26 Smooth Complex Orthogonal Decomposition Applied to Traveling Waves in Elastic Media 293 using symmetry KM 1ˆ T D ˆ T ƒ M 1ˆ T DK 1 ˆ T ƒ take the inverse transpose of both sides of the equation and using symmetry MˆDKˆƒ 1 MˆƒD Kˆ: Therefore, ‰and ˆare related through Mas ˆDM‰: (26.23) This relationship will be important in mathematical development of SCOD. In summary, for linear undamped, symmetric, multimodal free vibration systems, the two relationships for the eigenvectors of the mass spring system and smooth orthogonal decomposition were just established to be ‰ Dˆ T and ‰DMˆ. Furthermore, the SOD eigenvalues approximate the squares of the modal frequencies. These relationship will be approximated in experiments, since agreement with the model is an idealization. References 1. Berkooz, G., Holmes, P., Lumley J.L.: The proper orthogonal decomposition in analysis of turbulent flows. Annu. Rev. Fluid Mech. 25(539–575), 137–146 (1967) 2. Brincker, R., Zhang, L., Andersen, P.: Modal identification of output-only systems using frequency domain decomposition. Smart Mater. Struct. 10, 441 (2001). doi:10.1088/0964-1726/10/3/303 3. Caldwell Jr., R.A., Feeny, B.F.: Characterizing wave behavior in a beam experiment by using complex orthogonal decomposition. ASME J Vib Acoust (2016). doi:10.1115/1.4633268 4. Chelidze, D., Zhou, W.: Smooth orthogonal decomposition-based vibration mode identification. J. Sound Vib. 292, 461–473 (2006) 5. Farooq, U., Feeny, B.F.: Smooth orthogonal decomposition for modal analysis of randomly excited systems. J. Sound Vib. 316(1–5), 137–146 (2008) 6. Farooq, U., Feeny, B.F.: Smooth orthogonal decomposition for randomly excited systems. J. Sound Vib. 316(3–5), 137–146 (2008) 7. Farooq, U., Feeny, B.F: An experimental investigation of a state-variable modal decomposition method for modal analysis. J. Vib. Acoust. 132(2), 021017 (8 pages) (2012) 8. Feeny, B.F.: A complex orthogonal decomposition for wave motion analysis. J. Sound Vib. 310(1–2), 77–90 (2008) 9. Feeny, B.F.: Complex modal decomposition for estimating wave properties in one-dimensional media. J. Vib. Acoust. 135(3), 031010 (2013) 10. Graff, K.F.: Wave Motion in Elastic Solids. Courier Dover Publications, New York (1975) 11. Han, S., Feeny, B.F.: Application of proper orthogonal decomposition to structural vibration analysis. Mech. Syst. Signal Process. 17(5), 989–1001 (2003) 12. Ibrahim, S.R., Mikulcik, E.C.: A time domain modal vibration test technique. Shock Vib. Bull. 34(4), 21–37 (1973) 13. Ibrahim, S.R., Mikulcik, E.C.: A method for the direct identification of vibration parameters from the free response. Shock Vib. Bull. 47(4), 183–198 (1977) 14. Karhunen, K.: Zur Spektral theorie Stochastischer Prozesse. Ann. Acad. Sci. Fenn. A. 37, 1–34 (1946) 15. Kosambi, D.: Statistics in function space. J. Indian Math. Soc. 7, 76–88 (1943) 16. Lumley, J.: Stochastic Tools in Turbulence. Academic, New York (1970) 17. Önsay, T., Haddow, A.G.: Wavelet transform analysis of transient wave-propagation in a dispersive medium. J. Acoust. Soc. Am. 95(3), 1441–1449 (1994)
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