Flexible multibody systems Relative coordinates approach...Flexibility effects are modeled by spring...
Transcript of Flexible multibody systems Relative coordinates approach...Flexibility effects are modeled by spring...
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GraSMech – Multibody
Computer-aided analysis ofmultibody dynamics (part 2)
Flexible multibody systems-
Relative coordinates approach
Paul Fisette([email protected])
GraSMech – Multibody
Introduction
In terms of modeling, multibody scientists must develop a « critical mind »
=> What is the « real » problem to solve …. and thus the optimal model to build
From a « customer
problem »
statics
basic dynamics
advanceddynamics
flexible bodies advanced dynamics
« by hand », Matlab
Matlab
rigid MBS code
flexible MBS code
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GraSMech – Multibody
Introduction
It is particularily true when dealing with « flexibility »
=> What is the « real » problem to solve and thus the « minimal » model to build
Examples :Car suspension deformation :
Antenna deployment :
Flexible 2D mechanism :
Chassis torsion :
MBS + Finite segment / 2D beammodel
MBS + FEM (super-elements…)
MBS + static FEM (in post-process)
Full FEM or Lumped torsion (MBS-rigid) ?
GraSMech – Multibody
Introduction
FEM community MBS community
> 1990 : « handshake »
MBS community « credos » :• Simulation time efficiency …towards real time• Lumped and/or « macro » models are still very promising• High dynamics problems• Flexibility : small deformation – few modes• High interest in system control and optimization
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GraSMech – Multibody
Contents
Relative coordinates approach : review
MBS with flexible beams : finite segment approach
MBS with flexible beams : assumed mode approach =>chap. 9 of
Beam Model
Symbolic implementation
MBS with telescopic beams Kluwer Academic Publishers, 2003
GraSMech – Multibody
Contents
Relative coordinates approach : review
MBS with flexible beams : finite segment approach
MBS with flexible beams : assumed mode approach =>chap. 9 of
Beam Model
Symbolic implementation
MBS with telescopic beams Kluwer Academic Publishers, 2003
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GraSMech – Multibody
Concepts
joint
• Relative joint coordinates
Leaf body
base
• Tree-like structuresloop
• Closed structures
• Rigid bodies (+flexible beams)
body
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6
5
43
2
0
8
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• Topology (tree-like structure) :the « inbody » vector : inbody = [ 0 1 2 2 4 4 1 7]
Multibody structure
GraSMech – Multibody
Joints
Concepts - Definitions
Classical …
« Wheel on rail » joint (5 dof)« Knee » joint (1 … 6 dof)
More « Exotic »« Cam/Follower » joint
6 dof !
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GraSMech – Multibody
.. .
body i
body j
body k
Fi
Li
Fj
Lj
Fk
Lk
ωj
ωi
kin.
dyn.
Tree-like MBS: Recursive Newton-Euler
Origin : Robotics – inverse dynamics(O(Nbody) operations)
Forward kinematics :
ω j = ω i + Ω ij
Backward dynamics :
F i = F j + F k + m i x i …
L i = L j + L k + I i ω i + ...Joint projection :
...
GraSMech – Multibody
Tree-like MBS: Recursive Newton-Euler
Origin : Robotics – inverse dynamics (Luh, Walker, Paul, 1980)(O(Nbody) operations)
Objective : Recursive formulation for direct dynamics(O(Nbody
2) operations)
… to keep the advantage of the recursive formulation
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for unconstrained system
Closed-loop MBS
Kinematic loops
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Closed-loop MBS
Coordinate partitioning :
u
v
v
Velocity :
Acceleration :
Exact resolution of the constraints
Position :
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GraSMech – Multibody
Contents
Relative coordinates approach : review
MBS with flexible beams : finite segment approach
MBS with flexible beams : assumed mode approach =>chap. 9 of
Beam Model
Symbolic implementation
MBS with telescopic beams Kluwer Academic Publishers, 2003
GraSMech – Multibody
Finite segment formulation
A very simple idea
i jk ij
li lj
. . .. . .
Ω = 150 rad/sec
Ressorthélicoïdal
θGlissière
« Computer methods in flexible multibody dynamics», Huston R.L., IJNME 1991 (also … Amirouche, 1986)
Flexible rod
Flexibility effects are modeled by spring (and possible dampers) between bodies=> Lumped flexibility formulation
ex. bending :
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GraSMech – Multibody
Finite segment formulation
Motivation
• to avoid the « difficult » mariage between rigid body dynamics and structural dynamics
• intuitive and direct method – perfect for a first « pre-study »
• can be implemented in a rigid multibody code (easy to implement !)
• incorporate the flexibilty effects into the global dynamic equations
• not intrinsically limited to elastic systems (=> viscoelastic, … nonlinear elastic …)
i jk ij
li lj
. . .. . .
Limitations • restricted to slender bodies (beams, tapered bodies, rods, …)
• no prove to satisfy the « Rayleigh » vibration criteria (in terms of eigenvalues approx.)
• deformation coupling : sequence dependent (but ok for small deformation)
• could be used « erroneously » : requires skills, insight and intuition
• computer efficiency : OK in 2D, heavy in 3D
GraSMech – Multibody
Finite segment formulation
Computation of the equivalent stiffness coefficients
Equivalent stiffness coefficient are computed from basic principle
of structual mechanics applied to bending, torsion and extension
Resulting Force –Torque in the MBS equations
Extension (example) :
m
µ dxContinuous :
Lumped :
=> Joint force (for extension) or torque (for torsion/bending)
l
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GraSMech – Multibody
Finite segment formulation
Proposed combinations (not exhaustive)
GraSMech – Multibody
Finite segment formulation
Proposed combinations (not exhaustive) COMBINED TAPERED SEGMENTS
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GraSMech – Multibody
Finite segment formulation
Example : a flexible slider-crank
Ω = 150 rad/sec
Ressorthélicoïdal
θGlissière
A
yA
Rod modal analysis around equilibrium (horizontal configuration)
GraSMech – Multibody
Finite segment formulation
Example : a flexible slider-crank
Ω = 150 rad/sec
Ressorthélicoïdal
θGlissière
FEM FSM
A
yA
yA yA
Lateral deflection of the mid-point A in frame Y
Y
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GraSMech – Multibody
Finite segment formulation
Example : a flexible slider-crank
Ω = 150 rad/sec
Ressorthélicoïdal
θGlissière
FEM FSM
B
xBShortening of the rod (point B) in frame Y
xB xB
Y
GraSMech – Multibody
Finite segment formulation
___ : FEM (+)…. : monomials shape function (+)-.-.-. : FSM (+)-----:modal shape function (-)
Kane’s benchmark : 2D rotating beam
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GraSMech – Multibody
Contents
Relative coordinates approach : review
MBS with flexible beams : finite segment approach
MBS with flexible beams : assumed mode approach =>chap. 9 of
Beam Model
Symbolic implementation
MBS with telescopic beams Kluwer Academic Publishers, 2003
GraSMech – Multibody
MBS with flexible beams
Flexible beam model
Shape functions
Beam Kinematics
Multibody kinematics (forward)
Joint dynamic equations( backward)
Deformation equations (beam per beam)
Symbolic computation
Applications
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GraSMech – Multibody
MBS with flexible beams
Flexible beam model
Shape functions
Beam Kinematics
Multibody kinematics (forward)
Joint dynamic equations( backward)
Deformation equations (beam per beam)
Symbolic computation
Applications
GraSMech – Multibody
Flexible beam model : hypotheses
Geometry : prismatic beams - rectilinear centroidal axis
Material : homogeneous and isotropic - conforms to linear elasticity
Deformation model : Timoshenko 3D, conservation of plane cross sections, shear deformation and rotary inertia included
Kinematics : angular and curvature of the beam must remain small(rotation matrix linearized) - but still compatible to « capture »geometic stiffening effect
Topology : a beam has the same status as a rigid body in the MBS :
Rigid bodyJoint
Flexible beam
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GraSMech – Multibody
Flexible beam model : notations
Beam local rotation
Linearized rotation matrix
GraSMech – Multibody
Flexible beam model : notations
Centroidal axis deformation
Current = undeformed + deformed :
Displacement field v :
Vector position (section S) :
C
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GraSMech – Multibody
Flexible beam : shape functions
For the x, y, z, linear deformation of the centroidal axis C
θ Ez^
Ex^ v
For the x, y, z, angular deformation of the cross sections S
with :
with :
generalized coordinates (=amplitude of shape functions)
generalized coordinates (=amplitude of shape functions)
GraSMech – Multibody
Flexible beam : shape functions
Which kind of shape functions ?
θ Ez^
Ex^ v
• From a previous (FEM) modal analysis => assumed modes
• From a purely mathematical set of functions:
• cubic splines
• Legendre polynomials
• Monomials
• …
« global » shape functions
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GraSMech – Multibody
• They can « theoretically » approximate any plausible deformation
(think at a Taylor series which combines them)
• Being an invariable set of functions, there is no need for a prior modal analysis*• They are perfectly suitable for symbolic computation of integrals, ex.:
• But … they do not form a set of orthogonal functions (=> numerical unstabilities)
Flexible beam : monomials
Why monomials ?
*Monomials being not eigenmodes, the beam configuration (q,qd,qdd) may« move away » from the equilibrium state of a prior modal analysis
= !
GraSMech – Multibody
MBS with flexible beams
Flexible beam model
Shape functions
Beam Kinematics
Multibody kinematics (forward)
Joint dynamic equations( backward)
Deformation equations (beam per beam)
Symbolic computation
Applications
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GraSMech – Multibody
Flexible beam : kinematics
Angular velocity and acceleration
« Relative » kinematics
Linear velocity and acceleration
GraSMech – Multibody
MBS with flexible beams
Flexible beam model
Shape functions
Beam Kinematics
Multibody kinematics (forward)
Joint dynamic equations( backward)
Deformation equations (beam per beam)
Symbolic computation
Applications
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GraSMech – Multibody
MBS with flexible beam : kinematics
Forward kinematic recursion
etc. for accelerations…
GraSMech – Multibody
MBS with flexible beam : joint dynamics
Virtual velocity field :
MBS Virtual power principle:
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GraSMech – Multibody
MBS with flexible beam : joint dynamicsBackward dynamic recursion
Fi
Li
GraSMech – Multibody
MBS with flexible beams
Flexible beam model
Shape functions
Beam Kinematics
Multibody kinematics (forward)
Joint dynamic equations( backward)
Deformation equations (beam per beam)
Symbolic computation
Applications
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GraSMech – Multibody
Flexible beam : deformation dynamics
Beam deformation :
Displacement gradient :
Strain vector of the centroïdal axis :
Beam curvature vector :
with
where
GraSMech – Multibody
Flexible beam : deformation dynamics
… requires the relative derivatives of Γ and K :
MBS Virtual Power Principle:(see next slides)
Real :
Virtual :
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GraSMech – Multibody
Flexible beam : deformation dynamics
Virtual velocity field :
MBS Virtual power principle:
GraSMech – Multibody
Flexible beam : deformation dynamics
Local equations of motion (of the beam portion ds) :
Constitutive equations (linear elasticity) :
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GraSMech – Multibody
Flexible beam : deformation dynamics
By integrating by part the terms :
Final form :
… for « each » beam i
GraSMech – Multibody
Flexible beam : deformation dynamics
Example of computation :
… for « each » beam i
recall … :
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GraSMech – Multibody
Flexible beam : deformation dynamics
Example of computation :
…
…
Using monomials
Analytical integrals
GraSMech – Multibody
Flexible beam : deformation dynamics
Example of computation :
Interpretation :
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GraSMech – Multibody
Flexible beam : deformation dynamics
Example of computation :
!!! Second order terms required in Γ to be consistent with first order kinematics in the VPP) !!!
Example : pure bending (=> Γ1 = 0 (by definition))
Second order :
pure bending =>
where :
First order :
pure bending => = 0 !!
GraSMech – Multibody
MBS + flexible beams : eq. of motion
Joint equations of body i :
Deformation equation of beam i :
Implicit equations of motion of the MBS
Rigid : Flexible (beam) :
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GraSMech – Multibody
MBS with flexible beams
Flexible beam model
Shape functions
Beam Kinematics
Multibody kinematics (forward)
Joint dynamic equations( backward)
Deformation equations (beam per beam)
Symbolic computation
Applications
GraSMech – Multibody
MBS + flexible beams : eq. of motion
with
Symbolic computation of the tangent matrices M, G, K with ROBOTRAN => recursive (efficient) derivation !!!!
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GraSMech – Multibody
MBS + flexible beams : eq. of motion
Global computation scheme (Robotran) :
GraSMech – Multibody
MBS + flexible beams : symbolic generation
Input file (for flexible beams)
Input file (for rigid bodies)(standard file) ROBOTRAN Symbolic
model…
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GraSMech – Multibody
MBS + flexible beams : symbolic generation
Symbolic model (example - Matlab)
Implicit form :
GraSMech – Multibody
MBS with flexible beams
Flexible beam model
Shape functions
Beam Kinematics
Multibody kinematics (forward)
Joint dynamic equations( backward)
Deformation equations (beam per beam)
Symbolic computation
Applications
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GraSMech – Multibody
MBS + flexible beams : examples
A cantilevered L-shaped structure : modal analysis
For each beam :• FEM : 10 beam elements• FSM : 10 interconnected bodies• Monomials : 5 shape functions in x, y, θ
GraSMech – Multibody
MBS + flexible beams : examples
Kane’s benchmark : 2D rotating beam
CPU time reduction factor :14
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GraSMech – Multibody
MBS + flexible beams : examples
Fisette et al. benchmark : 3D rotating beam
CPU time reduction factor :38
GraSMech – Multibody
MBS + flexible beams : examples
Jahnke, Popp’s benchmark : flexible slider-crank
FEM
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GraSMech – Multibody
Contents
Relative coordinates approach : review
MBS with flexible beams : finite segment approach
MBS with flexible beams : assumed mode approach
Beam Model
Symbolic implementation
MBS with telescopic beams
GraSMech – Multibody
MBS + telescopic beam : principle
Sliding section S, frame t
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MBS + telescopic beam : principle
Position constraints :
Orientation constraints :
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Loop closure : pseudo-rotation constraints
… recall…(part 1)
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Let’s choose a subset of 3 independent constraints(general 3D rotation):
Loop closure : pseudo-rotation constraints
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if all the constraints are satisfied :
because : = E
Loop closure : pseudo-rotation constraints
=> Coordinate partitioning method can be used to reduce the system => ODE
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Conclusion 1 :
with :
Loop closure : pseudo-rotation constraints
Pseudo-gradientPseudo-rotationconstraints
GraSMech – Multibody
MBS + telescopic beam : example
A telescopicflexible slider-crank
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GraSMech – Multibody
Conclusions
Flexible multibody systems in relative coordinates
Finite segment method : a « pragmatic » technique
MBS with flexible beams
Floating frame approach – set of « problem-independent » shape functions
Beam Model : Timoshenko (Euler-Bernouilli would certainly be a bit more efficient)
Symbolic implementation : OK and « fully » in case of monomial shape functions
CPU time : very powerfull (but limitation in terms of flexibility modeling)
MBS with telescopic beams : closed loop approach – symbolic implementation