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The aim of our research is to develop mathematical models of metal cutting operations, machine tool vibrations and control. The science based digital models allow the virtual design of machine tools; simulation, optimization and online monitoring of machining operations. The model predicts the cutting forces, torque and power consumed in machining parts by considering material properties, cutter geometry, structural flexibilities, and cutting conditions along the tool path. The dynamics of servo drive control systems, and trajectory generation of machine tools are considered. An in-house developed virtual and real time CNC system allows the design and analysis of any five axis machine tool controller. The virtual machining system simulates the process and adjusts the feedrate along the tool path in CAM environment. The system is also used for on-line monitoring and control of machine tool and machining process by communicating with CNC in real time as a digital twin. The presentation will cover the overview of fundamental research we conducted at our laboratory and its application in industry.