Simulation of Flexible Multibody Systems Robert Altmann Technische - - PowerPoint PPT Presentation

Simulation of Flexible Multibody Systems

Robert Altmann

Technische Universit¨ at Berlin raltmann@math.tu-berlin.de

Trogir 2011, October 12th

• R. Altmann (TU Berlin)

Simulation of Flexible MBS Trogir 12.10.2011 1 / 6

Multibody Dynamics

Dynamics of multiple rigid bodies E.g. robots

• R. Altmann (TU Berlin)

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Multibody Dynamics

Dynamics of multiple rigid bodies E.g. robots, slider crank mechanism

X Y x1 y1 x2 y2 p1 p2

DAE of index 3 M(p)¨ p = f(p, ˙ p) − GTλ 0 = g(p)

• R. Altmann (TU Berlin)

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Flexible Multibody Systems

Assumption of rigid bodies not accurate Allow deformable bodies

• R. Altmann (TU Berlin)

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Flexible Multibody Systems

Assumption of rigid bodies not accurate Allow deformable bodies

X Y p1 p2 u ∈ H1

0(Ω)

Combine deformation and rigid motion Linear elasticity (PDE)

• R. Altmann (TU Berlin)

Simulation of Flexible MBS Trogir 12.10.2011 3 / 6

Equations of Motion

Euler-Lagrange Formalism d dt ∂L ∂ ˙ p

• − ∂L

∂p = 0 with strain energy Principle of virtual work

◮ Weak formulation of dynamic elasticity problem ◮ Moving reference frame ◮ Rigid body transformation u → r = y(t) + A(φ)[x + u(x, t)]

Langrange multipliers

◮ Weak formulation of dynamic elasticity problem ◮ Rigid motion as (weak) constraint

M¨ u + D ˙ u + Ku + B∗λu = F Bu = G

• R. Altmann (TU Berlin)

Simulation of Flexible MBS Trogir 12.10.2011 4 / 6

Simulation

Coupled system of ODEs, PDEs and algebraic constraints Different time scales for rigid motion / elastic deformation Standard procedure: Semi-discretization in

◮ Time (Rothe method) ◮ Space (Method of Lines) → DAE of index 3

Finite Element discretization

• R. Altmann (TU Berlin)

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Simulation

Coupled system of ODEs, PDEs and algebraic constraints Different time scales for rigid motion / elastic deformation Standard procedure: Semi-discretization in

◮ Time (Rothe method) ◮ Space (Method of Lines) → DAE of index 3

Finite Element discretization

My work Multiphysics Need adaptivity in space and time ! Method of Lines with mesh update, index reduction Possibility to switch between index-1 formulations

• R. Altmann (TU Berlin)

Simulation of Flexible MBS Trogir 12.10.2011 5 / 6

Literature

Peter Kunkel and Volker Mehrmann, Differential-algebraic equations, EMS Textbooks in Mathematics, European Mathematical Society (EMS), Z¨ urich, 2006, Analysis and numerical solution. MR 2225970 (2007e:34001)

• B. Simeon, Modelling a flexible slider crank mechanism by a

mixed system of DAEs and PDEs, Mathematical and Computer Modelling of Dynamical Systems 2 (1996), 1–18. , Numerische Simulation gekoppelter Systeme von partiellen und dierential-algebraischen Gleichungen der Mehrk¨

• rperdynamik, Rechnerunterst¨

utzte Verfahren, vol. 325, VDI Verlag Dusseldorf, 2000.

• R. Altmann (TU Berlin)

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