Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts
I can remember that, when I started out as a graduate student in the
late 1980’s, we were just coming to the end of the era where there was
a lack of communication between the science of the West and that of
the Soviet and Eastern European countries. This schism was especially
widely felt in dynamical systems. Techniques like the Melnikov method,
Shilnikov coordinates and the whole qualitative theory of dynamical
systems from the Andronov school were being rediscovered in the
West. There was also a strong tradition in mechanics, especially
nonsmooth mechanics of impacts and friction especially in Poland, the
former Czechoslovakia, Hungary etc. The Eastern European approach
combined mathematical rigor with flair in geometry, the qualitative
theory of dynamical systems and their associated approximation
methods, whereas the Western literature seemed, at least to me, to
have been overly mired in either functional analysis and so-called
rational mechanics, or in formal asymptotic expansions.
Of course, all that has changed in the intervening three decades.
Through the Internet, all science is available at the click of a
button (although sometimes behind expensive paywalls). Conferences
take place all over the world on topics that include many
cross-fertilizations between traditions and
specializations. International scientific barriers really don’t
exist. And yet, it seems to me that there is still a temptation to
revert to national or regional stereotypes and ways of thinking. We
can all accidentally be guilty of citing our friends and colleagues
from whom we first learned a technique or concept rather than the true
originator of the idea.
This monograph weaves an interesting path through theory and
application of methods from dynamical systems theory to systems with
friction and impact. The perspective is very much from that of the
Polish and Slovakian tradition and the large contribution to these
topics made by the authors, their collaborators and their forbears. It
thus provides a much needed alternative perspective to other books,
papers and review articles on related topics.
It is important in my view that these alternative perspectives are
added to the literature, because, the definitive theory of nonlinear
systems with so-called harsh nonlinearities (either multi-scale or
nonsmooth) has yet to emerge. This book provides an interesting
addition to the attempts to unify this topic, specifically from the
point of view of low degree-of-freedom rigid-body systems. There is
also a nice juxtaposition between practical experimental
implementation and mathematical rigor. The emphasis is on
understanding the origin of chaotic dynamics, and other irregular
behavior, and how this can be identified, analyzed and ultimately
controlled. The book moves from experiments, modeling, system
identification, analysis of ensuring motion, to control of systems
with discontinuities. This latter topic seems to me to be the most
novel. There are also numerous references to the literature, and an
attempt is made to maintain a historical narrative.
The main strength of the book is that, after the first general chapter
on friction modeling, each chapter reads like a self-contained
study. I would imagine that many of the chapters could form the basis
of an advanced student project or topic for independent study. That
being said, this is a monograph, and not primarily a text book.
However, this strength is also a potential weakness. It is episodic
rather than comprehensive; there is no attempt to present a general
theory. It ends abruptly, without any attempt to summarize or look to
the future. Some of the chapters appear a bit thin and only loosely
connected to the general thrust of the book. While Chapter 1 contains
many references to the literature on different friction models, some
of the later chapters miss the opportunity to survey what can be a
large literature. A striking example is the thin Chapter 5 on
Bifurcations in Planar Discontinuous Systems which makes no attempt to
be general or to mention the huge literature on this topic.
Also, despite the ample bibliography, the literature that is surveyed
is rather patchy. Perhaps, given the range of topics presented and the
popularity of their study across mathematics, engineering and applied
science, an exhaustive reference list would be impossible.
Nevertheless, I found the choices to be idiosyncratic. For example,
on friction laws, there is nothing on the so-called rate-and-state
formalism , from which many other friction laws can be derived. I also
think the ground-breaking discontinuity-mapping ideas due to Nordmark
deserve mention [2,3]. But maybe I am being guilty of my own bias. There
is certainly work here with which I was not familiar, and references I
was unaware of. And that, I guess, is the point.
In summary then, this monograph presents an enticing alternative view
of a field that is very much in progress. It is not the definitive
textbook, but would be a welcome addition to the bookshelf of
researchers and advanced students trying to make sense of mechanical
systems with discontinuities in the form of friction or impact.
- Cabboi, A., Putelat, T., and Woodhouse, J.,
The frequency response of dynamic friction: Enhanced rate-and-state models.
J. Mech. Phys. Solids, 92 , 210-236 (2016).
- Nordmark, A.B.,
Existence of periodic orbits in grazing bifurcations of impacting mechanical oscillators
Nonlinearity 14:6.1517-1542 (2001).
- Dankowicz, H, and Nordmark, A.B.,
On the origin and bifurcations of stick-slip oscillations,
Physica D 136:3-4,280-302 (2000).