By Bernd Simeon
This monograph, written from a numerical research viewpoint, goals to supply a finished therapy of either the mathematical framework and the numerical tools for versatile multibody dynamics. not just is that this box completely and speedily turning out to be, with a variety of functions in aerospace engineering, biomechanics, robotics, and motor vehicle research, its foundations is usually outfitted on quite validated mathematical versions. concerning genuine computations, nice strides were remodeled the final 20 years, as subtle software program applications are actually in a position to simulating hugely advanced constructions with inflexible and deformable elements. The procedure utilized in this publication may still gain graduate scholars and scientists operating in computational mechanics and similar disciplines in addition to these attracted to time-dependent partial differential equations and heterogeneous issues of a number of time scales. also, a few open matters on the frontiers of study are addressed via taking a differential-algebraic method and lengthening it to the inspiration of brief saddle aspect problems.
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Additional resources for Computational Flexible Multibody Dynamics: A Differential-Algebraic Approach
The full rank of the constraint Jacobian G(q 0 ) implies that, after an appropriate permutation of the columns of G(q 0 ) which we omit here for simplicity, the coordinate vector q can be partitioned into independent coordinates q I (t) ∈ Rnq −nλ and dependent coordinates q D (t) ∈ Rnλ such that q= qI qD and GD (q 0 ) := ∂g(q 0 ) ∈ Rnλ ×nλ is invertible. 88) Recall that ns = nq − nλ is the number of DOF, and these degrees of freedom are momentarily identified with q I . 4 Analysis of the Equations of Constrained Mechanical Motion 49 By the implicit function theorem, the constraint 0 = g(q I , q D ) defines formally a function q D = η(q I ) in a neighborhood of q 0 , and by computing the derivative and observing the chain rule, we obtain 0= ∂ ∂η(q I ) g(q I , q D ) = GI (q I , q D ) + GD (q I , q D ) ∂q I ∂q I where GI (q) := ∂g(q)/∂q I ∈ Rnλ ×ns .
42 2 Rigid Multibody Dynamics We point out that, in practice, the complexity of the equations of motion prohibits such explicit expressions as given here for the planar pendulum example. Instead, the multibody formalisms provide automatic procedures for numerically evaluating constraints and differentiated constraints. 2 Minimax Characterization of Constraints As a prelude to the topic of transient saddle point problems in Chap. 2). The above analysis steps demonstrate that the full rank of the constraint Jacobian G is a fundamental property.
70) nevertheless is satisfied. 29) for the Newton–Euler equations with quaternions. 69) for the acceleration v˙ and the Lagrange multiplier λ by block Gaussian elimination. This leads to the explicit expressions v˙ = M(q)−1 f (q, v, t) − GT (q)λ , λ = G(q)M(q)−1 GT (q) −1 G(q)M(q)−1 f (q, v, t) + κ(q, v) . 71), which leads to an ordinary differential equation for the velocity variables v. 71) is guaranteed and in turn, the multiplier λ is uniquely determined as well. 66c). 69) that allows to solve for the Lagrange multiplier as a function λ(q, v, t).