Complex orthogonal decomposition (COD) was used to determine the extracted dispersion relationship of a traveling wave in a mass chain. When COD extracts a wavenumber it will produce M values for each wavenumber, γi, and N values for each frequency, ωi; where M is the number of masses and N is the number of time samples. In this work, least squares and a simple mean of the M-γi’s and N-ωi’s extracted values were used to determine each γi and ωi, respectively. An analytical dispersion relationship for the mass-chain is derived in addition to an approximate dispersion relationship. The approximate derivation was found using Lindstedt-Poincare’s perturbation method. Lastly, the effects of the sampling rate on parameter´ extraction was studied. COD could accurately extract the wavenumber and frequency of a traveling wave in the mass chain. Using a simple mean provided marginally better results than that of least squares. Sampling at the Nyquist criterion gave accurate results which improved both marginally and asymptotically as the sampling rate increased.
Valente, Nicholas A. and Caldwell, Rickey A.
"Using Complex Orthogonal Decomposition to Extract Dispersion Relationships for Mass Chain,"
Across the Bridge: The Merrimack Undergraduate Research Journal: Vol. 1
, Article 9.
Available at: https://scholarworks.merrimack.edu/atb/vol1/iss1/9