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.

References

1.
Ciang
,
C. C.
,
Lee
,
J. R.
, and
Bang
,
H. J.
,
2008
, “
Structural Health Monitoring for a Wind Turbine System: A Review of Damage Detection Methods
,”
Meas. Sci. Technol.
,
19
(
12
), p.
122001
.10.1088/0957-0233/19/12/122001
2.
Mahajan
,
A. J.
,
Kaza
,
K. R. V.
, and
Dowell
,
E. H.
,
1993
, “
Semi-Empirical Model for Prediction of Unsteady Forces on an Airfoil With Application to Flutter
,”
J. Fluids Struct.
,
7
(
1
), pp.
87
103
.10.1006/jfls.1993.1007
3.
Janssens
,
K.
,
Gajdatsy
,
P.
,
Gielen
,
L.
,
Mas
,
P.
,
Britte
,
L.
,
Desmet
,
W.
, and
Auweraer
,
H. V.
,
2011
, “
OPAX: A New Transfer Path Analysis Method Based on Parametric Load Models
,”
Mech. Syst. Signal Process.
,
25
(
4
), pp.
1321
1338
.10.1016/j.ymssp.2010.10.014
4.
Qiao
,
B. J.
,
Zhao
,
T.
,
Chen
,
X. F.
, and
Liu
,
J. X.
, “
The Assessment of Active Vibration Isolation Performance of Rotating Machinery Using Power Flow and Vibrational Energy: Experimental Investigation
,”
Proc. Inst. Mech. Eng., Part C
(epub).10.1177/0954406215572434
5.
Uhl
,
T.
,
2007
, “
The Inverse Identification Problem and Its Technical Application
,”
Arch. Appl. Mech.
,
77
(
5
), pp.
325
337
.10.1007/s00419-006-0086-9
6.
Vogel
,
C. R.
,
2002
,
Computational Methods for Inverse Problems
,
SIAM
,
Philadelphia, PA
.
7.
Huang
,
C. H.
, and
Wang
,
S. P.
,
1999
, “
A Three-Dimensional Inverse Heat Conduction Problem in Estimating Surface Heat Flux by Conjugate Gradient Method
,”
Int. J. Heat Mass Transfer
,
42
(
18
), pp.
3387
3403
.10.1016/S0017-9310(99)00020-4
8.
Mojabi
,
P.
, and
LoVetri
,
J.
,
2009
, “
Enhancement of the Krylov Subspace Regularization for Microwave Biomedical Imaging
,”
Trans. IEEE Med. Imaging
,
28
(
12
), pp.
2015
2019
.10.1109/TMI.2009.2027703
9.
Hadamard
,
J.
,
1923
,
Lectures on the Cauchy Problem in Linear Partial Differential Equations
,
Yale University Press
,
New Haven, CT
.
10.
Mroczka
,
J.
, and
Szczuczyński
,
D.
,
2009
, “
Inverse Problems Formulated in Terms of First-Kind Fredholm Integral Equations in Indirect Measurements
,”
Metrol. Meas. Syst.
,
16
(
3
), pp.
333
357
.https://www.infona.pl/resource/bwmeta1.element.baztech-article-BSW1-0059-0001
11.
Sanchez
,
J.
, and
Benaroya
,
H.
,
2014
, “
Review of Force Reconstruction Techniques
,”
J. Sound Vib.
,
333
(
14
), pp.
2999
3018
.10.1016/j.jsv.2014.02.025
12.
Yu
,
L.
, and
Chan
,
T. H.
,
2003
, “
Moving Force Identification Based on the Frequency-Time Domain Method
,”
J. Sound Vib.
,
261
(
2
), pp.
329
349
.10.1016/S0022-460X(02)00991-4
13.
Liu
,
Y.
, and
Shepard
,
W. S.
, Jr.
,
2005
, “
Dynamic Force Identification Based on Enhanced Least Squares and Total Least-Squares Schemes in the Frequency Domain
,”
J. Sound Vib.
,
282
(
1
), pp.
37
60
.10.1016/j.jsv.2004.02.041
14.
Thite
,
A. N.
, and
Thompson
,
D. J.
,
2003
, “
The Quantification of Structure-Borne Transmission Paths by Inverse Methods, Part 1: Improved Singular Value Rejection Methods
,”
J. Sound Vib.
,
264
(
2
), pp.
411
431
.10.1016/S0022-460X(02)01202-6
15.
Thite
,
A. N.
, and
Thompson
,
D. J.
,
2003
, “
The Quantification of Structure-Borne Transmission Paths by Inverse Methods. Part 2: Use of Regularization Techniques
,”
J. Sound Vib.
,
264
(
8
), pp.
433
451
.10.1016/S0022-460X(02)01203-8
16.
Ewins
,
D. J.
,
2000
,
Modal Testing: Theory, Practice and Application
, 2nd ed.,
Research Studies Press
,
Baldock, UK
.
17.
Khoo
,
S. Y.
,
Ismail
,
Z.
,
Kong
,
K. K.
,
Ong
,
Z. C.
,
Noroozi
,
S.
,
Chong
,
W. T.
, and
Rahman
,
A. G. A.
,
2014
, “
Impact Force Identification With Pseudo-Inverse Method on a Lightweight Structure for Under-Determined, Even-Determined and Over-Determined Cases
,”
Int. J. Impact Eng.
,
63
, pp.
52
62
.10.1016/j.ijimpeng.2013.08.005
18.
Doyle
,
J. F.
,
1997
, “
A Wavelet Deconvolution Method for Impact Force Identification
,”
Exp. Mech.
,
37
(
4
), pp.
403
408
.10.1007/BF02317305
19.
Jacquelin
,
E.
,
Bennani
,
A.
, and
Hamelin
,
P.
,
2003
, “
Force Reconstruction: Analysis and Regularization of a Deconvolution Problem
,”
J. Sound Vib.
,
265
(
1
), pp.
81
107
.10.1016/S0022-460X(02)01441-4
20.
Wang
,
L. J.
,
Han
,
X.
,
Liu
,
J.
,
He
,
X. Q.
, and
Huang
,
F.
,
2011
, “
A New Regularization Method and Application to Dynamic Load Identification Problems
,”
Inverse Probl. Sci. Eng.
,
19
(
6
), pp.
765
776
.10.1080/17415977.2010.531468
21.
Lin
,
D. C.
,
2012
, “
Estimation Impulsive Loads in Duffing's Equation Using Two Methods
,”
ASME J. Vib. Acoust.
,
134
(
3
), p.
031001
.10.1115/1.4005653
22.
Liu
,
Y.
, and
Shepard
,
W. S.
, Jr.
,
2006
, “
An Improved Method for the Reconstruction of a Distributed Force Acting on a Vibrating Structure
,”
J. Sound Vib.
,
291
(
1
), pp.
369
387
10.1016/j.jsv.2005.06.013
23.
Hu
,
N.
,
Fukunagab
,
H.
,
Matsumotob
,
S.
,
Yana
,
B.
, and
Peng
,
X. H.
,
2007
, “
An Efficient Approach for Identifying Impact Force Using Embedded Piezoelectric Sensors
,”
Int. J. Impact Eng.
,
34
(
7
), pp.
1258
1271
.10.1016/j.ijimpeng.2006.05.004
24.
Gunawan
,
F. E.
, and
Homma
,
H.
,
2008
, “
A Solution of the Ill-Posed Impact-Force Inverse Problems by the Weighted Least Squares Method
,”
J. Solid Mech. Mater. Eng.
,
2
(
2
), pp.
188
198
.10.1299/jmmp.2.188
25.
Gunawan
,
F. E.
, and
Homma
,
H.
,
2008
, “
Impact-Force Estimation by Quadratic Spline Approximation
,”
J. Solid Mech. Mater. Eng.
,
2
(
8
), pp.
1092
1103
.10.1299/jmmp.2.1092
26.
Gunawan
,
F. E.
,
Homma
,
H.
, and
Kanto
,
Y.
,
2006
, “
Two-Step B-Splines Regularization Method for Solving an Ill-Posed Problem of Impact-Force Reconstruction
,”
J. Sound Vib.
,
297
(
1
), pp.
200
214
.10.1016/j.jsv.2006.03.036
27.
Li
,
Z.
,
Feng
,
Z. P.
, and
Chu
,
F. L.
,
2014
, “
A Load Identification Method Based on Wavelet Multi-Resolution Analysis
,”
J. Sound Vib.
,
333
(
2
), pp.
381
391
.10.1016/j.jsv.2013.09.026
28.
Qiao
,
B. J.
,
Zhang
,
X. W.
,
Luo
,
X. J.
, and
Chen
,
X. F.
,
2015
, “
A Force Identification Method Using Cubic B-Spline Scaling Functions
,”
J. Sound Vib.
,
337
, pp.
28
44
.10.1016/j.jsv.2014.09.038
29.
Hansen
,
P. C.
,
1992
, “
Analysis of Discrete Ill-Posed Problems by Means of the L-Curve
,”
SIAM Rev.
,
34
(
4
), pp.
561
580
.10.1137/1034115
30.
Choi
,
H. G.
,
Thite
,
A. N.
, and
Thompson
,
D. J.
,
2007
, “
Comparison of Methods for Parameter Selection in Tikhonov Regularization With Application to Inverse Force Determination
,”
J. Sound Vib.
,
304
(
3
), pp.
894
917
.10.1016/j.jsv.2007.03.040
31.
Ahmed
,
N.
,
Natarajan
,
T.
, and
Rao
,
K. R.
,
1974
, “
Discrete Cosine Transform
,”
IEEE Trans. Comput.
,
C-23
(
1
), pp.
90
93
.10.1109/T-C.1974.223784
32.
Hua
,
B.
, and
Lu
,
S.
,
2012
, “
Numerical Differentiation by a Tikhonov Regularization Method Based on the Discrete Cosine Transform
,”
Appl. Anal.
,
91
(
4
), pp.
719
736
.10.1080/00036811.2011.598862
33.
Hansen
,
P. C.
,
2002
, “
Deconvolution and Regularization With Toeplitz Matrices
,”
Numer. Algorithms
,
29
(
4
), pp.
323
378
.10.1023/A:1015222829062
34.
Hansen
,
P. C.
,
1998
,
Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion
,
SIAM
,
Philadelphia, PA
.
35.
Delves
,
L. M.
, and
Mohamed
,
J. L.
,
1985
,
Computational Methods for Integral Equations
,
Cambridge University Press
,
Cambridge, UK
.
36.
Maleknejad
,
K.
,
Sohrabi
,
S.
, and
Rostami
,
Y.
,
2007
, “
Numerical Solution of Nonlinear Volterra Integral Equations of the Second Kind by Using Chebyshev Polynomials
,”
Appl. Math. Comput.
,
188
(
1
), pp.
123
128
.10.1016/j.amc.2006.09.099
37.
Sun
,
R.
,
Chen
,
G.
,
He
,
H.
, and
Zhang
,
B.
,
2014
, “
The Impact Force Identification of Composite Stiffened Panels Under Material Uncertainty
,”
Finite Elem. Anal. Des.
,
81
, pp.
38
47
.10.1016/j.finel.2013.11.008
38.
Qiao
,
B. J.
,
Chen
,
X. F.
,
Xue
,
X. F.
,
Luo
,
X. J.
, and
Liu
,
R. N.
,
2015
, “
The Application of Cubic B-Spline Collocation Method in Impact Force Identification
,”
Mech. Syst. Signal Process.
(in press).10.1016/j.ymssp.2015.04.009
39.
Rashidinia
,
J.
,
Babolian
,
E.
, and
Mahmoodi
,
Z.
,
2011
, “
Spline Collocation for Fredholm Integral Equations
,”
Math. Sci.
,
5
(
2
), pp.
147
158
.http://mathscience.kiau.ac.ir/Content/Vol5No2/4.pdf
40.
Chesne
,
S.
,
Pezerat
,
C.
, and
Guyader
,
J. L.
,
2008
, “
Identification of Plate Boundary Forces From Measured Displacements
,”
ASME J. Vib. Acoust.
,
130
(
4
), p.
041006
.10.1115/1.2890398
41.
Boukria
,
Z.
,
Perrotin
,
P.
, and
Bennani
,
A.
,
2011
, “
Experimental Impact Force Location and Identification Using Inverse Problems: Application for a Circular Plate
,”
Int. J. Mech.
,
5
(
1
), pp.
48
55
.http://www.naun.org/multimedia/NAUN/mechanics/20-093.pdf
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