Analysis of the implicit Euler time-discretization of passive linear descriptor complementarity systems

Abstract

paper 7269 This article is largely concerned with the time-discretization of descriptor-variable systems coupled to with complementarity constraints. They are named descriptor-variable linear complementarity systems (DVLCS). More speci cally passive DVLCS with minimal state space representation are studied. The Euler implicit discretization of DVLCS is analysed: the one-step non-smooth problem (OSNSP), that is a generalized equation, is shown to be well-posed under some conditions. Then the convergence of the discretized solutions is studied. Several examples illustrate the applicability and the limitations of the developments.

Dynamics modeling of gear transmissions with asymmetric load-dependent friction

“…1 with expressions in which the inertia varies between the forward-driving and backdriving cases. Here, it is shown that their results are consistent with the DAI (14) and DI (33). Under the condition |ρ| < 1, if vϕ(f u , f v , ρ) > 0, the DI (33) reduces to…”

“…Although a thorough mathematical analysis of such ill-posed cases is left outside the scope of the paper, one may interpret the empty-valued case as the situation where the triplet {f u , f v , v} in the DI (33) is not permitted and thus v should instantaneously reach zero with the infinite acceleration v = ±∞. The dual-valued case may be more debatable, but seeing the structure of λ in Fig.…”

“…Remark 5. Because the constraint (28c) can be converted into a complementarity form as in [32], the expression (28), which is equivalent to the DAI ( 14) and the DI (33), can be seen as a differential algebraic equation with a nonlinear complementarity constraint. Results of some previous studies, e.g., [33][34][35][36], may apply to the DAI ( 14) or the DI (33) for further theoretical analysis.…”

“…It might be possible to say that the presented DI ( 33) is more natural than (38) in that the internal friction appears as a single term, instead of influencing the inertial term. A more important point compared to (38) is that the DI (33), as well as the DAI (14), is a unified representation that stands in all the cases of backdriving, forward-driving, and the static friction, without involving switching between different expressions. This property makes ( 14) and (33) suited for computation, especially with a fixed timestep, and for rigorous mathematical analysis.…”

“…For simulation purposes, the continuous-time DAI representation (14) or its equivalent DI form (33) needs to be approximated by a discrete-time representation. The explicit discretization, such as the forward Euler scheme, is not applicable because, with such a scheme, the setvaluedness in the DAI (14) or the DI (33) acts only as a discontinuity, which results in chattering around v = 0 and fails to realize the static friction.…”