Study on response mechanism and optimization of fractional-order NES in automotive vibration control

The fractional-order derivative is introduced into the nonlinear energy sink, which is called fractional-order NES. In this paper, the dynamical model of a fractional-order NES coupled automotive powertrain is established to control the vertical vibration generated by the harmonic excitation, and the saddle-node bifurcation, Hopf bifurcation, and the strongly modulated response (SMR) of the dynamical system under harmonic excitation are investigated. The analytical solution, the saddle-node bifurcation boundary, and the Hopf bifurcation boundary under the harmonic excitation are derived by the complexification-averaging method. The necessary and sufficient condition for the (SMR) is derived by the multi-scale method. The analytical solution and the numerical solution are compared to verify the correctness of the analytical solution. The influence of the system parameters on the saddle-node bifurcation boundary and Hopf bifurcation boundary are analyzed, and the stability of the equilibrium points are analyzed according to the Lyapunov theory. The process of SMR is illustrated by slow invariant manifolds (SIM), the interval of SMR is explored by one-dimensional mapping, and the reasons for disappearance of SMR are revealed by the phase trajectory of SIM. By selecting specific points and making the frequency-response curve of the specific points, it can be found that the saddle-node bifurcation and the Hopf bifurcation match well with the intervals corresponding to the bifurcation boundaries, and the superior vibration reduction effect of SMR stage is verified by selecting the energy spectrum as the evaluation index. Finally, the optimization objective is established according to the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>∞</mml:mi> </mml:msub> </mml:mrow> </mml:math> criterion. The Grey Wolf Optimizer is applied to optimize the parameters of the fractional-order NES. The results show that the optimized NES can effectively suppress broad frequency vibration.