, Volume 3, Issue 4, pp 331–348

On Some Reliability Applications of Rice's Formula for the Intensity of Level Crossings

  • Igor Rychlik


Let X be a stationary process with absolutely continuous sample paths. If \(E[|\dot X(0)|]\) is finite and if the distribution of X(0) is absolutely continuous, then, for almost all u, the crossing intensity μ(μ) of the level u by X(t) is given by the generalized Rice's formula \(\mu (\mu ) = E[|\dot X(0)||X(0) = u]f_{X(0)} (u)\). The classical Rice's formula for μ(μ), which is valid for a fixed level u, \(\mu (\mu ) = \smallint |z|f_{\dot X(0),X(0)} (z,u){\text{dz}}\), holds under more restrictive technical conditions that can be difficult to check in applications. In this paper it is shown that often in practice the weaker form of Rice's formula (valid for almost all u) is sufficient. Three engineering problems are discussed; prediction of fatigue life time; computing the average stress at slams and analysis of crest height of sea waves.

level crossings Rice's formula fatigue rainflow damage wave height 


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© Kluwer Academic Publishers 2000

Authors and Affiliations

  • Igor Rychlik
    • 1
  1. 1.Center for Mathematical SciencesLund UniversityLundSweden

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