H∞ and H2 optimizations of a dynamic vibration absorber for suppressing vibrations in plates

H∞ and H2 optimization problems with respect to a dynamic vibration absorber (DVA) in a single degree-of-freedom (sdof) system are classical optimization problems and solutions to them were found about half a century ago. Numerical solutions to the H∞ and H2 optimization problems with respect to DVA for a multi-degree-of-freedom (mdof) or continuous system can be found in the literature but their analytical solutions have not yet been found. In this article, we report the derivation of an analytical solution to the H∞ and H2 optimization problems of DVA applied to suppress random vibrations in plate structures. Analytical formulae are also proposed to express the optimal tuning frequency and damping ratios of the absorber. The established theory improves our understanding of the effects of different parameters including the mass, damping and tuning ratios and also the point of attachment of the absorber on the vibration absorption by the absorber. Numerical results show the usefulness of the optimization solutions in comparison to solutions suggested by other researchers based on other approaches to the problem.