Recursive and Residual Algorithms for the Efficient Numerical Integration of Multi-Body Systems

According to a recent paper by Laulusa and Bauchau [1], Maggi's formulation is a simple and stable way to solve the dynamic equations of constrained multibody systems. Among the difficulties of Maggi's formulation, Laulusa and Bauchau quoted the need for an appropriate choice (and change, when necessary) of independent coordinates, as well as the high cost of computing and updating the basis of the tangent null space of constraint equations.In this paper, index-1 Lagrange's equations are first considered, including the not-so-rare case of having a singular mass matrix and redundant constraints. The existence and uniqueness of solution for acceleration vector and Lagrange multipliers vector is studied in a very simple way. Then, following Von Schwerin [2], Maggi's formulation is described as the most efficient way (globally speaking) to solve these index-1 equations. Next, an improved double-step method, which implements the matrix transformations of Maggi's formulation in an efficient way, is described. Finally, two large real-life examples are presented.

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“…(5) shall be used, taking into account that, if matrix q Φ has not got full rank, the notation of Eqn. (15) should be used instead.…”

Section: Range Space and Null Space Methodsmentioning

confidence: 99%

Efficient Solution of Maggi’s Equations

́n

Callejo

Hidalgo

et al. 2011

Volume 4: 8th International Conference on Multibody Systems, Nonlinear Dynamics, and Control, Parts a and B

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“…This approach corresponds to a state-space formulation of the equations of motion via the embedding or matrix-R method [10,12,20]. Introducing these expressions into the first velocity transformation (Eq.…”

Section: Semi-recursive Dynamic Formulationmentioning

confidence: 99%

“…Other approaches for the improvement of computational efficiency are based on the advances of parallelization technology [1,2,5,17,18]. García de Jalón et al [10,12,20] developed an alternative formulation of the equations of motion in terms of independent coordinates. By introducing a second velocity transformation, a set of independent relative accelerations was extracted from the dependent ones, which then allowed one to integrate the equations of motion using Maggi's equations in a fairly efficient way.…”

Section: Introductionmentioning

confidence: 99%

Efficient and accurate modeling of rigid rods

et al. 2016

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Ten years ago, an original semi-recursive formulation for the dynamic simulation of large-scale multibody systems was presented by García de Jalón et al. (Advances in Computational Multibody Systems, pp. 1-23, 2005). By taking advantage of the cut-joint and rod-removal techniques through a double-step velocity transformation, this formulation proved to be remarkably efficient. The rod-removal technique was employed, primarily, to reduce the number of differential and constraint equations. As a result, inertia and external forces were applied to neighboring bodies. Those inertia forces depended on unknown accelerations, a fact that contributed to the complexity of the system inertia matrix. In search of performance improvement, this paper presents an approximation of rod-related inertia forces by using accelerations from previous time-steps. Additionally, a mass matrix partition is carried out to preserve the accuracy of the original formulation. Three extrapolation methods, namely, point, linear Lagrange and quadratic Lagrange extrapolation methods, are introduced to evaluate the unknown rod-related inertia forces. In order to assess the computational efficiency and solution accuracy of the presented approach, a general-purpose MATLAB/C/C++ simulation code is implemented. A 15-DOF, 12-rod sedan vehicle model with MacPherson strut and multi-link suspension systems is modeled, simulated and analyzed.

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“…On the other hand, the stiffness matrix is constant in the case of linear deformation models, while it is non-constant in the case of geometric or material non-linearities. The computational efficiency can be improved for systems with non-constant mass matrices by using implicit time integration methods and simplified Jacobians [3]. However, in the case of explicit integration, it is desirable to have a constant mass matrix whenever it is possible.…”

Section: Introductionmentioning

confidence: 99%

Analysis of Thin Beams and Cables Using the Absolute Nodal Co-ordinate Formulation

2006

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The purpose of this paper is to present formulations for beam elements based on the absolute nodal co-ordinate formulation that can be effectively and efficiently used in the case of thin structural applications. The numerically stiff behaviour resulting from shear terms in existing absolute nodal co-ordinate formulation beam elements that employ the continuum mechanics approach to formulate the elastic forces and the resulting locking phenomenon make these elements less attractive for slender stiff structures. In this investigation, additional shape functions are introduced for an existing spatial absolute nodal co-ordinate formulation beam element in order to obtain higher accuracy when the continuum mechanics approach is used to formulate the elastic forces. For thin structures where bending stiffness can be important in some applications, a lower order cable element is introduced and the performance of this cable element is evaluated by comparing it with existing formulations using several examples. Cables that experience low tension or catenary systems where bending stiffness has an effect on the wave propagation are examples in which the low order cable element can be used. The cable element, which does not have torsional stiffness, can be effectively used in many problems such as in the formulation of the sliding joints in applications such as the spatial pantograph/catenary systems. The numerical study presented in this paper shows that the use of existing implicit time integration methods enables the simulation of multibody systems with a moderate number of thin and stiff finite elements in reasonable CPU time.

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Recursive and Residual Algorithms for the Efficient Numerical Integration of Multi-Body Systems

Cited by 47 publications

References 23 publications

Efficient Solution of Maggi’s Equations

Efficient Solution of Maggi’s Equations

Efficient and accurate modeling of rigid rods

Analysis of Thin Beams and Cables Using the Absolute Nodal Co-ordinate Formulation

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