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suggestions about hysteresis loops in transient from lubricated to dry friction

Mike Ciavarella's picture



dear Collegue Tribologist
  a quick question from a relatively newcomer in this particular
topic.  I have measured the attached 3 slides of hysteresis cycles
(Friction force - vs - sliding velocity) in reciprocating sliding. A standard massless Coulomb friction element would have this diagram looking like this:standard coulomb
Initially we have a drop of lubrication and we think we are under full
film lubrication, then it becomes practically dry.

The dry friction loops resemble standard Coulomb friction (see
attached image) although there are interesting deviations, due to high
speed range, memory effects, or dynamics in different behaviour when
speed is growing or slowing down.

The wet ones (friction is much lower obviously), resemble instead in
the central part a viscous damper (force proportional to v), then they
have some "bubble", pseudo-coulomb in a sense, and again in the high
speed.   Here, I am much less expert.hysteresis evaporating



Is all this very routine? evaporating 2  How would fit these curves?  Are there
links to softwares for identification, perhaps ready to use in Matlab?

Sorry for disturb, and happy easter.



Mike Ciavarella's picture


See here for more discussion.... 

Michele Ciavarella, Politecnico di BARI - Italy, Rector's delegate.

Mike Ciavarella's picture

Dear Michele,

these are very interesting data. Concerning the analysis, we often prefer to use the
friction force vs. load data to get full insight.
There is some software from the group of Rob Carpick, which might help.
The velocity dependence is of course much more challanging.
You might have a look on the so called Stribeck-curves. At present, I do not know if this
helps for your data. Nic Spencer at ETHZ might be another contact, who has experience with bio-related

Best wishes and have nice eastern


Michele Ciavarella, Politecnico di BARI - Italy, Rector's delegate.

Mike Ciavarella's picture

I don't believe the area inside the Q-v loop has any physical
significance. The energy dissipated per cycle is clearly the integral of
Q du, but you can't differentiate this with respect to time, because
the cyclic integral is not a function of time.

I presume the reason for plotting Q as a function of v is the idea that
the friction force might be a (unique) function of v, as would be the
case if the damping were purely viscous (including non-linear, which
would mean the function of v would be non-linear).

However, a hysteretic curve doesn't seem to
support this conjecture, since if it were true, it would imply that the
friction force at a given velocity depends not merely on velocity, but
either on the previous velocity history, or maybe on the acceleration,
or at least the sign of the acceleration. It's not easy to see how the
physics might support this, whereas the combination of Coulomb, viscous
and spring elements can easily generate curves like those you show.


Michele Ciavarella, Politecnico di BARI - Italy, Rector's delegate.

Mike Ciavarella's picture

it is
better to integrate the curves F - d, not F-v, although the difference
I suspect is very small, and the "physical" significance that you
refer to is a very subtle point.   The two "ways" to compute
dissipation are certainly very close.   So if I were a lazy phd
student, I would use the computation I had already done, and come to
you saying that I repeated all the calculations.   You would certainly
not spot my cheating, and so I would publish the paper with your name
with the wrong results :)  Sorry to be diseducative here.

More seriously speaking, you are suggesting us a "combination of
Coulomb, viscous and spring elements" as opposed to "hysteretic
elements" --- I do not understand what you mean:  at present we are
not using any model at all!

I am not sure we can easily generate our results with such a simple
model, except maybe in a loose least square sense, which for an
engineer is ok.

More to the point, we are REALLY interested in modelling the KEY
FEATURES that are changing from LUBRICATED regime to DRY FRICTION.

For this, I agree with you a "combination of Coulomb, viscous and
spring elements" would be enough.

But, as you generated the dry friction case quite well (can you give
us please the mathematica code you used?), can you show us how you
generate the lubricated case (presumably with much damping instead of
dry friction?).


Michele Ciavarella, Politecnico di BARI - Italy, Rector's delegate.

Mike Ciavarella's picture

To take a simple case where there is no spring, no damper and simple
Coulomb friction. Then Q=fP when v>0 and Q=-fP when v<0 and in a
sinusoidal cycle, the Q-v curve would not show any hysteresis. Would your hypothetical 'lazy PhD student' conclude that there was therefore no dissipation?


Michele Ciavarella, Politecnico di BARI - Italy, Rector's delegate.

Mike Ciavarella's picture

A good example!   Yes, in your case the lazy student finds no power
dissipated, and you would spot he is cheating!!   Since I was worried, I
asked you about this, before the student would do so many calculations
wrongly directed by me !

If the cyclic integral is not a function of time, I presume I have unwillingly defined something interesting.

fact, the student  should compute the F- d integral, (2) in my attached
notes.  However, what we are really interested is the TRANSITION between
the energy dissipated, i.e. the CHANGE in time of the integral, when we
move from lubricated to dry friction.
So integral 2 is indeed a function of time, and 3 is the change in time of 2.

the operation I am doing under the integral correct?  In that case, the
change is indeed the integral we were computing (the area under the F -
v curve, second of the 3) plus the first of 3.    In the case of the
lazy student simple Coulomb law, the second of 3 is zero, and the first
of 3 maybe not, since Q is a step function, and Qdot is a delta
function.  In fact the energy dissipated  per cycle of your example
should be

W = f P delta_max

and W-dot , assuming P is contant, but f is changing, W_new = f_new P delta_max
and delta_max is constant since it is imposed,

W-dot =  (f_new - f_old) P delta_max

so I expect to come from the first integral.

In conclusion, since the student is no lazy student, and I am insisting the 2 diagrams are BOTH interesting, I suggest we look at hysteresis in both, with a view to DEFINE a new meaning for the second plot, of INTEREST for the transient.

What do you think?

Michele Ciavarella, Politecnico di BARI - Italy, Rector's delegate.

Mike Ciavarella's picture

I was thinking of a generic model as in the attached figure, where the
block on the left is forced to move sinusoidally u0*cos(wt) and the
corresponding force is Q(t). k,c are linear spring and damper, and M is
the mass of the block on the right. Coulomb friction occurs between the
block of mass M and the ground, with some limiting friction force F0.

If you omit the damper and assume the mass is zero, the response is as
shown in my earlier figure. I attach the Maple file. I think it would be
easy enough to make some minor changes so as to separate slightly the
increasing and decreasing velocity curves in the saturation region, as
you see in your experiments.

If you perform experiments with the same values of u0 but different
frequencies w, [AND plot the results as Q against u, rather than against
v] you should get a measure of how important are damping and mass

But you could also just do a curve fit on a range of data to get a best
fit for M,c,k,F0 in this model, and then see how much error it implies
in other kinds of experiments. If you then find a particular scenario
where the errors from the simple model are large, then we have some hope
of adding more physics to get a better fit and hence also a better
physical model.



Michele Ciavarella, Politecnico di BARI - Italy, Rector's delegate.

Mike Ciavarella's picture

The cyclic integral makes the tacit assumption that we are in a steady
state, so that W is not a function of time. It would then follow that
\dot{W}=0, so the two integrals in your equation (3) should sum to zero.

Now you can consider a transient case where successive cycles are
different for some reason, but to make it rigorous, I think you would
then need to identify the start and end of the cycle you are
considering. For example, if the length of the cycle is T and you start
the integral at t0, the integral would be

W = int_{t=t0}^{t=T+t0} Qdu = int_{t=t0}^{t=T+t0} Q v dt

Now there is still no 't' explicit in this expression, but we CAN differentiate with respect to t0, obtaining

\dot{W} (defined as) dW/dt0 = Q(T+t0)v(T+t0) - Q(t0)v(t0)

but I don't think that is what you mean.

Michele Ciavarella, Politecnico di BARI - Italy, Rector's delegate.

Mike Ciavarella's picture

Suppose I do not care the exact path between two states, but I have a gradual evolution of shape and size, as indeed we have.

can obviously plot W as a function of time, and this is what Na Fan
will have to do, with the new attention that W is defined in F - d and
not as F - v plane.

This plot will give us the evolution of energy dissipated, and it is
a quantity different from the evolution of friction coefficient.

theory was that squeak started when f is critical, and a new one could
be when W is critical.  Obviously if W dependend MAINLY on f, then the
two theories are IDENTICAL.

We know that the theory f-critical is good up to a certain extent,
as we plot also the EVOLUTION of NOISE, and we see only partly
reproducible results.

But since the SHAPE of hysteresis is changing, maybe some OTHER factors, other than f, are playing a role for NOISE.

Naturally, if we had a model, as you suggested in your previous
email, we would also PREDICT which factors affect W, and the SHAPE of hysteresis.

We could plot the EVOLUTION of ALL of these FACTORS, to see which one correlates better with the EVOLUTION of NOISE.

Now I have defined the REAL problem we want to study, how would you best approach the various items?


PS. The lazy student could say that PERHAPS it is the change of shape of hysteresis curve in the F - v plane which matters, but in fact, it seems that is tending to a paradox.

the "wrong dissipation" factor we were computing so far, we are saying
now is ZERO for pure COULOMB friction, which is NOT too remote in DRY

Hence, the "wrong dissipation" factor is in fact probably DECREASING, and it would be an interesting quantity to control.

How to define I do not know, but it seems to be DEVIATION from COULOMB behaviour. 

Unfortunately I haven't seen an equivalently simple model for the
SIMPLEST lubricated regime, to completely predict what will happen.

Michele Ciavarella, Politecnico di BARI - Italy, Rector's delegate.

Mike Ciavarella's picture


see the main post.


What do you think of these powerful results? Would they help us?

Michele Ciavarella, Politecnico di BARI - Italy, Rector's delegate.

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