A model of a one-dimensional mass-spring chain with mass or spring defects is investigated. With a mass defect, all oscillators except the central one have the same mass, and with a spring defect, all the springs except those connected to the central oscillator have the same stiffness constant. The motion is assumed to be one-dimensional and frictionless, and both ends of the chain are assumed to be fixed. The system vibrational modes are obtained analytically, and it is shown that if the defective mass is lighter than the others, then a high frequency mode appears in which the amplitudes decrease exponentially with the distance from the defect. In this sense, the mode is localized in space. If the defect mass is greater than the others, then there will be no localized mode and all modes are extended throughout the system. Analogously, for some values of the defective spring constant, there may be one or two localized modes. If the two defected spring constants are less than that of the others, there is no localized mode.
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March 2017
PAPERS|
March 01 2017
Normal modes of a defected linear system of beaded springs
Amir Aghamohammadi;
Amir Aghamohammadi
a)
Department of Physics,
Alzahra University
, Tehran 19938-93973, Iran
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M. Ebrahim Foulaadvand;
M. Ebrahim Foulaadvand
b)
Department of Physics,
University of Zanjan
, P. O. Box 45196-313, Zanjan, Iran and School of Nano-Science, Institute for Research in Fundamental Sciences (IPM), P. O. Box 19395-5531, Teheran, Iran
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Mohammad Hassan Yaghoubi;
Mohammad Hassan Yaghoubi
Complexity Science Group, Department of Physics and Astronomy,
University of Calgary
, Calgary, Alberta T2N 1N4, Canada
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Amir Hossein Mousavi
Amir Hossein Mousavi
Department of Physics,
University of Zanjan
, P.O. Box 45196-313, Zanjan, Iran
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a)
Electronic mail: mohamadi@alzahra.ac.ir
b)
Elecronic mail: foolad@znu.ac.ir
Am. J. Phys. 85, 193–201 (2017)
Article history
Received:
July 02 2015
Accepted:
November 23 2016
Citation
Amir Aghamohammadi, M. Ebrahim Foulaadvand, Mohammad Hassan Yaghoubi, Amir Hossein Mousavi; Normal modes of a defected linear system of beaded springs. Am. J. Phys. 1 March 2017; 85 (3): 193–201. https://doi.org/10.1119/1.4972176
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