Chapter 26 Smooth Complex Orthogonal Decomposition Applied to Traveling Waves in Elastic Media Rickey A. Caldwell Jr. and Brain F. Feeny Abstract A novel method called smooth complex orthogonal decomposition (SCOD) was applied to a simulated infinite beam and an experimental beam that emulated a semi-infinite beam. The beam was instrumented with accelerometers, and the accelerations were numerically integrated to compute displacements and velocities. These measurements were converted into complex analytic displacements and velocities ensembles, which were used to compute two correlation matrices. These correlation matrices formed a complex generalized eigenvalue problem whose eigenvalues and eigenvectors led to the extractions of the frequencies and wavenumbers of the constituent waves of the traveling pulse. SCOD directly extracts the frequencies of the traveling waves from the eigenvalues. SCOD was able to extract the geometric relationship, phase velocity, and group velocity, and agrees with analytical predictions. Keywords SCOD • COD • Dispersion relationship • Wavenumber • Frequency 26.1 Introduction During the middle of the twentieth century various researchers independently developed proper orthogonal decomposition (POD) [1, 14–16] which was applied to statistics and turbulent fluid flows. POD later caught the interest of structural engineers and was used to extract mode shapes from vibrating structures [2, 11–13]. Other generalizations of POD were developed, such as the smooth orthogonal decomposition (SOD) for finding the natural frequencies and linear normal modes [4, 6], and the state variable modal decomposition (SVMD) [7] for finding natural frequencies, normal modes, and (in theory) modal damping. Soon after POD, was expanded to measuring traveling waves modes using complex orthogonal decomposition [8]. Complex orthogonal decomposition (COD) [9] can extract nonsynchronous and standing waves of vibrating structures. When waves are traveling through and elastic media COD can be used to extract the wavenumbers, frequencies, and the dispersion relationship of the waves. In the application of COD the wavenumber is extracted from the eigenvector, and the frequency is extracted from the modal coordinate. A new generalization of COD is outlined here called the smooth complex orthogonal decomposition (SCOD). It will be shown that with SCOD the wavenumber and frequency can be extracted from the eigenvector and eigenvalue of the SCOD eigenvalue problem (EVP). A mathematical development for SCOD will be covered in Sect. 26.2. SCOD will be applied to a simulated infinite beam in Sect. 26.3, and experimentally to a rectangular beam in Sect. 26.4. A primer for SOD is reviewed in the Appendix. The application of SCOD involves gathering state measurements. Accelerometers were used in this body of work. After the accelerations are measured the data is post-processed to derive velocity and displacement. The velocities are collected in a velocity ensemble, V, and displacements are organized in a ensemble matrix, X. The ensembles are organized so each sensor is allocated to a row such that all the data from the ith sensor is in the ith row. Measurement samples are then in each column. For example, Xij contains the ith sensor’s jth sample, for i D1; ; Mandj D1; ; N, where Mis the number of sensors and n is the number of time samples. Typically, the mean of each row is subtracted from each element of that row. The two ensembles are converted to analytic matrices, Zand Zv. Next, two correlation matrices are computed, one from the analytic displacements R D ZZT =N and R.A. Caldwell Jr. ( ) • B.F. Feeny Department of Mechanical Engineering, Michigan State University, 428 South Shaw Lane, RM East Lansing, RM 2328C East Lansing, MI 48824, USA e-mail: caldwe20@msu.edu; feeny@egr.msu.edu © The Society for Experimental Mechanics, Inc. 2016 J. De Clerck, D.S. Epp (eds.), Rotating Machinery, Hybrid Test Methods, Vibro-Acoustics & Laser Vibrometry, Volume 8, Conference Proceedings of the Society for Experimental Mechanics Series, DOI 10.1007/978-3-319-30084-9_26 281
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