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Development and evaluation of a numerical method for the identification of a physical system described by a partial differential equation : a case study

Tuesday, 27 June, 2006 - 17:00
Campus: Brussels Humanities, Sciences & Engineering campus
Kathleen De Belder
phd defence

More and more engineering problems exploit mathematical models to get a better
insight in the behavior of a system. It is possible to improve the behavior of the
system thanks to knowledge retrieved from experiments. Mathematical models not
only decrease the number of experiments, they also allow doing simulations and
giving knowledge about the underlying physical processes. Linear models are
typically described by transfer functions or state equations to describe linear models.
To identify this class of models, a lot of commercial packages exist. Transfer function
models can be reduced to linear total differential equations. Despite their successful
applications, this class of models can be too restrictive. Partial differential equations
are a logical generalization. In a lot of scientific areas they form a natural framework
to describe physical systems. It seems useful to expand the known identification
techniques for transfer function models to the class of models described by partial
differential equations.

At this point this story starts. We have selected the flexural vibration of a beam as a
case study. This system is described by the partial differential equation of Euler-
Bernoulli. In this equation we can find physical properties as mass, density, stiffness
and damping; parameters we want to identify.

In this thesis, we have to cope with five different aspects: i) a partial differential
equation and numerical integration algorithms, ii) physical insight of material
properties like damping and stiffness, iii) implementation of system identification
techniques on a model described by a partial differential equation, iv) designing an
improved partial differential equation and v) performing experiments to validate the
proposed ideas.

We have come with two stable numerical solution algorithms to solve the equation of
Euler-Bernoulli. Those algorithms will still induce numerical errors on the solution.
We have shown by means of a simulation example (an optimization algorithm
performed on simulated data where we have added noise with a level realistic in
measurements) that the errors on the estimated parameters are dominated by the
noise contributions and not by the errors produced by the numerical integration

The frequency dependency of the dynamic elastic and damping properties of solid
materials in the linear range is mostly characterized by the concept of the complex
modulus. From broadband modal analysis experiments, we can identify Young’s
modulus for different types of material. The final result is a parametric model for
Young’s modulus over a broad frequency band together with an uncertainty bound.
The uncertainty bound takes into account the disturbing input/output (measurement)
noise and the stochastic nonlinear distortions. The parametric model can also be
used to determine an interpolation function for an equivalent Young’s modulus for
sandwich beams.

An extended Euler-Bernoulli partial differential equation is presented. This model is
validated with results coming from a flexural vibration experiment on a Plexiglas
beam. We can conclude that this model describes the experiment of the flexural
vibration of a Plexiglas beam much better, especially in the frequency band up to
800 Hz. It is clear that the model proposed is material dependent. Therefore, a more
general equation is also introduced.