With nonlinear systems extensively studied for applications in vibration isolation, energy harvesting, etc., most of the studies focused on the response of the nonlinear oscillator defined by some explicit equations, namely the typical oscillator. Rare studies or approaches are devoted to the non-typical nonlinear oscillator for which the stiffness profile is possible defined by an arbitrary curve without an explicit function. This paper proposes a novel design method for both typical and non-typical monostable hardening oscillators, especially for the latter one in order to achieve some arbitrarily specified response. The key idea is to construct a series of linear equivalent oscillators to reproduce the response of the nonlinear oscillator. A trial stiffness characteristic expression of the linear equivalent oscillators has been given as a function of several particular variables. It has been verified that the linear equivalent oscillators can obtain the response for some typical nonlinear oscillators. Moreover, the method can be also used in a reversed way to identify the stiffness characteristics from a given response for either typical or non-typical systems. In particular, this parameter identification feature can be utilized to design a customized oscillator with arbitrarily specified hardening response. By constructing the linear equivalent oscillators to reproduce the desired response, the linear equivalent stiffness expression can be determined, while the customized nonlinear stiffness characteristics can be obtained accordingly. Experiments on an elaborated example have validated effectiveness of the proposed method. It shows that the mapping method can be used to determine the proper stiffness characteristics for customized hardening response beyond the scope of typical nonlinear oscillators.

对于在振动隔离，能量收集等方面广泛应用的非线性系统，大多数研究集中在由一些明确的方程式（即典型振荡器）定义的非线性振荡器的响应上。非典型非线性振荡器的研究或方法很少，其刚度分布可能由任意曲线定义，而没有明确的函数。本文针对典型的和非典型的单稳态强化振荡器，特别是对于后者，提出了一种新颖的设计方法，以实现任意指定的响应。关键思想是构造一系列线性等效振荡器，以再现非线性振荡器的响应。已经给出了线性等效振荡器的试验刚度特性表达式，作为几个特定变量的函数。已经证实，线性等效振荡器可以获得某些典型非线性振荡器的响应。此外，该方法还可以以相反的方式用于从典型或非典型系统的给定响应中识别刚度特性。特别地，该参数识别特征可用于设计具有任意指定的硬化响应的定制振荡器。通过构造线性等效振荡器以再现期望的响应，可以确定线性等效刚度表达式，同时可以相应地获得定制的非线性刚度特性。在一个详细示例上的实验已经验证了该方法的有效性。它表明，映射方法可用于确定适当的刚度特性，以实现超出典型非线性振荡器范围之外的定制硬化响应。