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
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
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
5
Fachbereich
Mathematik
Part I: Mathematical Modeling
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
7
Fachbereich
Mathematik
Pendulum II
Revolute joint at tip → kinematic constraint
Ã
!
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
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
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
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
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
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
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
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
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
Ã
!
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
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!
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
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
⎛
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
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!
21
Fachbereich
Mathematik
Part II:
Structure of DAEs in
Multibody Dynamics
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
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).
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
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.
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
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 =???
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
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)
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
Fachbereich
Mathematik
Hidden Constraints Pendulum
⎛
⎞
⎛
⎞
v1 (t)
ṙ1 (t)
Velocity variables v(t) = ⎝ v2 (t) ⎠ := ⎝ ṙ2 (t) ⎠
v3 (t)
α̇(t)
Differentiation steps
⎞
⎛
Ã
! 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
Ã
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
¶
¶
32
16
22.09.2014
Fachbereich
Mathematik
Part III:
Stabilized Formulations for
Multibody Dynamics
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
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!
35
Fachbereich
Mathematik
„Flying Wheelset“
„Hunting motion“ – stable limit cycle
Si./Führer/Rentrop1991, Eich 1993
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
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
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!
39
Fachbereich
Mathematik
Part IV:
Hitchhiker‘s Guide to Time
Integration in Multibody Dynamics
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
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 )
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!
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
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)
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
46
23
22.09.2014
Fachbereich
Mathematik
Part V:
Specific Application Fields & Outlook
• Real-time integration
• Flexible multibody systems
• Transient saddle point problems
47
48
Fachbereich
Mathematik
Real-Time Simulation
24
22.09.2014
Fachbereich
Mathematik
Time Integration
49
Fachbereich
Mathematik
Vehicle-Trailer Coupling
50
25
22.09.2014
Fachbereich
Mathematik
DAE Real-Time Integrator
51
52
Fachbereich
Mathematik
Simulation Example
26
22.09.2014
Fachbereich
Mathematik
Drift-Off?
53
Fachbereich
Mathematik
Flexible Multibody Systems
elastic
elastic
54
27
22.09.2014
Fachbereich
Mathematik
Linear Elasticity
55
Fachbereich
Mathematik
Floating Frame of Reference
56
28
22.09.2014
Fachbereich
Mathematik
Slider Crank Revisited
57
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
59
30

Multibody Systems

Transcrição

Documentos relacionados

Calculus of Several Variables

2010-10-01 - 03 s101003cl - Session 3 - Final