Multi-component dry friction
and the problem statement correctness of a rigid body rolling
V.Ph.Zhuravlev
Institute for Problems in Mechanics of
RAS
k.1, 101, Vernadskogo pr.,
The question of correctness of dry friction mathematical models in non-holonomic
Mechanics was earlier treated in [1, 2]. In [3] they estimate the error which
occurs if we substitute a non-holonomic approximation for the exact model
(within the framework of the Coulomb dry friction law). In the current paper we
revise the problem stated in [1, 2], and we also discuss the details which are
not covered in [1-3].
It is possible to demonstrate if it
is correct to replace the dry friction with a mere condition of no sliding in
problems of a rigid body rolling by a simple example of a homogeneous ball
rolling. Since this relates to sliding friction only, the rolling friction can
be neglected.
In this work it is shown, that the model with
the point contact is incorrect because the case of an arbitrarily small contact
spot differs from the case of a point contact as much as desired.
The replacement of a circular contact by a
point contact for simplicity reasons only is too rough an approximation since a
non-holonomic model has a smaller dimension. As is seen from the above, such an
approximation gives a very poor answer to the question of the time of sliding.
The question of how close to each other are the
solutions in these models with respect to the angular and linear velocities is
studied in [3] by means of Tikhonov theorem. It is demonstrated that if e R 0 the angular and linear velocities
in the two models tend to each other on a finite segment of time. Such a
justification is often sufficient in practice.
With this in view, we have two other points to
add.
In the first place, it doesn't seem relevant to
study such issues as the first integrals, invariant measure, tensor invariants
and the like in problems of body rolling taking into account that the non-holonomic
approximation is rather rough unless these issues are studied for purely
educational purposes.
Secondly, it seems reasonable to change the
focus in the non-holonomic model justification fulfilled in [3]. As it follows
from property 2 the coefficient of a pseudo-viscosity (a derivative dF/dv
in the zero) is proportional to 1/u =
1/ewz. The
proximity to Coulomb law is determined by u
R 0. This can be achieved by either e R 0 (as is done in [3]) or by wz R 0 (that is in much more natural
way). Since in concrete problems a researcher cannot dispose of the value of e,
the condition e R 0 must be understood on an
artificial supposition that we have a set of balls with a sequence decreasing
value of e. On the contrary, the velocity wz is in our disposal and we can make it as small
as we need. In experiments with wheel robots [7] it is this very variable that
is small. This explains why the experiment and the theory constructed on the
non-holonomic model agree.
1.
Ju.I.Neimark,
N.A.Fufaev. Dynamics of a non-holonomic systems. English
2.
N.A.Fufaev. On an idealization of
contact surface as a point contact in a rolling problems. PMM (Journal of
applied mathematics and mechanics), v.30, Issue 1, 1966.
3.
A.P.Ivanov.
Comparative analysis of friction models in dynamics of a ball on a plane. Rus.
J.Nonlin.Dyn., 2010, 6(4), 907-912.
4.
P.Contensou. Couplage entre frottement de glissement et frottement de pivotement dans la
théorie de la toupie. Kreiselprobleme Gyrodynamics: IUTAM Symp. Celerina, 1962, Berlin etc.,
Springer, 1963, 201-216.
5.
V.Ph.Zhuravlev. On a
model of dry friction in a problem of a body rolling. PMM (Journal of applied
mathematics and mechanics), v.62, Issue 5, 1998.
6.
V.V.Andronov.
V.Ph.Zhuravlev. Dry
friction in mechanical problems. M.-I., SRC "Regular and chaotic dynamic",
Institute of computer investigations, 2010, 184.
7.
V.N.Belotelov, Ju.G.Martinenko. Space motion control of turned over
pendulum on a wheel set. MTT, v.6, 2006, 10-28.
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