Abstract This paper addresses the free-vibration response of structures coupled with periodically-distributed resonators, generally referred to as locally-resonant structures. The first step is to show that all exact natural frequencies can be calculated by a proper formulation of the Wittrick-Williams algorithm, involving a condensed dynamic stiffness matrix whose size depends only on the number of degrees of freedom of the structure and is independent of the number of degrees of freedom within every resonator. Indeed, the presence of resonators is accounted for in the condensed dynamic stiffness matrix via a pertinent frequency-dependent stiffness, readily obtainable from the resonator motion equations. Within this framework, a novel procedure is proposed to construct the condensed dynamic stiffness matrix of locally-resonant plates, applicable for any number of resonators. Numerical applications show the exactness of the proposed formulation.

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关于局部共振结构的自由振动

摘要 本文讨论了与周期性分布的谐振器耦合的结构的自由振动响应，通常称为局部谐振结构。第一步是证明所有精确的自然频率都可以通过 Wittrick-Williams 算法的适当公式计算，涉及一个压缩的动态刚度矩阵，其大小仅取决于结构的自由度数，并且与每个谐振器内的自由度数。事实上，谐振器的存在是通过相关的频率相关刚度在压缩动态刚度矩阵中考虑的，很容易从谐振器运动方程中获得。在这个框架内，提出了一种新的程序来构建局部共振板的凝聚动态刚度矩阵，适用于任意数量的谐振器。数值应用显示了所提出的公式的准确性。