Reducing spatial data using an optimized regressive discrete Fourier series

Abstract

With the development of optical measurement techniques it is possible to obtain vast amounts of data. In vibrometry applications in particular operational deflection shapes are often obtained with very high spatial resolution. Fortunately, many techniques exist to reduce (approximate) the measurement data. One of the most common techniques for evaluating optical measurement data is by means of a Fourier analysis. However, this technique suffers from what is known as leakage when a non-integer number of periods is considered. This gives rise to non-negligible errors, which will obviously hamper the accuracy of the synthesized shape. Another technique such as a discrete cosine transform, used in the widely spread -jpeg standard does not suffer these anomalies but can still prove erroneous at times. One of the more recent approaches is via a so-called regressive discrete Fourier series (introduced by Arruda) which suffers one disadvantage. The problem statement is nonlinear in the parameters and needs a priori information about the deflection shape. This can be resolved by using the optimized regressive discrete fourier series (ORDFS), introduced in this article, which uses a nonlinear least squares approach. In this article the method will be applied in particular to the reduction of data for laser vibrometer measurements performed on an inorganic phosphate cement (IPC) beam (1D), as well as on a car door (2D). The proposed technique will also be validated on simulations to illustrate the properties concerning compression ratio and synthesized mode shape error. The introduced method will be bench marked for compression ratio and synthesized deflection shape error with all prior mentioned techniques.