Mathematical Study of the Small Oscillations of a Pendulum Completely Filled with a Viscoelastic Fluid

We study the small oscillations of a pendulum completely filled by a viscoelastic fluid, restricting ourselves for the fluid to the simpler Oldroyd model. We establish the equations of motion of the system. Writing them in a suitable form, we obtain an existence and unicity theorem of the solution of the associated evolution problem by means of semigroup theory. Afterwards, we show the existence and symmetry of the spectrum and prove the stability of the system. We show the existence of two sets of positive real eigenvalues, of which the first has infinity, and the second a point of the real axis, as points of accumulation. Finally, we specify the location of the possible nonreal eigenvalues.