Structural modal interaction of a four degree-of-freedom bladed disk and casing model

Consideration is given to a very specific interaction phenomenon that may occur in turbomachines due to radial rub between a bladed disk and surrounding casing. These two structures, featuring rotational periodicity and axisymmetry, respectively, share the same type of eigenshapes, also termed nodal diameter traveling waves. Higher efficiency requirements leading to reduced clearance between blade-tips and casing together with the rotation of the bladed disk increase the possibility of interaction between these traveling waves through direct contact. By definition, large amplitudes as well as structural failure may be expected. A very simple two-dimensional model of outer casing and bladed disk is introduced in order to predict the occurrence of such phenomenon in terms of rotational velocity. In order to consider traveling wave motions, each structure is represented by its two $nd$-nodal diameter standing modes. Equations of motion are solved first using an explicit time integration scheme in conjunction with the Lagrange multiplier method, which accounts for the contact constraints, and then by the harmonic balance method (HBM). While both methods yield identical results that exhibit two distinct zones of completely different behaviors of the system, HBM is much less computationally expensive.