Uncertainty of the fatigue damage arising from a stochastic process with multiple frequency modes

• Analytical method for predicting the fatigue damage variance for any multimodal Gaussian process.
• Closed form solutions are available for special cases.
• Equations yield insight on correlation effects produced by different narrowband components.
• Proposed method compares well with simulation.

Predicting the variance of the fatigue damage due to a stochastic load process is a difficult classical problem that dates back to the 1960s. For many years, the available analytical methods for tackling this problem have been limited to the linear oscillator response under Gaussian white noise excitation. In a recent prior work, the author developed an improved method for calculating the damage variance for a general narrowband Gaussian process. From a fatigue uncertainty perspective, a narrowband process is particularly crucial as the amplitude correlation magnifies the variance. This paper extends the analysis to a multimodal process comprising two or more narrowband components. The proposed method is tractable, involving a single summation for an arbitrary spectral density. Moreover, closed form expressions are available for two special cases, i.e. the components are all linear oscillator responses or the spectral density of each is rectangular. The equations also yield insight on the multilayered correlation mechanisms produced by different narrowband components. The accuracy of the method is verified by rainflow counting of simulated time history stresses.