Abstract
Application of the MEBDFV code in multibody dynamics is discussed. The solver is based on the modified extended backward differentiation formulae of Cash. It is especially suited for the solution of differential-algebraic equations with time- and state-dependent mass matrix. Such a case can occur when motion of a multibody open-chain system is described within the classical Lagrangian formalism, which still has some advantages. An outline of the numerical algorithm is given. As an example, a simplified mathematical model of the human head-cervical spine system is presented. In a numerical experiment the model is used to analyze motion of the head–neck during rear-end impact which may lead to whiplash injury. The obtained results are compared to literature data. Performance of the MEBDFV solver is examined in terms of algorithmic energy conservation. The test is based on an appropriately formulated ‘kinetic energy–work’ relation.
Highlights
Multibody dynamics is one of the most rapidly developing branches of computational mechanics
Like other codes based on modified extended backward differentition formulae (MEBDF), designed for initial value problems in ODEs and differential-algebraic equations (DAEs) of various forms, the MEBDFV solver is available on the Internet [22]
The code is useful for open-chain systems modeled via the classical Lagrangian formulation, since such an approach leads to a set of implicit differential equations or DAEs with non-constant mass-matrix
Summary
Multibody dynamics is one of the most rapidly developing branches of computational mechanics. By the term ‘standard formulation’ the authors mean applying the Lagrange’s equations of the second kind, where all the generalized coordinates are independent and the dynamic equations do not involve constraint forces, e.g. the ones which act at the joints connecting rigid members. In such a case the standard approach can be regarded as an embedding technique. In the method of Lagrange multipliers, for example, the primary model can be extended by introducing constraints equations and undetermined multipliers which are related just to the new reaction forces [11,12] This procedure, in turn, corresponds to the augmented formulation. In such a case the initial conditions (4) refer to the multipliers λ too
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