The main challenge in control of modern industrial drive systems with elastic coupling is the ability to fulfill the inherent safety and physical constraints of the drive for a large variety of operating conditions and load disturbances. This paper presents a methodology for handling drive's input and shaft torque constraints in an optimal and effective way while guaranteeing desired speed response characteristics. The presented strategy is based on an explicit form of Model Predictive Control (MPC) which is computationally more suited for systems with demanding sampling rates. This paper demonstrates that the classical cascade control scheme with a MPC speed controller is unable to ensure the fulfillment of the drive's torque constraints leading to significant loss in the closed-loop performance. We propose to incorporate the inner torque loop control synthesis problem into the design of the outer speed controller thus yielding a single-loop MPC control law. The proposed explicit MPC controller is evaluated through simulation demonstrating feasibility and high effectiveness.