Dynamical integrity estimation in time delayed systems: A rapid iterative algorithm

• Published in: Journal of sound and vibration Vol. 571; p. 118045
• Main Authors: Szaksz, Bence, Stepan, Gabor, Habib, Giuseppe
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 17.02.2024
• Subjects: Basin of attraction; Dynamical integrity; Local integrity measure; Time-delay; Basin of attraction; Time-delay; Local integrity measure; Dynamical integrity
• ISSN: 0022-460X; 1095-8568
• DOI: 10.1016/j.jsv.2023.118045

The robustness of dynamical systems against external perturbations is crucial in engineering; however, it is often overlooked for the lack of methods for rapidly computing it. This paper proposes a novel algorithm for estimating the robustness of systems subject to time delay. More precisely, the algorithm iteratively estimates the so-called local integrity measure (LIM), that is, the radius of the largest hypersphere centered at the fixed point and located within its basin of attraction. Since time-delayed systems are infinite dimensional, initial conditions are restricted to a constrained type of initial functions. The semi-discretization method is used for rapidly simulating the dynamics of the systems, while trajectories are classified as converging or diverging using a subdivision of the reduced phase space into cells. The algorithm was tested on four different mechanical systems, and in all cases it very quickly provided an accurate estimation of the LIM. Moreover, it enabled the study of LIM trends in a multi-dimensional parameter space, which would have been unfeasible with alternative methods. This breakthrough in computational efficiency has important implications for engineering design, allowing for careful consideration of dynamical integrity and enhancing the safety and reliability of engineered systems, especially in the presence of time delays. •An algorithm for estimating the dynamical integrity of equilibria is developed.•The algorithm works for time-delayed dynamical systems.•The algorithm works iteratively, it is very quick and has no memory issues.•It enables one to consider dynamical integrity during engineering design.•Several case studies demonstrate the effectiveness of the algorithm.