Study on response mechanism of nonlinear energy sink with inerter and grounded stiffness

Novel and efficient vibration control units such as inerter and grounded stiffness have contributed significantly to structural vibration reduction. However, as the performance of vibration suppression systems gradually improves, their structures become more complex. And the coupling effects among complex structures, as well as the effects on the system dynamics, are hazy. This paper aims to investigate the influence of the combined structure of inerter and grounded stiffness on the response mechanism and damping effect of the nonlinear energy sink. The damping of the primary system, a parameter that has been neglected in the majority of studies, is also included in the model. The closed-form solutions of the system are derived by the complexification-averaging method and verified numerically. Then, the control equations for stability judgment, saddle-node bifurcation, and Hopf bifurcation are calculated. Combining the multi-scale method to analyze slow invariant manifolds, sufficient and necessary conditions for strongly modulated response are deduced. It is found that the primary system damping, inerter-mass ratio, and grounded stiffness ratio not only change the bifurcation areas but also shift the bifurcation boundaries on the ξ2,f\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( {\xi_{2} ,\,f} \right)$$\end{document} plane upwards, which means an increase in the amplitude of the external force required to excite bifurcations. There is a particular case that the grounded stiffness reduces the Hopf bifurcation area and the required external excitation amplitude. In addition, the introduction of the primary system damping complicates the Hopf bifurcation boundaries without affecting the range of NES damping ratio that generates the strongly modulated response. The primary system damping, inerter, and grounded stiffness all enlarge domains of attraction that may inhibit the production of strongly modulated response. The correctness of theoretical derivations is verified. In comparison, the optimized model shows a better damping effect and a wider damping bandwidth.