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(infinity ) 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(infinity ) 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(infinity ) criterion.
The fixed-points method for the dynamic vibration absorber (DVA) is widely accepted and the results are prevalent for practical applications. However, they usually have to fall back to a heuristic approach from the point of view of its optimization criterion. A typical design problem to minimize the maximum amplitude magnification factor of the primary system, for which the fixed-points method was originally developed, is an example of such common cases. In the present paper, a new algebraic formulation is developed to this classic problem and closed-form exact solutions to both the optimum tuning ratio and the optimum damping parameters are derived, on the assumption of undamped primary system. This algebraic approach is based on an observation of trade-off between two resonance amplitude magnification factors. Thus, the problem reduces to a solution of an algebraic equation, which is derived as a discriminant of quartic algebraic equation. In undamped case, it was proven that the optimum parameters, the minimum amplitude magnification factor, the resonance and antiresonance frequencies, and sensitivities of the amplitude magnification factors are totally algebraic. A numerical extension enables efficient solutions for the damped primary system and has more direct applicability.
Recently, Nishihara and Matsuhisa have proposed a new theory for attaining the H(infinity) optimization of a dynamic vibration absorber (DVA) in the linear vibratory systems. The H(infinity) optimization of DVA is a classical optimization problem, and already solved more than 50 years ago. All of us know the solution through the textbook written by Den Hartog. The new theory proposed them gives us the exact algebraic solution of the problem. In the first report, we have expounded the theory and showed the procedure of finding the algebraic solution to a typical performance index (compliance transfer function) of the viscous damped system. In this paper, we will apply this theory to another performance indexes: mobility and accelerance transfer functions for force excitation system, and the absolute and relative displacement responses to acceleration, velocity or displacement input to foundation for motion excitation system. We apply this theory not only the viscous damped system but also the hysteretic damped system. As a result, we found the closed-form exact solutions in every performance indexes when the primary system has no damping. The solutions obtained here are compared with the classical ones solved by the fixed-points theory. We further apply this theory to design of DVAs attached to damped primary systems, and found the closed-form exact solutions to some performance indexes of the hysteretic damped system.
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