Computation of quasi-periodic localised states in nonlinear cyclic and symmetric structures with harmonic balance methods

The emergence of localised vibrations in cyclic and symmetric rotating structures, such
as bladed disks of aircraft engines, has challenged engineers in the past decades. In the
linear regime, localised states emerge due to inhomogeneities induced by manufacturing
variability, and this phenomenon is well-known in the turbomachinery community as a
mistuning problem (see e.g. Ref. [1]). However, the linear behaviour has to be seen as
an approximation, and structures may deviate significantly from the linear assumption
due to e.g. frictional dampers, impacts, and large displacements. Within this nonlinear
regime even perfectly symmetric structures can localise vibrations due to bifurcations (see
e.g. [2]).
In this paper we develop a fully numerical methodology based on harmonic balance meth-
ods to compute localised vibrations in cyclic and symmetric structures. The main focus
is on quasi-periodic states induced by strong engine order excitations. In this case, peri-
odic solutions loose stability due to Neimark-Sacker bifurcations, and localised vibrations
detach from homogeneous states. We apply the numerical methodology for two minimal
models of bladed disks. The first one investigates the geometrical stiffening effect induced
by large deformations, while the second one models impacts induced soft joints. In both
cases the methodology is able to identify and trace multiple coexistent quasi-periodic solu-
tions. Future investigations will apply the present methodology in reduced-order models
computed from high-fidelity finite element codes. These results should clarify whether
bladed disks can localise vibrations in operational conditions due to nonlinearities.