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Research Papers

Efficient Nonlinear Vibration Analysis of the Forced Response of Rotating Cracked Blades

[+] Author and Article Information
Akira Saito

Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109-2125asakira@umich.edu

Matthew P. Castanier1

Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109-2125mpc@umich.edu

Christophe Pierre

Faculty of Engineering, McGill University, Montreal QC, H3A 2K6, Canadachristophe.pierre@mcgill.ca

Olivier Poudou2

Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109-2125poudou@umich.edu

1

Corresponding author.

2

Presently a Lead Engineer∕Technologist at GE Energy, 300 Garlington Rd., Greenville, SC 29615-4614.

J. Comput. Nonlinear Dynam 4(1), 011005 (Nov 11, 2008) (10 pages) doi:10.1115/1.3007908 History: Received August 08, 2007; Revised July 02, 2008; Published November 11, 2008

The efficient nonlinear vibration analysis of a rotating elastic structure with a crack is examined. In particular, the solution of the forced vibration response of a cracked turbine engine blade is investigated. Starting with a finite element model of the cracked system, the Craig–Bampton method of component mode synthesis is used to generate a reduced-order model that retains the nodes of the crack surfaces as physical degrees of freedom. The nonlinearity due to the intermittent contact of the crack surfaces, which is caused by the opening and closing of the crack during each vibration cycle, is modeled with a piecewise linear term in the equations of motion. Then, the efficient solution procedure for solving the resulting nonlinear equations of motion is presented. The approach employed in this study is a multiharmonic hybrid frequency∕time-domain technique, which is an extension of the traditional harmonic balance method. First, a simple beam model is used to perform a numerical validation by comparing the results of the new method to those from transient finite element analysis (FEA) with contact elements. It is found that the new method retains good accuracy relative to FEA while reducing the computational costs by several orders of magnitude. Second, a representative blade model is used to examine the effects of crack length and rotation speed on the resonant frequency response. Several issues related to the rotation are investigated, including geometry changes of the crack, shifts in resonant frequencies, and the existence of multiple solutions. For the cases considered, it is found that the nonlinear vibration response exhibits the jump phenomenon only when rotation is included in the model.

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Copyright © 2009 by American Society of Mechanical Engineers
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Figures

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Figure 1

(a) Normal vector at the ith node with Nei=4, (b) Definition of gap for the ith contact pair.

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Figure 2

Schematic of the cracked beam model used for the validation of the HFT method

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Figure 3

Finite element model of the beam

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Figure 4

Frequency response of the beam

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Figure 5

(a) FE model of the cracked blade under the static condition: (b) deformed equilibrium under the rotating condition

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Figure 6

Mode shapes of the cracked blade obtained by neglecting the nonlinear boundary condition: (a) sixth mode and (b) tenth mode. Thin lines show undeformed shapes.

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Figure 7

Frequency response under the (a) static condition and (b) rotating condition at 5000rpm for increasing values of ‖b‖=4.44N, 6.67N, 8.89N, 13.3N, 22.2N, and 44.4N

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Figure 8

Resonant frequency under the rotating condition versus amplitude of force

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Figure 9

Frequency response for ‖b‖=8.89N

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Figure 10

Frequency shift for the tenth mode: (a) resonant frequency versus crack length and (b) resonant frequency shift versus crack length

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Figure 11

Frequency shift for the sixth mode: (a) resonant frequency versus crack length and (b) resonant frequency shift versus crack length

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