We derive a model for a wind turbine tower in the plane of the turbine blades, which comprises an Euler-Bernoulli beam coupled with a nacelle (rigid body) and a two-mass drive-train model (with gearbox). This model has two possible control inputs: the torque created by the electrical generator and the force created by an electrically driven mass located in the nacelle. First, we consider the case of only torque control and a possibly non-uniform tower. Using the theory of coupled linear systems (one infinite dimensional and one finite dimensional) developed by us recently, we show that this wind turbine tower model is well-posed and regular on either the energy state space X-c or the domain of its generator on X-c, denoted by X-1(c). We also show that generically, this model is exactly controllable on X-1(c) in arbitrarily short time. More precisely, for every T >0, we show that if we vary a certain parameter in the model, then exact controllability in time T holds for all except three values of the parameter. In the case of using both force and torque control, we derive similar well-posedness, regularity and generic exact controllability results on a state space that is larger than X-1(c) but smaller than X-c. In this second case, we assume that the tower is uniform.