Efficient predictions of the vibratory response of mistuned bladed disks by reduced order modeling

The research work reported in this dissertation aims to provide computational methodologies and to further the understanding of physical phenomena that will aid and improve the design of bladed disk assemblies from a structural dynamics standpoint. Bladed disk structures are found in a wide array of applications, including small impeller pumps andautomotive turbo systems; large gas, steam, and hydro turbines for power generation; and jet engines for military and commercial aircraft propulsion. Based on the nominal design, a bladed disk assembly is a rotationally periodic structure. If it is assumed that each disk-blade sector is identical, then the theory of cyclic symmetry may be used to analyze the dynamics of the entire structure based on one fundamental disk-blade sector. In practice, however, there are always small differences among the structural properties of individual blades, which destroy the cyclic symmetry of the bladed disk assembly. These structural irregularities, commonly referred to as mistuning, may derive from manufacturing tolerances, deviations in material properties, or non-uniform operational wear. Mistuning is known to have a potentially dramatic effect on the vibratory behavior of the rotor, since it can lead to spatial localization of the vibration energy. Spatial localization implies that the vibration energy in a bladed disk becomes confined to one or a few blades rather than being uniformly distributed throughout the system. This phenomenon may be explained by viewing the vibration energy of the system as a circumferentially traveling wave. In a perfectly tuned system, the wave propagates through each identical disk-blade sector, yielding uniform vibration amplitudes that differ only in phase. In the mistuned case, however, the structural irregularities may cause the traveling wave to be partially reflected at each sector. This can lead to confinement of vibration energy to a small region of the assembly. As a result, certain blades may experience forced response amplitudes and stresses that are substantially larger than those predicted by an analysis of the nominal design. Hence, certain blades may exhibit much shorter lifespans than would be predicted by a fatigue life assessment based on the nominal assembly. In order to address this concern, some efficient computational methods have been developed that can predict the effects of mistuning on the vibratory response of a turbomachinery rotor stage. Furthermore, these techniques enable analyses of large numbers of randomly mistuned bladed disks in order to estimate the mistuned forced response statistics for a rotor design. However, at the outset of this research effort, no methods possessed the combination of accuracy and computational efficiency required to allow reliable statistical assessments of mistuning sensitivity to be included as an integral part of the turbomachinery rotor design process. Motivated by the turbomachinery community's need for practical design tools that incorporate mistuning effects, three distinct objectives are identified and addressed in this research effort: 1 - To develop highly efficient and accurate reduced order modeling techniques for the free and forced response of tuned and mistuned bladed disks, based on parent finite element representations of arbitrary complexity and detail in a consistent and systematic fashion. 2 - To broaden the scope of these reduced order modeling techniques by establishing linearized formulations for shrouded blade designs. 3 - To increase the understanding of the underlying physical mechanisms of the mistuning phenomenon, with particular emphasis on the role of disk flexibility and structural stage-to-stage coupling in determining a design's sensitivity to mistuning, leading to formulations for multi-stage synthesis and analysis.