In the oil drilling industry, a drill-tube is used for well-boring. It is a tube of outer diameter \(D_{o}\) and inner diameter \(D_{i}\) and is made of steel of density \(ho\) and modulus of rigidity \(G\). At the end of the drill-tube there is a drill bit that does the actual drilling. (The bit rotates, enabling it to cut through the toughest of material. The bits are generally made of manufactured diamonds.) Assuming a tube length \(l\) and a bit moment of inertia \(I_{b}\), derive an expression for the first two natural frequencies of torsional vibration, assuming that the tube is fixed at the upper end.