On damping models preserving the eigenfunctions of conservative systems: a port-Hamiltonian perspective
Résumé
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.