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A system-level model reduction technique for the efficient simulation of flexible multibody systems

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MULTIBODY DYNAMICS 2009, ECCOMAS Thematic ConferenceK. Arczewski, J. Fr ˛aczek, M. Wojtyra (eds.)Warsaw, Poland, 29 June–2 July 2009
A SYSTEM-LEVEL MODEL REDUCTION TECHNIQUE FOREFFICIENT SIMULATION OF FLEXIBLE MULTIBODY DYNAMICS
Gert H.K. Heirman
∗
, Olivier Brüls
†
and Wim Desmet
∗∗
Department of Mechanical EngineeringKatholieke Universiteit LeuvenCelestijnenlaan 300B, B-3001 Heverlee (Leuven), Belgiume-mail: gert.heirman@mech.kuleuven.be, wim.desmet@mech.kuleuven.be
†
Department of Aerospace and Mechanical Engineering (LTAS)University of LiègeChemin des Chevreuils 1, 4000 Liège, Belgiume-mail: o.bruls@ulg.ac.be
Keywords:
Non-linear Model Reduction, Flexible Multibody Dynamics, Moving ConnectionPoints, Global Modal Parametrization, Real-Time Simulation
Abstract.
In ﬂexible multibody dynamics, body-level model reduction is typically used to de-crease the computational load of a simulation. Body-level model reduction is generally per- formed by means of Component Mode Synthesis. This offers an acceptable solution for manyapplications, but does not result in signiﬁcant model reduction for systems with moving connec-tion points, e.g. due to a ﬂexible sliding joint. In this research, Global Modal Parametriza-tion, a model reduction technique initially proposed for real-time control of ﬂexible mech-anisms, is further developed to speed up simulation of multibody systems. The reduction isachieved by a system-level modal description, as opposed to the classic body-level modal de-scription. As the dynamics is conﬁguration-dependent, the system-level modal description ischosen conﬁguration-dependent in such a way that the system dynamics are optimally described with a minimal number of degrees of freedom. Moving connection points do not pose a prob-lem to the proposed model reduction methodology. The complexity of simulation of the reduced model equations is estimated. The applicability to systems with moving connection points ishighlighted. In a numerical experiment, simulation results for the srcinal model equationsare compared with simulation results for the model equations obtained after model reduction,showing a good match. The approximation errors resulting from the model reduction tech-niques are investigated by comparing results for different mode sets. The mode set affects theapproximation error similarly as it does in linear modal synthesis.
1
Gert H.K. Heirman, Olivier Brüls and Wim Desmet
1 INTRODUCTION
Body-level model reduction, such as linear modal synthesis, is used extensively in ﬂexiblemultibody dynamics. A modal description of a body’s ﬂexibility requires considering the con-tribution of the body’s dominant eigenmodes, and one static deformation pattern per interfacedegree of freedom (DOF): a DOF which could be loaded during the simulated scenario. Thisloading can be caused either by a constraint or by an external loading. The DOFs, on which aconstraint acts, can vary as a function of the relative position and orientation of the connectedbodies. Thisvariability holds foralmostallconstraintstypically encounteredin multibodymod-els, but for some types of joints it is more pronounced. Slider joints typically result in a highlyvariable connection interface between bodies. A classic positioning system typically consistsof bodies connected through slider joints (e.g. the system shown in Fig. 1). Systems with manyinterface DOFs due to constraints, will be further referred to as systems with moving connectionpoints.
yzx
framelinear motorscarriageflexible beamgripper
Figure 1: The FlexCell pick-and-place machine: an example of a system with moving connection points
If the ﬂexibility of bodies with variable connectivity needs to be taken into account, thisposes a problem for the body-level model reduction by modal synthesis. Loads can be ex-pected in many DOFs during the simulated scenario, and thus many static deformation patternsshould be included in the modal description of the body ﬂexibility. Ignoring the contributionof static deformation patterns can lead to a bad approximation in case of concentrated loads[1], such as typically encountered in multibody systems. Wasfy and Noor give an overview of modelling techniques for ﬂexible multibody systems [2]. Current approaches to model systemswith moving connection points are the aforementioned body-level modal synthesis [3], or usingexpensive non-reduced models [4]. To the authors’ knowledge, system-level model reductionhas not yet been used to tackle this problem.2
Gert H.K. Heirman, Olivier Brüls and Wim Desmet
Even after body-level model reduction, the resulting set of equations is still a rather largeset of differential-algebraic equations (DAE), especially for problems with moving connectionpoints. Both the DAE-character of the model equations, and the number of degrees of free-dom needed to accurately represent ﬂexibility, prohibit fast simulation of these systems. Thereis however an increasing demand for real-time and faster-than-real-time simulation of ﬂexiblemultibody systems, such as in Hardware/Human/Software-in-the-Loop and Model PredictiveControl applications. Formulations based on relative generalized coordinates may lead to efﬁ-cient simulation tools [5] but a number of loop closure kinematic constraints is then required tomodel complex parallel mechanisms, so that the resulting model is still computationally chal-lenging. Relative coordinates also allow body ﬂexiblity to be incorporated, e.g. through modalsynthesis, but this approach would also suffer from the multitude of required static deformationpatterns in case of moving connection points.Few techniques exist for system-level model reduction of ﬂexible multibody models. Mostsystem-level model reduction techniques for non-linear models build reduced models based ondata obtained from user-deﬁned numerical experiments of the srcinal unreduced model, e.g.techniques based on neural networks [6] and techniques based on projection on a ﬁxed vectorspace [7]. The resulting reduced models only offer an accurate approximation in scenariossimilar to the numerical experiments on which they are based. For these techniques, a goodapproximation for all possible states of the system requires many numerical experiments andlimits the computational efﬁciency of the reduced model equations.In this research, Global Modal Parametrization, a model reduction technique for ﬂexiblemultibody systems proposed by Brüls for controller design of ﬂexible mechatronic systems [8],is developed further for the purpose of speeding up simulation of ﬂexible multibody systems.The reduction of the model is achieved by projection on a curvilinear subspace [9] instead of the classically used ﬁxed vector space. Using a ﬁxed vector space for highly non-linear systemsrequires the inclusion of many vectors in the modal basis to span the state-dependent dominantdynamics of the system, i.e. the system’s dominant eigenmodes and relevant static deforma-tion patterns, for any point of the conﬁguration space. The curvilinear subspace, however, isdeﬁned by imposing that the tangent space spans, exactly and only, the state-dependent domi-nant eigenmodes, the rigid body modes and low-frequency elastic eigenmodes, and the relevantstatic deformation patterns. In this way, the dominant dynamics of a highly non-linear systemcan be represented by a minimal number of coordinates. The system-level eigenmodes andstatic deformation patterns automatically satisfy the constraints imposed by the joints; Movingconnection points do not induce additional difﬁculty.The next section deals with the methodology of GMP. Section 3 explains how this method-ology should be applied to minimize computational load during simulation of the system. Thecomplexity of simulation of the reduced model equations is estimated and compared to simula-tion of the srcinal model equations. Section 4 elaborates on the applicability of GMP to sys-tems with moving connection points. A validation of the methodology is done by a numericalexperiment in Section 5. The test case is tailored to show the possibilities of the methodologyin systems with moving connection points. Simulation results for the srcinal model equationsare compared with simulation results for the model equations obtained after the proposed modelreduction methodology. The latter require much less degrees of freedom to represent the studieddynamical phenomena as compaired to unreduced and body-level reduced models. The approx-imation errors resulting from the model reduction techniques are investigated by comparingresults for different mode sets.3
Gert H.K. Heirman, Olivier Brüls and Wim Desmet
2 GLOBAL MODAL PARAMETRIZATION
Irrespective of the type of coordinates used, the equations of motion of a (ﬂexible) multibodysystemcanbewrittenasa(non-linear)second-orderDAEintermsofitsgeneralizedcoordinates
q
:
M
qq
(
q
) ¨
q
+
h
q
(
q,
˙
q
) +
V
,q
+ Φ
T ,q
λ
=
g
q
(1)
Φ(
q
) = 0
(2)In these equations:
•
q
is a vector of the
n
generalized coordinates, the value of
q
deﬁnes the
conﬁguration
of the system.
•
M
qq
(
q
)
is the conﬁguration-dependent mass matrix.
• V
,q
is the gradient of the potential energy. Only potential energy due to structural defor-mation is considered in this term. The generalized forces due to other potential energysources (e.g. gravity) will be taken into account through the source term
g
q
.
•
g
q
denotes the generalized forces due to external loads on a mechanical component.
•
Φ(
q
) = 0
expresses
m
kinematic holonomic constraints
1
. Its gradient
Φ
T ,q
is assumed tobe of full rank for all conﬁgurations
q
.
•
λ
is a vector of
m
Lagrange multipliers associated with the constraints.
•
Φ
,q
is the matrix of constraint gradients,
Φ
T ,q
λ
represents the reaction forces and momentsenforcing the constraints.
•
h
q
gathers the centrifugal and Coriolis inertia forces which are quadratic in
˙
q
. Using theindex summation convention, we have
(
h
q
)
i
= (Γ
qqq
)
ijk
˙
q
j
˙
q
k
(3)where
(Γ
qqq
)
ijk
is the Christoffel symbol of the ﬁrst kind:
(Γ
qqq
)
ijk
= 12
∂
(
M
qq
)
ij
∂q
k
+
∂
(
M
qq
)
ik
∂q
j
−
∂
(
M
qq
)
jk
∂q
i
(4)The overal motion is decomposed into a large amplitude rigid body motion
q
r
and a smallamplitude elastic displacement
q
f
q
=
q
r
+
q
f
(5)Both
q
and
q
r
satisfy the constraint equations Eqn. (2).
q
r
represents a non-deformed con-ﬁguration.
q
r
will be further referred to as the
rigid body motion
. The term
V
,q
in Eqn. (1)represents generalized forces due to elastic deformation. As only the elastic displacement
q
f
results in a change of the potential energy and as the small elastic deformation allow a lineariza-tion around the undeformed conﬁguration
q
r
, this term can be rewritten as:
1
Non-holonomic constraints are not considered in this work.
4
Gert H.K. Heirman, Olivier Brüls and Wim Desmet
V
,q
=
K
qq
(
q
r
)
q
f
+
O
(
q
f
2
)
(6)The number of rigid body degrees of freedom of the multibody system is referred to as
s
. Therigid body motion can be represented by
θ
, which is a selection of
s
coordinates out of the setof coordinates
q
. This representation of the system inevitably leads to singularities in the deadpoints corresponding to the coordinates
θ
. If the system has dead points corresponding to thecoordinates
θ
, theanalysisofthesystemshouldbelimitedtoanareainwhichthisrepresentationis valid, i.e. away from the system’s dead points. It will further be assumed that this is the case.The invertible, sufﬁciently continuous coordinate transformation
ρ
, which maps
θ
to the rigidbody conﬁguration
q
r
, can be deﬁned:
q
r
=
ρ
(
θ
)
(7)Its Jacobian
ρ
,θ
(
θ
)
can be interpreted as the matrix of rigid-body modes
Ψ
qθ
(
θ
)
. Note that,for all values of
θ
and away from the system dead points, this matrix has maximal rank, itscolumns (the rigid-body modes) have ﬁnite length and are continuous for varying
θ
. A vector iscontinuous if all of its elements are continuous.
ρ
,θ
(
θ
) = Ψ
qθ
(
θ
)
(8)The rigid body modes
Ψ
qθ
(
θ
)
respect the constraint equations by deﬁnition:
Φ
,q
Ψ
qθ
≡
0
(9)The degrees of freedom (DOF)
q
can be partitioned in:
•
s
rigid body DOFs
θ
•
n
g
constrained DOFs
q
g
: the DOFs that don’t belong to
θ
and on which an externalloading will be applied during the intended simulation
•
the internal DOFs
q
i
: the remainder of the DOFsThe elastic deformation
q
f
is considered as a deviation from the undeformed conﬁguration
q
r
, which can be represented by
(
n
−
m
)
−
s
independent coordinates
ˆ
δ
through an invertible,sufﬁciently continuous coordinate transformation
σ
.
q
f
=
σ
(
θ,
ˆ
δ
)
σ
(
θ,
0)
≡
0
(10)
q
has to satisfy the constraint equations:
Φ(
ρ
(
θ
) +
σ
(
θ,
ˆ
δ
)) = 0
(11)Linearizing around an undeformed conﬁguration
(
θ,
ˆ
δ
) = (
θ,
0)
and ignoring second orderterms, results in:
Φ
,q
σ
,
ˆ
δ
(
θ,
0) = 0
(12)Under the small deformation assumption,
σ
(
θ,
ˆ
δ
)
can be approximated linearly:
q
f
=
σ
,q
(
θ,
ˆ
δ
) ˆ
δ
= Ψ
q
ˆ
δ
(
θ
) ˆ
δ
(13)5

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