Download e-book for iPad: Bifurcation and Chaos in Nonsmooth Mechanical Systems (World by J. Awrejcewicz, Claude-Henri Lamarque

By J. Awrejcewicz, Claude-Henri Lamarque

ISBN-10: 9812384596

ISBN-13: 9789812384591

This quantity provides the theoretical body for learning lumped nonsmooth dynamical platforms: the mathematical tools are recalled, and tailored numerical tools are brought (differential inclusions, maximal monotone operators, Filippov concept, Aizerman theory). instruments on hand for the research of classical gentle nonlinear dynamics (stability research, the Melnikov procedure, bifurcation eventualities, numerical integrators, solvers) are prolonged to the nonsmooth body. Many types and purposes coming up from mechanical engineering, electric circuits, fabric habit and civil engineering are investigated to demonstrate theoretical and computational advancements.

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Extra resources for Bifurcation and Chaos in Nonsmooth Mechanical Systems (World Scientific Series on Nonlinear Science: Series A) (v. 45)

Sample text

The friction and inertial forces act on each other, but they always have the same direction and the first tries to stop the body faster. Knowing that we can also discuss the slip -» slip, stick -> slip and slip -> stick phenomena. The following hypothesis can be formulated. If \£(rn\<\mTn]mT[iiTt)]\, (1-68) then x(rf) = 0 (transition slip -¥ stick). 67) holds and we have slip -* slip transition with exactly defined jumps 20 Bifurcation and Chaos in Nonsmooth Mechanical Systems of accelerations.

Such a frame is easily obtain from fundamental relation of dynamics and M is then derived from mass matrix. 7 V£ G D(d),3\X € W^fato + T ] , R P ) SO that { for almost every t G [t0, t0 + T},X(t) + Md

This is why, it is much more convenient to use the following numerical scheme. 2 (Partial Implicit Euler Scheme) The scheme is described by the following recursion: Xo = Xo, Y n > 0> -X (3"3) ^n l_^n+F{t^Xn)+A(Xn+i) = Q Here, one must simply compute the inverse graph (/ + hA)"1. 4) n > 0, Xn+l = (I + hA)-1 (Xn + hF(tn,Xn),) where the term Xn + hF(tn,Xn) is evaluated at each iteration. The key point for building these numerical schemes is to make the implicit choice A(Xn+i): the properties of A imply that (I + hA) is invertible.

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Bifurcation and Chaos in Nonsmooth Mechanical Systems (World Scientific Series on Nonlinear Science: Series A) (v. 45) by J. Awrejcewicz, Claude-Henri Lamarque

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