**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.