# Spindle Error Movements and Their Measurement
## 5 Publications Related to the Measurement of Spindle Error Movements
### Measurement Only in the Plane Perpendicular to the Axis  
In a two-part study [39, 40], the authors carried out a comprehensive analysis of issues related to the methods determining the error movement of the spindle in the radial direction (Figure 48).  
A novelty is the statement that, after eliminating the errors of circularity of the reference element (ball) from the data of taken measurements, movement of the centre of the sphere in the X-Y plane can be represented by a complex number dependent on rotation in the complex plane:  
\[v_{p}(\text{C}){=}\;x_{p}(\text{C})+jy_{p}(\text{C}) \tag{6}\]  
where \(x_{p}(\text{C})\;i\,j\!y_{p}(\text{C})\) are the motions of the point P in the X and Y directions, respectively, and \(j\) is an imaginary unit.  
As shown in Figure 48b, the X axis is assigned to the real value (Re) of the complex plane and the Y axis is assigned to its imaginary part. Further, by operating on complex numbers, the authors break down motion in the \(v_{p}(\text{C})\) plane similarly to the standard for synchronous and asynchronous movements.  
Figure 47: Forcing static and dynamic load [38].  
Figure 48: (**a**) Three-dimensional model of a typical setup and (**b**) view in the sensors’ axis plane [39].  
Then, by creating a numerical model of the spindle, they introduce the concepts of the axis of rotation and the axis average line (AVL) [39] (Figure 49).  
At a selected axial position (e.g., on the top surface of the spindle), as shown in Figure 49, the spindle is represented by the outer surface contour, the axis of rotation by the center of rotation R\({}_{1}\), and AVL by the mean point A\({}_{1}\).  
Accordingly, in the plane, we distinguish that the centre of rotation R\({}_{1}\) is the point of rotation of the spindle cross-section and that the centre of rotation of R\({}_{1}\) moves around the static point A\({}_{1}\). Thus, the radial error in the plane is the vector between points A\({}_{1}\) and R\({}_{1}\). After the above, from the positions of the rotation centres in two axial sections, e.g., R\({}_{1}\) and R\({}_{2}\), the tilt error can be calculated.  
Figure 50a shows the trajectory of the three points of a perfect spindle during one complete revolution. You can see that the trajectories of all points are concentric circles. The spindle's radial error movement is always 0, regardless of the eccentricity of the mounted ball standard.  
Experimental tests were also carried out for two types of spindles: a spindle with ball bearings and a spindle with aerostatic bearings.  
Calculations were carried out on the basis of the formulas in part 1 of the article. The test results showed that both types of spindles having a significant radial error was  
Figure 49: The concept of the axis of rotation and axis average line (AVL) [39].  
Figure 50: Comparison between the ideal spindle (**a**) and the spindle with significant radial error movement (**b**) [39].  
dominant. However, it was noticed that the radial error movement calculated according to the applicable standards is much smaller than that calculated with the described method. The authors argue that this is the result of not taking into account the essential (basic) component of error movement in the applicable standards.