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# Fontanela instability in non-linear oscillators with cyclic symmetry

I wanted to bring attention to the very important Fontanela et al recent results in vibrations of cyclic structures like in turbine engines.

It is well known mistuning occurs in a system which is linear but a little inhomogeneous (it is a form of Anderson localization for which Anderson got the Nobel prize, but was found independently in turbine engines by Whitehead in 1966). This has tremendous implication for the design of turbine engines, because very small deviations from identical mass or stiffness amplify the response by a large factor which needs to be taken into account in design, and after 50 years there has been no solution to this problem. It is even difficult to "measure" this effect of mistuning, and indeed most design people just assume Whitehead simple closed form equation for the worst case scenario, for which the amplification grows as the square root of the number of blades.

Instead Fontanela et al. have shown that transition from travelling wave modes to localised vibration can occur also for non-linearities (see attached), even in a perfectly homogeneous structure.

Whether this will make situation even worse, or better is unclear at present. Or maybe mistuning never really existed, and it is all instead about solitons and breathers?

The paper is about to appear in JSV.

Attachment | Size |
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JSV_2017.pdf | 2.41 MB |

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## a related problem is that of non-linear damping by rate friction

A related problem is that of a chain of oscillators where there is no forcing term (but a belt self-excites the chain), and non-linearity comes from rate-dependent friction. Here we haven't found solitons yet, but certainly multistability and a rich pattern resembling chaos. We are now in the process of studying more in details how fronts of vibrations propagate or stop.Multiple spatially localized dynamical states in friction-excited oscillator chains

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Highlights

A friction-excited oscillator chain is studied with periodic boundary conditions.

Multiple, spatially localized vibrating states have been found.

Multiplicity can explain the lack of repeatability of friction-induced vibrations.

Different initial conditions lead to different localized patterns.

In the bifurcation diagram “snaking-like” bifurcations have been obtained.

Abstract

Friction-induced vibrations are known to affect many engineering applications. Here, we study a chain of friction-excited oscillators with nearest neighbor elastic coupling. The excitation is provided by a moving belt which moves at a certain velocity vd while friction is modelled with an exponentially decaying friction law. It is shown that in a certain range of driving velocities, multiple stable spatially localized solutions exist whose dynamical behavior (i.e. regular or irregular) depends on the number of oscillators involved in the vibration. The classical non-repeatability of friction-induced vibration problems can be interpreted in light of those multiple stable dynamical states. These states are found within a “snaking-like” bifurcation pattern. Contrary to the classical Anderson localization phenomenon, here the underlying linear system is perfectly homogeneous and localization is solely triggered by the friction nonlinearity.

Keywords

https://authors.elsevier.com/c/1WBWO,Whn45Te