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Introducing sources of axial vibration into an oilwell drillstring has the potential to improve the drilling efficiency. Vibration generator tools, such as drillstring agitators, are under development or in current use to excite the bottom-hole assembly (BHA) axially in order to increase power and weight at the bit, improve the rate of penetration (ROP), reduce drillstring-wellbore friction, and accelerate the cutting removal process. Enhanced drilling under the effect of intentional imposed vibration is called “vibration-assisted rotary drilling” or VARD. While potentially enhancing the drilling process, VARD tools can also excite many unwanted vibration modes of the drillstring. These unwanted vibrations can cause fatigue damage and failure of BHA components such as “measurement while drilling” (MWD) tools, bit and mud motors, and consequently, inefficient drilling. This motivates a study of the complex dynamic behavior of an axially excited drillstring. Transverse vibration is the most destructive type of drillstring vibration, and the coupling between transverse and axial vibration of a drillstring subjected to an applied VARD force is of great interest to the experts in the field. In this study, the coupled axial-transverse vibration behavior of the entire drillstring under the effect of a VARD tool is investigated. A dynamic finite element method (FEM) model of the vertical drillstring assuming a multispan BHA is generated and validated with a coupled nonlinear axial-transverse elastodynamic mathematical model. The effects of mud damping, driving torque, multispan contact and spatially varying axial load are included. Geometry, axial stiffening and Hertzian contact forces are sources of nonlinearity in the model. A mesh sensitivity analysis is conducted to reduce computational time. The accuracy of the retained modes in the analytical equations is verified by extracting the total effective mass derived by the FEM model. There is agreement between the FEM and analytical models for coupled-transverse and axial vibration velocities, displacements, resonance frequencies and contact locations and behavior. While the analytical model has fast running time and symbolic solution, the FEM model enables easy reconfiguration of the drillstring for different boundary conditions, inclusion of additional elements such as shock subs, and changing the number and locations of stabilizers.