Introduction to SAMCEF MECANO

Transcription

1 Introduction to SAMCEF MECANO 1 Outline Introduction Generalized coordinates Kinematic constraints Time integration Description and paramerization of finite rotations Acknowledgements Michel Géradin (ULg) Doan D. Bao (GoodYear) Alberto Cardona (Intec, Argentina) 2 1

2 Fixed deformable structures Our Goal Articulated rigid systems Finite Element Method? Multibody Dynamics how to analyze in a general manner flexible articulated structures 3 Strategic choices Structural behavior Structural behavior Rigid/flexible Flexible?? Kinematic description Kinematic description Recursive? Absolute? Software architecture Time integration Time integration Explicit Implicit? Motion equations formulation Symbolic Numerical? 4 2

3 Finite rotation parameterization Finite motion description Measures of deformation and relative motion Rigid bodies zero deformation Joints relative motion + external work Flexible elements elastic strains + internal work Control system library Mechanical elements library rigid members flexible members rigid articulations flexible articulations active members FE structural substructuring Modelling Step 5 Control system library Mechanical elements library rigid members flexible members rigid articulations flexible articulations active members FE structural substructuring DAE system equations dynamic equilibrium kinematic constraints Post-processing kinematics internal forces element stresses Modelling Step 6 3

4 The MECANO software Preprocessing topological description (FE) (FE) mechanical properties response specification Static analysis equilibrium configuration static static forces forces Initial Assembly Kinetostatic analysis trajectories quasi-static forces forces Dynamic analysis trajectories dynamic forces forces Linearized analysis eigenvalues stability stability Preprocessing trajectories loads loads and and stresses stresses animation 7 The MECANO element library Some rigid joint elements Hinge joint Prismatic joint Spherical joint Gear joint Cylindrical joint Rack-and-pinion Kinematic constraints 8 4

5 The MECANO element library some nonholonomic joints Cam contact Wheel + tyre model Point-on-plane Cable + pulley Kinematic constraints 9 The MECANO element library The basic elastic elements Superelement Spring-damper-stop Bushing Nonlinear beam Kinematic constraints 10 5

6 The MECANO element library Some nonlinear and control features Nonlinear constraint Hydraulic actuator Nonlinear feedback Kinematic constraints 11 The MECANO element library Rail-wheel contact Kinematic constraints 12 6

7 The MECANO element library Flexible slider Kinematic constraints 13 The MECANO element library Symbolic element representation Kinematic constraints 14 7

8 Domains of application mechanical engineering automotive industry sport industry aeronautics space structures biomechanics Dynamic car behavior Adequacy of Mecano association of kinematic joints with finite elements open software through user element capability M. Ebalard, J. Mercier (PSA Citroën) Utilisation du logiciel Mecano pour le comportement routier à PSA 16 Examples of application 8

9 Landing gear analysis Research conducted in framework of ELGAR consortium 17 Examples of application Landing gear analysis Functional phases deployment/retraction ground impact rolling breaking taxiing Model components mechanism damper (internal hydraulic system) tyre: contact, deformation, friction 18 Examples of application 9

10 MEA antenna: result animation 19 Examples of application Simulation of Cluster satellite boom deployment 20 Examples of application 10

11 Generalized coordinates for mechanism analysis Generalized co-ordinates description of various types the 4-bar mechanism example The finite element method for articulated systems kinematics elasto-dynamics differential-algebraic equations of motion numerical integration by Newmark algorithm the double pendulum example 21 Generalized coordinates Introduction Classical approach: minimal number of DOF. Lagrangian coordinates: best suited to robotics open-tree structure + associated graph loop closure conditions lack of generality, difficult for flexible systems Cartesian coordinates: large scale simulation packages inherent topology description large sets of differential-algebraic equations (DAE) still difficult for flexible systems Nonlinear Finite element coordinates: inherent topology description simplification of kinematic constraints geometrically nonlinear effects naturally included 22 Generalized coordinates 11

12 The 4-bar 4 mechanism example Simplest planar, closed-loop mechanism one single variable: crank angle β Gruebler s formula F =3(N 1) mx p i i=1 m : number of joints N: number of bodies p : number of DOF removed by joint i F = 1 23 Generalized coordinates Minimal coordinates Expression in terms of minimal coordinate β θ j = f(β) j =2,...4 From geometry cos θ2 = d2 + s 2 +2r`cos β 2ds r 2 `2 (` r cos β) cos θ 3 + r sin β sin θ 3 = s d cos θ 2 θ 1 =2π β θ 2 θ 3 sin(θ 3 + φ) = c ρ 24 Generalized coordinates 12

13 Minimal coordinates: kinematic transformer Input-output relationship Complex mechanisms as assembly of transformers 25 Generalized coordinates Lagrangian coordinates 3 generalized coordinates q T = [θ 1 θ 2 θ 3 ] 2 closure conditions `1 cos θ 1 + `2 cos(θ 1 + θ 2 ) + `3 cos(θ 1 + θ 2 + θ 3 ) `4 =0 `1 sin θ 1 + `2 sin(θ 1 + θ 2 ) + `3 sin(θ 1 + θ 2 + θ 3 ) =0 Equations of motion: 3 differential equations linked by 2 constraints (DAE) 26 Generalized coordinates 13

14 Cartesian coordinates 9 generalized coordinates q T = [ x 1 y 1 θ 1 x 2 y 2 θ 2 x 3 y 3 θ 3 ] Equations of motion: 9 differential equations linked by 8 constraints (DAE) 8 kinematic constraints x 1 `1 2 cos θ 1 =0 y 1 `1 2 sin θ 1 =0 x 1 + `1 2 cos θ 1 x 2 + `2 2 cos θ 2 =0 y 1 + `1 2 sin θ 1 y 2 + `2 2 sin θ 2 =0 x 2 + `2 2 cos θ 1 x 2 + `3 2 cos θ 3 =0 y 2 + `2 2 sin θ 2 y 3 + `3 2 sin θ 3 =0 x 3 + `3 2 cos θ 3 = `4 y 3 + `3 2 sin θ 3 =0 27 Generalized coordinates Finite Element coordinates q =[x 1 y 1... x 6 y 6 ] T 13 generalized coordinates: 12 finite element DOF q 1 driving DOF β Strong form of kinematic constraints 3 zero strain constraints 4 boundary nodal constraints ² i = 1 `2i `2i,0 x 1 =0, y 1 =0 2 `2i,0 x 6 =0, y 6 = `4 ² 4 assembly nodal constraints 1 =0 ² 2 =0 ² 3 =0 1 driving constraints (implicit) x 2 = x 3 y 2 = y 3 x 4 = x 5 y 4 = y 5 y 2 cos β x 2 sin β =0 Φ D (q) =0 28 Generalized coordinates 14

15 Finite element description of a mechanical system q = DOF at structural level q e = DOF at element level L e = DOF localization operator(boolean) λ = lagrangian multipliers Boolean constraints: localization of DOF Implicit constraints: algebraic treatment System topology results implicitly from Boolean assembly and constraint description q e = L e q Φ(q e, q e,t) =0 Kinematic constraints 29 Classification of kinematic constraints Holonomic constraints ex: prismatic joint Restrict the number of DOF of the system Φ(q,t)=0 Non-holonomic constraints ex: rolling motion Restrict the system behavior Ψ(q, q,t) =0 Unilateral constraints Ψ(q, q,t) 0 Kinematic constraints 30 15

16 Classical mechanical pairs Only 3 lower pairs surface contact motion reversibility All other mechanical pairs are higher pairs revolute prismatic planar cylindrical screw spherical Other modes of classification: number of DOF (1 for all lower pairs) mode of closure 31 The algebraic constrained problem min F(q) q subject to Φ(q) =0 Possible methods of solution constraint elimination Lagrange multipliers penalty function augmented Lagrangian perturbed Lagrangian Jacobian matrix Formulated in terms of B such that B mj = Φ m q j Residual vector r(q) = F q Hessian matrix H such that H ij = 2 F q i q j 32 16

17 Constraint elimination method First variation of constraints Partitioning into independent and dependent variables δφ = Bδq =0 B I δq I + B D δq D =0 Elimination of dependent variables δq = Rδq I Projected residual equation I R = B 1 D B I reduction matrix R T F q = RT r(q) =0 exact verification of constraints complex and computationally expensive 33 Augmented Lagrangian method Augmented functional F p (q, λ) = F(q) + (kλ T Φ + p 2 ΦT Φ) Second-order solution method: by Newton-Raphson p penalty coefficient k scaling factor q λ = q? + q = λ? + λ H + pb T B kb T q r? B T (kλ? + pφ) = kb 0 λ kφ provides exact solution regardless of penalty and scaling factors quadratic term pb T B reinforces positive character of H scaling factor required for pivoting 34 17

18 Constrained dynamic problems Hamilton s principle Z t2 t 1 [δ(l kλ T Φ pφ T Φ) + δw]dt =0 Matrix form of constrained motion equations M q + B T (pφ + kλ) = g(q, q,t) Φ(q,t) =0 Differential - Algebraic nature (DAE) vanishing of penalty term at equilibrium exact scaling of constraints for numerical conditioning multipliers λ = forces needed to close constraints all remaining forces (internal, external, complementary inertia) in RHS Kinematic constraints 35 Constrained equations of motion M q + B T (pφ + kλ) = g(q, q,t) Φ(q,t) =0 Methods of solution Constraint reduction Constraint regularization m+n ODE + constraint stabilization second-order solution linearization But: specific problem of numerical stability of time integration Kinematic constraints 36 18

19 Linearization of motion equations M q C t + λ q K t λ + kb T 0 kb q r? = λ Φ? + O( 2 ) Residual vector of equilibrium r = g(q, q,t) M q B T (pφ + kλ) Tangent damping matrix C t = g q plays central role for convergence in iteration quality of approximation needed depends on penalty factor Tangent stiffness matrix K t = g (M q) + + (BT (pφ + kλ)) q q q approx. K t ' g q + pbt B + k (BT λ) q Kinematic constraints 37 Integration of motion equations: Implicit Method Motivations filtering of high frequencies (elasticity) independence of algorithm stability on time step size one-step method simplicity of software experience in structural dynamics. Newmark algorithm Application to multibody systems appropriate integration of rotation motion stability in presence of constraints (DAE) 38 19

20 Implicit integration Differential - algebraic equations r(q, q, q, λ) = M q + B T λ g(q, q,t) = 0 Φ(q,t) = 0 Correction of solution at time t q = q + q (q, q, q, λ ) q = q + q q = q + q λ = λ + λ Linearized equations of motion Tangent damping Tangent stiffness Linearized matrices C T = r q = g q K T = r q = g q + q (M q + B T λ ) M q + λ CT q KT λ + B T B 0 q r = λ Φ 39 Newmark formula Interpolation of displacements and velocities q n+1 = q n + (1 γ)h q n + γh q n+1 q n+1 = q n + h q n + ( 1 2 β)h2 q n ++βh 2 q n+1 Free parameters: average constant acceleration γ = 1 2 and β = 1 4 Introduction of numerical damping γ = α and β = 1 4 (γ )2 α >

21 Prediction at time t q 0 n+1 =0 q 0 n+1 = q n + (1 γ)h q n q 0 n+1 = q n + h q n + ( 1 2 β)h2 q n Correction q k+1 n+1 = qk n+1 + q = q k n βh q 2 q k+1 n+1 = q k n+1 + q = q k n+1 + γ βh q q k n+1 = q k n+1 + q Correction equation ST B T B 0 Iteration matrix q r = λ Φ Convergence check S T = K T + γ βh C T + 1 βh 2 M krk < ² and kφk < η Integration procedure 41 The double pendulum Description by Cartesian coordinates (redundant) q T = [ x 1 y 1 θ 1 x 2 y 2 θ 2 ] Constraint matrix B = m 1 = m 2 =1. l 1 = l 2 =1., (θ 1 = π/2, θ 2 =0 1 0 `1 cos θ `1 sinθ `2 cos θ `2 sin θ 2 External force vector and mass matrix are constant Tangent stiffness contains only geometric terms Kinetic and potential energies K = 1 2 m 1(ẋ ẏ2 1 ) m 2(ẋ ẏ2 2 ) P = m 1 gy 1 + m 2 gy 2 Kinematic constraints Φ = x 1 `1 sin θ 1 y 1 + `1 cos θ 1 x 2 x 1 `2 sin θ 2 y 2 y 1 + `2 cos θ 2 K T = diag [ 0 0 `1(λ 1 sin θ 1 λ 2 cos θ 1 ) 0 0 `2(λ 3 sin θ 2 λ 4 cos θ 2 ) ] 42 21

22 Newmark integration : undamped (γ = 1 2, β = 1 4 ) normal evolution of angular displacements and velocities but: numerical instability detected after 2 sec. 43 Newmark integration : undamped (γ = 1 2, β = 1 4 ) Weak instability: generated by kinematic constraints Numerical damping 44 22

23 Newmark integration with damping (γ = α, β = 1 4 (1 2 + γ)2, α =0.015) Normal aspect of response over 5 sec. period, but Newmark integration with damping (γ = α, β = 1 4 (1 2 + γ)2, α =0.015) Unacceptable amount of energy loss Search for better compromise between accuracy and stability 46 23

24 Kinematics of finite motion Spherical motion Eigenvalue analysis of rotation operator Euler theorem Explicit expressions of rotation operator Expression in terms of linear invariants Exponential map General motion of rigid body Velocity analysis of spherical motion Explicit expression of angular velocities Rotation parameterization Cartesian rotation vector Finite Motion 47 Spherical motion Current position: material configuration x Initial position: material configuration X Spatial coordinates x = [x 1 x 2 x 3 ] T material coordinates X = [X 1 X 2 X 3 ] T Base vectors of current configuration [~e 1 ~e 2 ~e 3 ] Base vectors of initial configuration [ ~ E 1 ~ E2 ~ E3 ] Linear transformation Spherical motion such that: length of position vector unaffected by pure rotation relative angle between 2 directions remain constant x = RX with R T = R 1 det(r) =1 Proper orthogonal Finite Motion 48 24

25 Euler theorem If a rigid body undergoes a motion leaving fixed one of its points, then a set of points of the body lying on a line that passes through that point, remains fixed as well. Finite Motion 49 Explicit expressions of rotation operator Orthonomality implies 6 constraints R = [ r 1 r 2 r 3 ] r T i r j = δ ij 3 independent rotation parameters R = R(α 1, α 2, α 3 ) Outer product expression Inner product expression R = X i e i E T i E T 1 e 1 E T 1 e 2 E T 1 e 3 R = E T 2 e 1 E T 2 e 2 E T 2 e 3 E T 3 e 1 E T 3 e 2 E T 3 e 3 Finite Motion 50 25

26 Expression in terms of linear invariants Rotation operator R= I cosφ+(1 cosφ)nn T +ñsinφ Invariants tr(r) = λ 1 +λ 2 +λ 3 vect(r) =1+2cosφ = nsinφ Finite Motion 51 Exponential map Differentiate with respect to φ verify that differential motion x = RX dr dφ RT = ñ integration dx ñx =0 with x(0) = X dφ x =exp(ñφ) X R =exp(ñφ) Finite Motion 52 26

27 General motion of rigid body Combination of translation and rotation x P = x 0 + RX P spatial motion of origin material coordinates of point P Finite Motion 53 Velocity analysis of spherical motion Differentiate Spatial description x P = RX P with R T = R 1 Material description spatial velocities material velocities v P = ẋ P = ṘX P skew-symmetry V P = R T v P = ṘR T + (ṘR T ) T =0 v P = ωx P with ω = ṘRT V P = ΩX P with Ω = R T Ṙ spatial angular velocities ω =vect(ṙr T T )) material angular velocities Ω =vect(r T T Ṙ) Ṙ) Finite Motion 54 27

28 Cartesian rotation vector defined as Rotation operator Ψ = nφ R = I + Trigonometric form sin kψ k kψ k eψ + 1 cos kψ k kψ k 2 e Ψ e Ψ Exponential map R =exp( e Ψ) Angular velocities Ω = T (Ψ) Ψ Tangent operator µ cos kψk 1 T (Ψ) = I + kψk 2 µ eψ + 1 ω = T T (Ψ) Ψ sin kψk Ψ e Ψ e kψk kψk 2 Limit properties lim kψk 0 T (Ψ) = lim kψk 0 R(Ψ) = I Finite Motion 55 28