Vibration suppression and P-bifurcation of a randomly excited fractional-order damping system controlled by nonlinear energy sink

Nonlinear energy sink (NES) is a device used for structural vibration reduction, which can not only achieve targeted energy transfer, but also has many advantages such as wide working frequency band, high reliability and strong robustness. In a fractional-order differential system, the historical memory of fractional-order differential operators enables us to more accurately simulate and study physical phenomena in actual systems. With this in mind, the vibration suppression performance of the fractional-order damping system coupled with NES is considered. However, random vibrations are everywhere and systems are susceptible to random excitations. Therefore, this paper conducts a numerical and analytical investigation into the dynamic response mechanism of a fractional-order damping system excited by random excitation, which coupled with a NES. Firstly, The Fokker-Planck-Kolmogorov equation and stationary probability density function expression are derived by using stochastic averaging method. Secondly, The effectiveness of this approach is confirmed by the concurrence between the analytical solutions and the numerical solutions obtained through Monte Carlo simulation. Subsequently, the effects of system parameters and noise intensity on the stationary response of the system are discussed, as well as stochastic P-bifurcation caused by system parameters. Ultimately, the time histories of displacement and the energy transformation of the primary structure with changes in system parameters under the condition of coupled with NES and without NES are analyzed to verify that NES can achieve the desired goal of vibration reduction.