Collisions of multibody systems

Abstract

This paper presents a computational procedure for studying collisions of multibody systems. It combines the procedures of impact analysis and the methods of modern multibody dynamics (including the use of Kane's equations, lower body arrays, generalized speeds, and differentiation algorithms). By assuming the duration of impact to be very short and that the con®gurations of the systems have only small changes during the colliding process, we can automatically generate the governing dynamical equations. By using Newton's impact law, the partial velocities of the contact points determine impulse force components. Then by back substituting into the governing equations, the changes of velocities during the collision, the components of internal impulses, and the subsequent motions of the systems after collision may be determined.

Dynamical Systems With Active Singularities of Elastic Type: A Modeling and Controller Synthesis Framework

“…[26, p. 71] . Therefore, there exists (24) satisfying the equality in (22). The inequality in (22) follows from the representation:…”

“…Then, the limit function is continuous with respect to all variables and Lipschitz in , uniformly on as , and for all sufficiently large there exists given by (22) such that as , Proof: Due to the continuity of it is obvious that As follows from (14), (20) satisfies the Lipschitz condition with respect to . The continuity of the ordinary differential equation solution with respect to the parameters (cf.…”

“…In order to formulate the next theorem, introduce equation (19) further referred to as the controlled infinitesimal dynamics equation, where (20) As it will be seen further in the paper, (19) and its solution constitute the key analytical/computational tools for the controller synthesis in the singular phase and full limit system representation, respectively. Introduce also two time instants in the fast time defined as (21) and (22) The quantity in (21) will be shown below to be the time moment, in the fast time , of the limit system (19) exit from the constraint. This quantity depends on the original system state at the time moment of the collision onset through setting initial conditions for (19) as and it is found by solving (19) until (21) is met.…”

“…Satisfying these conflicting requirements turns out, however, to be highly nontrivial. Indeed, examining the first requirement, for example, in the context of modeling of mechanical collisions, known to typically result in the velocity jumps in the system motion, one finds that the conventional techniques, such as complementarity, make use of either the so-called collision mapping or the restitution law, both of which give an expression for the velocity after the impact in terms of the velocity and position before the impact [2], [31], [17], [18], [22], [60]. These tools, however, do not provide mathematical structure rich enough to incorporate the impulsively controlled contact forces.…”

Numerical and experimental studies on impact dynamics of a planar flexible multibody system

“…When the impact happens, the integration of dynamic equations is halted and the momentum balance is performed to obtain the generalized velocity of the system after impact. Then the integration of dynamic equations is restarted [5][6][7][8][9]. Although the impulse momentum methods can achieve high computational efficiency, however, the assumption that the system configuration does not change during impact is no longer valid when the impact duration is enough long.…”