Multiple scale analysis of the forced response of a mistuned bladed disk with nonlinear friction forces

The unavoidable small blade-to-blade variations in a turbomachinery bladed disk are know as "mistuning". When compared with the ideal, "tuned", bladed-disk, the mistuning can produce a considerable increase of the vibratory forced response level of the blades. This situation can lead to high cycle fatigue failure, and, therefore, the correct estimation of the vibratory response is of crucial importance for the prediction of the operative life span of the bladed disk. The computation of the final amplitude of the limit cycle oscillation requires to solve a quite complicated problem; the mistuned bladed disk is not cyclic symmetric, and, consequently, the complete bladed disk has to be considered, increasing dramatically the computational cost of the numerical simulations. For this reason we discuss the possibility of applying an asymptotic multiple scale analysis to derive a simplified model that can be used to analyze the characteristics of the final vibration states at a much lower computational cost. The key idea for the simplification is the fact that all significant effects present (forcing, nonlinear friction damping, and mistuning) are, in most practical situations, small effects, that develop in a time scale that is much longer than that associated with the natural elastic vibration frequency of the tuned system. In this paper the bladed disk is described using a mass-spring system with nonlinear friction, and we consider the frequent case of forcing a blade dominated modal family, where all modes have very similar vibration frequencies. The derivation of the asymptotically simplified model is explained in detail, and the validity of its results is verified against the results from the original mass-spring model, for both tuned and mistuned configurations