Sensitivity of Lyapunov Exponents in Design Optimization of Nonlinear Dampers

Abstract

This work presents how the analytical sensitivity of Lyapunov characteristic exponents (LCEs) can be used in the design of nonlinear dampers, which are frequently utilized to stabilize the response of mechanical systems. The kinetic energy dissipated in the form of heat often induces nonlinearities, therefore reducing the reliability of standard stability evaluation methods. Owing to the difficulty of estimating the stability properties of equilibrium solution of the resulting 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 damper 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 that the problem of helicopter ground resonance (GR) and landing gear shimmy vibration with nonlinear dampers are implemented instead of linear ones. 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.