Abstract The present study theoretically investigates the free vibration problem of a discrete granular system. This problem can be considered as a simple model to rigorously study the effects of the microstructure on the dynamic behavior of the equivalent continuum structural model. The model consists of uniform grains confined by discrete elastic interactions, to take into account the lateral granular contributions. This repetitive discrete system can be referred to discrete Cosserat chain or a lattice elastic model with shear interaction. First for the simply supported granular beam resting on Winkler foundations, due to the critical frequencies which concern the nature of the dynamic results, the natural frequencies are exactly calculated, starting from the resolution of the linear difference eigenvalue problem. The natural frequencies of such a granular model are analytically calculated for whatever modes. It is shown that the difference equations governed to the discrete system converge to the differential equations of the Bresse-Timoshenko beam resting on Winkler foundation (also classified as a continuous Cosserat beam model) for an infinite number of grains. A gradient Bresse-Timoshenko model is constructed from continualization of the difference equations. This continuous gradient elasticity Cosserat model is obtained from a polynomial or a rational expansion of the pseudo-differential operators, stemming from the continualization process. Scale effects of the granular chain are captured by the continuous gradient elasticity model. The natural frequencies of the continuous gradient Cosserat models are compared with those of the discrete Cosserat model associated with the granular chain. The results clarify the dependency of the beam dynamic responses to the beam length ratio.

中文翻译：

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使用离散和梯度弹性 Cosserat 模型求解有限颗粒梁振动的精确解

摘要 本研究从理论上研究了离散颗粒系统的自由振动问题。这个问题可以看作是一个简单的模型来严格研究微观结构对等效连续结构模型的动力学行为的影响。该模型由离散弹性相互作用限制的均匀颗粒组成，以考虑横向颗粒的贡献。这种重复的离散系统可以称为离散 Cosserat 链或具有剪切相互作用的晶格弹性模型。首先对于Winkler基础上的简支粒状梁，由于临界频率关系到动力学结果的性质，所以从线性差分特征值问题的求解开始，精确计算了固有频率。这种颗粒模型的固有频率是针对任何模式进行分析计算的。结果表明，对于无限数量的晶粒，离散系统的差分方程收敛到位于 Winkler 基础上的 Bresse-Timoshenko 梁（也被归类为连续 Cosserat 梁模型）的微分方程。梯度 Bresse-Timoshenko 模型是根据差分方程的连续化构建的。这种连续梯度弹性 Cosserat 模型是从多项式或伪微分算子的有理扩展中获得的，源于连续化过程。颗粒链的尺度效应由连续梯度弹性模型捕获。将连续梯度 Cosserat 模型的固有频率与与颗粒链相关的离散 Cosserat 模型的固有频率进行比较。结果阐明了梁动态响应对梁长度比的依赖性。