Maximum amplification of blade response due to mistuning in multi-degree-of-freedom blade models

This paper focuses on the determination and study of the maximum amplification of the steady state forced response of bladed disks due to mistiming. A general multi-degree-of-freedom dynamic model is adopted for each blade/disk sector and optimization techniques are used to maximize a weighted quadratic norm of the response of the degrees-of-freedom of blade 1 (overall response of blade 1). First, a mathematical optimization effort is conducted in which the resonant mistimed mode shape(s) (1 for engine orders 0 and N/2 where N is the number of blades, 2 otherwise) is selected to maximize the overall response of blade 1. The form of these optimum mode shapes is derived for all weighting matrices. The specific mode shapes are also derived for two particular weights the first one of which depends on the tuned bladed disk mass matrix and for which the overall response is akin to the kinetic energy. A closed form solution is also derived when the analysis focuses solely on the response of a specific degree-of-freedom or a specific stress component. In these cases, the ratio of the corresponding overall response to its tuned counterpart, i.e. the amplification factor, is found to be the product of two terms. The first one is an amplification obtained by tuned variations of the blade properties/mode shapes and thus is referred to as the modal amplification factor. The second term is an amplification obtained by proper mistuning. Interestingly, the modal amplification factor may take on very large values while a representative value of the largest mistuned factor is often the Whitehead limit of (1 + √N)/ 2 as in the single-degree-of-freedom per blade model. The above formulation and results are readily extended to the optimization of the blade alone response (as opposed to blade and disk sector). Numerical optimization efforts were also undertaken on both a two-degree-of-freedom per blade disk model and a 24-blade blisk reduced order model. The results of these computational efforts not only confirm the assumptions and findings of the theoretical developments but also demonstrate that substantially larger amplification factors can be obtained with a general natural frequency mistiming as opposed to Young's modulus mistiming. Finally, an amplification due to mistiming (no tuned amplification) slightly larger than the Whitehead limit was obtained with relative variations in blade alone frequencies less than 0.5%.