Longitudinal and torsional vibrations in a drill string consisting of drill pipe and collars, and the boundary conditions at the ends of the string, are discussed. A trial-and-error method of solution for the natural frequencies is demonstrated. Since it is very lengthy, certain useful charts are then derived, and a graphical method of solution for the natural frequencies is developed. By use of this method the natural frequencies can be quickly and easily obtained. The influences of various parameters on the natural frequencies are considered.

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