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Research Papers

Efficient and Accurate Calculation of Discrete Frequency Response Functions and Impulse Response Functions

[+] Author and Article Information
Y. F. Xu

Department of Mechanical Engineering,
University of Maryland, Baltimore County,
1000 Hilltop Circle,
Baltimore, MD 21250
e-mail: yxu2@umbc.edu

W. D. Zhu

Professor
Fellow ASME
Division of Dynamics and Control,
School of Astronautics,
Harbin Institute of Technology,
P.O. Box 137,
Harbin 150001, China;
Department of Mechanical Engineering,
University of Maryland, Baltimore County,
1000 Hilltop Circle,
Baltimore, MD 21250
e-mail: wzhu@umbc.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 January 26, 2015; final manuscript received October 9, 2015; published online March 21, 2016. Assoc. Editor: Lei Zuo.

J. Vib. Acoust 138(3), 031003 (Mar 21, 2016) (17 pages) Paper No: VIB-15-1033; doi: 10.1115/1.4031998 History: Received January 26, 2015; Revised October 09, 2015

Modal properties of a structure can be identified by experimental modal analysis (EMA). Discrete frequency response functions (FRFs) and impulse response functions (IRFs) between response and excitation series are bases for EMA. In the calculation of a discrete FRF, the discrete Fourier transform (DFT) is applied to both response and excitation series, and a transformed series in the DFT is virtually extended to have an infinite length and be periodic with a period equal to the length of the series; the resulting periodicity can be physically incorrect in some cases, which depends on an excitation technique used. An efficient and accurate methodology for calculating discrete FRFs and IRFs is proposed here, by which fewer spectral lines are needed and accuracies of resulting FRFs and IRFs can be maintained. The relationship between an IRF from the proposed methodology and that from the least-squares (LS) method is shown. A coherence function extended from a new type of coherence functions is used to evaluate qualities of FRFs and IRFs from the proposed methodology in the frequency domain. The extended coherence function can yield meaningful values even with response and excitation series of one sampling period. Based on the extended coherence function, a fitting index is used to evaluate overall qualities of the FRFs and IRFs. The proposed methodology was numerically and experimentally applied to a two degrees-of-freedom (2DOF) mass–spring–damper system and an aluminum plate to estimate their FRFs and IRFs, respectively. In the numerical example, FRFs and IRFs from the proposed methodology agree well with theoretical ones. In the experimental example, an FRF and its associated IRF from the proposed methodology with a random impact series agreed well with benchmark ones from a single impact test.

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Figures

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Fig. 1

A 2DOF mass–spring–damper system

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Fig. 2

(a) Random impact series, (b) the response of m1 in Fig. 1 of one sampling period, (c) the pseudoperiodic excitation series, and (d) the pseudoperiodic response series of m1 in Fig.1

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Fig. 3

(a) Comparison of amplitudes of the analytical H1,1(s) (analytical), H1,1(s) in Eq. (8) using y1(t) and f(t) of the whole sampling period (complete), H1,1(s) in Eq. (11) using y1(t) and f(t) of the first three subsampling periods (averaged), and H1,1(s) from the proposed methodology (proposed); (b) comparison of their phase angles; (c) an enlarged view of amplitudes of the above four H1,1(s) in the neighborhood of the first natural frequency of the system; and (d) an enlarged view of amplitudes of the four H1,1(s) in the neighborhood of the second natural frequency of the system

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Fig. 4

(a) Conventional coherence function associated with H1,1(s) in Eq. (11) using y1(t) and f(t) of the first three subsampling periods and (b) the extended coherence function associated with H1,1(s) from the proposed methodology

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Fig. 5

(a) Comparison of IRFs associated with the analytical H1,1(s) (analytical), H1,1(s) in Eq. (11) using y1(t) and f(t) of the whole sampling period (complete), H1,1(s) in Eq. (11) using y1(t) and f(t) of the first three subsampling periods (averaged), and H1,1(s) from the proposed methodology (proposed); (b) an enlarged view of the IRFs in the first 0.05 s; and (c) an enlarged view of the IRFs between t = 7.2 s and t = 8 s

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Fig. 6

Enlarged views of the actual y1(t) (actual) and calculated ones by convolution between f(t) and IRFs associated with H1,1(s) in Eq. (8) using y1(t) and f(t) of the whole sampling period (complete), H1,1(s) in Eq. (11) using y1(t) and f(t) of the first three subsampling periods (averaged), and H1,1(s) from the proposed methodology (proposed) in different time intervals: (a) between t = 10.3 s and t = 10.9 s, (b) between t = 15.7 s and t = 16.3 s, and (c) between t = 28 s and t = 32 s

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Fig. 7

(a) Zero-mean white-noise excitation, (b) the response of m1 in Fig. 1 of five sampling periods, (c) the pseudoperiodic excitation of the first sampling period, and (d) the pseudoperiodic response of m1 in Fig. 1 of the first sampling period

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Fig. 8

(a) Comparison of amplitudes of the analytical H1,1(s) (analytical), H1,1(s) in Eq. (11) using y1(t) and f(t) of the first sampling period (complete), and H1,1(s) from the proposed methodology using y1(t) and f(t) of the first sampling period (proposed); (b) comparison of their phase angles; (c) an enlarged view of amplitudes of the above three H1,1(s) in the neighborhood of the first natural frequency of the system; and (d) an enlarged view of amplitudes of the three H1,1(s) in the neighborhood of the second natural frequency of the system

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Fig. 9

(a) Conventional coherence function associated with H1,1(s) in Eq. (11) using y1(t) and f(t) of the first sampling period and (b) the extended coherence function of H1,1(s) from the proposed methodology using y1(t) and f(t) of the first sampling period

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Fig. 10

(a) Comparison of amplitudes of the analytical H1,1(s) (analytical), H1,1(s) in Eq. (11) using y1(t) and f(t) of the first two sampling periods (complete), and H1,1(s) from the proposed methodology using y1(t) and f(t) of the first two sampling periods (proposed) and (b) comparison of their phase angles

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Fig. 11

(a) Comparison of amplitudes of the analytical H1,1(s) (analytical), H1,1(s) in Eq. (11) using y1(t) and f(t) of the first five sampling periods (complete), and H1,1(s) from the proposed methodology using y1(t) and f(t) of the first five sampling periods (proposed) and (b) comparison of their phase angles

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Fig. 12

(a) Conventional coherence function associated with H1,1(s) in Eq. (11) using y1(t) and f(t) of the first two sampling periods and (b) the extended coherence function of H1,1(s) from the proposed methodology using y1(t) and f(t) of the first two sampling periods

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Fig. 13

(a) Conventional coherence function associated with H1,1(s) in Eq. (11) using y1(t) and f(t) of the first five sampling periods and (b) the extended coherence function of H1,1(s) from the proposed methodology in Eq. (19) using y1(t) and f(t) of the first five sampling periods

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Fig. 14

(a) Comparison of IRFs associated with the analytical H1,1(s) (analytical), H1,1(s) in Eq. (11) using y1(t) and f(t) of the first sampling period (complete), and H1,1(s) from the proposed methodology using y1(t) and f(t) of the first sampling period (proposed); (b) an enlarged view of the IRFs in the first 0.05 s; and (c) an enlarged view of the IRFs between t = 7.2 s and t = 8 s

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Fig. 15

(a) Comparison of IRFs associated with the analytical H1,1(s) (analytical), H1,1(s) in Eq. (11) using y1(t) and f(t) of the first five sampling periods (complete), and H1,1(s) from the proposed methodology using y1(t) and f(t) of the first five sampling periods (proposed); (b) an enlarged view of the IRFs in the first 0.05 s; and (c) an enlarged view of the IRFs between t = 7.2 s and t = 8 s

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Fig. 16

Test setup of EMA on an aluminum plate

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Fig. 17

(a) Random impact series manually generated at the excitation point on the plate inFig. 16, (b) the response of the measurement point on the plate in Fig. 16, (c) the pseudoperiodic excitation of the first sampling period, and (d) the pseudoperiodic response of the measurement point of the first sampling period

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Fig. 18

(a) Comparison of amplitudes of an FRF from a single impact test (benchmark), that in Eq.(11) using z¨(t) and f(t) of the first sampling period (complete), and that from the proposed methodology using z¨(t) and f(t) of the first sampling period (proposed); (b) comparison of their phase angles; (c) an enlarged view of amplitudes of the FRFs between 75 and 76 Hz; and (d) an enlarged view of amplitudes of the FRFs between 245 and 246 Hz

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Fig. 19

(a) Comparison of amplitudes of an FRF from a single impact test (benchmark), that in Eq.(11) using z¨(t) and f(t) of the first two sampling periods (complete), and that from the proposed methodology using z¨(t) and f(t) of the first two sampling periods (proposed); (b) comparison of their phase angles; (c) an enlarged view of amplitudes of the FRFs between 75 and 76 Hz, and (d) an enlarged view of amplitudes of the FRFs between 245 and 246 Hz

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Fig. 20

(a) Comparison of amplitudes of an FRF from a single impact test (benchmark), that in Eq.(11) using z¨(t) and f(t) of the first five sampling periods (complete), and that from the proposed methodology using z¨(t) and f(t) of the first five sampling periods (proposed); (b) comparison of their phase angles; (c) an enlarged view of amplitudes of the FRFs between 75 and 76 Hz, and (d) an enlarged view of amplitudes of the FRFs between 245 and 246 Hz

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Fig. 21

(a) Conventional coherence function associated with the FRF in Eq. (11) using z¨(t) and f(t) of the first sampling period and (b) the extended coherence function associated with the FRF from the proposed methodology using z¨(t) and f(t) of the first sampling period

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Fig. 22

(a) Conventional coherence function associated with the FRF in Eq. (11) using z¨(t) and f(t) of the first two sampling periods and (b) the extended coherence function associated with the FRF from the proposed methodology using z¨(t) and f(t) of the first two sampling periods

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Fig. 23

(a) Conventional coherence function associated with the FRF in Eq. (11) using z¨(t) and f(t) of the first five sampling periods and (b) the extended coherence function associated with the FRF from the proposed methodology using z¨(t) and f(t) of the first five sampling periods

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Fig. 24

(a) Comparison of IRFs associated with an FRF from a single impact test (benchmark), that in Eq. (11) using z¨(t) and f(t) of the first sampling period (complete), and that from the proposed methodology using z¨(t) and f(t) of the first sampling period (proposed); (b) an enlarged view of the IRFs between t = 3.9 s and t = 3.94 s; and (c) an enlarged view of the IRFs between t = 25.96 s and t = 26 s

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Fig. 25

(a) Comparison of IRFs associated with an FRF from a single impact test (benchmark), that in Eq. (11) using z¨(t) and f(t) of the first five sampling periods (complete), and that from the proposed methodology using z¨(t) and f(t) of the first five sampling periods (proposed); (b) an enlarged view of the IRFs between t = 3.9 s and t = 3.94 s; and (c) an enlarged view of the IRFs between t = 25.96 s and t = 26 s

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Fig. 26

Enlarged views of the measured z¨(t) (measured) and calculated ones by convolution between f(t) and IRFs associated with the FRFs in Eq. (11) using z¨(t) and f(t) of the first five sampling periods (complete) and from the proposed methodology using z¨(t) and f(t) of the first five sampling periods (proposed) in different time intervals: (a) between t = 43.96 s and t = 44.05 s, (b) between t = 220.21 s and t = 220.30 s, and (c) between t = 257.68 s and t = 257.71 s

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