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.

1.
Asami
,
T.
,
Nishihara
,
O.
, and
Baz
,
A. M.
,
2002
, “
Analytical Solutions to H∞ and H2 Optimization of Dynamic Vibration Absorbers Attached to Damped Linear Systems
,”
ASME J. Vibr. Acoust.
,
124
(
2
), pp.
284
295
.
2.
Ormondroyd
,
J.
, and
Den Hartog
,
J. P.
,
1928
, “
The Theory of the Dynamic Vibration Absorber
,”
ASME J. Appl. Mech.
,
50
(
7
), pp.
9
22
.
3.
Crandall, S. H., and Mark, W. D., 1963, Random Vibration in Mechanical Systems, Academic Press.
4.
Yamaguchi
,
H.
,
1988
, “
Damping of Transient Vibration by a Dynamic Absorber
,”
Trans. Jpn. Soc. Mech. Eng., Ser. C
,
54
(
499
), Ser C, pp.
561
568
(in Japanese).
5.
Satoh
,
Y.
,
1991
, “
The Reductive Performance of the Dynamic Absorbers and Dynamic Properties of the Viscoelastic Absorber Elements
,”
Trans. Jpn. Soc. Mech. Eng., Ser. C
,
57
(
534
), Ser C, pp.
446
452
(in Japanese).
6.
Asami
,
T.
, and
Nishihara
,
O.
,
1999
, “
Analytical and Experimental Evaluation of an Air Damped Dynamic Vibration Absorber: Design Optimizations of the Three-Element Type Model
,”
ASME J. Vibr. Acoust.
,
121
(
3
), pp.
334
342
.
7.
Gatade, Y., Misawa, H., Seto, K., and Doi, F., 1996, “Optimal Design of Notch Type Dynamic Absorber,” Prepr. of Jpn. Soc. Mech. Eng., No. 95-5(I), Vol. B, pp. 569–572 (in Japanese).
8.
Kawashima
,
T.
,
1992
, “
Vibration Prevention by a Three-Element Dynamic Vibration Absorber
,”
Trans. Jpn. Soc. Mech. Eng., Ser. C
,
58
(
548
548
), Ser C, pp. 1024–1029 (in Japanese).
9.
Kreyszig, E., 1999, Advanced Engineering Mathematics, 8th ed., John Wiley & Sons, Inc.
10.
Warburton
,
G. B.
,
1982
, “
Optimum Absorber Parameters for Various Combinations of Response and Excitation Parameters
,”
Earthquake Eng. Struct. Dyn.
,
10
, pp.
381
401
.
11.
Kowalik, J., and Osborne, M. R., 1968, Methods for Unconstrained Optimization Problems, American Elsevier Publishing Company.