PSI - Issue 37

Arvid Trapp et al. / Procedia Structural Integrity 37 (2022) 622–631 Author name / Structural Integrity Procedia 00 (2019) 000 – 000 trispectrum 4 ( ) ( 1 , 2 , 3 ) displays the decomposition of the fourth-order central moment 4 over the three dimensional frequency space { 1 , 2 , 3 } , which for sampled series is restrained between the negative and the positive Nyquist-frequency . Because of the three frequency arguments and its function value, the trispectrum requires a multi-dimensional visualization. Hereby, the trispectrum is intersected by three planes = { 1 + 2 = 0 , 1 + 3 = 0 , 2 + 3 = 0 } holding essential properties (Picinbono, 1999), for which the sum of two of the three frequency arguments add up to zero. The entire set of individual – i.e. non-redundant – trispectral estimates can be found in the so-called primary zone which occupies a small prismatic part of the trispectrum and can be mirrored along symmetry planes (Peinelt, 1992) to fill the whole frequency space. Rivola and Collis (Rivola and White, 1999) use a three dimensional depiction of the trispectral space in which trispectral estimates are displayed as spheres whereby their value determines the spheres’ size and color. To care for suffi cient visibility, small-valued entries are disguised by a threshold. Their visualization approach provides an advantageous three-dimensional representation and is itself easily comprehensible. However, the drawback is that only high kurtosis values are emphasized and if there are many, the depiction becomes cluttered. Further, it is difficult to clearly identify the corresponding frequency arguments within the frequency set as this requires some skills to project them on each axis. Contributing to this effect is that the planes are not included. Trapp et al. (Trapp and Wolfsteiner, 2019) developed a way to represent the trispectrum by resolving the frequency set into two-dimensional slices. Therefore, the trispectrum is sliced along the 3 -axis and each slice is represented in the 1 , 2 -plane. Analogously to Collis ’ representation, high-valued entries can be distinguished from low-valued entries by their color. In contrast, it includes all entries of the trispectrum, independently of their values. This enables a detailed analysis of all estimates without requiring a threshold. However, this technique also resolves all symmetries and requires a lot of space when printed. For the work presented within this paper, a representation elaborating some of the previously mentioned advantages is implemented. For depicting trispectra in scientific publications the reviewed approaches are purposeful. Nevertheless, analyzing the non-Gaussian spectral decomposition of a time series can be facilitated by an appropriately programmed graphical user interface (GUI). This allows to simply adjust parameters such as the threshold, the scaling, symmetries and the choice between moment or cumulant spectra, but also to meaningfully compare different trispectra and transfer functions with another. Such a GUI was developed within the scope of this paper. It represents trispectral values by size and color within the three dimensional frequency space. Every data point stems from an averaged volume (cube) of 4 ( ) ( 1 , 2 , 3 ) with a bandwidth of Δ . The planes are included by dashed-line frames. An exemplary setting is shown in Figure 3 by comparing the primary zone of the HP and the CN load series. To highlight the deviations from a stationary Gaussian process, cumulant spectra are shown in Figure 4. A threshold (red bar in scale set at approx. 7% of the value range) is included to remove small values and error-noise from the visualization. Figure 3: Comparison of the primary zone of the (a) HP and (b) CN load series of same kurtosis and PSD, dashed frames depict planes 4 ( ) [ 2 4 ] 4 ( ) [ 2 4 ] (a) (b) 627 6