Identification of the Essential Features of the Transient Amplification of Mistuned Systems

The dynamic behavior of bladed disks in resonance crossing has been intensively investigated in the community of turbomachinery. In particular, the attention has been addressed towards two issues: (1) the transient-type response that appear when the resonance is crossed with a finite sweep rate and (2) the localization of the vibration in the disk due to the blade mistuning.

The transient effects produce a reduction of the maximum amplitude of the response and shift (upwards in run-up) the rotor speed at which the maximum response appears. Coherently with the common intuition, the reduction of the maximum amplitude of vibration is proportional to the rotor acceleration.

Blade mistuning produces the increment of the dynamic response of some blades when the disk is excited with a resonant or quasi-resonant force. Some analytical formulations developed in the quasi-steady regime, i.e. disregarding the transient effects due to the rotor acceleration, show that the increment of dynamic amplification compared to the tuned case is a function of the mistuning level and damping.

In real conditions, the two mentioned effects coexist and can interact in a complex manner. In particular, it was observed that the reduction of the dynamic response that is expected in resonant crossing due to the transient effects may vanish when dealing with mistuned disks. For some systems a weak over-amplification has been observed compared to the quasi-steady prediction, which appears quite counterintuitive. This phenomenon has been called Transient Amplitude Amplification of Mistuned Systems (TAMS) and has been investigated both numerically as well as experimentally.

This paper investigates the TAMS by means of analytic solutions obtained through asymptotic expansions, as well as numerical simulations. The problem is initially studied working on the simplest possible dynamical system able to produce TAMS, namely, a 2-dof linear system defined through the three parameters: damping ratio x, frequency mistuning D, rotor acceleration a. Then the system is generalized to represent the case of a bladed disk at resonance crossing for with two isolated active modes interact. It turns out that, as far as the TAMS is concerned,  the problem is governed by the two parameters a/x² and D/x. The analysis of the system response in the neighborhood of the point of the parameter space providing the maximum TAMS highlights the physical mechanism producing the transient amplification.