Normal Vibrations of a Finite Linear Lattice with Nearest- and Next-nearest-neighbor Interactions

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An exact normal mode analysis is carried out for a free-ended monatomic linear lattice with both nearest-neighbor and next-nearest-neighbor harmonic interactions. The method of Lagrange's equations with "extra" forces makes it possible to obtain exact normal modes of the system relatively easily. In this method, a part of the forces acting on the system is treated as an extra force that modifies Lagrange's equations of motion for the basic system whose solution is readily available. The correctness of the analysis is demonstrated by applying it to simple systems containing only several particles. It is also shown that the effect of the next-nearest-neighbor interaction is negligible when the system is large. The method of analysis used in this article can be applied to more complicated normal vibrational problems.

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© 1970

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Journal of Sound and Vibration