WORKSHOP: Resonance oscillations and stability of nonsmooth systems

有时，对一个机械或电气系统施加一个小的力，就会使稳定振荡的振幅大大增加。一个熟悉的例子是被推动的操场秋千：（平稳地）推动秋千（几乎）与其自然频率一致，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一个周期解（来自李雅普诺夫族）在时间周期扰动下变得渐近稳定，并且该解的振幅随着时间的增加而增加|w_0-w|减少。经典文献中共振振荡的严格证明是基于平均法，得到一个分岔函数M_w，其零点A_w对应于解，如果（M_w）的特征值有负的真实的部分，则渐近稳定。这里A_w表示振荡的幅度，其随着w接近w_0而增加。我们注意到，经典理论以本质的方式使用未扰动系统的光滑性和扰动。非光滑系统的共振振荡理论是在第二次世界大战期间，即1940年的TauritiNarrow桥倒塌之后开始发展起来的。直到1989年，格洛弗等人才证明坍塌是由于悬索从下面限制了桥梁的运动。这种单侧悬挂将不可微（非光滑）函数引入到相关的数学模型中，使得经典的平均理论不适用。这一直是一个efort的出发点扩展经典的结果在分歧理论的非光滑系统（通过放松光滑性假设）。非光滑系统中的共振振荡被认为能够表现出非光滑系统所特有的行为，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带上的刚性固定梁上的身体。如果物体和皮带之间的摩擦是干摩擦，并且摩擦系数和外力幅值之间的某些关系成立，那么物体可能周期性地粘在皮带上，并表现出所谓的粘滑振荡。采用符号函数建模的平滑近似，干摩擦破坏了这种现象。由于在A_w处不存在相应的导数（M_w）'，所以在这里（甚至形式上）不能应用类似平均的方法。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，就会发生共振振荡。稳定性和稳定性的变化是分叉理论的一个重要方面，也是一个活跃的研究领域，特别是当光滑系统的技术和概念不能应用时。对非光滑系统中共振振荡的数学研究最近受到了稳定的关注，并且由于最近的进展，关于非光滑机械和物理系统的共振的许多技术上相关的公开问题已经变得可以解决。研讨会将关注与工业相关的问题。也许是由于该主题的强劲增长，出现了这样一种情况，即研究非光滑系统和稳定性的不同学派在没有意识到对方的情况下取得了类似的结果。本次研讨会在创造意识，并激发讨论和不同的研究小组之间的合作非光滑系统的振荡和稳定性的实质性尝试。

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.