The article describes a systematic experimental study of a string vibrating nonlinearly. The string is tracked in real time using strategically located cameras; the video tracking enables a remote observation of the oscillator without perturbing its inherent nonlinearities. We show that our technique can help probe the parametrically excited oscillations and study phenomena such as elliptical and circular trajectories near resonance, resonance fold-over, jump, hysteresis, and subharmonic resonance. The experiment has been successfully employed in the advanced physics laboratory.
REFERENCES
1.
J. W.
Strutt
(Lord Rayleigh), “On maintained vibrations
,” Philos. Mag.
15
, 229
–235
(1883
).2.
O. O.
Reilly
and P. J.
Holmes
, “Non-linear, non-planar and non-periodic vibrations of a string
,” J. Sound Vib.
153
, 413
–435
(1992
).3.
G.
Weinreich
, “Coupled piano strings
,” J. Acoust. Soc. Am.
62
, 1474
–1484
(1977
).4.
B.
Bank
and L.
Sujbert
, “Generation of longitudinal vibrations in piano strings: From physics to sound synthesis
,” J. Acoust. Soc. Am.
117
, 2268
–2278
(2005
).5.
6.
J. A.
Elliott
, “Intrinsic nonlinear effects in vibrating strings
,” Am. J. Phys.
48
, 478
–480
(1980
).7.
J. A.
Elliott
, “Nonlinear resonance in vibrating strings
,” Am. J. Phys.
50
, 1148
–1150
(1982
).8.
T.
Tolenon
, V.
Valimaki
, and M.
Karjalainen
, “Modeling of tension modulation nonlinearity in plucked strings
,” IEEE Trans. Speech Audio Process.
8
, 300
–310
(2000
).9.
T. C.
Molteno
and N. B.
Tufillaro
, “An experimental investigation into the dynamics of a string
,” Am. J. Phys.
72
, 1157
–1169
(2004
).10.
E. W.
Lee
, “Nonlinear forced vibration of a stretched string
,” Br. J. Appl. Phys.
8
, 411
–413
(1957
).11.
S.
Parmley
, T.
Zobrist
, T.
Clough
, A.
Perez-Miller
, M.
Makela
, and R.
Yu
, “Vibrational properties of a loaded string
,” Am. J. Phys.
63
, 547
–553
(1995
).12.
L. A.
Pipes
and L. R.
Harvill
, Applied Mathematics for Engineers and Physicists
(McGraw-Hill
, Tokyo
, 2000
).13.
14.
D. W.
Jordan
and P.
Smith
, Nonlinear Ordinary Differential Equations
(Oxford U.P.
, Oxford
, 2007
).15.
N.
Kryloff
and N.
Bogoliuboff
, Introduction to Non-Linear Mechanics
(Princeton U.P.
, London
, 1947
).16.
C. M.
Bender
and S. A.
Orszag
, Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory
(Springer International Edition
, Delhi
, 2008
).17.
R. A.
Sousa
, R. M.
Souza
, F. P.
Figuerido
, and I. F. M.
Menezes
, “The influence of bending and shear stiffness and rotational inertia in vibrations of cables: An analytical approach
,” Eng. Struct.
33
, 689
–695
(2011
).18.
R. J.
Hanson
, J. M.
Anderson
, and H. K.
Macomber
, “Measurements of nonlinear effects in a driven vibrating wire
,” J. Acoust. Soc. Am.
96
, 1549
–1556
(1994
).19.
D.
Banerjee
and J. K.
Bhattacharjee
, “Analyzing jump phenomena and stability in nonlinear oscillators using renormalization group arguments
,” Am. J. Phys.
78
, 142
–149
(2010
).20.
E.
Delabaere
and F.
Pham
, “Unfolding the quartic oscillator
,” Ann. Phys.
261
, 180
–218
(1997
).21.
See supplementary material at http://dx.doi.org/10.1119/1.4740251 for a video of the string made to vibrate at twice the fundamental frequency.
22.
O. V.
Gendelman
, D. V.
Gorlov
, L. I.
Manevitch
, and A. I.
Musienko
, “Dynamics of coupled linear and essentially nonlinear oscillators with substantially different masses
,” J. Sound Vib.
286
, 1
–19
(2005
).© 2012 American Association of Physics Teachers.
2012
American Association of Physics Teachers
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.