Stochastic analysis and vibration suppression of a time-delayed system with nonlinear energy sink

Nonlinear energy sink (NES) is of great significance in attenuating structural vibration. Time-delay phenomenon in a mechanical system is inevitable. Different from existing works, the influences of time-delayed factors in a nonlinear oscillator attached with NES are considered in this paper under Gaussian white noise (GWN). Firstly, the time-delayed term is transformed into a linear combination of stiffness and damping terms, and an equivalent system without the time-delayed term is obtained. Stationary probability density function (SPDF) is deduced by leveraging stochastic averaging method. Secondly, good agreement is observed between analytical and numerical results, which verifies the effectiveness of the method. Furthermore, bifurcation conditions for the amplitude of SPDF of the primary system are obtained, indicating that delayed time, feedback gain and damping coefficients can cause stochastic P-bifurcation of the primary system. Finally, the vibration suppression performance of NES is analyzed via energy variation and the time–history diagram of displacement.