WORKSHOP: Resonance oscillations and stability of nonsmooth systems

研讨会：非光滑系统的共振和稳定性

• 批准号: EP/H000577/1
• 负责人: Jeroen Lamb
• 金额: $ 2.02万
• 依托单位: Imperial College London
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2009
• 资助国家: 英国
• 起止时间: 2009 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FH000577%2F1
• 关键词: WORKSHOP; Resonance; oscillations; stability; nonsmooth

Sometimes the application of a small force to a mechanical or electrical system causes a considerable growth in the amplitude of stable oscillations. A familiar example is a pushed playground swing: (smoothly) pushing a swing (almost) in tune with its natural frequency will make the swing's amplitude increase to some maximal amplitude oscillation, commonly known as a resonance oscillation. The corresponding mathematical model is that of a harmonic oscillator with frequency w_0 perturbed by a periodic force of frequency w, where |w_0-w| is small. The essence of the phenomenon is that one of the periodic solutions (from a Lyapunov family) of the harmonic oscillator becomes asymptotically stable under the time-periodic perturbation and the amplitude of this solution increases as |w_0-w| decreases. The rigorous justification of resonance oscillations in the classical literature is based on an averaging method yielding a bifurcation function M_w whose zeroes A_w correspond to solutions and asymptotic stability follows if the eigenvalues of (M_w)'(A_w) have negative real part. Here A_w represents the amplitude of the oscillation which increases as w approaches w_0. We note that the classical theory uses smoothness of the unperturbed system and the perturbations in an essential way. The theory of resonance oscillations in nonsmooth systems started to be developed around the second world war following the Tahoma Narrow Bridge collapse in 1940. Only in 1989, Glover et al. proved that the collapse was due to suspension cables which restrict the motion of the bridge from below. This one-sided suspension introduces a nondifferentiable (nonsmooth) function to the relevant mathematical model, rendering the classical averaging theory non-applicable. This has been the starting point of an efort to extend classical results in bifurcation theory to nonsmooth systems (by relaxation of smoothness assumptions). Resonance oscillations in nonsmooth systems are known to be able to exhibit behaviour that is unique to nonsmooth systems but disappears in smooth approximations. A mechanical system that demonstrates this is a body attached to a rigid fixed beam on a moving belt. If the friction between the body and the belt is dry and some relations between the friction coefficient and the amplitude of the external forcing hold true, then the body may stick to the belt periodically and exhibit so-called stick-slip oscillations. Taking a smooth approximation of the sign-function modelling the dry friction destroys the phenomenon. Averaging-like methods can not be applied here (even formally) since the corresponding derivative (M_w)' does not exist at A_w. This observation has lead to a substantial research effort, in particular to explain the stability of stick-slip motions.As discussed above in the example of the forced swing, the resonance oscillations occurs when some non-asymptotically stable periodic solution becomes asymptotically stable. Stability and change of stability is thus - as in smooth systems - an essential aspect of the bifurcation theory, and an active field of research, in particular when techniques and concepts from smooth systems cannot be applied.Research into the mathematics of resonance oscillations in nonsmooth systems has seen a steady growth in attention lately, and due to recent progress a number of technically relevant open problems concerning resonances of nonsmooth mechanical and physical systems have become in reach of being resolved. The workshop will pay attention to such problems with industrial relevance.Perhaps due to the strong growth in the subject, the situation has arisen where different schools working on nonsmooth systems and stability achieved similar results without being aware of each other. This workshop makes a substantial attempt in creating awareness and stimulate discussion and collaboration between different research groups working on oscillations and stability of nonsmooth systems.

有时，对机械或电气系统施加很小的力会导致稳定振荡的幅度显着增加。一个熟悉的例子是推动游乐场秋千：（平稳地）推动秋千（几乎）与其固有频率一致将使秋千的振幅增加到某个最大振幅振荡，通常称为共振振荡。相应的数学模型是频率为 w_0 的谐振子受到频率为 w 的周期力的扰动，其中 |w_0-w|很小。该现象的本质是谐振子的周期解之一（来自李亚普诺夫族）在时间周期扰动下变得渐近稳定，并且该解的幅度随着 |w_0-w| 增加。减少。经典文献中对共振振荡的严格证明是基于产生分岔函数 M_w 的平均方法，其零点 A_w 对应于解，并且如果 (M_w)'(A_w) 的特征值具有负实部，则渐近稳定性随之而来。这里A_w表示随着w接近w_0而增加的振荡幅度。我们注意到，经典理论本质上利用了未扰动系统的平滑性和扰动。非光滑系统的共振理论是在第二次世界大战前后 1940 年塔霍马窄桥倒塌后开始发展的。直到 1989 年，Glover 等人才提出了这一理论。事实证明，倒塌是由于悬索从下方限制了桥梁的运动造成的。这种单侧悬置向相关数学模型引入了不可微（非光滑）函数，使得经典平均理论不适用。这是将分岔理论的经典结果扩展到非光滑系统（通过放松光滑度假设）的起点。众所周知，非光滑系统中的共振能够表现出非光滑系统特有的行为，但在光滑近似中消失。演示这一点的机械系统是一个连接到移动皮带上的刚性固定梁上的物体。如果物体与皮带之间的摩擦是干摩擦，并且摩擦系数与外力幅度之间存在某些关系，则物体可能会周期性地粘在皮带上并表现出所谓的粘滑振荡。对干摩擦建模的符号函数进行平滑逼近会破坏这种现象。类似平均的方法不能在这里应用（即使是形式上的），因为相应的导数 (M_w)' 在 A_w 处不存在。这一观察结果导致了大量的研究工作，特别是解释粘滑运动的稳定性。正如上面在强迫摆动的例子中所讨论的，当一些非渐近稳定的周期解变得渐近稳定时，就会发生共振。因此，与平滑系统一样，稳定性和稳定性变化是分岔理论的一个重要方面，也是一个活跃的研究领域，特别是当平滑系统的技术和概念无法应用时。近来，对非光滑系统中共振振荡数学的研究受到了越来越多的关注，并且由于最近的进展，许多与非光滑机械和物理系统共振有关的技术相关的开放问题已经成为人们所关注的焦点。 正在解决。研讨会将关注此类具有工业相关性的问题。也许由于该学科的强劲增长，出现了以下情况：研究非光滑系统和稳定性的不同学校在彼此不知情的情况下取得了相似的结果。本次研讨会在提高认识并促进致力于非光滑系统振荡和稳定性的不同研究小组之间的讨论和合作方面做出了实质性的尝试。