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Mathematical Analysis of Flow-Induced Oscillations of a Spring-Mounted Body in a Navier-Stokes Liquid

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 $λ\,\bfe_1$, $λ>0$. We show that as long as $λ\in(0,λ_c)$, with $λ_c$ 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 $λ$, no oscillatory flow may occur. Successively we prove that if certain suitable spectral properties of the relevant linearized operator are met, there exists a $λ_0>λ_c> 0$ 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, $ω$, of the bifurcating solution, which may thus coincide with the natural structural frequency, $ω_{\sf n}$, 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 $ω$ becomes sufficiently close to $ω_{\sf n}$ the amplitude of oscillations can become very large in the limit when the density of $\mathscr L$ becomes negligible compared to that of $\mathscr B$.