Title:  Model reduction for nonlinear dynamics by projection on statedependent modal base 
Authors:  Heirman, Gert Desmet, Wim # 
Issue Date:  28Sep2009 
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 (nonlinear) dierentialalgebraic equations (DAE): secondorder dierential equations
express the equations of motion, while algebraic equations impose constraints on the motion of the
system. Both the DAEcharacter of the model equations, and the number of degrees of freedom needed
to accurately represent
exibility (typically up to 1000's of DOFs), prohibit realtime simulation of these
systems. Model reduction on subsystemlevel, e.g. modal representation of the
exibility of a mechanism
component, is used extensively in
exible mechanism models to limit the computational load. However,
subsystemlevel model reduction suers from the intrinsic drawback that it does not result in signicant
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 realtime running models of
exible mechanisms are mostly based on ad hoc simplications of the
model. Few techniques exploit the mathematical structure of the original model equations. Most systemlevel
model reduction techniques for nonlinear models build reduced models based on data obtained from
userdened numerical experiments of the original model. The resulting reduced models only oer 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 systemlevel model reduction technique is developed that exploits the mathematical properties of the
original model equations.
Projectionbased 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
(statedependent) dominant dynamics of the system, i.e. the 1) statedependent dominant eigenmodes,
1a) the rigid body modes and 1b) lowfrequency 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
nonlinear systems requires the inclusion of many vectors in the vector set, to ensure the statedependent
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 statedependent vector set instead of a
xed vector set. The statedependent 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 dened by the statedependent eigenmodes and static deformation patterns of the system. Dierent
mode sets meet these demands [1].
Using a statedependent vector set oers 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 systemlevel eigenmodes and static deformation patterns intrinsically satisfy the constraints,
such that the algebraic equations are automatically satied after model reduction; The model reduction
transforms the model equations from a set of dierentialalgebraic equations (DAE) into a set of ordinary
dierential 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 realtime
simulation, gain in simulation speed during online simulation justies an expensive oine preparation
phase. The assumption of small deformations allows to approximate the statedependent 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, nonlinearity arises from two eects: 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 nonlinear 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 nonlinearity the traditional subsystemlevel model reduction fails
to achieve a signicant dimension reduction of the problem, whereas the proposed systemlevel 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 slidercrank
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 dierent sources of approximation error are identied and explained. The computational load of both
simulations is quantied: 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 systemlevel 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, \Systemlevel 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

