Multibody Systems

Eich-Soellner, Edda; Führer, Claus

doi:10.1007/978-3-663-09828-7_1

Cited by 41 publications

(57 citation statements)

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“…Much of the literature on polynomial eigenvalue problems considers only polynomials whose leading coefficient matrix A k is nonsingular (or even the identity), so the issue of infinite eigenvalues doesn't even arise. But there are a number of applications, such as constraint multi-body systems [2], [16], circuit simulation [3], or optical waveguide design [17], where the leading coefficient is singular. In such cases one must choose a linearization with care, since not all linearizations properly reflect the Jordan structure of the eigenvalue ∞ [13].…”

mentioning

confidence: 99%

Vector Spaces of Linearizations for Matrix Polynomials

Mackey¹,

Mackey²,

Mehl³

et al. 2006

SIAM J. Matrix Anal. & Appl.

260

482

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Abstract. The classical approach to investigating polynomial eigenvalue problems is linearization, where the polynomial is converted into a larger matrix pencil with the same eigenvalues. For any polynomial there are infinitely many linearizations with widely varying properties, but in practice the companion forms are typically used. However, these companion forms are not always entirely satisfactory, and linearizations with special properties may sometimes be required.Given a matrix polynomial P , we develop a systematic approach to generating large classes of linearizations for P . We show how to simply construct two vector spaces of pencils that generalize the companion forms of P , and prove that almost all of these pencils are linearizations for P . Eigenvectors of these pencils are shown to be closely related to those of P . A distinguished subspace is then isolated, and the special properties of these pencils are investigated. These spaces of pencils provide a convenient arena in which to look for structured linearizations of structured polynomials, as well as to try to optimize the conditioning of linearizations [7], [8], [12].

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mentioning

confidence: 99%

Vector Spaces of Linearizations for Matrix Polynomials

Mackey¹,

Mackey²,

Mehl³

et al. 2006

SIAM J. Matrix Anal. & Appl.

260

482

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“…For ordinary differential equations (ODEs) with switching conditions and discontinuous right hand side a so called switch algorithm was formulated in [19]. We generalize this procedure here for hybrid systems of DAEs as in (2.6).…”

Section: Numerical Integration Of Hdaesmentioning

confidence: 99%

“…Consider first the constrained motion of the pendulum. Using the classical Euler-Lagrange formalism [19,23] T and the acceleration forces F (t, x 1 , x 2 ) = (F x 1 (t, x 1 ), F x 2 (t, x 2 )) T , one obtains the following system of DAEs.…”

mentioning

confidence: 99%

“…There exist many numerical solution methods and software for DAEs [8,18,19,23,36,38,40], but most of these cannot be directly applied to HDAEs and also the analysis of HDAEs has not yet been investigated in detail, see [3,4,28,42,43] for such approaches. In practical applications the model is usually simulated in one of the modes, the switching points are determined and then the system is simulated in the next mode.…”

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confidence: 99%

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Numerical solution of hybrid systems of differential-algebraic equations

Hamann

Mehrmann

2008

Computer Methods in Applied Mechanics and Engineering

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Abstract. We present a mathematical framework for general over-and underdetermined hybrid (switched) systems of differential-algebraic equations (HDAEs). We give a systematic formulation of HDAEs and discuss existence and uniqueness of solutions, the treatment of the switch points and how to perform consistent initialization at switch points.We show how numerical solution methods for DAEs can be adapted for HDAEs and present a numerical results for these methods for the real world example of simulating an automatic gearbox.

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“…The monograph [14] has also established a methodology for convergence proofs. In the special field of multibody dynamics, various attempts have been undertaken to construct methods that exploit the structure of the equations of motion to a high degree, such as the half-explicit one-step methods by Arnold [1] and Murua [21], the extrapolation methods by Lubich [19], and the projection methods by Eich [9] and Simeon [23]. Strategies contra the drift-off are discussed in Ascher et al [3].…”

Section: Introductionmentioning

confidence: 99%

Solving constrained mechanical systems by the family of Newmark and α‐methods

Lunk

Simeon

2006

Z Angew Math Mech

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The family of Newmark and generalized α-methods is extended to constrained mechanical systems by using simultaneous position and velocity stabilization as key ideas. In this way, the acceleration constraints need not be evaluated, and the overall algorithm is about as expensive as the application of a BDF method to the GGL-stabilized equations of motion. Moreover, the RATTLE method of molecular dynamics is included as special case. A convergence analysis of the presented α-RATTLE algorithm shows global second order in both position and velocity variables while the Lagrange multipliers are computed to first order accuracy. Additonally, the property of adjustable numerical dissipation carries over from the unconstrained case.

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Multibody Systems

Cited by 41 publications

References 0 publications

Vector Spaces of Linearizations for Matrix Polynomials

Vector Spaces of Linearizations for Matrix Polynomials

Numerical solution of hybrid systems of differential-algebraic equations

Solving constrained mechanical systems by the family of Newmark and α‐methods

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