Keywords: Port-Hamiltonian
systems, Modeling,
Distributed
parameter systems
Abstract:In this paper, a special class of
damping model is introduced for second order dynamical systems. This
class is built so as to leave the eigenfunctions invariant, while
modifying the dynamics: for mechanical systems, well-known examples are
the standard fluid and structural dampings.
In the finite-dimensional case, the so-called Caughey series are a
general extension of these standard damping models; the damping matrix
can be expressed as a polynomial of a matrix, which depends on the mass
and stiffness matrices. Damping is ensured whatever the eigenvalues of
the conservative problem if and only if the polynomial is positive for
positive scalar values.
This can be recast in the port-Hamiltonian framework by introducing a
port variable corresponding to internal energy dissipation (resistive
element). Moreover, this formalism naturally allows to cope with
systems including gyroscopic effects (gyrators).
In the infinite-dimensional case, the previous polynomial class can be
extended to rational functions and more general functions of operators
(instead of matrices), once the appropriate functional framework has
been defined. In this case, the resistive element is modelled by a
given static operator, such as an elliptic PDE. These results are
illustrated on several PDE examples: the Webster horn equation, the
Bernoulli beam equation; the damping models under consideration are
fluid, structural, rational and generalized fractional Laplacian or
bi-Laplacian.