Certain aspects and features relate to a system that selects an output value for controlling a drilling tool using dynamic force analysis coupled to fluid effects as part of a model that estimates projected torque and drag. A drilling model according to aspects and features of the present disclosure takes into account pipe axial elasticity as it relates to dynamic, time-based, force analysis and couples this relationship with drilling fluid effects over time. In some examples, a system calculates at least one dynamic sideforce and at least one dynamic, hydraulic force for each interval of time. An equilibrium solution for an output value using the dynamic sideforce and dynamic, hydraulic force for each time interval can be applied to the drilling tool for each time interval during drilling operations.