By Dr. Remco I. Leine, Professor Dr. Henk Nijmeijer (auth.)

ISBN-10: 3540443983

ISBN-13: 9783540443988

ISBN-10: 3642060293

ISBN-13: 9783642060298

This monograph combines the information of either the sector of Nonlinear Dynamcis and Non-smooth Mechanics and provides a framework for a category of non-smooth mechanical structures utilizing suggestions from either fields. over the past a long time, the Non-smooth Mechanics neighborhood has constructed a formula of non-smoth platforms, the mathematical necessities (Convex research) in addition to devoted numerical algorithms. "Dynamics and Bifurcations of Non-smooth Mechanical platforms" provides those advancements in a complete approach and opens the sphere to the Nonlinear Dynamics neighborhood. This ebook addresses researchers and graduate scholars in engineering and arithmetic drawn to the modelling, simulation and dynamics of non-smooth platforms and nonlinear dynamics.

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**Additional info for Dynamics and Bifurcations of Non-Smooth Mechanical Systems**

**Sample text**

Indicator function and support function [46]. This is exactly the deﬁnition of the normal cone at C. The subdiﬀerential of the indicator function at x ∈ C is therefore the normal cone of C at x, ∂ΨC (x) = NC (x). 13 (Conjugate Function). Let f be a convex function. The function f ∗ is called the conjugate function of f and is deﬁned as f ∗ (x∗ ) = sup{xT x∗ − f (x)}. 14 (Support Function). Let C be a convex set. The conjugate function of the indicator function ΨC is called support function, ΨC∗ (x∗ ) = sup{xT x∗ − ΨC (x)} x = sup{xT x∗ | x ∈ C}.

1 but with Coulomb friction at the contact point. We denote the relative velocity of point 1 with respect to point 2 along their tangent plane by γT . e. 2 Contact Laws 51 between the bodies imposes a force λT along the tangent plane of the contact point. 15) where μ is the friction coeﬃcient and λN is the normal contact force. e. γT = 0, then the bodies purely roll over each other without slip. Pure rolling, or no slip for locally ﬂat objects, is denoted by stick. When the bodies stick, then the friction force must lie in the interval −μλN ≤ λT ≤ μλN .

We start with a single hyper-surface. The state space Rn is split into two subspaces V− and V+ by a hyper-surface Σ such that Rn = V− ∪ Σ ∪ V+ . The hypersurface Σ is called the switching boundary and is deﬁned by a scalar switching boundary function h(x) 2 . The state x is in Σ when h(x) = 0 ⇐⇒ x ∈ Σ. 10) The subspaces V− and V+ and switching boundary Σ can be formulated as V− = {x ∈ Rn | h(x) < 0}, Σ = {x ∈ Rn | h(x) = 0}, V+ = {x ∈ Rn | h(x) > 0}. 11) The switching boundary is assumed to be continuous but can be allowed to be non-smooth.

### Dynamics and Bifurcations of Non-Smooth Mechanical Systems by Dr. Remco I. Leine, Professor Dr. Henk Nijmeijer (auth.)

by John

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