The azimuthal-radial pendulum is a coupled pendulum-beam system where a pendulum and a supporting string are attached to the free end of a cantilever beam with its other end clamped. Conversion between radial (parallel to cantilever) and azimuthal (perpendicular to cantilever) modes of oscillations is observed when the radial frequency is twice the azimuthal frequency. A theoretical Lagrangian formulation of the system is presented and numerical predictions are compared to experimental data, showing good agreement. A non-linear coupling term arising from the inextensibility of the cantilever beam is shown, via simplifying assumptions that reduce the equations of motion to the Mathieu equation for parametric oscillators, which is responsible for the parametric resonance observed in the system. Experimental work involved measurements of the dynamic trajectory of the pendulum bob in the lab frame, and systematic investigation into the maximum conversion amplitude, beat frequency, and parametric resonance regime boundaries as a function of bob mass, string length, cantilever length, initial release amplitude, and other relevant parameters, shows good agreement with numerical results.

1.
W. B.
Case
, “
Parametric instability: An elementary demonstration and discussion
,”
Am. J. Phys.
48
,
218
–
221
(
1980
).
https://doi.org/10.1119/1.12156
Google Scholar
Crossref
Search ADS
 
2.
L.
Falk
, “
Student experiments on parametric resonance
,”
Am. J. Phys.
47
,
325
–
328
(
1979
).
https://doi.org/10.1119/1.11837
Google Scholar
Crossref
Search ADS
 
3.
I.
Grosu
 and
D.
Ursu
, “
Simple apparatus for obtaining parametric resonance
,”
Am. J. Phys.
50
,
561
(
1982
).
https://doi.org/10.1119/1.12803
Google Scholar
Crossref
Search ADS
 
4.
L.
Adler
 and
M. A.
Breazeale
, “
Parametric phenomena in physics
,”
Am. J. Phys.
39
,
1522
–
1527
(
1971
).
https://doi.org/10.1119/1.1976707
Google Scholar
Crossref
Search ADS
 
5.
N.
Fameli
,
F. L.
Curzon
, and
S.
Mikoshiba
, “
Floquet's theorem and matrices for parametric oscillators: Theory and demonstrations
,”
Am. J. Phys.
67
,
127
–
132
(
1999
).
https://doi.org/10.1119/1.19207
Google Scholar
Crossref
Search ADS
 
6.
R.
Berthet
,
A.
Petrosyan
, and
B.
Roman
, “
An analog experiment of the parametric instability
,”
Am. J. Phys.
70
,
744
–
749
(
2002
).
https://doi.org/10.1119/1.1477428
Google Scholar
Crossref
Search ADS
 
7.
W. B.
Case
, “
The pumping of a swing from the standing position
,”
Am. J. Phys.
64
(
3
),
215
–
220
(
1996
).
https://doi.org/10.1119/1.18209
Google Scholar
Crossref
Search ADS
 
8.
T. I.
Fossen
 and
H.
Nijmeijer
,
Parametric Resonance in Dynamical Systems
(
Springer
,
New York
,
2011
).
Google Scholar
 
9.
A. H.
Nayfeh
 and
D. T.
Mook
,
Nonlinear Oscillations
(
Wiley
,
New York
,
1979
).
Google Scholar
 
10.
H.
Yabuno
,
Y.
Ide
, and
N.
Aoshima
, “
Nonlinear analysis of a parametrically excited cantilever beam
,”
JSME Int. J.
41
(
3
),
555
–
562
(
1998
).
https://doi.org/10.1299/jsmec.41.555
Google Scholar
Crossref
Search ADS
 
11.
P.
Roura
 and
J. A.
González
, “
Towards a more realistic description of swing pumping due to the exchange of angular momentum
,”
Eur. J. Phys.
31
(
5
),
1195
–
1207
(
2010
).
https://doi.org/10.1088/0143-0807/31/5/020
Google Scholar
Crossref
Search ADS
 
12.
J. L.
Bogdanoff
 and
S. J.
Citron
, “
Experiments with an inverted pendulum subject to random parametric excitation
,”
J. Acoust. Soc. Am.
38
(
3
),
447
–
452
(
1965
).
https://doi.org/10.1121/1.1909711
Google Scholar
Crossref
Search ADS
 
13.

International Young Physicist's Tournament, “Problem 11,” in Problems for the 31st IYPT 2018, Beijing, China, 2018.

14.
M.
Plesch
,
M.
Badin
, and
N.
Ruzickova
, “
The International Young Physicists' Tournament 2017
,”
Eur. J. Phys.
39
(
6
),
064003
(
2018
).
https://doi.org/10.1088/1361-6404/aaa889
Google Scholar
Crossref
Search ADS
 
15.
H.
Yabuno
,
T.
Murakami
,
J.
Kawazoe
, and
N.
Aoshima
, “
Suppression of parametric resonance in cantilever beam with a pendulum (effect of static friction at the supporting point of the pendulum)
,”
J. Vib. Acoust.
126
(
1
),
149
–
162
(
2004
).
https://doi.org/10.1115/1.1596554
Google Scholar
Crossref
Search ADS
 
16.
J.
Xu
 and
J.
Tang
, “
Multi-directional energy harvesting by piezoelectric cantilever-pendulum with internal resonance
,”
Appl. Phys. Lett.
107
(
21
),
213902
(
2015
).
https://doi.org/10.1063/1.4936607
Google Scholar
Crossref
Search ADS
 
17.
B.
Pratiher
 and
S. K.
Dwivedy
, “
Parametric instability of a cantilever beam with magnetic field and periodic axial load
,”
J. Sound. Vib.
305
(
4–5
),
904
–
917
(
2007
).
https://doi.org/10.1016/j.jsv.2007.04.039
Google Scholar
Crossref
Search ADS
 
18.
H.
Broer
 and
C.
Simo
, “
Hill's equation with quasi-periodic forcing: Resonance tongues, instability pockets and global phenomena
,”
Bol. Soc. Bras. Mat.
29
(
2
),
253
–
293
(
1998
).
https://doi.org/10.1007/BF01237651
Google Scholar
Crossref
Search ADS
 
19.
J.
Mixer
 and
F. W.
Schafke
, “
Anwendungen der mathieuschen funktionen und der sphäroidfunktionen
,” in
Mathieusche Funktionen und Sphäroidfunktionen
(
Springer
,
Berlin
,
1954
), Vol.
71
, pp.
324
–
392
.
Google Scholar
 
20.
F.
Verhulst
, “
Autoparametric resonance of relaxation oscillations
,”
J. Appl. Math. Mech.
85
(
12
),
122
–
131
(
2005
).
https://doi.org/10.1002/zamm.200410159
Google Scholar
Crossref
Search ADS
 
21.
Iván
Delgado-Velázquez
, “
Nonlinear vibration of a cantilever beam
,” doctoral thesis (
Rochester Institute of Technology
,
2007
).
Google Scholar
 
22.
E.
Dowell
 and
K.
McHugh
, “
Equations of motion for an inextensible beam undergoing large deflections
,”
J. Appl. Mech.
83
(
5
),
53
–
60
(
2016
).
https://doi.org/10.1115/1.4032795
Google Scholar
Crossref
Search ADS
 
23.
H.
Goldstein
,
C.
Poole
, and
J.
Safko
,
Classical Mechanics
, 3rd ed. (
Addison Wesley
,
San Francisco, CA
,
2002
).
Google Scholar
 
24.
O. A.
Bauchau
,
J. I.
Craig
, “
Euler-Bernoulli beam theory
,” in
Structural Analysis
, Solid Mechanics and its Applications Vol. 163 (
Springer
,
Dordrecht
,
2009
), pp.
173
–
221
.
Google Scholar
Crossref
Search ADS
 
25.
E.
Reissner
, “
On one-dimensional large-displacement finite-strain beam theory
,”
Stud. Appl. Math.
52
,
87
–
95
(
1973
).
https://doi.org/10.1002/sapm197352287
Google Scholar
Crossref
Search ADS
 
26.
S.
Timoshenko
,
History of Strength of Materials
(
McGraw-Hill
,
New York
,
1953
).
Google Scholar
 
27.
Tarsicio
Beléndez
,
Cristian
Neipp
, and
Augusto
Beléndez
, “
Large and small deflections of a cantilever beam
,”
Eur. J. Phys.
23
(
3
),
371
–
379
(
2002
).
https://doi.org/10.1088/0143-0807/23/3/317
Google Scholar
Crossref
Search ADS
 
28.
B.
Fornberg
, “
Generation of finite difference formulas on arbitrarily spaced grids
,”
Math. Comput.
51
,
699
–
706
(
1988
).
https://doi.org/10.1090/S0025-5718-1988-0935077-0
Google Scholar
Crossref
Search ADS
 
29.
J. C.
Butcher
,
Numerical Methods for Ordinary Differential Equations
(
John Wiley & Sons
,
New York
,
2003
).
Google Scholar
Crossref
Search ADS
 
30.
E.
Hairer
,
C.
Lubich
, and
G.
Warner
, “
Geometric numerical integration illustrated by the Störmer–Verlet method
,”
Acta Numer.
12
,
399
–
450
(
2003
).
https://doi.org/10.1017/S0962492902000144
Google Scholar
Crossref
Search ADS
 
31.
tracker is an open source video-analysis and modeling program that is freely available online at <http://physlets.org/tracker/>.
32.
A. E.
Baroudi
 and
F.
Razafimahery
, “
Transverse vibration analysis of Euler-Bernoulli beam carrying point masses submerged in fluid media
,”
Int. J. Eng. Technol.
4
(
2
),
369
–
380
(
2015
).
https://doi.org/10.14419/ijet.v4i2.4570
Google Scholar
Crossref
Search ADS
 
33.
I. S.
Sokolnikoff
,
Mathematical Theory of Elasticity
, 2nd ed. (
McGraw-Hill
,
New York
,
1956
).
Google Scholar
 
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