$H2$ Optimization of the Three-Element Type Dynamic Vibration Absorbers

The dynamic vibration absorber (DVA) is a passive vibration control device which is attached to a vibrating body (called a primary system) subjected to exciting force or motion. In this paper, we will discuss an optimization problem of the three-element type DVA on the basis of the $H2$ optimization criterion. The objective of the $H2$ optimization is to reduce the total vibration energy of the system for overall frequencies; the total area under the power spectrum response curve is minimized in this criterion. If the system is subjected to random excitation instead of sinusoidal excitation, then the $H2$ optimization is probably more desirable than the popular $H∞$ optimization. In the past decade there has been increasing interest in the three-element type DVA. However, most previous studies on this type of DVA were based on the $H∞$ optimization design, and no one has been able to find the algebraic solution as of yet. We found a closed-form exact solution for a special case where the primary system has no damping. Furthermore, the general case solution including the damped primary system is presented in the form of a numerical solution. The optimum parameters obtained here are compared to those of the conventional Voigt type DVA. They are also compared to other optimum parameters based on the $H∞$ criterion.