This paper reveals the theoretical (upper and lower) bounds of the performance of an on-off damping vibration absorber attached to an undamped multi-degrees-of-freedom (MDOF) system under harmonic excitation. The solution reduces to the maximization or minimization problem of a simple single-variable function. Among the class of on-off damping controller, which switches the damping level from high to low and back at fixed times every half period, the revealed solutions produce the highest and lowest amplitude–frequency curves. These curves are the good theoretical benchmarks to measure how good or bad an on-off damping controller is. To demonstrate the usefulness of the theoretical bound solutions, two versions of power flow-driven controller are introduced to produce the amplitude–frequency curves tracing the lowest-amplitude curve. A case study of a four-mass system is discussed.

References

References
1.
Asami
,
T.
, and
Nishihara
,
O.
,
2003
, “
Closed-Form Exact 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
.
2.
Ozer
,
M. B.
, and
Royston
,
T. J.
,
2005
, “
Extending Den Hartog's Vibration Absorber Technique to Multi Degree of Freedom Systems
,”
ASME J. Vib. Acoust.
,
127
(
4
), pp.
341
350
.
3.
Ozer
,
M. B.
, and
Royston
,
T. J.
,
2005
, “
Application of Sherman–Morrison Matrix Inversion Formula to Damped Vibration Absorbers Attached to Multi-Degree of Freedom Systems
,”
J. Sound Vib.
,
283
(
3–5
), pp.
1235
1249
.
4.
Petit
,
F.
,
Loccufier
,
M.
, and
Aeyels
,
D.
,
2009
, “
On the Attachment Location of Dynamic Vibration Absorbers
,”
ASME J. Vib. Acoust.
,
131
(
3
), p.
034501
.
5.
Nematipoor
,
N.
,
Ashory
,
M. R.
, and
Jamshidi
,
E.
,
2012
, “
Imposing Nodes for Linear Structures During Harmonic Excitations Using SMURF Method
,”
Arch. Appl. Mech.
,
82
(
5
), pp.
631
642
.
6.
Noori
,
B.
, and
Farshidianfar
,
A.
,
2013
, “
Optimum Design of Dynamic Vibration Absorbers for a Beam, Based on H∞ and H2 Optimization
,”
Arch. Appl. Mech.
,
83
(
12
), pp.
1773
1787
.
7.
Casciati
,
F.
,
Magonette
,
G.
, and
Marazzi
,
F.
,
2006
,
Technology of Semi-Active Devices and Applications in Vibration Mitigation
,
Wiley
, Chichester,
UK
.
8.
Savaresi
,
S. M.
,
Poussot-Vassal
,
C.
,
Spelta
,
C.
,
Sename
,
O.
, and
Dugard
,
L.
,
2010
,
Semi-Active Suspension Control, Design for Vehicles
,
Butterworth-Heinemann
, Oxford,
UK
.
9.
Koo
,
J. H.
,
Ahmadian
,
M.
,
Setareh
,
M.
, and
Murray
,
T. M.
,
2004
, “
In Search of Suitable Control Methods for Semi-Active Tuned Vibration Absorbers
,”
J. Vib. Control
,
10
(
2
), pp.
163
174
.
10.
Viet
,
L. D.
,
Nghi
,
N. B.
,
Hieu
,
N. N.
,
Hung
,
D. T.
,
Linh
,
N. N.
, and
Hung
,
L. X.
,
2014
, “
On a Combination of Ground-Hook Controllers for Semi-Active Tuned Mass Dampers
,”
J. Mech. Sci. Technol.
,
28
(
6
), pp.
2059
2064
.
11.
Shen
,
Y.
, and
Ahmadian
,
M.
,
2013
, “
Nonlinear Dynamical Analysis on Four Semi-Active Dynamic Vibration Absorbers With Time Delay
,”
Shock Vib.
,
20
(
4
), pp.
649
663
.
12.
Couillard
,
M.
,
Micheau
,
P.
, and
Masson
,
P.
,
2008
, “
Improved Clipped Periodic Optimal Control for Semi-Active Harmonic Disturbance Rejection
,”
J. Sound Vib.
,
318
(
4–5
), pp.
737
756
.
13.
Shen
,
Y. J.
,
Wang
,
L.
,
Yang
,
S. P.
, and
Gao
,
G. S.
,
2013
, “
Nonlinear Dynamical Analysis and Parameters Optimization of Four Semi-Active On-Off Dynamic Vibration Absorbers
,”
J. Vib. Control
,
19
(
1
), pp.
143
160
.
14.
Potter
,
J. N.
,
Neild
,
S. A.
, and
Wagg
,
D. J.
,
2010
, “
Generalisation and Optimisation of Semi-Active, On-Off Switching Controllers for Single Degree-of-Freedom Systems
,”
J. Sound Vib.
,
329
(
13
), pp.
2450
2462
.
15.
Viet
,
L. D.
,
2012
, “
Semi-Active On–Off Damping Control of a Dynamic Vibration Absorber Using Coriolis Force
,”
J. Sound Vib.
,
331
(15), pp.
3429
3436
.
16.
Viet
,
L. D.
, and
Adam
,
C.
,
2018
, “
General On-Off Damping Controller for Semi-Active Tuned Liquid Column Damper
,”
J. Vib. Control
,
24
(
23
), pp.
5487
5501
.
17.
Morselli
,
R.
, and
Zanasi
,
R.
,
2008
, “
Control of Port Hamiltonian Systems by Dissipative Devices and Its Application to Improve the Semi-Active Suspension Behaviour
,”
Mechatronics
,
18
(
7
), pp.
364
369
.
18.
Koopman
,
J.
,
Jeltsema
,
D.
, and
Verhaegen
,
M.
,
2011
, “
Port-Hamiltonian Description and Analysis of the LuGre Friction Model
,”
Simul. Modell. Pract. Theory
,
19
(
3
), pp.
959
968
.
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