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Research Papers

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

[+] Author and Article Information
Mathias Legrand

Institut de Recherche en Génie civil et Mécanique (GeM), UMR CNRS 6183, École Centrale de Nantes, BP 92101 44321, Nantes Cedex 3, Francemathias.legrand@mcgill.ca

Christophe Pierre

McGill University, McDonald Engineering Building, 817 Sherbrooke West, Montreal H3A 2K6, Canadachristophe.pierre@mcgill.ca

Bernard Peseux

Institut de Recherche en Génie civil et Mécanique (GeM), UMR CNRS 6183, École Centrale de Nantes, BP 92101 44321, Nantes Cedex 3, Francebernard.peseux@ec-nantes.fr

J. Comput. Nonlinear Dynam 5(4), 041013 (Aug 17, 2010) (9 pages) doi:10.1115/1.4001903 History: Received March 20, 2007; Revised March 24, 2009; Published August 17, 2010; Online August 17, 2010

## Abstract

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.

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## Figures

Figure 1

Figure 2

The two three-nodal diameter modes of the casing and bladed disk

Figure 3

Schematic of the ith predicted gap distance between blade i and casing

Figure 4

Time evolution of uc during interaction for h=10−6 s(⋯⋅), h=5×10−7 s (- -) and h=10−7 s (—) for nd=3 and Ω=500 rad/s

Figure 5

Distances between the casing and the 22 blade-tips for nd=3 and Ω=500 rad/s: two blades 6 (- - -) and 21 (—) are in permanent contact with the casing during the interaction phenomenon

Figure 6

Modal vibrations of the structures for Ω=500 rad/s and nd=3: At t≃0.08 s, a nd nodal diameter forward traveling wave starts to propagate in the casing while the blade disk simultaneously reaches a purely static deformation state

Figure 7

Modal vibrations of the structures for Ω=500 rad/s and nd=4: At t≃0.08 s, a nd nodal diameter forward traveling wave starts to propagate in the casing while the blade disk simultaneously reaches a purely static deformation state

Figure 8

Zone of interaction in the “Initial Pulse Amplitude—Rotational Velocity Ω” space

Figure 9

Contact forces λ1 and λ2 versus Ω

Figure 10

Frequency-domain amplitudes of both structures with respect to Ω

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