This paper investigates on the \(H_{\infty }\) and \(H_2\) control performances for a novel grounded inerter-based dynamic vibration absorber configuration (GIDVA) by entirely replacing both damper and stiffness of the classic dynamic vibration absorber with grounded inerter-based mechanical network. First, the equations of motion of the coupled system are derived and the analytical solution is established in terms of primary system displacement under harmonic excitation. Then, the optimum tuning parameters are found by first using the extended fixed points theory (EFPT) to get analytical design. It is found that this EFPT yields the approximate but accurate analytical solutions, so this method can be commonly used to design inerter-based DVAs with four fixed points in the frequency response curves of the primary system. Further, to evaluate the control performance of the proposed GIDVA, the numerical \(H_{\infty }\) and \(H_{2}\) optimizations are performed in MATLAB for the case of harmonic and random excitation, respectively, using the Newton–Raphson algorithm with the starting points, the analytical approximate optimal parameters based on the EFPT. Based on the results comparison of the control performance, one can remark that with the same small mass ratio, the proposed GIDVA can decrease the peak vibration amplitude of primary system more than 64% and 50% and enlarge the suppression bandwidth more than 64% and 57% when compared with the classic DVA and the recent published non-traditional inerter-based DVAs (NIDVA-\(C_i\), i=3, 4, 6), respectively, for both optimal designs \(H_{\infty }\) and \(H_2\). However, one can also notice that this improvement decreases with the increase in the mass ratio. Moreover, under random excitation, the GIDVA also shows good control performance with 56% and 44% reduction in terms of mean square and time history responses of primary system. This result reveals the potential of a novel grounded inerter-based dynamic vibration absorber, which provides a reference for the design index of the novel vibration absorber in engineering practice.

本文通过将经典动态吸振器的阻尼器和刚度完全替换为基于惯性的接地机械网络。首先，推导了耦合系统的运动方程，并根据谐波激励下的主系统位移建立了解析解。然后，首先使用扩展不动点理论（EFPT）来获得解析设计，找到最佳调整参数。结果发现，该 EFPT 产生近似但准确的解析解，因此该方法通常可用于设计主系统频率响应曲线中具有四个固定点的基于惯量的 DVA。此外，为了评估所提出的 GIDVA 的控制性能，在 MATLAB 中分别针对谐波和随机激励的情况进行数值\(H_{\infty }\)和\(H_{2}\)优化，使用Newton-Raphson 算法以 EFPT 为起点，解析近似最优参数。根据控制性能的结果比较，可以看出，在相同的小质量比下，所提出的GIDVA可以将主系统的峰值振幅降低64%和50%以上，并将抑制带宽扩大64%以上，与经典 DVA 和最近发布的非传统惰性 DVA (NIDVA- \(C_i\) , i=3, 4, 6) 相比，两种最优设计分别提高了57% \(H_{\infty } \)和\(H_2\)。然而，人们也可以注意到，这种改善随着质量比的增加而减弱。此外，在随机激励下，GIDVA还表现出良好的控制性能，一次系统的均方响应和时程响应分别降低了56%和44%。该结果揭示了新型接地惯性体动态吸振器的潜力，为工程实践中新型吸振器的设计指标提供了参考。