By Ngoc Son Nguyen
The extension of collision types for unmarried affects among our bodies, to the case of a number of affects (which occur while a number of collisions happen even as in a multibody method) is a problem in sturdy Mechanics, end result of the complexity of such phenomena, even within the frictionless case. This monograph goals at offering the most a number of collision principles proposed within the literature. Such collisions quite often happen in granular fabrics, the easiest of that are made from chains of aligned balls. those chains are used during the ebook to investigate numerous a number of influence principles which expand the classical Newton (kinematic restitution), Poisson (kinetic restitution) and Darboux-Keller (energetic or kinetic restitution) ways for influence modelling. The surprise dynamics in a variety of different types of chains of aligned balls (monodisperse, tapered, adorned, stepped chains) is thoroughly studied and proven to depend upon a number of parameters: restitution coefficients, touch stiffness ratios, elasticity coefficients (linear or nonlinear strength/ indentation relation), and kinetic angles (that rely on the mass ratios). The dissipation and the dispersion of kinetic strength in the course of a a number of influence are necessary modelling, and are quantified with appropriate indices. specific awareness is paid to the facility of the provided legislation to properly expect the wave results within the chains. Comparisons among many numerical and experimental effects are proven, in addition to comparisons among 4 diversified influence legislation when it comes to their respective skills to properly version dissipation and dispersion of energy.
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Extra info for Multiple Impacts in Dissipative Granular Chains (Lecture Notes in Applied and Computational Mechanics)
As a consequence, Moreau’s impact law spans the same horizontal axis in the admissible domain in Figure 5 Other cases may be considered as pathological initial conditions, realizable only with speciﬁc experimental apparatus. 6 as Newton’s impact law, that’s why this line is called ”Moreau’s line”. 6). The limitation of Moreau’s impact law when applied to the rocking block system have been studied in detail in , where it is shown that the so-called rocking motion cannot be described6 . 1. How can the global dissipation index e be identiﬁed from independent experiments?
It is clear that the energetical consistency will imply some kind of smallness of the restitution matrix. However, the Delassus’ matrix introduces some distortion compared to the single impact case in which one concludes that the restitution coeﬃcient must be less than or equal to one. Actually, as noted in , even if E is a diagonal matrix with nonnegative entries ei , imposing 0 ≤ ei ≤ 1 is not suﬃcient to guarantee that −E T (W T M −1 W )−1 E + (W T M −1 W )−1 is positive semi deﬁnite. It is noteworthy that the copositivity of this matrix is suﬃcient to guarantee that TL ≤ 0 since γ − ≤ 0.
As a conclusion, the assumptions considered in Newton’s and Poisson’s restitution laws are not justiﬁed when the wave eﬀect in a multibody system is high. the wave propagation phenomenon makes hard to justify. 1. Newton’s impact law clearly belongs to class (i), whereas Poisson’s impact law belongs to class (ii) since it distinguishes compression and expansion phases. The above calculations prove that the models in class (ii) encapsulate more information than those in class (i), as expected. When the mass matrix is constant and there is no friction, the Darboux-Keller dynamics can be explicitly integrated and take the form of algebraic impact dynamics.