Fast assessment of non-Gaussian inputs in structural dynamics exploiting modal solutions

TL;DR

The paper tackles the challenge of evaluating non-Gaussian, non-stationary loading in linear structural dynamics, where traditional PSD-based Gaussian assumptions can be non-conservative. It introduces a modal-based approach that derives response kurtosis and higher-order statistics from a single modal solution by scaling with mode shapes, avoiding element-wise time-domain simulations. A multivariate fourth-order moment and kurtosis tensor are developed, with a Voigt-projected representation to manage rank-4 statistics efficiently; transforms from modal to physical coordinates enable rapid, full-structure statistical analysis. Validation on an L-shaped FE model under mixed sine-random excitation demonstrates accurate capture of non-Gaussian effects and dramatic computational savings, supporting quick fatigue assessment and design optimization under complex loading. The work lays a foundation for future data-driven damage estimation and load-spectrum analysis of non-stationary stress states in structural dynamics.

Abstract

In various technical applications, assessing the impact of non-Gaussian processes on responses of dynamic systems is crucial. While simulating time-domain realizations offers an efficient solution for linear dynamic systems, this method proves time-consuming for finite element (FE) models, which may contain thousands to millions of degrees-of-freedom (DOF). Given the central role of kurtosis in describing non-Gaussianity - owing to its concise, parametric-free and easily interpretable nature - this paper introduces a highly efficient approach for deriving response kurtosis and other related statistical descriptions. This approach makes use of the modal solution of dynamic systems, which allows to reduce DOFs and responses analysis to a minimum number in the modal domain. This computational advantage enables fast assessments of non-Gaussian effects for entire FE models. Our approach is illustrated using a simple FE model that has found regular use in the field of random vibration fatigue.

Figures (9)

• Figure 1: Schematic overview of different paths to obtain response kurtoses: Using linear systems theory, via a) spectral densities including the trispectrum, b) the non-stationarity matrix (NSM), and c) response time series and subsequent moment estimation, alternatively using mode shape scaling, via d) response time series and e) second- and fourth-order moments of the modal solution, presented within this paper
• Figure 2: Sine-on-random load exciting the edges of the L-shaped specimen in $z$ direction
• Figure 3: Transfer functions from clamped edges to modal solution $\bm{q}(t)$ with universal damping coefficient $\zeta = 0.01$
• Figure 4: Distribution of kurtosis over nodes for maximum of stress components
• Figure 5: Maximum stationary fourth-order moment of shear stress (proportional to variance)