Spectrum of an operator arising elastic system with local K-V damping

In this paper, we analyze the spectrum of an operator arising in elastic system with local K‐V damping, whose vibration is modelled as the Euler–Bernoulli beam. Suppose that the beam is clamped at both ends, at its internal bonded a patch made of smart material that produces the Kelvin–Voigt damping. Our aim is to study the spectrum of the operator determined by such a system. By a detail analysis, we show that the spectrum consists of the point spectrum and continuous spectrum, in which the point spectrum is a denumerable set, and the continuous spectrum is a segment on the real axis. Under certain conditions, the continuous spectrum is possibly the whole half line, and also degenerates possibly one point. For a special case, we give the asymptotic expression of the point spectrum. In this paper, the spectrum of an operator arising in elastic system with local K‐V damping is analyzed, whose vibration is modelled as the Euler–Bernoulli beam. Suppose that the beam is clamped at both ends, at its internal bonded a patch made of smart material that produces the Kelvin–Voigt damping. It is shown that the spectrum spectrum consists of the point spectrum and continuous spectrum, in which the point spectrum is a denumerable set, and the continuous spectrum is a segment on the real axis. Under certain conditions, the continuous spectrum is possibly the whole half line, and also degenerates possibly one point. For a special case, the asymptotic expression of the point spectrum is given.