This paper considers the optimal design of double-mass dynamic vibration absorbers (DVAs) attached to an undamped single degree-of-freedom system. Three different optimization criteria, the H optimization, H2 optimization, and stability maximization criteria, were considered for the design of the DVAs, and a performance index was defined for each of these criteria. First, the analytical models of vibratory systems with double-mass DVAs were considered, and seven dimensionless parameters were defined. Five of these parameters must be optimized to minimize or maximize the performance indices. Assuming that all dimensionless parameters are non-negative, the optimal value of one parameter for a double-mass DVA arranged in series (series-type DVA) was proven to be zero. The optimal adjustment conditions of the other four parameters were derived as simple algebraic formulae for the H2 and stability criteria and numerically determined for the H criterion. For a double-mass DVA arranged in parallel (parallel-type DVA), all five parameters were found to have nonzero optimal values, and these values were obtained numerically by solving simultaneous algebraic equations. Second, the performance of these DVAs was compared with a single-mass DVA. The result revealed that for all optimization criteria, the performance of the series-type DVA is the best among the three DVAs and that of the single-mass DVA is the worst. Finally, a procedure for deriving the algebraic or numerical solutions for H2 optimization is described. The derivation procedure of other optimal solutions will be introduced in the future paper.

References

References
1.
Ormondroyd
,
J.
, and
Den Hartog
,
J. P.
,
1928
, “
The Theory of the Dynamic Vibration Absorber
,”
ASME J. Appl. Mech.
,
50
, pp.
9
22
.
2.
Crandall
,
S. H.
, and
Mark
,
W. D.
,
1963
,
Random Vibration in Mechanical Systems
,
Academic Press
,
New York
, p.
71
.
3.
Yamaguchi
,
H.
,
1988
, “
Damping of Transient Vibration by a Dynamic Absorber
,”
Trans. JSME, Ser. C
,
54
(
499
), pp.
561
568
, (in Japanese).
4.
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. JSME, Ser. C
,
63
(
614
), pp.
3438
3445
, (in Japanese).
5.
Nishihara
,
O.
, and
Asami
,
T.
,
2002
, “
Closed-Form Solutions to the Exact Optimizations of Vibration Absorbers (Minimizations of the Maximum Amplitude Magnification Factors)
,”
ASME J. Vib. Acoust.
,
124
(
4
), pp.
576
582
.
6.
Asami
,
T.
, and
Nishihara
,
O.
,
2003
, “
Closed-Form Solution to H∞ Optimization of Dynamic Vibration Absorbers (Application to Different Transfer Functions and Damping Systems)
,”
ASME J. Vib. Acoust.
,
125
(
3
), pp.
398
405
.
7.
Warburton
,
G. B.
,
1982
, “
Optimum Absorber Parameters for Vibration Combinations of Response and Excitation Parameters
,”
Earthquake Eng. Struct. Dyn.
,
10
(
3
), pp.
381
401
.
8.
Nishihara
,
O.
, and
Matsuhisa
,
H.
,
1997
, “
Design and Tuning of Vibration Control Devices Via Stability Criterion
,”
Dynamics and Design Conference’97 (in Japanese)
, No. 97-10-1, pp.
165
168
.
9.
Febbo
,
M.
, and
Vera
,
S. A.
,
2008
, “
Optimization of a Two Degree of Freedom System Acting as a Dynamic Vibration Absorber
,”
ASME J. Vib. Acoust.
,
130
(
1
), p.
011013
.
10.
Chun
,
S.
,
Baek
,
H.
, and
Kim
,
T.
,
2014
, “
Discussion: Optimization of a Two Degree of Freedom System Acting as a Dynamic Vibration Absorber
,”
ASME J. Vib. Acoust.
,
136
(
2
), p.
025502
.
11.
Sinha
,
A.
,
2015
, “
Optimal Damped Vibration Absorber: Including Multiple Modes and Excitation Due to Rotating Unbalance
,”
ASME J. Vib. Acoust.
,
137
(
6
), p.
064501
.
12.
Tursun
,
M.
, and
Eskinat
,
E.
,
2014
, “
H2 Optimization of Damped-Vibration Absorbers for Suppressing Vibrations in Beams With Constrained Minimization
,”
ASME J. Vib. Acoust.
,
136
(
2
), p.
021012
.
13.
Iwanami
,
K.
, and
Seto
,
K.
,
1984
, “
An Optimum Design Method for the Dual Dynamic Damper and Its Effectiveness
,”
Bull. JSME
,
27
(
231
), pp.
1965
1973
.
14.
Kamiya
,
K.
,
Kamagata
,
K.
,
Matsumoto
,
S.
, and
Seto
,
K.
,
1996
, “
Optimal Design Method for Multi Dynamic Absorber
,”
Trans. JSME, Ser. C
,
62
(
601
), pp.
3400
3405
, (in Japanese).
15.
Yasuda
,
M.
, and
Pan
,
G.
,
2003
, “
Optimization of Two-Series-Mass Dynamic Vibration Absorber and Its Vibration Control Performance
,”
Trans. JSME, Ser. C
,
69
(
688
), pp.
3175
3182
, (in Japanese).
16.
Pan
,
G.
, and
Yasuda
,
M.
,
2005
, “
Robust Design Method of Multi Dynamic Vibration Absorber
,”
Trans. JSME, Ser C
,
71
(
712
), pp.
3430
3436
, (in Japanese).
17.
Zuo
,
L.
,
2009
, “
Effective and Robust Vibration Control Using Series Multiple Tuned-Mass Dampers
,”
ASME J. Vib. Acoust.
,
131
(
3
), p.
031003
.
18.
Zuo
,
L.
, and
Cui
,
W.
,
2013
, “
Dual-Functional Energy Harvesting and Vibration Control: Electromagnetic Resonant Shunt Series Tuned Mass Dampers
,”
ASME J. Vib. Acoust.
,
135
(
5
), p.
051018
.
19.
Den Hartog
,
J. P.
,
1956
,
Mechanical Vibrations
,
4th ed.
,
McGraw-Hill
,
New York
, pp.
96
100
.
20.
Issa
,
J. S.
,
2013
, “
Optimal Design of a Damped Single Degree of Freedom Platform for Vibration Suppression in Harmonically Forced Undamped Systems
,”
ASME J. Vib. Acoust.
,
135
(
5
), p.
051003
.
21.
Tong
,
X.
,
Liu
,
Y.
,
Cui
,
W.
, and
Zuo
,
L.
,
2015
, “
Analytical Solutions to H2 and H∞ Optimizations of Resonant Shunted Electromagnetic Tuned Mass Damper and Vibration Energy Harvester
,”
ASME J. Vib. Acoust.
,
138
(
1
), p.
011018
.
22.
Argentini
,
T.
,
Belloli
,
M.
, and
Borghesani
,
P.
,
2015
, “
A Closed-Form Optimal Tuning of Mass Dampers for One Degree-of-Freedom Systems Under Rotating Unbalance Forcing
,”
ASME J. Vib. Acoust.
,
137
(
3
), p.
034501
.
23.
Tamura
,
S.
, and
Shiotani
,
T.
,
2012
, “
Design of Dynamic Absorber for Two DOF System by Fixed Points Theory (1st Report, Case of Two DOF System With Identical Mass and Stiffness)
,”
Trans. JSME, Ser. C
,
78
(
786
), pp.
372
381
, (in Japanese).
24.
Tomimuro
,
T.
, and
Tamura
,
S.
,
2015
, “
Design of Dynamic Absorber for Two DOF System by Fixed Points Theory (2nd Report, Case of the System with Different Mass and Stiffness, and Excited Secondary Mass)
,”
Trans. JSME
,
81
(
825
), p.
14-00622
, (in Japanese).
25.
Weisstei
, and
Eric
,
W.
, 2016, “
Parseval's Theorem
,” from MathWorld—A Wolfram Web Resource, http://mathworld.wolfram.com/ParsevalsTheorem.html.
26.
Hahnkamm
,
E.
,
1932
, “
Die Dämpfung von Fundamentschwingungen bei veränderlicher Erregergrequenz
,”
Ing. Arch.
,
4
, pp.
192
201,
(in German).
27.
Brock
,
J. E.
,
1946
, “
A Note on the Damped Vibration Absorber
,”
ASME J. Appl. Mech.
,
13
(
4
), p.
A-284
.
28.
Asami
,
T.
,
Nishihara
,
O.
, and
Baz Amr
,
M.
,
2002
, “
Analytical Solutions to H∞ and H2 Optimization of Dynamic Vibration Absorbers Attached to Damped Linear Systems
,”
ASME J. Vib. Acoust.
,
124
(
2
), pp.
284
295
.
29.
Kreyszig
,
E.
,
1999
,
Advanced Engineering Mathematics
,
8th ed.
,
Wiley
,
New York
, p.
784
.
30.
Binmore
,
K.
, and
Davies
,
J.
,
2007
,
Calculus Concepts and Methods
,
Cambridge University Press
,
Cambridge
, p.
190
.
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