base number of nodal diameters, which contaminate and distort the appearance of certain doublet modes. The manner in which the natural frequency and wavenumber content of such modes shift and split as functions of the number of blades and their span angle is discussed in the light of a companion perturbation analysis for rotationally periodic structures. Resonance conditions are established and verified through simultaneous measurements made with a spin test stand using sensors that are placed in the rotating (structure) and stationary (excitation) frames of reference. The traveling wave response components of a repeated frequency doublet mode are shown to propagate either in the same or opposite direction as the excitation source, depending on whether certain algebraic relationships between the excitation order, the base number of nodal diameters, and the contamination wavenumbers are satisfied. To the extent that such components can travel at different phase speeds and directions relative to one another, the placement of sensors on the structure can be optimized to best measure the response amplitude. Conversely, other placements can result in sub-maximal measurement of peak vibration amplitude over the structure.

Free in-plane vibration of rectangular plates undergoing plane stress deformation is investigated through Ritz discretization of the Rayleigh quotient. Of particular interest is the manner in which the plate's natural frequencies and coupled longitudinal–lateral vibration modes evolve as its length-to-width aspect ratio is varied. Vibration modes are grouped into families or classes based on (1) common asymptotic behavior of natural frequency loci with increasing aspect ratio, (2) reflective symmetry of the displacement field about the plate's centerline, and (3) the presence of nodal points on or off the plate's centerline. The applicability of the traditional strength-of-materials bending and longitudinal vibration models for predicting the lower in-plane modes is also discussed, and the range of aspect ratios for which those simpler models provide acceptable accuracy is quantified. In-plane twisting and higher order modes having nodal points off the plate's centerline are not predicted by those traditional theories, and these are also discussed in several parameter studies.

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 shape's 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 structure's 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 mode's constituent wavenumber components, and the phase speeds of those components are discussed in illustrative case studies.