**6.2. Dynamic analysis**

The most valuable advantage of this system is the possibility of calculating dynamic stiffness at different revolving frequencies of the spindle. The given mathematical model was verified on a number of spindles with programs which enabled the calculation of natural frequencies (COSMOS).The results were in good compliance [24].

The verified spindle, which complied with research findings, was reduced to a three discrete parts. The dynamic mathematical model described above was used to calculate the natural frequencies and dynamic deflections. Table 3 compares calculated and experimental values.


One of the main requirements in designing new spindle housing systems is the ability of the design to be quickly applied to real world practice. The methodologies of calculation that were created must be verified, and models must be adapted into a suitable user friendly, computerized format. The models must illustrate the real characteristics of a spindle

In this design process, only one variable or parameter was changed and the optimal configuration was identified. The results calculated for a static analysis of the SBL Headstock are presented in Table 2 and Figure 33. The dynamic analysis results are presented in Table 3 and Figure 35. The calculated results were verified with experimental measurements. The difference between measured and calculated values is relatively small.

There is no doubt that the re-design has been a success story, and has proven to be highly effective in the identification of optimal SBS design. More detailed information can be read in [22], [23] and its application can be seen in the machine tools made by TRENS Inc., The SBL Lathe was presented in the Mechanical Engineering Exhibition in Nitra in 2010 and in

**6.3. Conclusion** 

housing system.

**7. Nomenclature** 

N - high-speed ability δ – elastic compression

i – number of bearings α – contact angle D, d – diameter

F – external load P - roller loading

K - stiffness γ – pitch angle O – centre

**INDEXES** 

a – axial direction r radial direction z – built-in state

l – referring to its bearing

0 – roller loaded to the maximum

the EMO Exhibition in Düseldorf in 2011.

E - modulus of elasticity of the material

J – quadratic moments of inertia

n – high spindle revolutions l – distance of curvature centre


The results can be considered as correct, in spite of the relatively large difference in values (28 %) in the first frequency. This is as a result of the fact that the dimensions of the additional rotating parts are not included. If these parts were included in the calculation, the values of the calculated natural frequencies would be smaller.

An example of the graphic output of calculated values is shown in Figure 35, [23]. The chart shows the dynamic deflection of the spindle reduced to three masses. The first two resonant frequencies of the optimized spindle are marked on the chart.

DEFLECTIONS UNDER MASSES Y1=f(f), Y2=f(f), Y3=f(f)

**Figure 35.** Dynamic deflections of the spindle according to research findings [7]
