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The main challenge in control of modern industrial drive systems with elastic coupling is the ability to fulfill the inherent safety and operational limitations of the drive despite severe torsional vibrations and load disturbances. This study 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 relies on an explicit version of model predictive control (MPC) that is computationally more suited for systems with demanding sampling rates. The resulting explicit MPC controller achieves the same level of performance as the conventional MPC, but requires only a fraction of the real-time computational machinery, thus leading to fast and reliable implementation. The proposed explicit MPC controller is evaluated experimentally on a pilot-scale two-mass drive system demonstrating feasibility and high effectiveness.