Abstract

This paper describes a primarily experimental study in which a nonlinear structural component (a slender, mechanically buckled panel) is subject to probing. That is, equilibrium configurations are explored when a specific location on the panel is subject to the application of a (variable) displacement constraint and characterized by a corresponding probe force. This probe force (in this study located at the center of the rectangular panels) is measured using a load cell and the resulting shape(s), taken up by the panel, measured using digital image correlation (DIC). Although the probe is only applied at a single location, this arrangement supplies considerable information about the changing equilibrium landscape including revealing co-existing equilibrium configurations using large perturbations and associated hysteresis phenomena. In addition, monitoring the probing force, and specifically when it drops to zero, provides a window into “free” equilibria that would otherwise be unstable and unobservable. Finally, it is shown that the probed equilibrium configurations provide the “landscape” within which any dynamically induced trajectories evolve including snap-through oscillations.

References

References
1.
Spottswood
,
S. M.
,
Beberniss
,
T. J.
,
Eason
,
T. G.
,
Perez
,
R. A.
,
Donbar
,
J. M.
,
Ehrhardt
,
D. A.
, and
Riley
,
Z. B.
,
2019
, “
Exploring the Response of a Thin, Flexible Panel to Shock-Turbulent Boundary-Layer Interactions
,”
J. Sound. Vib.
,
443
, pp.
74
89
. 10.1016/j.jsv.2018.11.035
2.
Virgin
,
L. N.
,
2007
,
Vibration of Axially-Loaded Structures
,
Cambridge University Press
,
New York
.
3.
Timoshenko
,
S. P.
, and
Gere
,
J.
,
1961
,
Theory of Elastic Stability
,
Dover
,
New York
.
4.
Thompson
,
J. M. T.
, and
Hunt
,
G. W.
,
1984
,
Elastic Instability Phenomena
,
Wiley
,
Chicester
.
5.
Wiebe
,
R.
, and
Virgin
,
L. N.
,
2016
, “
On the Experimental Identification of Unstable Static Equilibria
,”
Proc. R. Soc. London A: Math., Phys. Eng. Sci.
,
472
(
20160172
), pp.
1
15
.
6.
Thompson
,
J. M. T.
,
Hutchinson
,
J. W.
, and
Sieber
,
J.
,
2017
, “
Probing Shells Against Buckling: A Non-Destructive Technique for Laboratory Testing
,”
Int. J. Bifurcat. Chaos
,
27
(
14
), p.
1730048
. 10.1142/S0218127417300488
7.
Neville
,
R. M.
,
Groh
,
R. M. J.
,
Pirrera
,
A.
, and
Schenk
,
M.
,
2018
, “
Shape Control for Experimental Continuation
,”
Phys. Rev. Lett.
,
120
, p.
254101
. 10.1103/PhysRevLett.120.254101
8.
Ross
,
S. D.
,
Bozorgmagham
,
A. E.
,
Naik
,
S.
, and
Virgin
,
L. N.
,
2018
, “
Experimental Validation of Phase Space Conduits of Transition Between Potential Wells
,”
Phys. Rev. E
,
98
, p.
052214
. 10.1103/PhysRevE.98.052214
9.
van Iderstein
,
T.
, and
Wiebe
,
R.
,
2018
,
Experimental Path Following of Unstable Equilibria for Snap-Through Buckling
, Vol.
1
,
G.
Kerschen
, ed.,
Springer
,
Cham
, pp.
17
22
.
10.
Xu
,
Y.
, and
Virgin
,
L. N.
,
2019
, “
Probing the Force Field to Identify Potential Energy
,”
ASME J. Appl. Mech.
,
86
(
10
), p.
101008
. 10.1115/1.4044305
11.
Stoll
,
F.
,
1994
, “
Analysis of the Snap Phenomenon in Buckled Plates
,”
Int. J. Non-Linear Mech.
,
29
(
2
), pp.
123
138
. 10.1016/0020-7462(94)90031-0
12.
Bryan
,
G. H.
,
1891
, “
On the Stability of a Plane Plate Under Thrust in Its Own Plane with Application to the Buckling of the Side of a Ship
,”
Proc. London Math. Soc.
,
22
(
1
), pp.
54
67
.
13.
Levy
,
S.
,
Bending of rectangular plates with large deflections
.
Technical report, NASA Report 737, 1942
.
14.
Coan
,
J. M.
,
1951
, “
Large Deflection Theory for Plates With Small Initial Curvature Loaded in Edge Compression
,”
Trans. ASME
,
73
, pp.
143
151
.
15.
Yamaki
,
N.
,
1959
, “
The Post-Buckling Behaviour of Rectangular Plates With Small Initial Curvature Loaded in Edge Compression
,”
ASME J. Appl. Mech.
,
26
(
2
), pp.
407
414
.
16.
Timoshenko
,
S. P.
, and
Woinowsky-Krieger
,
S.
,
1959
,
Theory of Plates and Shells
,
McGraw-Hill
,
New York
.
17.
Sundara Raja Iyengar
,
K. T.
, and
Matin Naqvi
,
M.
,
1966
, “
Large Deflections of Rectangular Plates
,”
Int. J. Non-Linear Mech.
,
1
(
2
), pp.
109
122
. 10.1016/0020-7462(66)90024-2
18.
Bulson
,
P. S.
,
1970
,
The Stability of Flats Plates
,
Chatto and Windus
,
London
.
19.
Walker
,
A. C.
,
1984
, “A Brief Review of Plate Buckling Research,”
Behaviour of Thin-walled Structures
,
Rhodes
,
J.
, and
Spence
,
J.
, eds.,
Elsevier
.
20.
Ugural
,
A. C.
,
1999
,
Stresses in Plates and Shells
,
McGraw-Hill
,
New York
.
21.
Ng
,
C. F.
,
1989
, “
Nonlinear and Snap-through Responses of Curved Panels to Intense Acosutic Excitation
,”
J. Air.
,
26
(
3
), pp.
281
. 10.2514/3.45757
22.
Blevins
,
R. D.
,
Holehouse
,
I.
, and
Wentz
,
K. R.
,
1993
, “
Thermoacosutic Loads and Fatigue of Hypersonic Vehicle Skin Panels
,”
J. Air.
,
30
(
6
), pp.
971
978
. 10.2514/3.46441
23.
Dhainaut
,
J. M.
,
Mei
,
C.
,
Spottswood
,
S. M.
, and
Wolfe
,
H. F.
,
2002
, “
Sonic Fatigue Design and Nonlinear Panel Response to Flight Nonwhite Pressure Fluctuations
,”
43rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference
,
Denver, CO
,
April
.
24.
Gordon
,
R. W.
, and
Hollkamp
,
J. J.
,
2006
, “
Nonlinear Random Response of a Clamped Plate: a Well-characterized Experiment
,”
47th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference
,
Newport, RI
,
May
.
25.
Cui
,
D.
, and
Hu
,
H.
,
2014
, “
Thermal Buckling and Natural Vibration of a Rectangular Thin Plate with in-plane Stick-slip-stop Boundaries
,”
J. Vib. Control
,
22
(
7
), pp.
1950
1966
. 10.1177/1077546314546394
26.
Murphy
,
K. D.
,
Virgin
,
L. N.
, and
Rizzi
,
S. A.
,
1996
, “
Experimental Snap-through Boundaries for Acoustically Excited, Thermally Buckled Plates
,”
Exp. Mech.
,
36
, pp.
312
317
. 10.1007/BF02328572
27.
Virgin
,
L. N.
,
2000
,
Introduction to Experimental Nonlinear Dynamics
,
Cambridge University Press
,
Cambridge UK
.
28.
Stein
,
M.
,
Loads and deformation of buckled rectangular plates
.
Technical report, NASA Technical Report R-40, 1959
.
29.
Chen
,
H.
, and
Virgin
,
L. N.
,
2006
, “
Finite Element Analysis of Postbuckling Dynamics in Plates: Part I: An Asymptotic Approach
,”
Int. J. Solids. Struct.
,
43
(
13
), pp.
3983
4007
. 10.1016/j.ijsolstr.2005.04.036
30.
Schaeffer
,
D.
, and
Golubitsky
,
M.
,
1979
, “
Boundary Conditions and Mode Jumping in the Buckling of a Rectangular Plate
,”
Commun. Math. Phys.
,
69
, pp.
209
236
. 10.1007/BF01197444
31.
Chen
,
H.
, and
Virgin
,
L. N.
,
2006
, “
Finite Element Analysis of Postbuckling Dynamics in Plates: Part II: A Nonstationary Analysis
,”
Int. J. Solids. Struct.
,
43
(
13
), pp.
4008
4027
. 10.1016/j.ijsolstr.2005.04.037
32.
Taffetani
,
M.
,
Jiang
,
X.
,
Holmes
,
D. P.
, and
Vella
,
D.
,
2018
, “
Static Bistability of Spherical Caps
,”
Proc. R. Soc. A
,
474
(
2213
). 10.1098/rspa.2017.0910
33.
Young
,
W.
, and
Bufynas
,
R.
,
2011
,
Roark’s Formulas for Stress and Strain
,
McGraw-Hill
.
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