Analytical study on the dynamic response of single-DOF mechanical systems to stationary non-Gaussian random loads

The frequency domain approach for fatigue analysis is becoming increasingly important for efficiently assessing damage caused by random loads, saving time, raw materials, and design costs. However, the most common random loads in mechanical systems are non-Gaussian, necessitating corrective formulas to calculate damage using frequency domain fatigue methods. To address this, the authors previously conducted an extensive series of numerical simulations to predict the kurtosis and skewness values of a structural component's stress response from a known non-Gaussian input signal, avoiding the need for costly time-domain dynamic analyses. However, in the realm of random signals, stochastic differential equations (SDEs) are encountered instead of the more familiar ordinary differential equations (ODEs). Consequently, the use of commonly employed numerical integration methods to calculate the behavior of mechanical systems may be mathematically imprecise and potentially incorrect. This study aims to evaluate the results of the aforementioned numerical mapping and demonstrate their reliability by providing a robust analytical foundation. Specifically, the dynamic response of a simple oscillator subjected to a non-Gaussian load, obtained through a polynomial transformation of an Ornstein–Uhlenbeck process, was examined. This was achieved using both classical numerical integration methods and by analytically determining the moments of the probability distribution of the response through the use of Îto calculus and Moment Equations. The comparison between numerical and analytical results provides a rigorous validation of the reliability of the numerical mapping of dynamic system responses to non-Gaussian stochastic loads and allows the confirmation and analytical demonstration of some significant findings previously observed, like the strong dependence of response's non-Gaussianity from the damping of the system and the robustness of the Normalized Bandwidth Factor as the frequency-based invariant parameter.