Conditions for the convergence of proper orthogonal modes to linear vibration modes
Considerable attention has been given to the use of proper orthogonal decomposition (POD) as a tool for modal analysis. Much of this work hinges on early theoretical developments that conclude that the resulting proper orthogonal modes (POMs) converge to the linear vibration modes (LVMs) as the number of samples increases. The purpose of the present work is to clarify the conditions under which the POMs of a linear, undamped, and unforced mechanical system converge to the system's LVMs. To that end, it presents a rigorous proof of convergence that demonstrates that increasing the number of samples (i.e., the sampling rate) alone is insufficient; instead, convergence is shown to require increasing both the sampling rate and the duration of the sampling.
© 2023 Elsevier. The author was supported by a NASA Space Technology Research Fellowship . He thanks Charles J. Dorn and J. Nelson Wright for helpful discussions and reviewing drafts of this manuscript. Data availability. Data will be made available on request.