# Optimization of the Control Algorithm for an Active Bearing Support
## 2 Natural Frequencies of the Spindle System  
In mechanical structures, such as spindles or electro spindles of machine tools, the basic construction element is the shaft, which, from the point of view of mechanics, should be understood as a rod or beam element. In these types of element, their dynamics are a very important issue. For this reason, each time when designing a spindle, one should remember about vibrations, which, caused by even small forces in resonance conditions, may lead to permanent destruction or damage to the structure. The simplest example of a periodically variable load is the harmonic excitation, which is graphically presented in Figure 2 and can be described by the following formula. \[y(t)=A\sin(\omega t+\varphi_{0})\]  
where:  
\(A\)--vibration amplitude;  
\(\omega\)--circular frequency of vibrations;  
\(\varphi_{0}\)--phase shift angle;  
\(t\)--time. Each spindle system has a specific stiffness, which results from many factors. In this work, the author focuses on the possibilities of changing the stiffness of the spindle system during its operation. The change in stiffness is also related to the change in the value of the system's resonance frequencies. At this frequency, even a small periodic force can cause resonant vibrations of significant amplitude. Therefore, you should avoid working around these frequencies. In dynamic systems with many degrees of freedom, there are many natural frequencies of the system, and resonance may occur for each of them. Resonant vibrations with the largest amplitudes occur at the lowest frequencies, which often occur in the spindle working area. The spindle assembly is a system composed of an elastic beam (spindle) and elastic supports (bearings). There are two types of natural vibration and forms of vibrations accompanying them. These are flexural vibrations of the spindle with small vibration amplitudes in the bearing supports, and spindle rocking, during which the spindle shaft behaves almost like a rigid body, and large amplitudes occur in the bearings. This second type of natural vibration is the main interest of this work, because its frequency and amplitude are determined by the elastic properties of the bearings. Actively changing the stiffness of the bearing supports is the action that can most influence the spindle rocking and the related natural frequencies and forms of vibrations. However, it has a small impact on the spindle's flexural vibrations. To initially determine the frequency and mode of the natural vibrations of the tested spindle, a Dassault System software called Abaqus 6.14 was used. Due to its computational capabilities, this program introduced a simplification consisting in modeling only the spindle shaft with two rings that imitated bearings. These rings were fixed on the outer surface, thus modeling the fixation in the spindle body. The research assumed that the modeled spindle would be an example of a grinding spindle. The model used C3D8R finite elements. The mesh size was set at 0.005 m. During the calculations, the bearings were tensioned with a force of 412 N, which corresponded to the catalog middle preload value. This preload was identical for both bearing supports. For this reason, a replacement ring was mounted on the front end of the spindle, the mass of which corresponded to the approximate mass of the grinding wheel for the grinding spindle. The stiffness of the bearing supports was changed by changing the properties of the material from which the replacement bearing rings were made. To correctly determine the stiffness, a number of model tests were carried out, the aim of which was to determine the value of Young's modulus for replacement rings. The model developed in this way allowed us to initially estimate the level of the three lowest frequencies and forms of natural vibrations. These frequencies were as follows: shaft A--265 Hz, 1147 Hz and 2191 Hz; and shaft B--278 Hz, 979 Hz and 1874 Hz, respectively. The visualization of the results is shown in Figure 3 for shafts A (left column) and B (right column).