An oil well drilling tool is modeled as a bit attached to the end of a long
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An oil well drilling tool is modeled as a bit attached to the end of a long shaft, unrestrained from rotation at its fixed end.
(a) Determine the equation defining the natural frequencies of the drilling tool.
(b) For a particular operation, the shaft ( \(ho=7500 \mathrm{~kg} / \mathrm{m}^{3}, G=80 \times 10^{9} \mathrm{~N} / \mathrm{m}^{2}\) ) is \(20 \mathrm{~m}\) long with a \(20-\mathrm{cm}\) diameter. The tool operates at a speed of \(400 \mathrm{rad} / \mathrm{s}\). What are the limits on the moment of inertia of the drill bit such that the two lowest nonzero natural frequencies of the tool are not within 20 percent of the operating speed?
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