By Bernd Simeon

This monograph, written from a numerical research standpoint, goals to supply a complete therapy of either the mathematical framework and the numerical tools for versatile multibody dynamics. not just is that this box completely and swiftly starting to be, with numerous functions in aerospace engineering, biomechanics, robotics, and motor vehicle research, its foundations can be outfitted on kind of confirmed mathematical versions. relating to genuine computations, nice strides were remodeled the final twenty years, as refined software program programs are actually in a position to simulating hugely advanced constructions with inflexible and deformable elements. The method utilized in this publication may still profit graduate scholars and scientists operating in computational mechanics and comparable disciplines in addition to these drawn to time-dependent partial differential equations and heterogeneous issues of a number of time scales. also, a few open concerns on the frontiers of analysis are addressed by means of taking a differential-algebraic procedure and increasing it to the thought of brief saddle aspect problems.

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**Additional info for Computational Flexible Multibody Dynamics: A Differential-Algebraic Approach**

**Sample text**

A fundamentally different aspect comes into play by adopting the viewpoint of differential equations on manifolds, as introduced by Rheinboldt [Rhe84]. 48b) of index 2 where the constraint 0 = b(y), assuming sufficient differentiability, defines the manifold M := y ∈ Rny : b(y) = 0 . 49) for the matrix product ∂b/∂y · ∂a/∂z implies that the Jacobian B(y) = ∂b(y)/∂y ∈ Rnz ×ny possesses also full rank nz . 55) spans the kernel of B and has the same dimension ny − nz as the manifold M. 48b), which, starting from a consistent initial value, is required to proceed on the manifold.

36b) is consistent if the constraint is satisfied, 0 = b(y 0 , z0 ). 49) in a neighborhood of the solution. 50) and 0 = B(y) ∂a d ∂a (y, z)˙z + B(y) (y, z)a(y, z) + B(y) · a(y, z). 48b) is thus k = 2. As a rule of thumb, differential-algebraic equations of index 2 or higher are generally more difficult to analyze and to solve numerically than ordinary differential equations or DAEs of index 1. 50). , 0 = b(y 0 ), 0 = B(y 0 )a(y 0 , z0 ). 52) In practice, finding such consistent initial values may constitute a challenging problem of its own [BCP96, ST00].

34). An exception is the case k = 1 where N 1 is the zero matrix. 34) in a stable way, the computation of the Jordan canonical form and consequently also of the Kronecker form are notoriously ill-conditioned problems. 40), we introduce new variables and right hand side vectors V −1 x =: y , z U c =: δ θ . 43a) N z˙ + z = θ . 43a) follows by integrating and results in an expression based on the matrix exponential exp(−C(t − t0 )), Eq. 43b) for z can be solved recursively by differentiating. More precisely, we have N z¨ + z˙ = θ˙ ⇒ N 2 z¨ = −N z˙ + N θ˙ = z − θ + N θ˙ .