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
).
2.
L.
Falk
, “
Student experiments on parametric resonance
,”
Am. J. Phys.
47
,
325
328
(
1979
).
3.
I.
Grosu
and
D.
Ursu
, “
Simple apparatus for obtaining parametric resonance
,”
Am. J. Phys.
50
,
561
(
1982
).
4.
L.
Adler
and
M. A.
Breazeale
, “
Parametric phenomena in physics
,”
Am. J. Phys.
39
,
1522
1527
(
1971
).
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
).
6.
R.
Berthet
,
A.
Petrosyan
, and
B.
Roman
, “
An analog experiment of the parametric instability
,”
Am. J. Phys.
70
,
744
749
(
2002
).
7.
W. B.
Case
, “
The pumping of a swing from the standing position
,”
Am. J. Phys.
64
(
3
),
215
220
(
1996
).
8.
T. I.
Fossen
and
H.
Nijmeijer
,
Parametric Resonance in Dynamical Systems
(
Springer
,
New York
,
2011
).
9.
A. H.
Nayfeh
and
D. T.
Mook
,
Nonlinear Oscillations
(
Wiley
,
New York
,
1979
).
10.
H.
Yabuno
,
Y.
Ide
, and
N.
Aoshima
, “
Nonlinear analysis of a parametrically excited cantilever beam
,”
JSME Int. J.
41
(
3
),
555
562
(
1998
).
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
).
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
).
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
).
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
).
16.
J.
Xu
and
J.
Tang
, “
Multi-directional energy harvesting by piezoelectric cantilever-pendulum with internal resonance
,”
Appl. Phys. Lett.
107
(
21
),
213902
(
2015
).
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
).
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
).
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
.
20.
F.
Verhulst
, “
Autoparametric resonance of relaxation oscillations
,”
J. Appl. Math. Mech.
85
(
12
),
122
131
(
2005
).
21.
Iván
Delgado-Velázquez
, “
Nonlinear vibration of a cantilever beam
,” doctoral thesis (
Rochester Institute of Technology
,
2007
).
22.
E.
Dowell
and
K.
McHugh
, “
Equations of motion for an inextensible beam undergoing large deflections
,”
J. Appl. Mech.
83
(
5
),
53
60
(
2016
).
23.
H.
Goldstein
,
C.
Poole
, and
J.
Safko
,
Classical Mechanics
,
3rd ed.
(
Addison Wesley
,
San Francisco, CA
,
2002
).
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
.
25.
E.
Reissner
, “
On one-dimensional large-displacement finite-strain beam theory
,”
Stud. Appl. Math.
52
,
87
95
(
1973
).
26.
S.
Timoshenko
,
History of Strength of Materials
(
McGraw-Hill
,
New York
,
1953
).
27.
Tarsicio
Beléndez
,
Cristian
Neipp
, and
Augusto
Beléndez
, “
Large and small deflections of a cantilever beam
,”
Eur. J. Phys.
23
(
3
),
371
379
(
2002
).
28.
B.
Fornberg
, “
Generation of finite difference formulas on arbitrarily spaced grids
,”
Math. Comput.
51
,
699
706
(
1988
).
29.
J. C.
Butcher
,
Numerical Methods for Ordinary Differential Equations
(
John Wiley & Sons
,
New York
,
2003
).
30.
E.
Hairer
,
C.
Lubich
, and
G.
Warner
, “
Geometric numerical integration illustrated by the Störmer–Verlet method
,”
Acta Numer.
12
,
399
450
(
2003
).
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
).
33.
I. S.
Sokolnikoff
,
Mathematical Theory of Elasticity
,
2nd ed.
(
McGraw-Hill
,
New York
,
1956
).
AAPT members receive access to the American Journal of Physics and The Physics Teacher as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.