Purpose

The authors analyze the regular and distinctive patterns of the free motion of a ball type tuned mass damper.

Methods

The governing differential system modeling movement of a heavy ball rolling inside a spherical cavity is formulated and investigated; six degrees of freedom with three non-holonomic constraints and no slipping are assumed. Predominance of the Appell-Gibbs approach over the conventional Lagrangian procedure is pointed out when complicated non-holonomic systems are in question. General properties of the differential system in the normal form are discussed and possibilities of further investigation using semi-analytical methods are outlined. Simultaneously, a wide program of numerical simulation is presented concerning the homogeneous system with a number of initial condition settings and other parameter variants.

Results

Seven types of limit solutions (or limit trajectories) have been found and physically interpreted together with their neighborhood. The set of limit trajectories represents boundaries separating solution groups of a certain character. The shape and general character of regular solutions within particular domains delimited by these limit solutions were analyzed to facilitate a practical application of this theoretical background.

Conclusions

The paper represents a contribution to the theoretical background of the ball type tuned mass damper used in Civil Engineering. The analysis provides an insight to the possible character of a free motion of a ball type tuned mass damper under various configurations; this way it helps to analyze possible critical parameters of the system or interaction with the structure. Assumptions of further investigation are outlined.

中文翻译：

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在球形腔内移动的非完整球系统的极限轨迹

目的

作者分析了球形调谐质量阻尼器自由运动的规律性和独特性。

方法

制定并研究了控制球形系统在球腔内滚动的微分系统建模运动；假设六个自由度，且具有三个非完整约束且没有打滑。当对复杂的非完整系统提出疑问时，指出了Appell-Gibbs方法相对于传统拉格朗日方法的优势。讨论了正常形式的微分系统的一般特性，并概述了使用半分析方法进行进一步研究的可能性。同时，提出了涉及均匀系统的大量数值模拟程序，该程序具有许多初始条件设置和其他参数变量。

结果

找到了七种类型的极限解（或极限轨迹），并对其邻域进行了物理解释。极限轨迹的集合表示将某个特征的解决方案组分开的边界。分析了由这些极限解界定的特定域内正则解的形状和一般特征，以促进该理论背景的实际应用。

结论

本文代表了对土木工程中使用的球形调谐质量阻尼器的理论背景的贡献。该分析提供了对球型调谐质量阻尼器在各种配置下的自由运动的可能特征的洞察力。这样，它有助于分析系统的可能关键参数或与结构的交互作用。概述了进一步调查的假设。