The Nature of Epistemic Uncertainty in Linear Dynamical Systems
Adhikari, S. and Sarkar, A.
Proceedings of the 25th International Modal Analysis Conference
(IMAC-XXV), February 2007, Orlando, Florida, USA.
In the modeling of complex dynamical systems, high-resolution
finite element models are routinely adopted to reduce the
discretization error. This is often implemented by exploiting
cost-effective computing hardware through parallel processing to
solve the resulting large scale linear systems. Such an approach
fails to enhance confidence in simulation-based predictions when
the dynamical systems exhibit significant variability in their
data and model leading to so-called data and modeling uncertainty.
When substantial statistical information is available, data
uncertainty can be tackled using probabilistic methods by modeling
the parameters of the data as random variables or stochastic
processes. Model uncertainty poses significant challenges as no
parameter is available a priori as opposed to the case of
data uncertainty. In modelling complex systems such marine (e.g.
ships, submarines) and aerospace systems (e.g. helicopters and
space shuttles) modeling uncertainty arises naturally due to the
lack of complete knowledge and even presence of many subsystems
attached to the main structural components. In the low frequency
regions, effect of such substructure may be modeled by rigid
masses attached to the primary structures. In higher frequency,
the mechanics of energy flow among the primary and secondary
systems may not be captured by these rigid masses alone as
dynamics of the subsystems becomes more important. The additional
degree-of-freedom arising from the subsystems should be
incorporated to model the entire system. A sprung-mass models are
adopted in the current study to investigate the effect of such
subsystem on the vibration of a thin steel plate. The location of
the attachments of these sprung-mass systems and their natural
frequencies are assumed to be uncertain while the constitutive and
geometric properties of the steel plate (e.g. the primary
structure) are known. In contrast to the case of data uncertainty
(traditionally modeled in the framework of stochastic finite
element method), the model uncertainty arising from the
sprung-masses (attached randomly to the plate) gives rise to
entirely different variety of dynamical system for each sample.
This can be observed from the variation in sparsity
structure of the mass, stiffness and damping matrices of the
total system from sample to sample. Clearly such change in
sample-wise sparsity pattern can not be modeled by data
uncertainty alone. In the case of data uncertainty, the actual
configuration of dynamical system remains unchanged, just its
local parameters change from sample to sample and therefore,
sparsity structure of the system matrices for each sample remains
the same. In this study, we investigate the feasibility of
adopting a global probabilistic model to represent such entire
ensemble of different dynamical systems derived from perturbing
the model of a baseline system. In the current study, the
baseline dynamical system is just the thin plate (without any
sprung-mass attachment). A range of dynamical systems is then
generated from the random attachment topology of the sprung-masses
with the thin plate. As mentioned before, each of these dynamical
systems possesses mass, stiffness and damping matrices for which
sparsity pattern differ from sample to sample. The objective of
this investigation is to represent uncertainty arising from model
perturbation. We explore the possibility of stochastic
representation of this entire variety derived from the baseline
system with model perturbation (in contrast to data perturbation).
More specifically, in the framework of random matrix theory, we
fit the parameters of Wishart random matrices to model uncertainty
in the mass, stiffness and damping matrices of the total system,
namely the plate having randomly attached sprung-masses.
BiBTeX Entry
@INPROCEEDINGS{cp28,
AUTHOR={S. Adhikari and A. Sarkar},
TITLE={The nature of epistemic uncertainty in linear dynamical systems},
BOOKTITLE={Proceedings of the 25th International Modal Analysis Conference (IMAC-XXV)},
YEAR={2007},
Address={Orlando, Florida, USA},
Month={February},
Note={}
}
by Sondipon Adhikari