Abstract

In this paper, a simple and efficient method to enforce nodes, or points of zero vibration, on an arbitrarily supported rectangular plate subjected to multiple harmonics with distinct excitation frequencies is developed. This vibration suppression is achieved by attaching a chain of properly tuned oscillators, configured in series, to the host plate. The governing equations of the combined system are first obtained using the assumed-modes method. Enforcing the node conditions for the given excitation frequencies, a set of constraint equations are formulated, from which the sprung mass parameters can be determined. Two special cases are considered: when the attachment and node locations coincide (or collocated) and when they do not (or non-collocated). For the former case, determining the required oscillator parameters requires one to solve an inverse eigenvalue problem, and for the latter case, it requires one to use a generic optimization algorithm. Procedures to tune the parameters of the oscillator chains are outlined in detail, and numerical experiments validate the proposed method of enforcing nodes to mitigate excess vibration when the plate is subjected to multiple harmonic excitations.

References

1.
Kukla
,
S.
,
2003
, “
Frequency Analysis of a Rectangular Plate With Attached Discrete Systems
,”
J. Sound Vib.
,
264
(
1
), pp.
225
234
. 10.1016/S0022-460X(02)01180-X
2.
Zhou
,
D.
, and
Ji
,
T.
,
2006
, “
Free Vibration of Rectangular Plates With Continuously Distributed Spring-Mass
,”
Int. J. Solids Struct.
,
43
(
21
), pp.
6502
6520
. 10.1016/j.ijsolstr.2005.12.005
3.
Cha
,
P. D.
, and
Ren
,
G.
,
2006
, “
Inverse Problem of Imposing Nodes to Suppress Vibration for a Structure Subjected to Multiple Harmonic Excitations
,”
J. Sound Vib.
,
290
(
1–2
), pp.
425
447
. 10.1016/j.jsv.2005.04.025
4.
Cha
,
P. D.
, and
Zhou
,
X.
,
2006
, “
Imposing Points of Zero Displacements and Zero Slopes Along Any Linear Structure During Harmonic Excitations
,”
J. Sound Vib.
,
297
(
1–2
), pp.
55
71
. 10.1016/j.jsv.2006.03.032
5.
Cha
,
P. D.
, and
Yoder
,
N. C.
,
2007
, “
Applying Sherman-Morrison-Woodbury Formulas to Analyze the Free and Forced Responses of a Linear Structure Carrying Lumped Elements
,”
ASME J. Vib. Acoust.
,
129
(
3
), pp.
307
316
. 10.1115/1.2730537
6.
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
. 10.1007/s00419-011-0578-0
7.
Issa
,
J. S.
,
2014
, “
Reduction of the Transient Vibration of Systems Using the Classical and a Modified Vibration Absorber Setup
,”
J. Vib. Control
,
20
(
10
), pp.
1475
1487
. 10.1177/1077546312469423
8.
Pasternak
,
E.
,
Dyskin
,
A. V.
, and
Sevel
,
G.
,
2014
, “
Chains of Oscillators With Negative Stiffness Elements
,”
J. Sound Vib.
,
333
(
24
), pp.
6676
6687
. 10.1016/j.jsv.2014.06.045
9.
Cha
,
P. D.
, and
Buyco
,
K.
,
2015
, “
An Efficient Method for Tuning Oscillator Parameters in Order to Impose Nodes on a Linear Structure Excited by Multiple Harmonics
,”
ASME J. Vib. Acoust.
,
137
(
3
), p.
031018
. 10.1115/1.4029612
10.
Patil
,
S. S.
, and
Awasare
,
P. J.
,
2016
, “
Vibration Isolation of Lumped Masses Supported on Beam by Imposing Nodes Using Multiple Vibration Absorber
,”
Mech. Eng. Res.
,
6
(
1
), pp.
88
95
. 10.5539/mer.v6n1p88
11.
Zuo
,
L.
,
2009
, “
Effective and Robust Vibration Control Using Series Multiple Tuned-Mass Dampers
,”
ASME J. Vib. Acoust.
,
131
(
3
), p.
031003
. 10.1115/1.3085879
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
. 10.1115/1.4026246
13.
Jin
,
X.
,
Chen
,
Z. Q.
, and
Huang
,
Z.
,
2016
, “
Minimization of the Beam Response Using Inerter-Based Passive Vibration Control Configurations
,”
Int. J. Mech. Sci.
,
119
(
Dec.
), pp.
80
87
. 10.1016/j.ijmecsci.2016.10.007
14.
Charlemagne
,
S.
,
Lamarque
,
C. H.
, and
Ture Savadkoohi
,
A.
,
2017
, “
Vibratory Control of a Linear System by Addition of a Chain of Nonlinear Oscillators
,”
Acta Mech.
,
228
(
9
), pp.
3111
3133
. 10.1007/s00707-017-1867-7
15.
Javidialesaadi
,
A.
, and
Wierschem
,
N. A.
,
2018
, “
Three-Element Vibration Absorber-Inerter for Passive Control of Single-Degree-of-Freedom Structures
,”
ASME J. Vib. Acoust.
,
140
(
6
), p.
061007
. 10.1115/1.4040045
16.
Asami
,
T.
,
2017
, “
Optimal Design of Double-Mass Dynamic Vibration Absorbers Arranged in Series or in Parallel
,”
ASME J. Vib. Acoust.
,
139
(
1
), p.
011015
. 10.1115/1.4034776
17.
Yang
,
X.
,
Zhou
,
B.
, and
Lam
,
J.
,
2019
, “
Bounded Control of Feedforward Time-Delay Systems With Linearized Systems Consisting of Chain of Oscillators
,”
Int. J. Robust Nonlinear Control
,
29
(
6
), pp.
283
305
. 10.1002/rnc.4389
18.
Bruant
,
I.
,
Gallimard
,
L.
, and
Nikoukar
,
S.
,
2010
, “
Optimal Piezoelectric Actuator and Sensor Location for Active Vibration Control, Using Genetic Algorithm
,”
J. Sound Vib.
,
329
(
10
), pp.
1615
1635
. 10.1016/j.jsv.2009.12.001
19.
Julai
,
S.
, and
Tokhi
,
M. O.
,
2010
, “
Vibration Suppression of Flexible Plate Structures Using Swarm and Genetic Optimization Techniques
,”
J. Low. Freq. Noise V. A.
,
29
(
4
), pp.
293
318
. 10.1260/0263-0923.29.4.293
20.
Tairidis
,
G.
,
Foutsitzi
,
G.
,
Koutsianitis
,
P.
, and
Stavroulakis
,
G. E.
,
2016
, “
Fine Tuning of a Fuzzy Controller for Vibration Suppression of Smart Plates Using Genetic Algorithms
,”
Adv. Eng. Software
,
101
(
Nov.
), pp.
101
135
. 10.1016/j.advengsoft.2016.01.019
21.
Shen
,
Y.
,
Zhou
,
X.
, and
Cha
,
P. D.
,
2017
, “
Imposing Points of Zero Displacement and Zero Slopes on a Plate Subjected to Steady-State Harmonic Excitation
,”
J. Vib. Control
,
24
(
20
), pp.
4904
4920
. 10.1177/1077546317738616
22.
Shen
,
Y.
,
Zhou
,
X.
, and
Cha
,
P. D.
,
2018
, “
An Efficient Method to Quench Excess Vibration for a Harmonically Excited Damped Plate
,”
Int. J. Mech. Sci.
,
141
(
June
), pp.
372
385
. 10.1016/j.ijmecsci.2018.04.016
23.
Meirovitch
,
L.
,
2001
,
Fundamentals of Vibrations
,
McGraw-Hill
,
Boston, MA
.
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