Statistical investigation of the free vibration of mistuned blade system

A statistical investigation of the effects of uncertainty in root fixity on the free vibration of turbine blades is made. Emphasis is particularly placed on the statistical properties of the random eigenvalues and essentially on their standard deviations. These are evaluated using the direct product technique between matrices [1] and validated by Monte Carlo SImulations (MCS). The studied system is a simplified model of a shrouded blade assembly under the conditions of weak interblade coupling. It essentially consists of a cyclic chain of continuous beams with identical properties, fixed at one end via rotational springs with random stiffnesses representing the uncertain roots stiffnesses and coupled via linear springs at their tips. Finite Element Method is used as a discretization technique to obtain the equations of motion of the tuned and mistuned systems and the corresponding random eigenvalue problem.Numerical simulations show that small differences between the rotational springs stiffnesses spoilt the natural frequencies that were in pairs, increase the width of each frequency-cluster and strongly localizes the vibration around one blade. This strong localization has been shown to occur in a chain of single-degree-of-freedom, nearly identical, coupled oscillators if the coupling frequency between the subsystems is of order of, or smaller than the spread in the natural frequencies [2]However, for the multi-degree-of-freedom and randomly mistuned system considered her, multiple realizations are required to capture the behaviour of the eigenvalues appearing in frequency-clusters. It is found that for each frequency-cluster, when the standard deviations of the eigenvalues are plotted against the mode number, they form a U-shaped curve. For the particular case when the coupling frequency line crosses a curve, this essentially shows that the vibration localization is stronger at the first and last modes than at the mid frequencies, which belong to one passband in the tuned system