In this paper, the dynamic oscillation of a rectangular thin plate under parametric and external excitations is investigated and controlled. The motion of a rectangular thin plate is modeled by coupled second-order nonlinear ordinary differential equations. The formulas of the thin plate are derived from the von Kármán equation and Galerkin's method. A control law based on negative acceleration feedback is proposed for the system. The multiple time scale perturbation technique is applied to solve the nonlinear differential equations and obtain approximate solutions up to the second-order approximations. One of the worst resonance case of the system is the simultaneous primary resonances, where Ω1ω1andΩ2ω2. From the frequency response equations, the stability of the system is investigated according to the Routh–Hurwitz criterion. The effects of the different parameters are studied numerically. It is also shown that the system parameters have different effects on the nonlinear response of the thin plate. The simulation results are achieved using matlab 7.0 software. A comparison is made with the available published work.

References

References
1.
Feng
,
Z. C.
, and
Sethna
,
P. R.
,
1993
, “
Global Bifurcations in the Motion of Parametrically Excited Thin Plates
,”
Nonlinear Dyn.
,
4
(
4
), pp.
389
408
.
2.
Chang
,
S. I.
,
Bajaj
,
A. K.
, and
Krousgrill
,
C. M.
,
1993
, “
Non-Linear Vibrations and Chaos in Harmonically Excited Rectangular Plates With One-to-One Internal Resonance
,”
Nonlinear Dyn.
,
4
(
5
), pp.
433
460
.
3.
Ostiguy
,
G. L.
,
Samson
,
L. P.
, and
Nguyen
,
H.
,
1993
, “
On the Occurrence of Simultaneous Resonances in Parametrically-Excited Rectangular Plates
,”
ASME J. Vib. Acoust.
,
115
(
3
), pp.
344
352
.
4.
Lai
,
S. K.
,
Lim
,
C. W.
,
Xiang
,
Y.
, and
Zhang
,
W.
,
2009
, “
On Asymptotic Analysis for Large Amplitude Nonlinear Free Vibration of Simply Supported Laminated Plates
,”
ASME J. Vib. Acoust.
,
131
(
5
), p.
051010
.
5.
Zhang
,
W.
,
2001
, “
Global and Chaotic Dynamics for a Parametrically Excited Thin Plate
,”
J. Sound Vib.
,
239
(
5
), pp.
1013
1036
.
6.
Zhang
,
W.
, and
Liu
,
Z.
,
2001
, “
Global Dynamics of a Parametrically and Externally Excited Thin Plate
,”
Nonlinear Dyn.
,
24
(
3
), pp.
245
268
.
7.
Kim
,
C. H.
,
Lee
,
C. W.
, and
Perkins
,
N. C.
,
2005
, “
Nonlinear Vibration of Sheet Metal Plates Under Interacting Parametric and External Excitation During Manufacturing
,”
ASME J. Vib. Acoust.
,
127
(
1
), pp.
36
43
.
8.
Shiau
,
L.-C.
, and
Kuo
,
S.-Y.
,
2006
, “
Free Vibration of Thermally Buckled Composite Sandwich Plates
,”
ASME J. Vib. Acoust.
,
128
(
1
), pp.
1
7
.
9.
Hegazy
,
U. H.
,
2010
, “
Nonlinear Vibrations of a Thin Plate Under Simultaneous Internal and External Resonance
,”
ASME J. Vib. Acoust.
,
132
(
5
), p.
051004
.
10.
Zhang
,
W.
, and
Li
,
S. B.
,
2010
, “
Resonant Chaotic Motions of a Buckled Rectangular Thin Plate With Parametrically and Externally Excitations
,”
Nonlinear Dyn.
,
62
(
3
), pp.
673
686
.
11.
Sorokin
,
S. V.
, and
Ershov
,
O. A.
,
2003
, “
Forced and Free Vibrations of Rectangular Sandwich Plates With Parametric Stiffness Modulation
,”
J. Sound Vib.
,
259
(
1
), pp.
119
143
.
12.
Lei
,
Y.
,
Xu
,
W.
,
Shen
,
J.
, and
Fang
,
T.
,
2006
, “
Global Synchronization of Two Parametrically Excited Systems Using Active Control
,”
Chaos, Solitons Fractals
,
28
(
2
), pp.
428
436
.
13.
Anlas
,
G.
, and
Elbeyli
,
O.
,
2002
, “
Nonlinear Vibrations of a Simply Supported Rectangular Metallic Plate Subjected to Transverse Harmonic Excitation in the Presence of a One-to-One Internal Resonance
,”
Nonlinear Dyn.
,
30
(
1
), pp.
1
28
.
14.
Sayed
,
M.
, and
Mousa
,
A. A.
,
2012
, “
Second-Order Approximation of Angle-Ply Composite Laminated Thin Plate Under Combined Excitations
,”
Commun. Nonlinear Sci. Numer. Simul.
,
17
(
12
), pp.
5201
5216
.
15.
Guo
,
X. Y.
, and
Zhang
,
W.
,
2011
, “
Nonlinear Dynamics of Composite Laminated Thin Plate With 1:2:3 Inner Resonance
,”
2nd International Conference on Mechanic Automation and Control Engineering
(
MACE'11
), Hohhot, China, July 15–17, pp.
7479
7482
.
16.
Udwadia
,
F. E.
, and
Wanichanon
,
T.
,
2014
, “
Control of Uncertain Nonlinear Multibody Mechanical Systems
,”
ASME J. Appl. Mech.
,
81
(4), p.
041020
.
17.
Udwadia
,
F. E.
, and
Wanichanon
,
T.
,
2014
, “
A New Approach to the Tracking Control of Uncertain Nonlinear Multi-Body Mechanical Systems
,”
Nonlinear Approaches in Engineering Applications 2
,
Springer
,
New York
, pp.
101
136
.
18.
Udwadia
,
F. E.
,
Wanichanon
,
T.
, and
Cho
,
H.
,
2014
, “
Methodology for Satellite Formation-Keeping in the Presence of System Uncertainties
,”
J. Guid. Control Dyn.
,
37
(
5
), pp.
1611
1624
.
19.
Udwadia
,
F. E.
, and
Wanichanon
,
T.
,
2013
, “
On General Nonlinear Constrained Mechanical Systems
,”
Numer. Algebra, Control Optim
.,
3
(
3
), pp.
425
443
.
20.
Udwadia
,
F. E.
, and
Wanichanon
,
T.
,
2012
, “
Explicit Equations of Motion of Constrained Systems With Applications to Multi-Body Dynamics
,”
Nonlinear Approaches in Engineering Applications
,
Springer
,
New York
, pp.
315
348
.
21.
Udwadia
,
F. E.
, and
Kalaba
,
R. E.
,
2000
, “
Nonideal Constraints and Lagrangian Dynamics
,”
J. Aerosp. Eng.
,
13
(
1
), pp.
17
22
.
22.
Udwadia
,
F. E.
,
2008
, “
Optimal Tracking Control of Nonlinear Dynamical Systems
,”
Proc. R. Soc. London, Ser. A
,
464
(
2097
), pp.
2341
2363
.
23.
Udwadia
,
F. E.
,
1996
, “
Equations of Motion for Mechanical Systems: A Unified Approach
,”
Int. J. Nonlinear Mech.
,
31
(
6
), pp.
951
958
.
24.
Udwadia
,
F. E.
, and
Schutte
,
A. D.
,
2010
, “
Equations of Motion for General Constrained Systems in Lagrangian Mechanics
,”
Acta Mech.
,
213
, pp.
111
129
.
25.
Nayfeh
,
A. H.
,
1973
,
Perturbation Methods
,
Wiley
,
New York
.
26.
Nayfeh
,
A. H.
, and
Mook
,
D. T.
,
1979
,
Nonlinear Oscillations
,
Wiley-Interscience
,
New York
.
27.
Chia
,
C. Y.
,
1980
,
Non-Linear Analysis of Plate
,
McGraw-Hill
,
New York
.
You do not currently have access to this content.