A generalized van der Pol nonlinear model of vortex-induced vibrations of bridge decks with multistability

The mathematical model of vortex-induced vibrations (VIV) on long-span bridges is important to predict nonlinear structural responses. Such models can be divided into two categories: wake-oscillator and single-degree-of-freedom (SDOF) models. The SDOF model is widely used for wind-induced vibration calculations. However, the traditional SDOF model based on the standard van der Pol oscillator cannot simulate VIVs with multistability. In this study, a newly generalized van der Pol model is proposed to incorporate the limit-cycle oscillation (LCO) with multiple amplitudes, and the nonlinear damping is expressed by polynomial expansion. Next, the multiple LCO amplitudes can be determined from the energy evolution formula derived from the averaging method. Similarly, the evolution of the vibration amplitude during the transient response is also derived by the same method. Subsequently, nonlinear parameter identification methods based on constraint optimization are derived according to both the LCO amplitude and transient responses. In the last part of this study, the "energy map" is proposed to present the energy extracted from the fluid-structure interaction with different wind speeds and vibration amplitudes, and it is constructed by the parameters identified in the lock-in range of VIV. The "energy map" can provide a complete picture of the evolution of the energy of VIVs on bridge decks.