Adjoint-Based Stochastic Analysis for Fatigue of Aeroelastic and Frictional Systems

Abstract

As engineering systems are optimized to reduce weight and improve performance, vibration behavior, including nonlinear effects, is becoming increasingly important to analyze. While methods such as harmonic balance are popular for analyzing periodic nonlinear vibration behavior, these analyses often only consider deterministic loading. However, real applications experience additional noise in the excitation, and alternative approaches are frequently limited to smaller models. This work expands on a recent approach to calculate a time-varying covariance using a linearized system around a periodic nonlinear response. Specifically, numerical conditioning is addressed and improved for cases increasing from a few degrees of freedom to hundreds of degrees of freedom. The algorithm is then extended to capture correlations across both space and time via a period-spanning covariance matrix. Model reduction is proposed for this calculation to maintain computational efficiency. The period-spanning covariance matrix is used to sample full stochastic cycles and estimate fatigue-relevant statistics up to 1393 times faster than stochastic time integration. Importantly, the present methodology uses the same gradient information as stability analyses and continuation solutions, so it can easily be applied for many nonlinear analyses. The methodology is demonstrated for two beam systems: one with unsteady aerodynamic forces and another with a Bouc-Wen frictional tip force. For the aeroelastic model, statistics are accurately estimated via the present approach. For the frictional system, some errors are introduced compared to stochastic time integration and are attributed to the linear time-periodic nature of the system used in the present stochastic calculations. However, estimated stochastic statistics remain useful given the significant reduction in computation time.