Analytical Solutions to $H∞$ and $H2$ Optimization of Dynamic Vibration Absorbers Attached to Damped Linear Systems

$H∞$ and $H2$ optimization problems of the Voigt type dynamic vibration absorber (DVA) are classical optimization problems, which have been already solved for a special case when the primary system has no damping. However, for the general case including a damped primary system, no one has solved these problems by algebraic approaches. Only the numerical solutions have been proposed until now. This paper presents the analytical solutions for the $H∞$ and $H2$ optimization of the DVA attached to the damped primary systems. In the $H∞$ optimization the DVA is designed such that the maximum amplitude magnification factor of the primary system is minimized; whereas in the $H2$ optimization the DVA is designed such that the squared area under the response curve of the primary system is minimized. We found a series solution for the $H∞$ optimization and a closed-form algebraic solution for the $H2$ optimization. The series solution is then compared with the numerical solution in order to check the accuracy in connection with the truncation error of the series. The exact solution presented in this paper is too complicated to handle by a hand-held calculator, so we proposed an approximate solution for the practical object.