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Can you imagine a mechanical system that is stable without friction, but suffers a flutter instability when friction is 'added'?

Davide Bigoni's picture

Have you ever seen a structure subject to flutter instability?

Can flutter instability be connected to dry friction?

We provide positive answers to the above questions, watch the video at

More information about my research activity can be found in
More information about our experiments can be found in

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Zhigang Suo's picture

Thank you, Davide, for this fascinating video.  Phenomena of dynamic structural instability are photogenic.  I look forward to watching your next production.

Davide Bigoni's picture

Nice to hearing from you Zhigang! Thanks for your interest in my research activity. I hope to meeting you soon, somewhere!

This mechanical system is subjected to an axial friction load. This problem may be linked to the flutter instability of a beam subjected to a follower load.

Davide Bigoni's picture

It is exactly this! It is a two-degree-of-freedom beam subject to a follower laoad induced by friction.

Mike Ciavarella's picture

Paradoxes of dissipation-induced destabilization or who opened Whitney's umbrella? O.N. Kirillov,  F. Verhulst

ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik

Volume 90, Issue 6, pages 462–488, June 2010


The paradox of destabilization of a conservative or non-conservative system by small dissipation, or Ziegler's paradox (1952), has stimulated an ever growing interest in the sensitivity of reversible and Hamiltonian systems with respect to dissipative perturbations. Since the last decade it has been widely accepted that dissipation-induced instabilities are closely related to singularities arising on the stability boundary. What is less known is that the first complete explanation of Ziegler's paradox by means of the Whitney umbrella singularity dates back to 1956. We revisit this undeservedly forgotten pioneering result by Oene Bottema that outstripped later findings for about half a century. We discuss subsequent developments of the perturbation analysis of dissipation-induced instabilities and applications over this period, involving structural stability of matrices, Krein collision, Hamilton-Hopf bifurcation, and related bifurcations.

Davide Bigoni's picture

Hi Michele! Nice to hear from you. Thanks for the suggestion.

Hi everyone:

My first post.

Sometimes, a decent model for a wheel is a no slip constraint orthogonal to the wheel and zero friction along the wheel, as opposed to a follower force.

So I wonder how the following perhaps unconventional model for this system will behave: model the wheel in the double pendulum, not as a follower force orthogonal to the wheel, but as a no-slip constraint orthogonal to the wheel. This would result in a non-holonomic system, with \dot{theta_1} and \dot{\theta_2} being related by a non-integrable velocity constraint.

Does the above picture help the stability analysis? Is it still meaningful? etc.

Manoj Srinivasan
Mechanical and Aerospace Engineering
The Ohio State University

PS. On modeling wheels as non-holonomic constraint, see for instance:

Non-Holonomic Stability Aspects of Piecewise-Holonomic Systems. Ruina, A. Reports on Mathematical Physics, V. 42, No 1/2, P. 91-100, 1998

Davide Bigoni's picture

Sorry, but I do not understand the point. The system has been designed to provide a follower force. For this reason, the wheel in the experiment is forced to slip against a plate and the slip occurs in reality, as it can be easily understood from the acoustic emission during the test in the video (but we have much more experimental proof of this, as can be found in our paper). Therefore, I do not see where the slip constraint enters the problem.

Davide - I entirely agree with you that the no-slip limit I posed does not really directly model the system you are considering, which I agree does slip (hence the follower force). I was just posing what appeared to me a related problem, which might perhaps have some interesting features on it's own, which may or not be related to your follower force problem in some limit. I should have made that clearer.


Davide Bigoni's picture

Hi Manoj, now I understand. Yes, it might be that working on related probelms with the tool we have invented something new can be shown.


It recently occured to me that the no-slip constraint (orthogonal to the wheel-direction) I suggested as an alternate simple model of the wheel will also have a follower friction force.

But this follower force --

(1) will not have a constant magnitude; the magnitude will be whatever it takes to enforce the no-slip constraint.

(2) will perform 'exactly' zero work on the system, neither positive or negative, because the slip is zero in the direction of the force. That is, no dissipation.

I just thought I would point out these properties. But I have not thought about their consequences.


Davide Bigoni's picture

The force you are speaking about is responsible of the so-called "Shimmy instability", you may find calculations on this in the Ziegler's book mentioned in my article.

Oleg Kirillov's picture

Since the system experiences both flutter and divergence the force is not pure tangential follower force, right?

Why not to include more links and do experiments with m>2 link pendulum?

Why also not to take a flexible rod with the wheel as an analogue of the Beck column?

Then with the m-link pendulum with tunable stiffnesses in the joints or with the rods of different shape one can do experimental structural optimization, which is still a challenge in non-conservative case, see e.g.


Oleg Kirillov's picture

There should be a connection with the shimmy phenomenon

Davide Bigoni's picture

Your question: "Since the system experiences both flutter and divergence the force is not pure tangential follower force, right?"
With pure tangential follower force there is a flutter region, terminated at higher load by a divergence semi-infinite interval. In our experiment the only possibility
of having an orthogonal force is related to the rotational inertia of the wheel. We have evaluated this effect in different ways: 1) with numerics and 2) changing
several wheels. The results seem unaffected. Moreover, if one adds a small orthogonal force in the Ziegler column, the scenario remains unchanged.
Your Questions: "Why not to include more links and do experiments with m>2 link pendulum?" and "Why also not to take a flexible rod with the wheel as an analogue of the Beck column?"
We can certainly do more. Our first attempt was motivated by the desire of proving in an indisputable way that friction is related to flutter, so that we have selected
the simplest setting. We have tryied a Beck's column version with a beam of rectangular cross section, but we had a mixture of flutter plus flexural-torsional
instability. However, it should be interesting to experiment on the Beck's column (where there is the advantage of not having hinges).
One thing that we are trying now is to experimentally proof the "paradox". Our published results are already in favour of this effect (since the version with
viscous hinges fits better the experiments), but a real proof requires much more.
Now we are trying with a system where a viscous damper can be added/removed, but it's a difficult stuff.
Our experiments are more difficult than they appear!
Finally, there should be some connection with shimmy instability, except that there is no sliding in that case, plus the instability depends on the velocity (which is
not the case for flutter).

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