Pointwise Stabilization in the Middle of the Span for Second Order Systems, Nonuniform and Uniform Exponential Decay of Solutions

Two vibrating strings are coupled at the connecting point, where a damping device is installed. This device, modeled by one of two sets of intermediate nodal conditions, causes the vibration to dissipate energy. A good design should satisfy the requirement that all modes be uniformly damped. In this paper, we show a “simultaneous diagonalization procedure” which can determine the rate of damping for certain coupled strings by solving a simple matrix eigenvalue problem. In particular, we show how to choose the “impedance coefficient” so that all of the vibration energy is absorbed within finite time duration. We can also see that for certain systems with symmetry, there exist modes which are not damped at all; therefore the vibration energy does not decay uniformly. We then use the Legendre spectral method to compute the eigenvalues of the damped operators. Numerical results have confirmed the theoretical estimates and eigenvalue distributions patterns. Possible mechanical designs of those damping devices are also illustrated.