Analytical Solutions for DVA Optimization Based on the Lyapunov Equation

Abstract

A novel method is proposed to obtain the optimum configuration of dynamic vibration absorber (i.e., DVA) attached to an undamped or damped primary structure. The performance index, i.e., the quadratic integration, includes two types of controls, i.e., velocity and displacement controls of the primary mass. Based on the Lyapunov equation, the performance indices are simplified into matrix quadratic forms. With the help of the Kronecker product and matrix column expansion, the closed-form solutions of optimum parameters for undamped primary structure are finally presented. Moreover, in some cases, the method can produce perturbation solutions in simple forms for damped primary structure. Especially, from these solutions, the classical optimum H2 designs under external force or base acceleration excitation can be derived out.