  We study the motion of a rigid body $\mathscr B$ subject to an undamped
elastic restoring force, in the stream of a viscous liquid $\mathscr L$. The
motion of the coupled system $\mathscr B$-$\mathscr L\equiv\mathscr S$ is
driven by a uniform flow of $\mathscr L$ at spatial infinity, characterized by
a given, constant dimensionless velocity $\lambda\,\bfe_1$, $\lambda>0$. We
show that as long as $\lambda\in(0,\lambda_c)$, with $\lambda_c$ a distinct
positive number, there is a uniquely determined time-independent state of
$\mathscr S$ where $\mathscr B$ is in a (locally) stable equilibrium and the
flow of $\mathscr L$ is steady. Moreover, in that range of $\lambda$, no
oscillatory flow may occur. Successively we prove that if certain suitable
spectral properties of the relevant linearized operator are met, there exists a
$\lambda_0>\lambda_c> 0$ where an oscillatory regime for $\mathscr S$ sets in.
More precisely, a bifurcating time-periodic branch stems out of the
time-independent solution. The significant feature of this result is that {\em
no} restriction is imposed on the frequency, $\omega$, of the bifurcating
solution, which may thus coincide with the natural structural frequency,
$\omega_{\sf n}$, of $\mathscr B$, or any multiple of it. This implies that a
dramatic structural failure cannot take place due to resonance effects.
However, our analysis also shows that when $\omega$ becomes sufficiently close
to $\omega_{\sf n}$ the amplitude of oscillations can become very large in the
limit when the density of $\mathscr L$ becomes negligible compared to that of
$\mathscr B$.
