# Optimization of the Control Algorithm for an Active Bearing Support
## 1 Introduction  
The spindle is the basic working element of the machine tool, on which the accuracy of the entire machine tool largely depends. In order to perform its functions properly, it must have the highest possible stiffness [1]. This depends on the dimensions of the spindle, the spacing of the supports, their number and the type of bearings used. In addition, efforts should be made to limit the spindle tip overhang as much as possible, which can reduce the radial displacements resulting from the applied external forces. Modern spindle systems have a very complex structure, a number of sensors and complex lubrication and/or cooling systems. In high-precision machine tools, systems providing cooling through the spindle are standard, which allows cooling and lubrication to be delivered directly to the cutting zone. If for some reasons (usually structural) the tip overhang must be significant, special attention should be paid to the spacing of the spindle bearing supports and their radial and axial stiffness. The spacing of the supports clearly affects the level of the lowest (so-called translational) natural frequencies of the spindle. The greater the overhang of the spindle tip, the greater its susceptibility to vibration. This type of regularity occurs outside machine tool spindles, e.g., in industrial robots, where, however, it is possible to use special dampers that can reduce the vibration amplitudes of the robot arms [2]. The constant increase in spindle speeds results from the fact that electrospindles are now widely used in main drives in precision and super-precision machine tools. Their bearing method is based almost exclusively on rolling bearings. Spindle systems of high-speed machine tools, due to their characteristics (the need to transfer forces both radially and axially), use angular contact ball bearings. This is due to the low power losses for these types of bearings, their quiet operation, high failure-free operation and simplicity of design. Of course, there are spindle systems of machine tools equipped with other typesof bearings, e.g., sliding bearings [3], but these are very old structures that are no longer applicable today, or prototype solutions used for experimental research, not in industrial production. In the case of high-speed cutting (HSC) machines, you can find, for example, solutions based on aerostatic bearings, the main disadvantage of which, however, is their very low stiffness. Modern development trends lead to maximizing the accuracy of manufactured products while maintaining high machining efficiency [4]. These types of activities are closely related to the design of machine tool spindles in such a way that they can meet modern market requirements [5]. A very important element of the operation of a machine tool spindle is its stiffness and resistance to vibrations [6], which are highly unfavorable, especially when using machining machines for finishing with an accuracy of less than 1 \(\upmu\)m. This level of accuracy is only achieved by machines equipped with high-precision electrospindles. They are used in the support systems of machine tools such as milling machines, grinders or machining centers. In this type of machine, vibration resistance is very important [7] from the point of view of the stability of the machining process and is directly related to the stiffness of the spindle and the type of machining being carried out at a given moment. In addition to the general advantages of rolling bearings (quiet running, compact design, high failure-free operation, maintenance-free operation), the choice of angular contact bearings for machine tool spindles is based on their ability to transfer loads both axially and radially. Stiffness in the selected direction is directly related to the so-called bearing operating angle (15-40 degrees) and is defined as the relative displacement (in \(\upmu\)m) of the bearing rings as a result of the external force.