Euler-Bernoulli Beam Equation with Boundary Energy Dissipation

摘要

Many problems in structural dynamics involve stabilizing the elastic energy of partial differential equations such as the Euler-Bernoulli beam equation by boundary conditions. Exponential stability is a very desirable property such elastic systems. The energy multiplier method has been successfully applied by several people to establish exponential stability for various PDEs and boundary conditions. However, it has also been found that for certain boundary conditions the energy multiplier method is not effective in proving the exponential stability property. A recent theorem of F. L. Huang introduces a frequency domain method to study such exponential decay problems. In this paper, we derive estimates of the resolvent operator on the imaginary axis and apply Huang's theorem to establish an exponential decay result for an Euler-Bernoulli beam with rate control of the bending moment only. We also derive asymptotic limits of eigenfrequencies, which was also done earlier by P. Rideau. Finally, we indicate the realizability of these boundary feedback stabilization schemes by illustrating some mechanical designs of passive damping devices.