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Research Papers

A Novel Method for Force Identification Based on the Discrete Cosine Transform

[+] Author and Article Information
Baijie Qiao, Xinjie Luo, Xiaofeng Xue

The State Key Laboratory
for Manufacturing Systems Engineering,
Xi'an 710061, China
School of Mechanical Engineering,
Xi'an Jiaotong University,
Xi'an 710049, China

Xuefeng Chen

The State Key Laboratory
for Manufacturing Systems Engineering,
Xi'an 710061, China
School of Mechanical Engineering,
Xi'an Jiaotong University,
Xi'an 710049, China
e-mail: chenxf@mail.xjtu.edu.cn

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received January 20, 2014; final manuscript received May 5, 2015; published online June 15, 2015. Assoc. Editor: Carole Mei.

J. Vib. Acoust 137(5), 051012 (Oct 01, 2015) (15 pages) Paper No: VIB-14-1020; doi: 10.1115/1.4030616 History: Received January 20, 2014; Revised May 05, 2015; Online June 15, 2015

Force identification is a classical inverse problem, in which the measured data and the mathematical models of mechanical structures are used to determine the applied force. However, the identified force may seriously diverge from the true solution due to the unknown noise contaminating the measured data and the inverse of the ill-posed transfer matrix characterizing the mechanical structure. In this paper, a novel method based on the discrete cosine transform (DCT) in the time domain is proposed for force identification, which overcomes the deficiency of the ill-posedness of the transfer matrix. The unknown force is expanded by a set of cosine basis functions and then the original governing equation is reformulated to find the coefficient of each cosine basis function. Furthermore, a modified generalized cross-validation (GCV) criterion for determining the regularization parameter is developed for the truncated singular value decomposition (TSVD), Chebyshev polynomial, and DCT solutions. Numerical simulation reveals that compared with the L-curve criterion, the modified GCV criterion is quite robust in the presence of noise. Finally, a clamped-free shell structure that is excited by an impact hammer is selected as an example to validate the performance of the proposed method. Experimental results demonstrate that compared with the TSVD-based and Chebyshev-based methods, the DCT-based method combined with the modified GCV criterion can reconstruct the force time history and identify the peak of impact force with high accuracy. Additionally, the identification of force location using the DCT-based method is also discussed.

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References

Figures

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Fig. 1

The first eight right singular vectors vi for an SDOF system with N=256

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Fig. 2

The first eight Chebyshev polynomials Ck(t) with N=256

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Fig. 3

The first eight orthonormal basis functions φi of the DCT with N=256

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Fig. 4

The solutions to the SDOF model with data length 512 and SNR = 50 dB: (a) the true and MI solutions and (b) the true, DCT, Chebyshev, and TSVD solutions

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Fig. 5

The RPE values of three solutions with data length 512 and SNR = 50 dB: (a) the DCT and Chebyshev solutions along with the number of basis functions and (b) the TSVD solution along with the number of singular values

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Fig. 6

The solutions to the SDOF model with data length 512: (a) SNR = 60 dB, (b) SNR = 70 dB, and (c) SNR = 80 dB

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Fig. 7

The solutions to the SDOF model with data length 256: (a) SNR = 60 dB, (b) SNR = 70 dB, and (c) SNR = 80 dB

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Fig. 8

Three criteria for choosing the regularization parameter of the DCT-based method with data length 512 and SNR = 50 dB: (a) modified GCV criterion, (b) L-curve criterion, and (c) RPE criterion

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Fig. 9

Three criteria for choosing the regularization parameter of the DCT-based method with data length 512 and SNR = 90 dB: (a) modified GCV criterion, (b) L-curve criterion, and (c) RPE criterion

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Fig. 10

The clamped-free shell structure applied for impact force identification: (a) the experimental setup and (b) the schema of the placement of accelerometers and exciting positions

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Fig. 11

The flow chart of force identification based on the DCT and TSVD methods

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Fig. 12

Case (F1, R1): the measured data from the force transducer and accelerometer—(a) the impact force and (b) the acceleration response

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Fig. 13

Case (F1, R1): the modified GCV criterion for choosing the optimal parameter—(a) the DCT-based and Chebyshev-based methods and (b) the TSVD-based method

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Fig. 14

Case (F1, R1): the measured force, the DCT solution with M=1060, the Chebyshev solution with M=1151, and the TSVD solution with K=1284—(a) the identified impact forces in the time domain and (b) zoomed-in impact force

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Fig. 15

Case (F2, R2): the measured data from the force transducer and accelerometer—(a) the impact force and (b) the acceleration response

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Fig. 16

Case (F2, R2): the modified GCV criterion for choosing the optimal parameter—(a) the DCT-based and Chebyshev-based methods and (b) the TSVD-based method

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Fig. 17

Case (F2, R2): the measured force, the DCT solution with M=835, the Chebyshev solution with M=1171, and the TSVD solution with K=1145—(a) the identified impact force in the time domain and (b) zoomed-in impact force

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Fig. 18

The location identification of unknown forces: (a) the unknown impact force acting on F1 and (b) the unknown impact force acting on F2

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