Sensitivity of Lyapunov Exponents in Design Optimization

[+] Author and Article Information
Aykut Tamer

Postdoc Researcher, Department of Aerospace Science and Technology, Politecnico Di Milano, Milano, Italy 20156

Pierangelo Masarati

Professor, Department of Aerospace Science and Technology, Politecnico Di Milano, Milano, Italy 20156

1Corresponding author.

ASME doi:10.1115/1.4041827 History: Received June 07, 2018; Revised October 21, 2018


This work presents how the analytical sensitivity of Lyapunov Characteristic Exponents can be used in the design of nonlinear dynamical systems. Owing to the complexity of estimating the stability properties of equilibrium solution of nonlinear time-dependent systems, engineers usually tend to linearize and time-average the governing equations. However, the solutions of nonlinear and time-dependent dynamical systems may exhibit unique properties, which are lost when they are simplified. When a system is designed based on a simplified model, the cost associated with neglecting nonlinearities can be significantly high, in terms of safety margins that are needed as a safeguard with respect to model uncertainties. Therefore, in those cases, a generalized stability measure, with its parametric sensitivity, can replace usual model simplifications in engineering design, especially when a system is dominated by specific, non-negligible nonlinearities and time-dependencies. The estimation of the characteristic exponents and their sensitivity is illustrated. A practical application of the proposed methodology is presented, considering the problem of helicopter ground resonance with nonlinear blade dampers. Exploiting the analytical sensitivity of the Lyapunov Exponents within a continuation approach, the geometric parameters of the damper are determined. The mass of the damper and the largest characteristic exponent of the system are used as the objective function and the inequality or equality constraint in the design of the viscous dampers.

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