Geometric methods and formulations in computational multibody system dynamics

Abstract

Multibody systems are dynamical systems characterized by intrinsic symmetries and invariants. Geometric mechanics deals with the mathematical modeling of such systems and has proven to be a valuable tool providing insights into the dynamics of mechanical systems, from a theoretical as well as from a computational point of view. Modeling multibody systems, comprising rigid and flexible members, as dynamical systems on manifolds, and Lie groups in particular, leads to frame-invariant and computationally advantageous formulations. In the last decade, such formulations and corresponding algorithms are becoming increasingly used in various areas of computational dynamics providing the conceptual and computational framework for multibody, coupled, and multiphysics systems, and their nonlinear control. The geometric setting, furthermore, gives rise to geometric numerical integration schemes that are designed to preserve the intrinsic structure and invariants of dynamical systems. These naturally avoid the long-standing problem of parameterization singularities and also deliver the necessary accuracy as well as a long-term stability of numerical solutions. The current intensive research in these areas documents the relevance and potential for geometric methods in general and in particular for multibody system dynamics. This paper provides an exhaustive summary of the development in the last decade, and a panoramic overview of the current state of knowledge in the field.

A Geometric Numerical Integration of Lie-Poisson System for Ideal Compressible Isentropic Fluid

“…This algorithm can be applied in similar situation, i.e., problems with semidirect product phase spaces. For basic mathematical background we refer the reader to [7]; and for some geometric numerical approaches to [8,9]. In Section 2 we have discussed some preliminaries and in section 3 the formulation of ideal compressible isentropic fluid problem is presented in detail.…”

The Lie group Euler methods of multibody system dynamics with holonomic constraints

“…With the successful developments and applications of symplecticity and energy methods in simulations of dynamic systems, the investigation of Lie group methods has become another research focus. [5][6][7][8][9][10] The Lie group method 11,12 was originally employed in computational solid dynamics to realize specialized orthogonal group (orientation) preserving for numerical stability. 13,14 It had been studied systematically in 1990s for matrix ordinary differential equations (ODEs) using the canonical coordinates of the first kind (CCFK) and canonical coordinates of the second kind (CCSK), based on which the Munthe-Kaas 15 (MK) method and the Crouch-Grossman 16 (CG) method were proposed creatively.…”

“…Arnold and coworkers [5][6][7] have extended the generalized-alpha method to solve DAEs on Lie group with indices 3 and 2. Mu¨ller and coworkers [8][9][10] have extended the explicit Runge-Kutta method with constraint projection method to solve DAEs with index 1 on Lie group.…”

Untersuchung der Strömungsformen im Spalt zwischen einer rotierenden Scheibe und einer ruhenden Gegenscheibe

“…Mit dieser Annahme hat Soo [3] den Zusammenhang für das dimensionslose Antriebmoment C M und der Reynold-Zahl Re entwickelt. In einer ähnlichen Arbeit hat Müller [5] 1971 die Geschwindigkeitsverteilung im laminaren Bereich zwischen einer rotierenden und einer ruhenden Scheibe berechnet. In einer ähnlichen Arbeit hat Müller [5] 1971 die Geschwindigkeitsverteilung im laminaren Bereich zwischen einer rotierenden und einer ruhenden Scheibe berechnet.…”

“…Köhler [4] beschreibt in seinen Untersuchungen eine Berechnung der Radial-und Axialgeschwindigkeitsverteilung für die Strömung zwischen zwei parallelen, rotierenden Scheiben. In einer ähnlichen Arbeit hat Müller [5] 1971 die Geschwindigkeitsverteilung im laminaren Bereich zwischen einer rotierenden und einer ruhenden Scheibe berechnet. Um die Einzelverluste der Turbomaschinen berechnen zu können, untersuchte Schultz [6] den Reibungswiderstand der rotierenden Scheibe im Gehäuse.…”