A system and method for monitoring underground drilling in which vibration is monitored by creating a model of the drill string using finite element techniques or finite difference techniques and (i) predicting vibration by inputting real time values of operating parameters into the model, and then adjusting the model to agree with measured vibration data, (ii) predicting the weight on bit and drill string and mud motor speeds at which resonance will occur, as well as when stick-slip will occur, so that the operator can avoid operating regimes that will result in high vibration, (iii) determining vibration and torque levels along the length of the drill string based on the measured vibration and torque at one or more locations, (iv) determining the remaining life of critical components of the drill string based on the history of the vibration to which the components have been subjected, and (v) determining the optimum drilling parameters that will avoid excessive vibration of the drill string.