A mathematical model for the drilling process is derived and solved numerically as an initial value problem. The equations of motion are nonlinear differential equations for longitudinal, lateral, and rotational motion of the pipe as well as for the rate of flow and pressure of the mud The model comprises a mud (Moineau) motor which rotates the bit relative to the lower end of the pipe. The model accounts for buckling of the pipe doe to excessive torque and longitudinal forces, as well as for the effect of hydraulic pressure on the deformed pipe. Weighs on bit and torque on bit are computed from characteristic curves which are functions of the penetration of the bit into the rock and the angular velocity of the bit. Numerical simulations show self-excited oscillations of the drillstring, including bit take-off from the bottom hole and large amplitudes in the bit's angular velocity. [S0195-0738(00)00602-6].