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.

1.
Frahm, H., 1911, “Device for Damping Vibrations of Bodies,” U.S. Patent, No. 989, 958, pp. 3576–3580.
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.
Hahnkamm
,
E.
,
1932
, “
Die Da¨mpfung von Fundamentschwingungen bei vera¨nderlicher Erregergrequenz
,”
Ing. Arch.
,
4
, pp.
192
201
, (in German).
4.
Brock
,
J. E.
,
1946
, “
A Note on the Damped Vibration Absorber
,”
ASME J. Appl. Mech.
,
13-4
, p.
A-284
A-284
.
5.
Den Hartog, J. P., 1956, Mechanical Vibrations, 4th ed., McGraw-Hill, New York.
6.
Nishihara, O., and Matsuhisa, H., 1997, “Design and Tuning of Vibration Control Devices via Stability Criterion,” Prepr. of Jpn. Soc. Mech. Eng., No. 97-10-1, pp. 165–168, (in Japanese).
7.
Ikeda
,
K.
, and
Ioi
,
T.
,
1978
, “
On the Dynamic Vibration Damped Absorber of the Vibration System
,”
Bull. JSME
,
21
-151, pp.
64
71
.
8.
Randall
,
S. E.
,
Halsted
, III,
D. M.
, and
Taylor
,
D. L.
,
1981
, “
Optimum Vibration Absorbers for Linear Damped Systems
,”
ASME J. Mech. Des.
,
103-4
, pp.
908
913
.
9.
Thompson
,
A. G.
,
1981
, “
Optimum Tuning and Damping of a Dynamic Vibration Absorber Applied to a Force Excied and Damped Primary System
,”
J. Sound Vib.
,
77-3
, pp.
403
415
.
10.
Soom
,
A.
, and
Lee
,
M.
,
1983
, “
Optimal Design of Linear and Nonlinear Vibration Absorbers for Damped Systems
,”
ASME J. Vibr. Acoust.
,
105-1
, pp.
112
119
.
11.
Sekiguchi
,
H.
, and
Asami
,
T.
,
1984
, “
Theory of Vibration Isolation of a System with Two Degrees of Freedom
,”
Bull. JSME
,
27
-234, pp.
2839
2846
.
12.
Asami
,
T.
, and
Hosokawa
,
Y.
,
1995
, “
Approximate Expression for Design of Optimal Dynamic Absorbers Attached to Damped Linear Systems (2nd Report, Optimization Process Based on the Fixed-Points Theory)
,”
Trans. Jpn. Soc. Mech. Eng., Ser. C
,
61-583
, pp.
915
921
, (in Japanese).
13.
Crandall, S. H., and Mark, W. D., 1963, Random Vibration in Mechanical Systems, Academic Press.
14.
Iwata, Y., 1982, “On the Construction of the Dynamic Vibration Absorbers,” Prepr. of Jpn. Soc. Mech. Eng., No. 820-8, pp. 150–152, (in Japanese).
15.
Asami
,
T.
et al.
,
1991
, “
Optimum Design of Dynamic Absorbers for a System Subjected to Random Excitation
,”
JSME Int. J., Ser. III
,
34-2
, pp.
218
226
.
16.
Asami
,
T.
,
Momose
,
K.
, and
Hosokawa
,
Y.
,
1993
, “
Approximate Expression for Design of Optimal Dynamic Absorbers Attached to Damped Linear Systems (Optimization Process Based on the Minimum Variance Criterion)
,”
Trans. Jpn. Soc. Mech. Eng., Ser. C
,
59-566
, pp.
2962
2967
, (in Japanese).
17.
Yamaguchi
,
H.
,
1988
, “
Damping of Transient Vibration by a Dynamic Absorber
,”
Trans. Jpn. Soc. Mech. Eng., Ser. C
,
54-499
, pp.
561
568
, (in Japanese).
18.
Nishihara
,
O.
, and
Matsuhisa
,
H.
,
1997
, “
Design of a Dynamic Vibration Absorber for Minimization of Maximum Amplitude Magnification Factor (Derivation of Algebraic Exact Solution)
,”
Trans. Jpn. Soc. Mech. Eng., Ser. C
,
63-614
, pp.
3438
3445
, (in Japanese).
19.
Meirovitch, L., 1986, Elements of Vibration Analysis, McGraw-Hill, New York.
You do not currently have access to this content.