Multibody Systems
Transcrição
Multibody Systems
22.09.2014 Fachbereich Mathematik Multibody Systems Bernd Simeon TU Kaiserslautern - FB Mathematik Felix-Klein-Zentrum [email protected] GAMM Junior‘s School Applied Mathematics & Mechanics Elgersburg, 17. September 2014 Fachbereich Mathematik Point of Departure Application fields for multibody dynamics Robotics Aerospace engineering Biomechanics Automotive Machinery Wind turbines … TESIS Dynaware SIMPACK AG Fraunhofer ITWM Bernd Simeon: Multibody Systems, SAMM Elgersburg Sept. 2014 2 1 22.09.2014 Fachbereich Mathematik Historical Remarks Euler-Lagrange equations 1788 ……….. Baumgarte stabilization 1972 DAEs are not ODEs, Petzold 1983 Gear-Gupta-Leimkuhler-Stabilization 1985 Oberwolfach Conference 1993 ……….. Today: Most software packages in multibody dynamics use DAE models Bernd Simeon: Multibody Systems, SAMM Elgersburg Sept. 2014 3 Fachbereich Mathematik Outline Mathematical modeling in multibody dynamics Examples: pendulum, slider crank, wheel suspension Structure of DAEs in multibody dynamics - BREAK – Stabilized formulations Hitchhiker‘s guide to numerical integration Specific application fields & outlook on future research topics Bernd Simeon: Multibody Systems, SAMM Elgersburg Sept. 2014 4 2 22.09.2014 Fachbereich Mathematik References Eich-Söllner & Führer: Numerical Methods in Multibody Dynamics, 1998 Hairer & Wanner: Solving ODEs II, 1996 Roberson & Schwertassek: Dynamics of Multibody Systems, 1988 Shabana: Dynamics of Multibody Systems, 1998 Si.: Computational Flexible Multibody Dynamics, 2013 Chapters 2 & 7 Bernd Simeon: Multibody Systems, SAMM Elgersburg Sept. 2014 5 Fachbereich Mathematik Part I: Mathematical Modeling Bernd Simeon: Multibody Systems, SAMM Elgersburg Sept. 2014 6 3 22.09.2014 Fachbereich Mathematik Planar Pendulum Rigid rod, planar motion Kinematics: Cartesian (absolute) coordinates ⎛ ⎞ r1(t) q(t) = ⎝ r2(t) ⎠ α(t) r1, r2 : coordinates of centroid α: angle between inertial reference frame and body-fixed frame placed in centroid Bernd Simeon: Multibody Systems, SAMM Elgersburg Sept. 2014 7 Fachbereich Mathematik Pendulum II Revolute joint at tip → kinematic constraint à ! l r1 − 2 cos α 0= =: g(q) r2 − 2l sin α l: length of pendulum Constraint Jacobian ¶ µ 1 0 2l sin α G(q) = 0 1 − 2l cos α is rectangular matrix, rank 2 Bernd Simeon: Multibody Systems, SAMM Elgersburg Sept. 2014 8 4 22.09.2014 Fachbereich Mathematik Pendulum III Selection of minimal coordinate α: Since r 1 = 2l cos α, r 2 = 2l sin α ⎞ ⎛ l cos α 2 ⇒ q(α) = ⎝ 2l sin α ⎠ α Choices for describing the motion (i) Full set of nq = 3 coordinates q plus nλ = 2 kinematic constraints 0 = g(q) (ii) Minimal coordinate α Bernd Simeon: Multibody Systems, SAMM Elgersburg Sept. 2014 9 Fachbereich Mathematik Multibody System Assumptions: multibody system = {rigid bodies} ∪ {interconnections} • rigid bodies have mass and geometry • springs, dampers, actuators: compliant elements, massless • joints: constrain relative motion of pairs of bodies, massless Bernd Simeon: Multibody Systems, SAMM Elgersburg Sept. 2014 10 5 22.09.2014 Fachbereich Mathematik Kinematics q(t) ∈ Rnq : coordinates for position + orientation Holonomic constraint equations 0 = g(q) ∈ Rnλ Well-defined model: G(q) := nλ < nq and Jacobian ∂g(q) ∈ Rnλ ×nq ∂q has full rank nλ Bernd Simeon: Multibody Systems, SAMM Elgersburg Sept. 2014 11 Fachbereich Mathematik Minimal Coordinates Implicit Function Theorem ⇒ minimal coordinates s(t) ∈ Rns exist such that q = q(s) and g(q(s)) ≡ 0 Number of degrees of freedom (DOF) ns = nq − nλ Orthogonality relation G(q(s)) N (s) = 0 with null space matrix N (s) := ∂q(s)/∂s ∈ Rnq ×ns Bernd Simeon: Multibody Systems, SAMM Elgersburg Sept. 2014 12 6 22.09.2014 Fachbereich Mathematik Dynamics Lagrange Equations of 1st Kind M (q) q̈ = f (q, q̇, t) − G(q)T λ 0 = g(q) where M (q) ∈ Rnq ×nq : mass matrix f (q, q̇, t) ∈ Rnq : applied and internal forces λ(t) ∈ Rnλ : Lagrange multipliers Conservative system: Hamilton’s Principle applies Z t1 ³ ´ T T − U − g(q) λ dt → stationary ! t0 Bernd Simeon: Multibody Systems, SAMM Elgersburg Sept. 2014 13 Fachbereich Mathematik Variational Principle Principle of Least Action (Hamilton) Z t1 ³ ´ T − U − g(q)T λ dt → stationary ! t0 with kinetic energy T (q, q̇) = 12 q̇ T M (q)q̇, potential energy U (q) Non-conservative case ³ ´ d ∂ T (q, q̇) − ∂ T (q, q̇) = f (q, q̇, t) − G(q)T λ a ∂q dt ∂ q̇ 0 = g(q) with applied forces fa and ∂ f (q, q̇, t) := fa (q, q̇, t) + ∂q µ ¶ ¶ µ 1 T d q̇ M (q)q̇ − M (q) q̇ 2 dt Bernd Simeon: Multibody Systems, SAMM Elgersburg Sept. 2014 14 7 22.09.2014 Fachbereich Mathematik State Space Form Insert q = q(s(t)) into variational principle ⇒ C(s) s̈ = h(s, ṡ, t) Lagrange Equations of 2nd Kind where d2 ∂N (s) (ṡ, ṡ) q(s) = N (s)s̈ + 2 dt ∂s d q(s) = N (s)ṡ, dt and C(s) = N (s)T M (q(s))N (s) ∈ Rns ×ns , h(s, ṡ, t) = N (s)T f (q(s), N (s)ṡ, t) − N (s)T M (q(s)) ∂N (s) (ṡ, ṡ) ∂s 15 Bernd Simeon: Multibody Systems, SAMM Elgersburg Sept. 2014 Fachbereich Mathematik Pendulum Example DAE model - Lagrange 1st kind: ⎛ m ⎝ 0 0 ⎞⎛ ⎞ 0 0 r̈1 m 0 ⎠ ⎝ r̈2 ⎠ l2 α̈ 0 m 12 0 ⎛ ⎞ ⎛ 1 0 ⎜ ⎟ ⎜ 0 = ⎝ −mγ ⎠ − ⎝ l 0 2 sin α à ! r1 − 2l cos α = r2 − 2l sin α 0 1 − 2l cos α ⎞ ⎟ ⎠ µ λ1 λ2 ¶ where m: mass, γ: gravitation constant Minimal coordinate α: state space form - Lagrange 2nd kind: l J α̈ = −mγ cos α 2 with moment of inertia J = l2 m/3 Bernd Simeon: Multibody Systems, SAMM Elgersburg Sept. 2014 16 8 22.09.2014 Fachbereich Mathematik Pros and Cons State space form is ODE, but existence in general guaranteed only locally closed kinematic loops DAE model is gobally defined, bypasses topological analysis Generation of eqs. of motion proceeds automatically by means of multibody formalisms. Newton-Euler eqs. instead of energy expressions Most software packages use DAE models (either completely or partially for closed loop systems) Need for DAE analysis and numerical methods! Bernd Simeon: Multibody Systems, SAMM Elgersburg Sept. 2014 17 Fachbereich Mathematik Remark on Hamiltonian Systems Conservative case allows modeling based on Hamiltonian H =T +U with momenta p := M (q)q̇ Hamiltonian system with constraints ∂ H(p, q) ∂p ∂ ṗ = − H(p, q) − G(q)T λ ∂q 0 = g(q) q̇ = −→ too restrictive for engineering applications Bernd Simeon: Multibody Systems, SAMM Elgersburg Sept. 2014 18 9 22.09.2014 Fachbereich Mathematik Slider Crank Example 3 rigid bodies: crank, connecting rod, slider 3 revolute joints 1 sliding joint ⎞ ⎛ ⎞ ⎞ ⎛ l1 cos α1 l1 sin α1 −1/2 l1 l2 m2 α̇22 sin(α1 − α2 ) µ ¶ α̈1 λ1 2 ⎠ ⎝ ⎝ ⎠ ⎠ ⎝ α̈2 l2 cos α2 l2 sin α2 1/2 l1 l2 m2 α̇1 sin(α1 − α2 ) − M = λ2 r̈3 −F (t) 0 1 ¶ µ l1 sin α1 + l2 sin α2 0 = r3 − l1 cos α1 − l2 cos α2 3rd revolute joint nq = 3, nλ = 2, ns = 1 ⎛ Bernd Simeon: Multibody Systems, SAMM Elgersburg Sept. 2014 19 Fachbereich Mathematik Wheel Suspension Hiller & Frik 1992 High comfort rear suspension 7 rigid bodies (5 rods between wheel carrier and chassis, wheel carrier, wheel) 14 relative coordinates q = (ϕ1 , ψ1 , α, β, γ, ϕ2 , ψ2 , ϕ3 , ψ3 , ϕ4 , ψ4 , ϕ5 , ψ5 , δ)T 12 constraints due to universal/spherical joints and closed kinematic loops Eqs. of motion spread over 7000 lines Bernd Simeon: Multibody Systems, SAMM Elgersburg Sept. 2014 20 10 22.09.2014 Fachbereich Mathematik Newton-Euler Equations Spatial motion of rigid body −→ Newton-Euler eqs. mr̈ = ff (t) ~ J~ω̇ + ω̃ Jω = fm (t) with angular velocity ω −→ no system of 2nd order! Euler parameters (quaternions) θ ∈ R4 lead to mr̈ = ff (t) ^ ~ θ̇ JQ(θ) θ̇ − θλ Q(θ)T J~ Q(θ)θ̈ = Q(θ)T fm (t) − Q(θ)T Q(θ) T 0 = θ θ−1 −→ complies with 2nd order system! Bernd Simeon: Multibody Systems, SAMM Elgersburg Sept. 2014 21 Fachbereich Mathematik Part II: Structure of DAEs in Multibody Dynamics Bernd Simeon: Multibody Systems, SAMM Elgersburg Sept. 2014 22 11 22.09.2014 Fachbereich Mathematik DAEs Refer to Brenan, Campbell & Petzold 1996, Hairer & Wanner 1996, Kunkel & Mehrmann 2006, Lamour, März & Tischendorf 2013, Rabier & Rheinboldt 2002 Consider fully implicit system F (ẋ, x, t) = 0 with state variables x(t) ∈ Rnx Assumption: nx × nx Jacobian ∂F /∂ ẋ is singular Linear-implicit system Semi-explicit system E ẋ = φ(x, t) with singular E ∈ Rnx ×nx ẏ = a(y, z) 0 = b(y, z) with differential variables y and algebraic variables z Bernd Simeon: Multibody Systems, SAMM Elgersburg Sept. 2014 23 Fachbereich Mathematik The Differential Index (Gear 1988) The differential index k of F (ẋ, x, t) = 0 is defined as k = 0: ⇔ ∂F /∂ ẋ is non-singular k > 0: Otherwise, consider F (ẋ, x, t) = 0, d ∂ F (ẋ, x, t) = F (ẋ, x, t) x(2) + . . . = 0, dt ∂ ẋ .. . ds ∂ F (ẋ, x, t) x(s+1) + . . . = 0 F (ẋ, x, t) = dts ∂ ẋ in ẋ, x(2) , . . . , x(s+1) , with x and t as independent variables. Then k := smallest s such that ODE ẋ = ψ(x, t) can be extracted (by algebraic manipulations only). Bernd Simeon: Multibody Systems, SAMM Elgersburg Sept. 2014 24 12 22.09.2014 Fachbereich Mathematik Semi-Explicit Systems ẏ = a(y, z) 0 = b(y, z) b (y, z) ∈ Rnz ×nz Assume Jacobian ∂∂z Consider ⇒ 0= is invertible d ∂b ∂b b(y, z) = (y, z)ẏ + (y, z)ż dt ∂y ∂z Thus ż = − µ ∂b (y, z) ∂z ¶−1 ∂b (y, z) · a(y, z) ∂y ẏ = a(y, z) and eq. for ż form the underlying ODE ⇒ Index k = 1 Bernd Simeon: Multibody Systems, SAMM Elgersburg Sept. 2014 25 Fachbereich Mathematik Perturbation Index (Hairer/Lubich/Roche 1989) System F (ẋ, x, t) = 0 has perturbation index k ≥ 1 along x(t) on [t0 , t1 ] if k is the smallest integer such that for all x̂ ˙ x̂, t) = δ(t) there exists on [t0 , t1 ] an estimate with defect F (x̂, ³ kx̂(t) − x(t)k ≤ c kx̂(t0 ) − x(t0 )k + maxt0 ≤ξ≤t kδ(ξ)k + . . . ´ (k−1) + maxt0 ≤ξ≤t kδ (ξ)k with sufficiently small bound on rhs. Constant c depends on F and on [t0 , t1 ], not on δ. Bernd Simeon: Multibody Systems, SAMM Elgersburg Sept. 2014 26 13 22.09.2014 Fachbereich Mathematik „Never Trust Authorities“ Conjecture PI ≤ DI + 1 proved by Gear 1990 Counter example by ⎛ 0 ⎝ 0 0 Campbell & Gear 1995 ⎞⎛ ⎞ ⎛ ⎞ y3 0 ẏ1 y1 0 y3 ⎠ ⎝ ẏ2 ⎠ + ⎝ y2 ⎠ = 0 0 0 ẏ3 y3 DI = Differential Index = 1, PI = Perturbation Index = 3! I think you should be more explicit here in step two! Cartoon by S. Harris Bernd Simeon: Multibody Systems, SAMM Elgersburg Sept. 2014 27 Fachbereich Mathematik Analysis of Equations of Constrained Mechanical Motion Rewrite as system of first order q̇ = v M (q) v̇ = f (q, v, t) − G(q)T λ 0 = g(q) v(t) ∈ Rnq : velocity variables Semi-explicit DAE - index k =??? Bernd Simeon: Multibody Systems, SAMM Elgersburg Sept. 2014 28 14 22.09.2014 Fachbereich Mathematik Hidden Constraints Differentiation of holonomic constraints d 0 = g(q) = G(q) q̇ = G(q) v dt Constraints at velocity level d2 0 = 2 g(q) = G(q) v̇ + κ(q, v), dt κ(q, v) := ∂ G(q ) (v, v) ∂q Constraints at acceleration level Bernd Simeon: Multibody Systems, SAMM Elgersburg Sept. 2014 29 Fachbereich Mathematik Eliminating the Lagrange Multipliers Dynamic equation and acceleration constraint yield ¶ ¶µ ¶ µ µ f (q, v, t) v̇ M (q) G(q)T = G(q) 0 λ −κ(q, v) µ ¶ M (q) G(q)T Assumption: is invertible G(q) 0 Thus ¢ ¡ v̇ = M (q)−1 f (q, v, t) − GT (q)λ ¡ ¢−1 ¡ ¢ λ = G(q)M (q)−1 GT (q) G(q)M (q)−1 f (q, v, t) + κ(q, v) −→ ODE q̇ = v, v̇ = φ(q, v, t) Bernd Simeon: Multibody Systems, SAMM Elgersburg Sept. 2014 30 15 22.09.2014 Fachbereich Mathematik Index Index of equations of motion k = 3 Consistent initial values satisfy 0 = g(q 0) , 0 = G(q 0 ) v 0 while λ0 = λ(q 0 , v 0, t0 ) completely determined Assumptions made: Sufficient smoothness + M is s.p.d. + G full rank 31 Bernd Simeon: Multibody Systems, SAMM Elgersburg Sept. 2014 Fachbereich Mathematik Hidden Constraints Pendulum ⎛ ⎞ ⎛ ⎞ v1 (t) ṙ1 (t) Velocity variables v(t) = ⎝ v2 (t) ⎠ := ⎝ ṙ2 (t) ⎠ v3 (t) α̇(t) Differentiation steps ⎞ ⎛ à ! µ ¶ v l 1 l r1 − 2 cos α 1 0 2 sin α ⎝ v2 ⎠ = 0= d dt 0 1 − 2l cos α r2 − 2l sin α v3 d2 0= 2 dt à r1 − r2 − l 2 l 2 cos α sin α ! = µ 1 0 0 1 l 2 sin α − 2l cos α ¶ ⎛ ⎞ µ v̇1 ⎝ v̇2 ⎠ + v32 v̇3 Lagrange multipliers ¶ µ µ ¶ µ mγ 3 sin α cos α λ1 2 + mv =− 3 λ2 1 + 3 sin2 α 4 l 2 l 2 cos α sin α l 2 l 2 cos α sin α ¶ ¶ Bernd Simeon: Multibody Systems, SAMM Elgersburg Sept. 2014 32 16 22.09.2014 Fachbereich Mathematik Part III: Stabilized Formulations for Multibody Dynamics Bernd Simeon: Multibody Systems, SAMM Elgersburg Sept. 2014 33 Fachbereich Mathematik Equations of Motion q̇ = v M (q) v̇ = f (q, v, t) − G(q)T λ 0 = g(q) is DAE of index k = 3 Issues • Well-posedness • Differentiation corresponds to difference quotient in numerical methods • Lack of reliable numerical methods Bernd Simeon: Multibody Systems, SAMM Elgersburg Sept. 2014 34 17 22.09.2014 Fachbereich Mathematik Formulation of Index 1 Replace original constraint by constraint at acceleration level q̇ = v M (q) v̇ = f (q, v, t) − G(q)T λ 0 = G(q) v̇ + κ(q, v) Drift off Consider w(t) := g(q(t)) and ODE ẅ = 0 with perturbed init. values ẅ = ζ a , ẇ(t0 ) = ζ v , w(t0 ) = ζ p Integration yields w(t) = 12 (t − t0 )2 ζ a + (t − t0 )ζ v + ζ p −→ quadratic growth - original constraints violated! Bernd Simeon: Multibody Systems, SAMM Elgersburg Sept. 2014 35 Fachbereich Mathematik „Flying Wheelset“ „Hunting motion“ – stable limit cycle Si./Führer/Rentrop1991, Eich 1993 Bernd Simeon: Multibody Systems, SAMM Elgersburg Sept. 2014 36 18 22.09.2014 Fachbereich Mathematik Baumgarte Stabilization (1972) q̇ = v M (q) v̇ = f (q, v, t) − G(q)T λ 0 = G(q) v̇ + κ(q, v) + 2αG(q)v + β 2 g(q) Still index k = 1. Select α, β ∈ R such that 0 = ẅ + 2αẇ + β 2 w becomes asymptotically stable. E.g., α = β > 0 yields ẅ + 2αẇ + α2 w = ζ a , ẇ(t0 ) = ζ v , w(t0 ) = ζ p and ¢ ¡ ζ w(t) = ζ p + (t − t0 )(ζ p + αζ v ) exp(−α(t − t0 )) + a2 α Bernd Simeon: Multibody Systems, SAMM Elgersburg Sept. 2014 37 Fachbereich Mathematik Formulation of Index 2 Use constraint at velocity level q̇ = v M (q) v̇ = f (q, v, t) − G(q)T λ 0 = G(q) v Index k = 2, still feasible for numerical methods like BDF Bernd Simeon: Multibody Systems, SAMM Elgersburg Sept. 2014 38 19 22.09.2014 Fachbereich Mathematik Gear/Gupta/Leimkuhler Formulation q̇ M (q) v̇ 0 0 It holds 0= = = = = (1985) v − G(q)T μ f (q, v, t) − G(q)T λ G(q) v g(q) with extra multipliers μ(t) ∈ Rnλ d g(q) = G(q)q̇ = G v − GGT μ = −GGT μ dt ⇒ μ = 0, index k = 2, original constraint included! Bernd Simeon: Multibody Systems, SAMM Elgersburg Sept. 2014 39 Fachbereich Mathematik Part IV: Hitchhiker‘s Guide to Time Integration in Multibody Dynamics Bernd Simeon: Multibody Systems, SAMM Elgersburg Sept. 2014 40 20 22.09.2014 Fachbereich Mathematik Basics Assume consistent initial values q 0 , v 0 0 = g(q 0 ) , 0 = G(q 0) v 0 Time grid t0 < t1 < . . . < tn with stepsize τi = ti+1 − ti q n : numerical approximation of q(tn ), analogously v n and λn Bernd Simeon: Multibody Systems, SAMM Elgersburg Sept. 2014 41 Fachbereich Mathematik Index 1 – Formulation and Drift Off Solve linear system µ ¶µ ¶ µ ¶ f (q, v, t) M (q) G(q)T Ψ = G(q) 0 Υ −κ(q, v) for Ψ ∈ Rnq and Υ ∈ Rnλ Given q and v, this defines rhs of q̇ = v, v̇ = Ψ(q, v, t) −→ any standard ODE integrator can be applied Expectation for method of order k: q(tn ) − q n = O(τ k ), v(tn ) − v n = O(τ k ), λ(tn ) − λn = O(τ k ) Bernd Simeon: Multibody Systems, SAMM Elgersburg Sept. 2014 42 21 22.09.2014 Fachbereich Mathematik Drift Off Formulation of index 1 is integrated by a method of order k kg(q n )k ≤ τ k (A(tn − t0 ) + B(tn − t0 )2 ), ⇒ kG(q n )v n k ≤ τ k C(tn − t0 ) with constants A, B, C Original constraints = invariants, not satisfied by numerical solution! Bernd Simeon: Multibody Systems, SAMM Elgersburg Sept. 2014 43 Fachbereich Mathematik Projection Methods q n+1 , v n+1 : numerical solution computed from consistent q n , v n Projection (Lubich 1989, Eich 1990) ½ 0 = M (q̃ n+1 )(q̃ n+1 − q n+1 ) + G(q̃ n+1 )T μ, solve 0 = g(q̃ n+1 ) for q̃ n+1 , μ solve ½ 0 = M (q̃ n+1 )(ṽ n+1 − v n+1 ) + G(q̃ n+1 )T η, 0 = G(q̃ n+1 ) ṽ n+1 for ṽ n+1 , η Bernd Simeon: Multibody Systems, SAMM Elgersburg Sept. 2014 44 22 22.09.2014 Fachbereich Mathematik Half-Explicit Methods Half-explicit Euler q n+1 = q n + τ v n M (q n ) v n+1 = M (q n ) v n + τ f (q n , v n , tn ) − τ G(q n )T λn 0 = G(q n+1 ) v n+1 Leads to linear system ¶µ ¶ µ ¶ µ v n+1 M (q n )v n + τ f (q n , v n , tn ) M (q n ) G(q n )T = τ λn 0 G(q n+1 ) 0 Basis for extrapolation and Runge-Kutta methods (Arnold/Murua 1996, Lubich 1992) Bernd Simeon: Multibody Systems, SAMM Elgersburg Sept. 2014 45 Fachbereich Mathematik Implicit Methods Write eqs. of motion (GGL) as linear-implicit system E ẋ = φ(x, t) with singular matrix E Prototype of an implicit method: implicit Euler E xn+1 − xn = φ(xn+1 , tn+1 ) τ −→ solve system of nonlinear eqs. with Jacobian ∂φ 1 E− (xn , tn ) τ ∂x Invertible for regular matrix pencil μE − ∂φ/∂x Bernd Simeon: Multibody Systems, SAMM Elgersburg Sept. 2014 46 23 22.09.2014 Fachbereich Mathematik Part V: Specific Application Fields & Outlook • Real-time integration • Flexible multibody systems • Transient saddle point problems Bernd Simeon: Multibody Systems, SAMM Elgersburg Sept. 2014 47 Bernd Simeon: Multibody Systems, SAMM Elgersburg Sept. 2014 48 Fachbereich Mathematik Real-Time Simulation 24 22.09.2014 Fachbereich Mathematik Time Integration Bernd Simeon: Multibody Systems, SAMM Elgersburg Sept. 2014 49 Fachbereich Mathematik Vehicle-Trailer Coupling Bernd Simeon: Multibody Systems, SAMM Elgersburg Sept. 2014 50 25 22.09.2014 Fachbereich Mathematik DAE Real-Time Integrator Bernd Simeon: Multibody Systems, SAMM Elgersburg Sept. 2014 51 Bernd Simeon: Multibody Systems, SAMM Elgersburg Sept. 2014 52 Fachbereich Mathematik Simulation Example 26 22.09.2014 Fachbereich Mathematik Drift-Off? Bernd Simeon: Multibody Systems, SAMM Elgersburg Sept. 2014 53 Fachbereich Mathematik Flexible Multibody Systems elastic elastic Bernd Simeon: Multibody Systems, SAMM Elgersburg Sept. 2014 54 27 22.09.2014 Fachbereich Mathematik Linear Elasticity Bernd Simeon: Multibody Systems, SAMM Elgersburg Sept. 2014 55 Fachbereich Mathematik Floating Frame of Reference Bernd Simeon: Multibody Systems, SAMM Elgersburg Sept. 2014 56 28 22.09.2014 Fachbereich Mathematik Slider Crank Revisited Bernd Simeon: Multibody Systems, SAMM Elgersburg Sept. 2014 57 Bernd Simeon: Multibody Systems, SAMM Elgersburg Sept. 2014 58 Fachbereich Mathematik 29 22.09.2014 Fachbereich Mathematik Concluding Remarks Mechanical multibody systems have been driving force in development of DAE methodology Trend to multiphysics and inclusion of PDE models Example wind turbine: Tower is classical mbs Turbine blades are flexible bodies Fluid-structure interaction with windfield Fatigue analysis and prediction of lifespan Bernd Simeon: Multibody Systems, SAMM Elgersburg Sept. 2014 59 30