Exact stability chart of an elastic beam subjected to delayed feedback

The stability of an elastic beam is studied when the beam is subjected to a longitudinal force governed by a feedback loop. The setup of the simplified mechanical model is motivated by a basic problem of electroacoustics. The corresponding governing equation is the 1D wave equation with delayed boundary conditions. By means of the D׳Alembert solution, the system can be transformed into a delay differential equation of neutral type that includes two time delays. The intricate stability chart is constructed analytically in the parameter plane of the gain parameter and the ratio of the time delays. The complete chart extends the numerical results of the literature, while it also draws attention to the numerical difficulties of finite degree of freedom approximations and to the extreme sensitivity of the dynamics with respect to the time delay parameters of the system.