Hyperbolic–Parabolic Coupling and the Occurrence of Resonance in Partially Dissipative Systems - Fluid-Structure Interaction and Biomedical Applications - page 250

Fig. 3.3 Time evolution of the structure kinetic energy E s for different values of f ; f and for
different functions g 1 .t/, g 101 .t/ defining the oscillating load

which represent the fluid kinetic energy, structure kinetic energy, and stored energy,
respectively.
Figure 3.3 shows the temporal evolution of E s for different fluid properties,
while the structure mechanical parameters remain unchanged. In particular we
adopt s D 100; s D s , while the fluid density and viscosity decrease from
f D 1; f D 1 to f D 10 4 ; f D 10 4 . The force f is characterized by a
single vibration mode, basically we choose f Df0;g 1 .t/ sin.x/g,whereg 1 .t/ is

proportional to cos 2n T . The simulations confirm that the structure is subject to
a periodic oscillatory motion under the action of the forcing term f .However,the
amplitude of the oscillations is not constant in time, but it features a periodic trend
as well. Indeed, this is due to the competing role of the conservation and dissipation
properties of structure and fluid, respectively. The energy of the structure tends to
monotonically increase under the action of the force. In absence of dissipation, the
structure displacement will eventually tend to an unbounded motion, that is the
occurrence of resonance. The dissipation due to the viscous liquid acts against the
latter, in such a way that the larger is the displacement of the structure, the more
relevant becomes the dissipative effect. Figure 3.3 suggests that the system reaches
a dynamic equilibrium between these opposing trends. The numerical investigation
also confirms that the displacement of the structure significantly increases when the
density and viscosity of the fluid decrease. Again, this confirms the fundamental
role of viscous effects in the control of resonance. Finally, we study the case where
the number of vibration modes contributing to f is increased up to N D 101.
In practice, we switch from g 1 .t/ to g 101 .t/ in equation ( 3.126 ), as illustrated in
Fig. 3.2 . The perturbation due to the high frequency modes doesn't introduce any
significant changes in the time evolution of E s reported in Fig. 3.3 . Unfortunately,
this result does not inform us on the impact of Theorem 5.4 on the occurrence of
resonance, because it is open to two possible interpretations. On the one hand, it is
possible that the sequence g N .t/ is not an appropriate example to capture the effect