Explicit decay rate for coupled string-beam system with localized frictional damping

Abstract

The large-time behavior of a 1-d coupled string-beam system is considered. Combining a detailed spectral analysis with resolvent estimation, we obtain two kinds of energy decay rates of the string-beam system with different locations of the frictional damping. On one hand, if the frictional damping is only actuated in the beam part, the system lacks exponential decay. Specifically, the optimal polynomial decay rate t−1 is obtained under smooth initial conditions. On the other hand, if the frictional damping is only effective in the string part, the exponential decay of energy is presented. Some numerical simulations are given to support these results.