introduction to samcef mecano - cours dispensés … system library fe structural substructuring 6...

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1Introduction to SAMCEF - MECANO

Introduction to Introduction to

SAMCEF MECANOSAMCEF MECANO


OutlineOutline

IntroductionGeneralized coordinatesKinematic constraintsTime integrationDescription and paramerization of finite rotations

AcknowledgementsMichel Géradin (ULg)

Doan D. Bao (GoodYear)Alberto Cardona (Intec, Argentina)

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Fixed deformable structures

Fixed deformable structures

Finite Element Method

Finite Element Method

Articulated rigid systemsArticulated

rigid systems

MultibodyDynamics

MultibodyDynamics

?

how to analyze in a general manner flexible articulated

structures

Our GoalOur Goal


Strategic choicesStrategic choicesKinematic descriptionKinematic description

Recursive?Absolute?Absolute

Structural behaviorStructural behavior

Rigid ?Flexible ?Rigid/flexible

Motion equationsformulation

Motion equationsformulationSymbolic ?Numerical ?Numerical Time integrationTime integration

Implicit ? Explicit ? Implicit

Software architecture

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Modelling Modelling StepStep

Finite rotation parameterization

Finite motion description

Measures of deformation and relative motion

Rigid bodieszero deformation

Flexible elementselastic strains

+ internal work

Jointsrelative motion + external work

Mechanical elements library

• rigid members• flexible members• rigid articulations• flexible articulations• active members

Control systemlibrary

FE structuralsubstructuring


Modelling Modelling StepStep

Mechanical elements library

• rigid members• flexible members• rigid articulations• flexible articulations• active members

Control systemlibrary

FE structuralsubstructuring

DAE system equations• dynamic equilibrium• kinematic constraints

Post-processingkinematics

internal forceselement stresses

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The MECANO The MECANO softwaresoftware

Preprocessing• topological description (FE)

• mechanical properties• response specification

Preprocessing• topological description (FE)

• mechanical properties• response specification

Initial Assembly

Static analysisequilibrium configuration

static forces

Static analysisequilibrium configuration

static forces

Kinetostatic analysistrajectories

quasi-static forces

Kinetostatic analysistrajectories

quasi-static forces

Dynamic analysistrajectories

dynamic forces

Dynamic analysistrajectories

dynamic forces

Linearized analysiseigenvalues

stability

Linearized analysiseigenvalues

stability

Preprocessing• trajectories

• loads and stresses• animation

Preprocessing• trajectories

• loads and stresses• animation


The MECANO The MECANO element libraryelement library

Some rigid joint elements

Hinge joint

Gear joint

Prismatic joint

Cylindrical joint

Spherical joint

Rack-and-pinion

Kinematic constraints

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The MECANO element libraryThe MECANO element library

Cam contact

Point-on-plane

Wheel + tyre model

Cable + pulley

some non-holonomic joints



The MECANO The MECANO element libraryelement library

BushingSpring-damper-stop

Nonlinear beam

Superelement

The basic elastic elements


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Hydraulic actuator

Nonlinear constraint

Nonlinear feedback

Some nonlinear and control features




Rail-wheel contact


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Flexible slider




Symbolic element representation


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Domains of applicationDomains of application

• mechanical engineering• automotive industry• sport industry• aeronautics• space structures• biomechanics• ...


Dynamic car Dynamic car behaviorbehavior

M. Ebalard, J. Mercier (PSA Citroën)Utilisation du logiciel Mecano pour le comportement routier à PSA

Adequacy of Mecano• association of kinematicjoints with finite elements• open software through user element capability

Examples of application

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Landing gear analysisLanding gear analysis

Research conducted in framework of ELGAR consortium



Landing gear analysisLanding gear analysisFunctional phases• deployment/retraction• ground impact• rolling• breaking• taxiing


Model components• mechanism• damper (internal hydraulic system)• tyre: contact, deformation, friction

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MEA antenna: result animationMEA antenna: result animation



Simulation of Cluster satellite boom deploymentSimulation of Cluster satellite boom deployment


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Generalized coordinates for mechanism analysisGeneralized coordinates for mechanism analysis• Generalized co-ordinates

description of various typesthe 4-bar mechanism example

• The finite element method for articulated systemskinematics

elasto-dynamicsdifferential-algebraic equations of motionnumerical integration by Newmark algorithmthe double pendulum example

Generalized coordinates


IntroductionIntroduction• 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


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The 4The 4--bar mechanism examplebar mechanism example

• Simplest planar, closed-loop mechanism• one single variable: crank angle β

Gruebler’s formula m : number of jointsN: number of bodiesp : number of DOF removed by joint i

F = 1F = 3(N − 1)−mXi=1

pi



Minimal coordinatesMinimal coordinates Expression in terms of minimal coordinate β

cosθ2 =d2 + s2 + 2r` cosβ

2ds− r2 − `2

sin(θ3 + φ) =c

ρ

θj = f(β) j = 2, . . . 4

From geometry

(`− r cosβ) cos θ3 + r sinβ sin θ3 = s− d cos θ2

θ1 = 2π − β − θ2 − θ3


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Minimal coordinates: Minimal coordinates: kinematickinematic transformertransformer

Input-output relationship

Complex mechanisms as assembly of transformers



LagrangianLagrangian coordinatescoordinates

3 generalized coordinates

2 closure conditions

Equations of motion: 3 differential Equations of motion: 3 differential equations linked by 2 constraintsequations linked by 2 constraints(DAE)(DAE)

qT = [θ1 θ2 θ3 ]

`1 cosθ1 + `2 cos(θ1 + θ2) + `3 cos(θ1 + θ2 + θ3)− `4 = 0`1 sin θ1 + `2 sin(θ1 + θ2) + `3 sin(θ1 + θ2 + θ3) = 0


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Cartesian coordinatesCartesian coordinates

8 kinematic constraints

9 generalized coordinates qT = [ x1 y1 θ1 x2 y2 θ2 x3 y3 θ3 ]

x1 −`12 cosθ1 = 0 y1 −

`12 sin θ1 = 0

x1 +`12 cosθ1 − x2 +

`22 cos θ2 = 0 y1 +

`12 sin θ1 − y2 +

`22 sin θ2 = 0

x2 +`22 cosθ1 − x2 +

`32 cos θ3 = 0 y2 +

`22 sin θ2 − y3 +

`32 sin θ3 = 0

x3 +`32 cosθ3 = `4 y3 +

`32 sin θ3 = 0

Equations of motion: 9 Equations of motion: 9 differential equations linked differential equations linked by 8 constraintsby 8 constraints (DAE)(DAE)



Finite Element coordinatesFinite Element coordinates

Strong formStrong form of of kinematickinematic constraintsconstraints• 4 boundary nodal constraints

• 4 assembly nodal constraints

13 generalized coordinates:• 12 finite element DOF q• 1 driving DOF βq = [x1 y1 . . . x6 y6 ]

T

• 3 zero strain constraints

• 1 driving constraints (implicit)

x1 = 0, y1 = 0x6 = 0, y6 = `4

x2 = x3 y2 = y3x4 = x5 y4 = y5

²1 = 0 ²2 = 0 ²3 = 0

y2 cosβ − x2 sinβ = 0ΦD(q) = 0

²i =1

2

`2i − `2i,0

`2i,0


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Finite element description of a mechanical systemFinite element description of a mechanical system

q = DOF at structural level

qe = DOF at element level

Le = DOF localization operator(boolean)

λ = lagrangian multipliers

qe = Leq

Φ(qe, qe, t) = 0

• Boolean constraints: localization of DOF• Implicit constraints: algebraic treatment

System topology results implicitly from Boolean assembly and

constraint description



Classification of Classification of kinematickinematic constraintsconstraints

• Holonomic constraints

• Restrict the number of DOF of the system

Φ(q, t) = 0

• Non-holonomic constraints

• Restrict the system behavior

• Unilateral constraints

ex: prismatic joint

ex: rolling motion

Ψ(q, q, t) ≥ 0

Ψ(q, q, t) = 0


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Classical mechanical pairsClassical mechanical pairs

revolute

cylindrical

prismatic

screw

planar

spherical

Only 3 lower pairs• surface contact• motion reversibility

• All other mechanical pairs are higher pairs

Other modes of classification:• number of DOF (1 for all lower pairs)• mode of closure


The algebraic constrained problemThe algebraic constrained problem⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

minqF(q)

subject to

Φ(q) = 0

Possible methods of solution• constraint elimination• Lagrange multipliers• penalty function• augmented Lagrangian• perturbed Lagrangian

Formulated in terms of

Residual vector

r(q) =∂F

∂q

Jacobian matrix

Hessian matrix

Bmj =∂Φm∂qj

B such that

H such that Hij =∂2F

∂qi∂qj

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Constraint elimination methodConstraint elimination method• First variation of constraints

δΦ =Bδq = 0• Partitioning into independent and dependent variables

BIδqI +BDδqD = 0

• Elimination of dependent variables

δq =RδqIreduction

matrix R =

⎡⎣ I

−B−1D BI

⎤⎦

RT ∂F

∂q=RT r(q) = 0

• Projected residual equation

• exact verification of constraints• complex and computationally expensive


• provides exact solution regardless of penalty and scaling factors

• quadratic term reinforces positive character of H

• scaling factor required for pivoting

Augmented Augmented LagrangianLagrangian methodmethod

F∗p (q,λ) = F(q) + (kλTΦ+

p

2ΦTΦ)

⎡⎣H + pBTB kBT

kB 0

⎤⎦⎡⎣∆q∆λ

⎤⎦ =⎡⎣−r? −BT (kλ? + pΦ)

kΦ

⎤⎦

p penalty coefficientk scaling factor

Augmented functional

Second-order solution method: by Newton-Raphson

q = q? +∆qλ = λ? +∆λ

pBTB

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Constrained dynamic problemsConstrained dynamic problems

Z t2

t1[δ(L− kλTΦ− pΦTΦ) + δW]dt = 0

Mq +BT (pΦ+ kλ) = g(q, q, t)Φ(q, t) = 0

Hamilton’s principle

Matrix form of constrained motion

equations

• 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



Constrained equations of motionConstrained equations of motion

Mq +BT (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


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Linearization of motion equationsLinearization of motion equationsM 00 0

¸∆q∆λ

¸+Ct 00 0

¸∆q∆λ

¸+Kt kBT

0 kB

¸∆q∆λ

¸=r?

−Φ?

¸+O(∆2)

Residual vector of equilibrium r = g(q, q, t)−Mq −BT (pΦ+ kλ)

Ct = −∂g

∂qKt = −

∂g

∂q+∂(Mq)

∂q+∂(BT (pΦ+ kλ))

∂q

Kt ' −∂g

∂q+ pBTB + k

∂(BTλ)

∂q

Tangent damping matrix Tangent stiffness matrix

approx.• plays central role for convergence in iteration• quality of approximation needed depends on penalty factor



Integration of motion equations: Integration of motion equations: Implicit MethodImplicit Method

Motivationsfiltering 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)

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Implicit integration Implicit integration Differential - algebraic equations

Correction of solution at time t

q = q∗ +∆qq = q∗ +∆qq = q∗ +∆qλ = λ∗ +∆λ

Linearized equations of motion

Linearized matrices

r(q, q, q,λ) =Mq+BTλ− g(q, q, t) = 0Φ(q, t) = 0

(q∗, q∗, q∗, λ∗)

KT =∂r

∂q= −

∂g

∂q+

∂

∂q(Mq∗ +BTλ∗)

CT =∂r

∂q= −

∂g

∂q

Tangent damping

Tangent stiffness

M 00 0

¸∆q∆λ

¸+

CT 00 0

¸∆q∆λ

¸+

KT BT

B 0

¸∆q∆λ

¸=

r∗

−Φ∗

¸


NewmarkNewmark formula formula

• Interpolation of displacements and velocities

qn+1 = qn + (1− γ)hqn + γhqn+1qn+1 = qn + hqn + (

12− β)h2qn ++βh

2qn+1

γ =1

2and β =

1

4

• Free parameters: average constant acceleration

γ =1

2+ α and β =

1

4(γ +

1

2)2 α > 0

• Introduction of numerical damping

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Integration procedure Integration procedure q0n+1 = 0q0n+1 = qn + (1− γ)hqnq0n+1 = qn + hqn + (

12 − β)h2qn

qk+1n+1 = qkn+1 +∆q = q

kn+1 +

1βh2∆q

qk+1n+1 = qkn+1 +∆q = q

kn+1 +

γβh∆q

qkn+1 = qkn+1 +∆q

ST BT

B 0

¸∆q∆λ

¸=

r∗

−Φ∗

¸

ST =KT +γ

βhCT +

1

βh2M

krk < ² and kΦk < η

• Prediction at time t

• Correction

• Correction equation

Iteration matrix

• Convergence check


The double pendulum The double pendulum

qT = [x1 y1 θ1 x2 y2 θ2 ]

• Description by Cartesian coordinates (redundant)

K = 12m1(x

21 + y

21) +

12m2(x

22 + y

22)

P = m1gy1 +m2gy2

• Kinetic and potential energies

Φ =

⎡⎢⎢⎣x1 − `1 sinθ1y1 + `1 cos θ1

x2 − x1 − `2 sin θ2y2 − y1 + `2 cos θ2

⎤⎥⎥⎦• Kinematic constraints

B =

⎡⎢⎢⎣1 0 −`1 cos θ1 0 0 00 1 −`1 sinθ1 0 0 0−1 0 0 1 0 −`2 cos θ20 −1 0 0 1 −`2 sin θ2

⎤⎥⎥⎦• Constraint matrix

• External force vector and mass matrix are constant• Tangent stiffness contains only geometric terms

KT = diag [ 0 0 `1(λ1 sin θ1 − λ2 cos θ1) 0 0 `2(λ3 sin θ2 − λ4 cos θ2) ]

m1 = m2 = 1.l1 = l2 = 1.,(θ1 = π/2, θ2 = 0

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(γ = 12 , β =

14 )NewmarkNewmark integration : integration : undampedundamped

• normal evolution of angular displacements and velocities• but: numerical instability detected after 2 sec.


(γ = 12 , β =

14 )NewmarkNewmark integration : integration : undampedundamped

Weak instability: generated by kinematic constraints

Numerical damping

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((γ = 1

2 + α, β = 14 (12 + γ)2, α = 0.015)

NewmarkNewmark integration with damping integration with damping

Normal aspect of response over 5 sec. period, but ...


NewmarkNewmark integration with damping integration with damping ((γ = 1

2 + α, β = 14 (12 + γ)2, α = 0.015)

Unacceptable amount of energy loss

Search for better compromise between accuracy and stability

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Kinematics of finite motionKinematics of finite motion

• Spherical motionEigenvalue analysis of rotation operatorEuler theorem Explicit expressions of rotation operator Expression in terms of linear invariantsExponential map

• General motion of rigid body• Velocity analysis of spherical motion• Explicit expression of angular velocities

Rotation parameterizationCartesian rotation vector

Finite Motion


Spherical motionSpherical motion

Initial position: material configuration X

Current position: material configuration x

Spatial coordinates

x = [x1 x2 x3]T

X = [X1 X2 X3]T

material coordinates

[~E1 ~E2 ~E3]

Base vectors of initial configuration

Base vectors of current configuration [~e1 ~e2 ~e3]

Spherical motion such that:• length of position vector unaffected by pure rotation• relative angle between 2 directions remain constant

Linear transformation

x = RX RT =R−1 det(R) = 1withProper orthogonal

Finite Motion

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Euler theorem 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


Explicit expressions of rotation operator Explicit expressions of rotation operator

Outer product expression

Orthonomality implies 6 constraints

rTi rj = δijR = [ r1 r2 r3 ]

R =R(α1,α2,α3)3 independent rotation parameters

R =Xi

eiETi

Inner product expression

R =

⎡⎣ET1 e1 ET1 e2 ET1 e3

ET2 e1 ET2 e2 ET2 e3

ET3 e1 ET3 e2 ET3 e3

⎤⎦

Finite Motion

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Expression in terms of linear invariantsExpression in terms of linear invariants

R=Icosφ+(1−cosφ)nnT +nsinφ

Rotation operator

Invariants

tr(R) =λ1 +λ2 +λ3 =1+2cosφvect(R) =nsinφ

Finite Motion


Exponential mapExponential map

• Differentiate with respect to φ

x =RX

• verify thatdR

dφRT = n

dx

dφ− nx = 0 with x(0) =X

• differential motion

• integration

x = exp(nφ) X R = exp(nφ)

Finite Motion

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General motion of rigid bodyGeneral motion of rigid body

xP = x0 +RXP

Combination of translation and rotation

spatial motion of origin

material coordinates of

point P

Finite Motion


Velocity analysis of spherical motionVelocity analysis of spherical motion

xP =RXP RT =R−1

ω = RRTvP = ωxP

ω = vect(RRT )ω = vect(RRT )

V P = ΩXP Ω =RT R

Ω = vect(RT R)Ω = vect(RT R)

vP = xP = RXP V P =RT vP

Spatial description Material descriptionDifferentiate

with

= RRT + (RRT )T = 0

with with

spatial angular velocities material angular velocities

spatial velocities material velocities

skew-symmetry

Finite Motion

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Cartesian rotation vectorCartesian rotation vector

Ψ = nφdefined as

R = I +sin kΨ k

kΨ keΨ +

1− cos kΨ k

kΨ k2eΨ eΨ

Rotation operator

Trigonometric form

R = exp( eΨ)Exponential map

Ω = T (Ψ)Ψ ω = T T (Ψ)Ψ

T (Ψ) = I +

µcos kΨk − 1

kΨk2

¶ eΨ+ µ1 − sin kΨkkΨk

¶ eΨ eΨkΨk2

Angular velocities

Tangent operator limkΨk→0

T (Ψ) = limkΨk→0

R(Ψ) = I

Limit properties

Finite Motion

introduction to samcef mecano - cours dispensés … system library fe structural substructuring 6...

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