The theory of linear and nonlinear dynamic vibration absorbers is a well-established topic for many years. However, many recent contributions paid attention to the nonlinear vibration absorbers and different practical realizations of corresponding devices. Here, we propose a mechanical system constituted of the inerter-based nonlinear energy sink attached to the main body that is resting on an elastic foundation and is grounded through the fractional type electromagnetic damper. The two-degree-of-freedom system is described via two coupled differential equations with one of them having a fractional-order derivative term and the other one containing cubic stiffness nonlinearity. The incremental harmonic balance (IHB) method is employed to solve the equations and studies the strongly nonlinear periodic responses of the system. Applied approximated solution methodology is validated by the numerical Newmark method adapted to deal with the system of nonlinear fractional-order differential equations. The appropriate and necessary number of harmonics used in the IHB solution is commented and validated. This study can be a first step in understanding the dynamics and giving directions for the future design of vibration-isolating platforms based on inerter-based nonlinear vibration absorbers and electromagnetic dampers.

中文翻译：

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通过分数型电磁阻尼器和基于 Inerter 的非线性能量阱对平台进行振动抑制


线性和非线性动态减振器的理论多年来一直是一个公认的话题。然而，最近的许多贡献都关注了非线性减振器和相应设备的不同实际实现。在这里，我们提出了一个机械系统，该系统由连接到主体的基于惰性的非线性能量汇构成，该能量汇位于弹性基础上，并通过分数型电磁阻尼器接地。二自由度系统通过两个耦合微分方程来描述，其中一个方程具有分数阶导数项，另一个包含三次刚度非线性。采用增量谐波平衡 （IHB） 方法求解方程并研究系统的强非线性周期响应。应用近似解方法通过适用于处理非线性分数阶微分方程组的数值 Newmark 方法进行了验证。对 IHB 解决方案中使用的适当和必要的谐波数量进行了注释和验证。这项研究可以成为理解动力学并为未来基于惯性非线性减振器和电磁阻尼器的隔振平台设计提供方向的第一步。