Chen Yandong, Tai Yongpeng, Xu Jun, Xu Xiaomei, Chen Ning
College of Mechanical and Electronic Engineering, Nanjing Forestry University, Nanjing 210037, China.
College of Intelligent Equipment Engineering, Wuxi Taihu University, Wuxi 214151, China.
Sensors (Basel). 2022 Aug 25;22(17):6408. doi: 10.3390/s22176408.
The fluid inerter described by the fractional derivative model is introduced into the traditional nonlinear energy sink (NES), which is called fractional-order NES in this paper. The slowly varying dynamic equation (SVDE) of the system coupled with fractional-order NES is obtained by the complex averaging method, in which the fractional derivative term is treated using the fractional Leibniz theorem. Then, the discriminants (Δ, Δ, and Δ) of the number of equilibrium points are derived. By using the variable substitution method, the characteristic equation for judging the stability is established. The results show: (1) the approximate SVDE is sufficient to reflect the slowly varying characteristics of the system, which shows that the mathematical treatment of the fractional derivative term is reliable; (2) the discriminant conditions (Δ, Δ) can accurately reflect the number of equilibrium points, and the corresponding range of nonlinear parameter can be calculated when the system has three equilibrium points. The expressions of Δ, Δ are simpler than Δ, which is suitable for analysis and design parameters; (3) the stability discrimination methods of schemes 1 and 2 are accurate. Compared with scheme 2, scheme 1 is more prone to various responses, especially various strongly and weakly modulated responses. In scheme 2, the inertia effect of mass can be completely replaced by integer order inerter. Compared with integer order inerter, the introduction of fractional order inerter, whether in series or in parallel, means that the amplitude of the equilibrium point on the NES vibrator is smaller, but it is also for this reason that it is not easy to produce a modulated response with scheme 2, and the vibration suppression effect of the main structure is not good.
将分数阶导数模型描述的流体惯性器引入传统的非线性能量阱（NES），本文将其称为分数阶NES。采用复平均法得到了耦合分数阶NES系统的慢变动力学方程（SVDE），其中分数阶导数项采用分数阶莱布尼茨定理进行处理。然后，推导了平衡点个数的判别式（Δ、Δ和Δ）。利用变量代换法建立了判断稳定性的特征方程。结果表明：（1）近似的SVDE足以反映系统的慢变特性，这表明分数阶导数项的数学处理是可靠的；（2）判别条件（Δ、Δ）能够准确反映平衡点个数，当系统有三个平衡点时，可以计算出非线性参数的相应范围。Δ、Δ的表达式比Δ简单，适用于参数分析与设计；（3）方案1和方案2的稳定性判别方法是准确的。与方案2相比，方案1更容易出现各种响应，尤其是各种强调制和弱调制响应。在方案2中，质量的惯性效应可以完全由整数阶惯性器代替。与整数阶惯性器相比，分数阶惯性器无论是串联还是并联引入，都意味着NES振子上平衡点的幅值较小，但也正因如此，方案2不容易产生调制响应，对主结构的振动抑制效果不佳。
