Research Papers

Characterizing Wave Behavior in a Beam Experiment by Using Complex Orthogonal Decomposition

[+] Author and Article Information
Rickey A. Caldwell, Jr.

Department of Mechanical Engineering,
Michigan State University,
East Lansing, MI 48824
e-mail: caldwe20@msu.edu

Brian F. Feeny

Fellow ASME
Department of Mechanical Engineering,
Michigan State University,
East Lansing, MI 48824
e-mail: feeny@egr.msu.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 November 11, 2014; final manuscript received March 19, 2016; published online May 23, 2016. Assoc. Editor: Mahmoud Hussein.

J. Vib. Acoust 138(4), 041007 (May 23, 2016) (7 pages) Paper No: VIB-14-1436; doi: 10.1115/1.4033268 History: Received November 11, 2014; Revised March 19, 2016

Complex orthogonal decomposition (COD) is applied to an experimental beam to extract the dispersive wave properties from response measurements. The beam is made of steel and is rectangular with a constant cross section. One end of the beam is free and is hung by a soft elastic cord. An impulse is applied to the free-end. The other end is buried in sand to absorb the wave as it travels from the impact site on the free-end; this effectively prevents reflections of the wave off the buried end and emulates a semi-infinite beam. The beam response is measured with an array of accelerometers, whose signals are integrated to obtain an ensemble of displacement signals. Acceleration responses are also compared in the frequency domain to predictions from the Euler–Bernoulli model. COD is applied to the displacement ensemble to obtain complex modal vectors and associated complex modal coordinates (COCs). The spatial whirl rates of nearly harmonic modal vectors are used to extract the modal wave numbers, and the temporal whirl rates of the modal coordinates are used to estimate the modal frequencies. The dispersion relationship between the frequencies and wave numbers compare favorably to those of the theoretical infinite Euler–Bernoulli beam.

Copyright © 2016 by ASME
Your Session has timed out. Please sign back in to continue.


Feeny, B. F. , 2008, “ A Complex Orthogonal Decomposition for Wave Motion Analysis,” J. Sound Vib., 310(1–2), pp. 77–90. [CrossRef]
Lumley, J. , 1970, Stochastic Tools in Turbulence, Academic Press, New York.
Berkooz, G. , Holmes, P. , and Lumley, J. L. , 1993, “ The Proper Orthogonal Decomposition in the Analysis of Turbulent Flows,” Ann. Rev. Fluid Mech., 25(1), pp. 539–575. [CrossRef]
Cusumano, J. C. , and Bai, B. , 1993, “ Period-Infinity Periodic Motion, Chaos, and Spatial Coherence in a 10-Degree-of-Freedom Impact Oscillator,” Chaos, Solitons Fractals, 3(5), pp. 515–535. [CrossRef]
FitzSimons, P. , and Rui, C. , 1993, “ Determining Low Dimensional Models of Distributed Systems,” Advances in Robust and Nonlinear Control Systems, Vol. DSC-53, pp. 9–15.
Azeez, M. F. A. , and Vakakis, A. F. , 2001, “ Proper Orthogonal Decomposition (POD) of a Class of Vibroimpact Oscillations,” J. Sound Vib., 240(5), pp. 859–889. [CrossRef]
Lee, E.-T. , and Eun, H.-C. , 2015, “ Damage Identification Based on the Proper Orthogonal Mode Energy Curvature,” ASME J. Vib. Acoust., 137(4), p. 041018. [CrossRef]
Feeny, B. F. , and Kappagantu, R. , 1998, “ On the Physical Interpretation of Proper Orthogonal Modes in Vibrations,” J. Sound Vib., 211(4), pp. 607–616. [CrossRef]
Riaz, M. S. , and Feeny, B. F. , 2003, “ Proper Orthogonal Decomposition of a Beam Sensed With Strain Gages,” ASME J. Vib. Acoust., 125(1), pp. 129–131. [CrossRef]
Kerschen, G. , and Golinval, J. C. , 2002, “ Physical Interpretation of the Proper Orthogonal Modes Using the Singular Value Decomposition,” J. Sound Vib., 249(5), pp. 849–865. [CrossRef]
Feeny, B. F. , 2002, “ On Proper Orthogonal Coordinates as Indicators of Modal Activity,” J. Sound Vib., 255(5), pp. 805–817. [CrossRef]
Biglieri, E. , and Yao, K. , 1989, “ Some Properties of Singular Value Decomposition and Their Applications to Digital Signal Processing,” Signal Process., 18(3), pp. 277–289. [CrossRef]
Liu, W. , Gao, W. , and Sun, Y. , 2009, “ Application of Modal Identification Methods to Spatial Structure Using Field Measurement Data,” ASME J. Vib. Acoust., 131(3), p. 034503. [CrossRef]
Liang, Y. C. , Lee, H. P. , Lim, S. P. , Lin, W. Z. , Lee, K. H. , and Wu, C. G. , 2002, “ Proper Orthogonal Decomposition and Its Applications—Part 1: Theory,” J. Sound Vib., 252(3), pp. 527–544. [CrossRef]
Yadalam, V. K. , and Feeny, B. F. , 2011, “ Reduced Mass Weighted Proper Decomposition for Modal Analysis,” ASME J. Vib. Acoust., 133(2), p. 024504. [CrossRef]
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. [CrossRef]
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.
Chelidze, D. , and Zhou, W. , 2006, “ Smooth Orthogonal Decomposition-Based Vibration Mode Identification,” J. Sound Vib., 292(3–5), pp. 461–473. [CrossRef]
Farooq, U. , and Feeny, B. F. , 2008, “ Smooth Orthogonal Decomposition for Modal Analysis of Randomly Excited Systems,” J. Sound Vib., 316(1–5), pp. 137–146. [CrossRef]
Farooq, U. , and Feeny, B. F. , 2012, “ An Experimental Investigation of a State-Variable Modal Decomposition Method for Modal Analysis,” ASME J. Vib. Acoust., 132(2), p. 021017. [CrossRef]
Oppenheim, A. V. , and Schafer, R. W. , 1989, Discrete-Time Signal Processing, Prentice Hall, Englewood Cliffs, NJ.
Schmidt, R. O. , 1986, “ Multiple Emitter Location and Signal Parameter Estimation,” IEEE Trans. Antennas Propag., 34(3), pp. 276–280. [CrossRef]
Roy, R. , and Kailath, T. , 1989, “ ESPRIT—Estimation of Signal Parameters Via Rotational Invariance Techniques,” IEEE Trans. Acoust., Speech Signal Process., 37(7), pp. 984–995. [CrossRef]
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. [CrossRef]
Feeny, B. F. , and Feeny, A. K. , 2013, “ Complex Modal Analysis of the Swimming Motion of a Whiting,” ASME J. Vib. Acoust., 135(2), p. 021004. [CrossRef]
Feeny, B. F. , 2013, “ Complex Modal Decomposition for Estimating Wave Properties in One-Dimensional Media,” ASME J. Vib. Acoust., 135(3), p. 031010. [CrossRef]
Lee, H. S. , and Kwon, S. H. , 2003, “ Wave Profile Measurement by Wavelet Transform,” Ocean Eng., 30(18), pp. 2313–2328. [CrossRef]
McDaniel, J. G. , and Shepard, W. S. , 2000, “ Estimation of Structural Wave Numbers From Spatially Sparse Response Measurements,” J. Acoust. Soc. Am., 108(4), pp. 1674–1682. [CrossRef] [PubMed]
Grosh, K. , and Williams, E. G. , 1993, “ Complex Wavenumber Decomposition of Structural Vibration,” J. Acoust. Soc. Am., 93(2), pp. 836–848. [CrossRef]
Hull, A. J. , and Hurdis, D. A. , 2003, “ A Parameter Estimation Method for the Flexural Wave Properties of a Beam,” J. Sound Vib., 262(2), pp. 187–197. [CrossRef]
Manktelow, K. L. , Leamy, M. J. , and Ruzzene, M. , 2014, “ Analysis and Experimental Estimation of Nonlinear Dispersion in a Periodic String,” ASME J. Vib. Acoust., 136(3), p. 031016. [CrossRef]
Graff, K. F. , 1975, Wave Motion in Elastic Solids, Courier Dover Publications, Columbus, OH.
Büssow, R. , 2008, “ Applications of the Flexural Impulse Response Functions in the Time Domain,” Acta Acust. Acust., 94(2), pp. 207–214. [CrossRef]
Önsay, T. , and Haddow, A. G. , 1994, “ Wavelet Transform Analysis of Transient Wave-Propagation in a Dispersive Medium,” J. Acoust. Soc. Am., 95(3), pp. 1441–1449. [CrossRef]
Beck, J. , and Arnold, K. , 1977, Parameter Identification in Engineering and Science, Wiley, New York.


Grahic Jump Location
Fig. 1

Schematic diagram of the experimental setup: top view

Grahic Jump Location
Fig. 2

Acceleration data versus position along the beam

Grahic Jump Location
Fig. 3

Accelerations from sensors 1, 16, and 31

Grahic Jump Location
Fig. 4

Displacements from sensors 1, 16, and 31

Grahic Jump Location
Fig. 5

FFT of sensors: (a) sensor 1, (b) sensor 16, and (c) sensor 31

Grahic Jump Location
Fig. 6

FFT of the modal impact hammer signal

Grahic Jump Location
Fig. 7

Selected COMs. Top left: second modal COM. Top right: sixth COM which is less circular than second. This trends continues and is illustrated in the lower left: seventh COM. Lower right: tenth COM.

Grahic Jump Location
Fig. 8

Selected COCs. Top left: second modal COC. Top right: sixth COC which shows attenuation after some initial oscillations. This trends continues and is illustrated in the lower left: seventh COC. Lower right: tenth COC.

Grahic Jump Location
Fig. 9

Real part of complex orthogonal coordinate number 4

Grahic Jump Location
Fig. 10

Real part of complex orthogonal coordinate number 8

Grahic Jump Location
Fig. 11

Experimental results. Theoretical dispersion relationship for the Euler–Bernoulli beam (solid line). COD extracted data points (○).

Grahic Jump Location
Fig. 14

COD extracted modal amplitude versus frequency (○) compared to theory



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In