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Research Papers

Generalized Bilinear Amplitude Approximation and X-Xr for Modeling Cyclically Symmetric Structures With Cracks

[+] Author and Article Information
Meng-Hsuan Tien

Department of Mechanical and
Aerospace Engineering,
The Ohio State University,
Columbus, OH 43210
e-mail: tien.36@osu.edu

Tianyi Hu

Department of Mechanical and
Aerospace Engineering,
The Ohio State University,
Columbus, OH 43210
e-mail: hu.629@osu.edu

Kiran D'Souza

Department of Mechanical and
Aerospace Engineering,
The Ohio State University,
Columbus, OH 43210
e-mail: dsouza.60@osu.edu

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received September 7, 2017; final manuscript received February 5, 2018; published online March 14, 2018. Assoc. Editor: Julian Rimoli.

J. Vib. Acoust 140(4), 041012 (Mar 14, 2018) (10 pages) Paper No: VIB-17-1409; doi: 10.1115/1.4039296 History: Received September 07, 2017; Revised February 05, 2018

The analysis of the influence of cracks on the dynamics of bladed disks is critical for design, failure prognosis, and structural health monitoring. Predicting the dynamics of cracked bladed disks is computationally challenging for two reasons: (1) the model size is quite large and (2) the piecewise-linear nonlinearity caused by contact eliminates the use of linear analysis tools. Recently, a technique referred to as the X-Xr approach was developed to efficiently reduce the model size of the cracked bladed disks. The method employs relative coordinates to describe the motion of crack surfaces such that an effective model reduction can be achieved using single sector calculations. More recently, a method referred to as the generalized bilinear amplitude approximation (BAA) was developed to approximate the amplitude and frequency of piecewise-linear nonlinear systems. This paper modifies the generalized BAA method and combines it with the X-Xr approach to efficiently predict the dynamics of the cracked bladed disks. The combined method is able to construct the reduced-order model (ROM) of full disks using single-sector models only and estimate the amplitude and frequency with a significantly reduced computational effort. The proposed approach is demonstrated on a three degrees-of-freedom (DOF) spring–mass system and a cracked bladed disk.

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Figures

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Fig. 1

(a) A bladed disk with a cracked blade and (b) crack area and contact pairs

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Fig. 2

Conceptual structure of the mass matrix M¯, where M¯xh,i represents the pristine component of the ith sector. The damping matrix C¯ and stiffness matrix K¯ have the same structure.

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Fig. 3

(a) One steady-state vibration cycle. Note that x̃r is precisely 0 when the system is in state (1), and x̃r is greater than 0 when the system is in state (2). (b) The relationship between the two modal coordinate systems.

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Fig. 4

The overall computational process of the combined X-Xr–BAA method

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Fig. 5

(a) 3DOF system with contacting masses, (b) the system in its open state, and (c) the system in its sliding state

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Fig. 6

Comparison of amplitudes computed by the X-Xr–BAA ((−−) for m1, (···) for m2, and (−) for m3) and time integration ((+) for m1, (◯) for m2, and (◻) for m3)

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Fig. 7

(a) Finite element model of the bladed disk with a cracked blade and driving forces and (b) close-up of blade 1 with crack indicated

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Fig. 8

(a) The linear natural frequencies of the ROM and the original FE model ((+) for the ROM and (◯) for the original FE model). (b) The linear displacement amplitude at the tip node of the cracked blade with EO 1 excitation ((−) for the ROM and (◯) for the original FE model).

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Fig. 9

Linear displacement amplitudes of each mode at the tip node of the cracked blade: (a) responses of each open mode with EO 1 excitation, (b) responses of each sliding mode with EO 1 excitation, (c) responses of each open mode with EO 11 excitation, and (d) responses of each sliding mode with EO 11 excitation

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Fig. 10

Comparison of nonlinear vibration responses computed by the X-Xr–BAA (−) and time integration (◯) for the cases where the system is excited by (a) EO 1, (b) EO 5, and (c) EO 11. The linear vibration response of the system in its sliding state is denoted by (···). The linear vibration response of the system in its open state is denoted by (−−).

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