We derive a model for a wind turbine tower in the plane of the turbine blades, consisting of a non-uniform SCOLE beam system coupled with a two-mass drive-train model (with gearbox). The control input of this wind turbine tower model is the torque created by the electrical generator. Using the theory of coupled linear systems 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 Xc, denoted by X 1c. We also show that generically, this model is exactly controllable on X1c 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.