During drilling operations, drill string interacts continuously with rock formation, which result in severe shock and vibrations. Lateral, torsion and axial vibration modes often cause failures of Bottom Hole Assembly (BHA), drill pipe abrasive wear, drill bit and wall borehole damages. It also leads to reduction in Rate of Penetration (ROP) and consequently incur unnecessarily high costs. The Lagrangian approach has been used in this study to attain drill pipe lateral and torsional vibration coupling equations of motion. The mathematical model is expressed in terms of four independent degrees of freedom. The effects of bending and torsion vibrations, and whirling motion of the drill string are incorporated in the developed model. A set of nonlinear equations are solved numerically to obtain the response of the system. In this work, we also present a brief description of an inhouse constructed experimental setup. The setup has the capability to imitate the downhole axial, lateral and torsional vibration modes and mechanisms. Experimental investigations for the drill pipe fatigue failure due to lateral and torsional cyclic stresses induced in the drill string are also presented. Such investigations are essential for oil/gas industry as they provide solutions for very common problems such as drill string fatigue failure. The performance of the setup was validated. Numbers of tests were performed to investigate the effects of rotational speeds on the vibration amplitudes of different drill string sizes.