The complex modes of an end-damped cantilevered beam are studied as an experimental example of a nonmodally damped continuous system. An eddy-current damper is applied, for its noncontact and linear properties, to the end of the beam, and is then characterized to obtain the effective damping coefficient. The state-variable modal decomposition (SVMD) is applied to extract the modes from the impact responses in the cantilevered beam experiments. Characteristics of the mode shapes and modal damping are examined for various values of the end-damper damping coefficient. The modal frequencies and mode shapes obtained from the experiments have a good consistency with the results of the finite element model. The variation of the modal damping ratio and modal nonsynchronicity with varying end-damper damping coefficient also follow the prediction of the model. Over the range of damping coefficients studied in the experiments, we observe a maximum damping ratio in the lowest underdamped mode, which correlates with the maximum modal nonsynchronicity. Complex orthogonal decomposition (COD) is applied in comparison to the modal identification results obtained from SVMD.

References

References
1.
Caughey
,
T. K.
, and
O'Kelly
,
M.
,
1965
, “
Classical Normal Modes in Damped Linear Systems
,”
ASME J. Appl. Mech.
,
32
(
3
), pp.
583
588
.
2.
Singh
,
R.
, and
Prater
,
G.
,
1989
, “
Complex Eigensolution for Longitudinally Vibrating Bars With a Viscously Damped Boundary
,”
J. Sound Vib.
,
133
(
2
), pp.
364
367
.
3.
Prater
,
G.
, and
Singh
,
R.
,
1990
, “
Eigenproblem Formulation, Solution and Interpretation for Non-Proportionally Damped Continuous Beams
,”
J. Sound Vib.
,
143
(
1
), pp.
125
142
.
4.
Hull
,
A. J.
,
1992
, “
A Closed Form Solution of a Longitudinal Bar With a Viscous Boundary Condition
,” Naval Undersea Warfare Center, Newport, RI, Technical Report No. 92-10022.
5.
Oliveto
,
G.
,
Santini
,
A.
, and
Tripodi
,
E.
,
1997
, “
Complex Modal Analysis of a Flexural Vibrating Beam With Viscous End Conditions
,”
J. Sound Vib.
,
200
(
3
), pp.
327
345
.
6.
Krenk
,
S.
,
2004
, “
Complex Modes and Frequencies in Damped Structural Vibrations
,”
J. Sound Vib.
,
270
(
4–5
), pp.
981
996
.
7.
Xing
,
X.
, and
Feeny
,
B. F.
,
2015
, “
Complex Modal Analysis of a Nonmodally Damped Continuous Beam
,”
ASME J. Vib. Acoust.
,
137
(
4
), p.
041006
.
8.
Sodano
,
H. A.
,
2005
, “
Development of Novel Eddy Current Dampers for the Suppression of Structural Vibrations
,”
Ph.D. thesis
, Virginia Polytechnic Institute and State University, Blacksburg, VA.http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.729.167&rep=rep1&type=pdf
9.
Feynman
,
R. P.
,
Leighton
,
R. B.
, and
Sands
,
M.
,
1977
,
The Feynman Lectures on Physics
,
Vol. 2, Addison-Wesley Publishing Company
, Boston, MA.
10.
Sodano
,
H. A.
,
Bae
,
J.
,
Inman
,
D. J.
, and
Belvin
,
W. K.
,
2005
, “
Concept and Model of Eddy Current Damper for Vibration Suppression of a Beam
,”
J. Sound Vib.
,
288
(4–5), pp.
1177
1196
.
11.
Davis
,
L.
, and
Reitz
,
J.
,
1971
, “
Eddy Currents in Finite Conducting Sheets
,”
J. Appl. Phys.
,
42
(
11
), pp.
4119
4127
.
12.
Wiederick
,
H.
,
Gauthier
,
N.
,
Campbell
,
D.
, and
Rochon
,
P.
,
1987
, “
Magnetic Braking: Simple Theory and Experiment
,”
Am. J. Phys.
,
55
(
6
), pp.
500
503
.
13.
Simeu
,
E.
, and
Georges
,
D.
,
1996
, “
Modeling and Control of an Eddy Current Brake
,”
Control Eng. Pract.
,
4
(
1
), pp.
19
26
.
14.
Fredrick
,
R.
, and
Darlow
,
M.
,
1994
, “
Operation of an Electromagnetic Eddy Current Damper With a Supercritical Shaft
,”
ASME J. Vib. Acoust.
,
116
(
4
), pp.
578
580
.
15.
Fung
,
R.
,
Sun
,
J.
, and
Hsu
,
S.
,
2002
, “
Vibration Control of the Rotating Flexible-Shaft/Multi-Flexible-Disk System With the Eddy-Current Damper
,”
ASME J. Vib. Acoust.
,
124
(
4
), pp.
519
526
.
16.
Juang
,
J.
, and
Pappa
,
R.
,
1985
, “
An Eigensystem Realization Algorithm for Modal Parameter Identification and Model Reduction
,”
J. Guid., Control, Dyn.
,
8
(
5
), pp.
620
627
.
17.
Ibrahim
,
S.
, and
Mikulcik
,
E.
,
1977
, “
A Method for the Direct Identification of Vibration Parameters From the Free Response
,”
Shock Vib. Bull.
,
47
(
4
), pp.
183
198
.
18.
Kerschen
,
G.
,
Poncelet
,
F.
, and
Golinval
,
J.
,
2007
, “
Physical Interpretation of Independent Component Analysis in Structural Dynamics
,”
Mech. Syst. Signal Process.
,
21
(
4
), pp.
1561
1575
.
19.
Poncelet
,
F.
,
Kerschen
,
G.
,
Golinval
,
J.
, and
Verhelst
,
D.
,
2007
, “
Output-Only Modal Analysis Using Blind Source Separation Techniques
,”
Mech. Syst. Signal Process.
,
21
(
6
), pp.
2335
2358
.
20.
Richardson
,
M.
, and
Formenti
,
D.
,
1982
, “
Parameter Estimation From Frequency Response Measurements Using Rational Fraction Polynomials
,”
International Modal Analysis Conference
(
IMAC
), Orlando, FL, Nov. 8–10.http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.458.3592&rep=rep1&type=pdf
21.
Shih
,
C.
,
Tsuei
,
Y.
,
Allemang
,
R.
, and
Brown
,
D.
,
1988
, “
Complex Mode Indication Function and Its Application to Spatial Domain Parameter Estimation
,”
Mech. Syst. Signal Process.
,
2
(
4
), pp.
367
377
.
22.
Brincker
,
R.
,
Zhang
,
L.
, and
Andersen
,
P.
,
2001
, “
Modal Identification of Output-Only Systems Using Frequency Domain Decomposition
,”
Smart Mater. Struct.
,
10
(
3
), pp.
441
445
.
23.
Farooq
,
U.
, and
Feeny
,
B. F.
,
2012
, “
An Experimental Investigation of State-Variable Modal Decomposition for Modal Analysis
,”
ASME J. Vib. Acoust.
,
134
(
2
), p.
021017
.
24.
Caldwell
,
R. A.
, Jr.
, and
Feeny
,
B. F.
,
2014
, “
Output-Only Modal Identification of a Nonuniform Beam by Using Decomposition Methods
,”
ASME J. Vib. Acoust.
,
136
(
4
), p.
041010
.
25.
Feeny
,
B. F.
,
2008
, “
A Complex Orthogonal Decomposition for Wave Motion Analysis
,”
J. Sound Vib.
,
310
(1–2), pp.
77
90
.
26.
Feeny
,
B. F.
,
2013
, “
Complex Modal Decomposition for Estimating Wave Properties in One-Dimensional Media
,”
ASME J. Vib. Acoust.
,
135
(
3
), p.
031010
.
27.
Feeny
,
B. F.
,
Sternberg
,
P. W.
,
Cronin
,
C. J.
, and
Coppola
,
C. A.
,
2013
, “
Complex Orthogonal Decomposition Applied to Nematode Posturing
,”
ASME J. Comput. Nonlinear Dyn.
,
8
(
4
), p.
041010
.
28.
Caldwell
,
R. A.
, Jr.
, and
Feeny
,
B. F.
,
2016
, “
Characterizing Wave Behavior in a Beam Experiment by Using Complex Orthogonal Decomposition
,”
ASME J. Vib. Acoust.
,
138
(
4
), p.
041007
.
29.
Bae
,
J. S.
,
Kwak
,
M. K.
, and
Inman
,
D. J.
,
2005
, “
Vibration Suppression of a Cantilever Beam Using Eddy Current Damper
,”
J. Sound Vib.
,
284
(3–5), pp.
805
824
.
30.
Farooq
,
U.
,
2009
, “
Orthogonal Decomposition Methods for Modal Analysis
,”
Ph.D. thesis
, Michigan State University, East Lansing, MI.https://search.proquest.com/docview/304948086
31.
Main
,
J. A.
, and
Krenk
,
S.
,
2005
, “
Efficiency and Tuning of Viscous Dampers on Discrete Systems
,”
J. Sound Vib.
,
286
(1–2), pp.
97
122
.
32.
Preumont
,
A.
,
2013
,
Twelve Lectures on Structural Dynamics
,
Springer
, Dordrecht,
The Netherlands
.
33.
Allemang
,
R. J.
,
2003
, “
The Modal Assurance Criterion—Twenty Years of Use and Abuse
,”
Sound Vib.
,
8
(1), pp.
14
20
.http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.582.30&rep=rep1&type=pdf
You do not currently have access to this content.