Cycle Range Distributions for Gaussian Processes Exact and Approximative Results

Abstract

Wave cycles, i.e., pairs of local maxima and minima, play an important role in many engineering fields. Many cycle definitions are used for specific purposes, such as crest-trough cycles in wave studies in ocean engineering and rainflow cycles for fatigue life predicition in mechanical engineering. The simplest cycle, that of a pair of local maximum and the following local minimum is also of interest as a basis for the study of more complicated cycles. This paper presents and illustrates modern computational tools for the analysis of different cycle distributions for stationary Gaussian processes with general spectrum. It is shown that numerically exact but slow methods will produce distributions in almost complete agreement with simulated data, but also that approximate and quick methods work well in most cases.Of special interest is the dependence relation between the cycle average and the cycle range for the simple maximumYminimum cycle and its implication for the range distribution. It is observed that for a Gaussian process with rectangular box spectrum, these quantities are almost independent and that the range is not far from a Rayleigh distribution. It will also be shown that had there been a Gaussian process where exact independence hold then the range would have had an exact Rayleigh distribution. Unfortunately no such Gaussian process exists.

Slepian models for the stochastic shape of individual Lagrange sea waves

“…in wave analysis; see [11]. The technique can be generalised to multivariate processes and fields to produce Slepian models also in the Lagrange wave model, as we will now show.…”

“…One of the main advantages of the stationary Gaussian model is that the stochastic properties are completely determined by the correlation structure or, alternatively, by the frequency energy content, as defined by the power spectral density. The statistical distribution of important wave characteristics, such as wave period and amplitude, steepness, wave front velocities, etc., can be studied, both in theoretical detail and numerically, using efficient algorithms or, in some cases, simple approximations; see, for example, [1], [2], [11], and [18]. The numerical algorithms give the exact distribution of the wave characteristics under the Gaussianity assumption.…”

“…This is the key to the success of the numerical algorithms. For a recent survey of techniques to calculate wave-characteristic distributions, see [11].…”

Gaussian Integrals and Rice Series in Crossing Distributions—to Compute the Distribution of Maxima and Other Features of Gaussian Processes

“…Conditioning on the height of the maximum, X max = u, one also can see how the joint distribution of H u = u−X min and T depends on u. More examples of cycle distributions can be found in (Lindgren and Broberg, 2004). 5.3 Period and amplitude by Rice/Slepian type arguments Lindgren (1972) used a combination of Rice type arguments and Slepian models to approximate the max-min period and amplitude distribution for Gaussian processes with medium width spectrum. The derivation proceeds in the following steps.…”

Uncertainty in fatigue life prediction of structures subject to Gaussian loads

“…In the variable amplitude experiments with MCTP simulated loads by Agerskov [1] a related method to approximate the Markov matrix, proposed by Krenk and Gluver [16], was used. The accuracy of MCTP approximation has been studied by Lindgren and Broberg [18].…”