Nonlinear stiffness and hysteresis phenomena of harbor oscillations

Abstract

This study investigates the nonlinear stiffness and hysteresis of harbor oscillations using a fully nonlinear Boussinesq wave model. Numerical results confirm the presence of two nonlinear phenomena in harbor oscillations: hardening stiffness, where the resonant frequency increases with the response amplitude, and hysteresis, where the harbor's response to incoming waves is influenced by its previous oscillatory state.

For gravity waves neglecting surface tension, the restoring force is gravity. The nonlinear restoring term in the wave equations is typically expressed by the surface gradient term g(h+η)▽η. As η increases, the nonlinear restoring term intensifies, thereby providing a basis for the emergence of nonlinear stiffness.

The nonlinear stiffness and hysteresis phenomena are identified as case-specific. The Duffing oscillator model is adopted to explain these characteristics. It is determined that the nonlinear stiffness parameter and damping coefficient of oscillatory patterns significantly influence the nonlinear stiffness and hysteresis observed in harbor oscillations. A larger stiffness parameter and smaller damping coefficient make nonlinear stiffness more likely to occur. Conversely, if the stiffness parameter is small and the damping is large, nonlinear stiffness is less likely to manifest. Furthermore, a high damping coefficient indicates that the harbor quickly releases the energy from past oscillations to the open sea, preventing the system from retaining a 'memory' of its previous states and thus minimizing hysteresis. The case-specific nature of these nonlinear phenomena highlights the importance of considering specific oscillatory pattern when assessing harbor dynamic response.