Nonlinear Vibration Localization in Cyclic Structures

工程和技术中的许多机械结构，如转子、涡轮机、压气机等，都是以几乎相同的部件的形式循环组装的。运行负载通常是不稳定的，并导致振动，这决定或限制了系统的功能或寿命。振动的结果往往不是均匀地分布在整个系统中，而是仅局限于某些空间区域：局部振动。在线性系统模型的背景下，对称性的轻微不均匀性或扰动可以很好地解释局域化。然而，最近越来越明显的是，这种循环系统也具有明显的非线性性质。产生了新的非线性振动现象。一类这样的非线性振动是非线性局部振动，这是本项目的主题。从一般的非线性动力学来看，我们都知道存在某些机制可以导致非线性局域动力学。虽然在许多科学领域，这种非线性局部化已经研究了很长时间，并且其相关性得到了广泛认可，但对于非线性机械振动的理解仍然相对有限。在非线性振动中，单个非线性振子的振幅相关频移可能导致与邻近振子的耦合减少，这在某些条件下可能导致振动的完全空间局部化。在本项目中，起点将是所谓的离散呼吸。基于模型系统的概念、数值和实验研究，将进行表征和量化与非线性振动局部化相关的现象。我们的目标是回答，在什么系统和负载条件下出现非线性振动局部化，以及它的鲁棒性或稳定性。进一步的研究主题是局部振动扩展域的出现，强振动和弱振动或无振动区域之间的过渡区域，以及线性和非线性局部化机制之间的相互作用。

Many mechanical structures in engineering and technology, like rotors, turbines, compressors etc, are cyclically assembled in the form of nearly identical components. The operational loads are often unsteady and lead to vibrations that determine or limit the function or lifetime of the systems. Often vibrations result that are not homomogeneously spread out over the whole system, but are confined to certain spatial zones only: localized vibrations. In the context of linear system models, slight inhomogeneities or perturbations of the symmetry explain localization very well. Rather recently, however, it has become more obvious that such cyclic systems also bear marked nonlinear properties. Which bring along novel nonlinear vibration phenomena. One class of such nonlinear vibrations are nonlinearly localized vibrations, which is the topic of the present project. From nonlinear dynamics in general it is well known that there are certain mechanisms that may lead to nonlinearly localized dynamics. While for many fields in science such nonlinear localization has been studied for a long time, and its relevance is widely acknowledged, for nonlinear mechanical vibrations the understanding is still comparatively limited. In nonlinear vibrations, the amplitude-dependent frequency shifts of single nonlinear oscillators may lead to a reduced coupling to neighbouring oscillators, which under certain conditions may result in full spatial localization of the vibration. In the present project the starting point will be the so-called discrete breathers. Conceptual, numerical, and experimental studies, based on a model system, are to be conducted to characterize and quantify the phenomena related with nonlinear vibration localization. We will aim at answering, under what system and loading conditions nonlinear vibration localization appears, and how robust or stable it is. Further topics of the study are the emergence of extended domains of localized vibration, the transition regeion between zones of strong and weak or no vibration, and the interplay between linear and nonlinear localization mechanisms.