Maximum amplification of blade response in mistuned multi stage assemblies

This paper focuses on the determination of the maximum amplification of blade response due to mistuning in multi stage assemblies. The modal optimization strategy developed earlier in connection with single stage models is extended here to multi stage configurations. Theoretical developments are carried out first and lead to the new upper limit of (1+√N ₁+N₂(g₂/g₁)+⋯) where N _i are the number of blades on the stages i and g_i = F _-i^T M_i^-1 F_-1with F _-i the force vector applied on a sector of stage _i and M_i its mass matrix. For identical stages, this maximum equals the Whitehead limit observed with single stages but with the number of blades equal to sum of the numbers of blades of the coupled stages. A computational validation of the theoretical results is achieved next on both a single degree of freedom per blade model and a reduced order model of a blisk. These numerical optimization efforts confirm the theoretical developments and demonstrate that such high amplification factors can indeed be achieved with small levels of mistuning. The effects of the number of blades on the different stages, damping in the system, stage coupling strength, etc are discussed in details.