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Title: Model reduction for non-linear dynamics by projection on state-dependent modal base
Authors: Heirman, Gert
Desmet, Wim #
Issue Date: 28-Sep-2009
Host Document: Euromech Colloquium 503: Nonlinear Normal Modes, Dimension Reduction and Localization in Virating Systems
Conference: Euromech Colloquium 503: Nonlinear Normal Modes, Dimension Reduction and Localization in Virating Systems date:28 September - 2 October 2009
Abstract: Irrespective of the type of generalized coordinates used, the modeling of the motion of a (
exible) mechanism
results in a set of (non-linear) di erential-algebraic equations (DAE): second-order di erential equations
express the equations of motion, while algebraic equations impose constraints on the motion of the
system. Both the DAE-character of the model equations, and the number of degrees of freedom needed
to accurately represent
exibility (typically up to 1000's of DOFs), prohibit real-time simulation of these
systems. Model reduction on subsystem-level, e.g. modal representation of the
exibility of a mechanism
component, is used extensively in
exible mechanism models to limit the computational load. However,
subsystem-level model reduction su ers from the intrinsic drawback that it does not result in signi cant
dimension reduction if the interface between the subsystems is (highly) variable. Furthermore, algebraic
equations will still be present in the resulting set of equations.
Current real-time running models of
exible mechanisms are mostly based on ad hoc simpli cations of the
model. Few techniques exploit the mathematical structure of the original model equations. Most systemlevel
model reduction techniques for non-linear models build reduced models based on data obtained from
user-de ned numerical experiments of the original model. The resulting reduced models only o er an
accurate approximation in scenarios similar to the numerical experiments on which they are based. For
these techniques, a good approximation for all possible states of the system requires many numerical
experiments and limits the computational eciency of the reduced model equations. In this research,
a system-level model reduction technique is developed that exploits the mathematical properties of the
original model equations.
Projection-based model reduction is done by projecting the instantaneous change of the state the system on
a vector set. To accurately model the change of the system, this vector set should span the instantaneous
(state-dependent) dominant dynamics of the system, i.e. the 1) state-dependent dominant eigenmodes,
1a) the rigid body modes and 1b) low-frequency elastic eigenmodes, and 2) the relevant static deformation
patterns. Using a xed vector set, i.e. the same vector set regardless of the state of the system, for
non-linear systems requires the inclusion of many vectors in the vector set, to ensure the state-dependent
dominant dynamics of the system are spanned for each state of the system. This research proposes to
project the instantaneous change of the state of the system on a state-dependent vector set instead of a
xed vector set. The state-dependent vector set consists of a set of modes that, exactly and only, span the
dominant dynamics. The system is thus expressed in a curvilinear coordinate system  of which the axes
are de ned by the state-dependent eigenmodes and static deformation patterns of the system. Di erent
mode sets meet these demands [1].
Using a state-dependent vector set o ers important advantages over a xed vector set. First, the dominant
dynamics are described by a minimal number of vectors, which will result in a minimal number of degrees of
freedom in the resulting reduced model equations, while only minimally decreasing the accuracy. Secondly,
the selected system-level eigenmodes and static deformation patterns intrinsically satisfy the constraints,
such that the algebraic equations are automatically sati ed after model reduction; The model reduction
transforms the model equations from a set of di erential-algebraic equations (DAE) into a set of ordinary
di erential equations (ODE). Both advantages result in faster simulation of the reduced model equations.
However, composing the matrices and tensors of the reduced model equations is expensive and can only
be done numerically. The continuity of these matrices and tensors allows to cheaply interpolate them from
their values at discrete states of the system. The overall process is thus decomposed in two steps: 1) a
preparation phase in which the reduced model matrices and tensors are computed and stored for a discrete
set of states, and 2) a simulation phase. At each time step in the simulation phase, the reduced model
matrices and tensors are interpolated out of the previously computed data, then the values of the DOFs 
at the next time point are computed, after which the simulation results can be expressed again in terms of
the original DOFs. The combined interpolation, solving and backtransformation is considerably cheaper
than solving the original model equations. In the applications envisioned in this research, such as real-time
simulation, gain in simulation speed during on-line simulation justi es an expensive oine preparation
phase. The assumption of small deformations allows to approximate the state-dependent mode set by the
mode set obtained at the corresponding undeformed state. This strongly limits the number of discrete
states for which the reduced model matrices and tensors need to be computed and stored during the
preparation phase, while only minimally decreasing the accuracy. A more elaborate explanation on the
methodology can be found in [1], [2] and [3].
In
exible mechanism dynamics, non-linearity arises from two e ects: 1) variable connectivity between
bodies, e.g. moving connection points, and 2) the variability of the mass matrix due to large rotations. In
a rst test case, the methodology is applied to a system with non-linear dynamics resulting from variability
in the connection between bodies: a
exible beam being clamped by a sliding joint at a continuously
changing location. For this type of non-linearity the traditional subsystem-level model reduction fails
to achieve a signi cant dimension reduction of the problem, whereas the proposed system-level model
reduction produces accurate simulation results for a low number of modes in the mode set, and thus for
a small dimension of the reduced model equations. The approximation errors of several mode sets are
compaired and the sources of error are explained. For a more complete overview of this test case the
reader is referred to [2].
In a second numerical experiment, a system with large rotations is considered: a
exible slider-crank
mechanism. Due to the large rotations, the fast transformation from the DOFs used in the reduced
model equations back to the original DOFs needs to be performed by interpolation, which is an additional
diculty compared to the rst test case. Again, the simulation results of the reduced model give a good
approximation of the simulation results of the original (unreduced) model for a very limited mode set.
The di erent sources of approximation error are identi ed and explained. The computational load of both
simulations is quanti ed: the proposed methodology proves to be more ecient. For a more complete
overview of this test case the reader is referred to [3].
References
[1] G.H.K. Heirman, O. Bruls and W. Desmet, \Coordinate transformation techniques for ecient model
reduction in
exible multibody dynamics", Proceedings of the ISMA2008 International Conference on
Noise and Vibration Engineering, Leuven, Belgium, September 15{17, 2008.
[2] G.H.K. Heirman, O. Bruls and W. Desmet, \A system-level model reduction technique for ecient simulation
of
exible multibody dynamics", Proceedings of the ECCOMAS Thematic Conference in Multibody
Dynamics (accepted), Warsaw, Poland, June 29 { July 2, 2009.
[3] G.H.K. Heirman and W. Desmet, \System-level modal representation of
exible multibody systems",
Proceedings of the 7th International Conference on Multibody Systems, Nonlinear Dynamics, and Control, San Diego, Ca., USA, August 30 { September 2, 2009.
Publication status: published
KU Leuven publication type: IMa
Appears in Collections:Production Engineering, Machine Design and Automation (PMA) Section
# (joint) last author

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