The essential mathematical model of parametric excitation appeared first in a paper of Mathieu in 1868. It took more than 60 years to construct the corresponding stability chart in the parameter plane of the system stiffness and the amplitude of the parametric excitation by Strutt and van der Pol. The governing equations of the delayed oscillator showed up first in the papers of Minorsky and von Schlippe related to ship stabilization and rolling wheel stability problems, respectively, in 1942. The corresponding stability charts in the parameter plane of the system stiffness and feedback gain appeared first correctly in 1966 only by Hsu and Bhatt in the ASME Journal of Applied Mechanics. Since 1960, when the mathematical model of machine tool vibrations appeared in the works of Tobias, it has been clear that the milling processes are governed by delay differential equations subjected to parametric excitation, but another 40 years were needed to construct the exact 3D stability chart in the space of the system stiffness, feedback gain and excitation amplitude in the ASME Journal of Dynamic Systems, Measurement, and Control in 2003.

The need for the stability chart of the delayed Mathieu equation has been induced by the continuous development of the high-performance milling technology in the 1990s, where the parametric excitation in the delayed oscillatory system is not negligible. The lecture will summarize the route to the intricate stability lobe diagrams, their experimental validation, and the actual results and challenges in the hardware-in-the-loop based emulation of high-speed milling processes.