Dynamic analysis and modeling of multi-coupled nearly periodic structures.

Abstract

This thesis investigates localization and dissipation phenomena in disordered nearly periodic structures, as well as the efficient dynamic modeling of a particular class of periodic structures, turbomachinery rotors. The effect of structural damping on the dynamics of periodic and nearly periodic structures is examined. Decay rates due to disorder and damping are calculated, and the interaction of the localization and dissipation mechanisms in the forced response is investigated. It is shown that damping increases the overall decay rate but reduces localization. Lyapunov exponents of a system wave transfer matrix are employed to analyze localization in multi-coupled (and as a special case, mono-coupled) disordered structures. The largest Lyapunov exponent is calculated numerically for a representative mono-coupled system, and found to compare well with localization factors found by Monte Carlo methods and perturbation approximations. The Lyapunov exponents are also computed for an example of a disordered bi-coupled system, and compared with wave amplitude decay and wave conversion observed in this system. The physical significance of the Lyapunov exponents is discussed. The standard deviations of the Lyapunov exponents are calculated and used to determine frequency ranges in which localization effects, rather than off-resonance or dissipation effects, are strongest. The modeling of bladed disks in turbomachines is considered. An order reduction method is presented which is capable of generating reasonably accurate, very low order models of mistuned bladed disks, so that forced response Monte Carlo simulations may be performed at a reasonable cost. This technique is based on selected component modes of vibration from a finite element analysis of a single disk-blade sector. An example reduced order model is generated, and the mistuned frequencies and localized mode shapes are found to match closely those of the mistuned finite element model. A transfer matrix is formulated, and a Lyapunov exponent analysis is shown to predict a frequency range where localized modes are found.