*2.1.2. Stiffness*

The total static stiffness of machine tools is, in almost all cases, limited by the stiffness of the weakest parts. Amongst all the elements, the Spindle-Bearings System of the machine tool plays the most important role.

From results of structural analyses, the headstock can be considered as the heart of the whole machine tool. The design and quality of the machine tool must respect the quality of the drives and their features.

The headstock (as tool, or workpiece carrier), has a direct influence on the static and dynamic properties of the cutting process. The Spindle-Bearing System's stiffness also influences the final surface quality, profile, and dimensional accuracy of the workpiece.

The problem here is how quickly the headstock stiffness can be calculated with sufficient precision. The headstock stiffness must be calculated according to the deflection at the front end of the spindle, because the deflection at this point directly affects the precision of the finished product. The deflection at the spindle front end is the accumulation of various other, more or less important, partial distortions. The radial headstock stiffness can be calculated as follows:

$$K\_{rc} = \frac{F\_r}{y\_{rc}} \tag{1}$$

The individual headstock parts, (spindle and bearing arrangement), create a serial spring arrangement and it is evident that the resulting stiffness *Krc* is limited by the stiffness of the weakest part. An expert can see which part should be improved, and which partial distortions need to be minimized.

### **2.2. Simplified method of calculation**

52 Performance Evaluation of Bearings

issues.

bearing nodes

> running accuracy durability stiffness

high - speedability temperature

1. Externally - by shortening working time - within a working cycle

**Figure 3.** Modular structure of theoretical research, [5]

clamping elements

spindle noses

and cooling conditions.

plays the most important role.

*2.1.2. Stiffness* 

on tool life and on the dynamic stability of the cutting process.

2. Internally - by reducing machining times (increasing the cutting width) - technological

Matematical models

**THEORETICAL RESEARCH** 

statical models

statical stiffness dyn. stiffness

dynamical models

> natural frequencies spectrum

**MOUNTING ELEMENTS RESEARCH SH SYSTEM RESEARCH**

supporting elements

The philosophy of intelligent manufacturing systems applied to production processes minimise lost time. Further reducing lost time is expensive and has limited effectiveness at current levels of technological development. It has been shown that increased productivity can be achieved for example by changing the cutting speed. However this has a direct effect

The cutting speeds in machining processes depend on the technology applied, the cutting tool, and the workpiece material. The cutting speed also relates directly to the high-speed capability, and average diameter, of the bearings, the so-called factor *N = nmax.dmid..* Thus, from the point of view of the required cutting speed, the most important factor is the

The calculation of the headstock's maximum revolving speed is relatively simple. The highest revolving speed of a bearing node is calculated on the basis of the highest revolving speed of one bearing, multiplied by various coefficients reflecting the influence on the bearings, the bearing arrangement, bearing precision, their preloaded value, and lubrication

The total static stiffness of machine tools is, in almost all cases, limited by the stiffness of the weakest parts. Amongst all the elements, the Spindle-Bearings System of the machine tool

revolving frequency capacity of a spindle which is supported on a bearing system.

The calculation of spindle front end deflection, which takes into consideration all the important parameters, can only be achieved by using powerful computers. The analysis can be carried out by standard or custom software programs.

Calculating the many combinations of SBS arrangements is very demanding on time and money. Undertaking stiffness analysis using standard programs depends on the engineers' experience. The results can be open to questionable even when a suitable mathematical method is used (finite element method, boundary element method, Castilian's theorem, graphic Mohr's method, etc). This is because the headstock box, bearings or bearings nodes are statically indefinite systems which produce a nonlinear deformation of the node when under load.

Special software programmes are very expensive. They are developed using the most up-todate theoretical and practical knowledge. These programs have been developed by research institutions and bearing producers and the possibility of using such programs significantly influences their position on the SBS market. Taking the above into account, engineers would benefit from the existence of a simplified method of static analysis. Such a methodology would enable the engineer at the preliminary design stage to limit the number of possible spindle-bearing variants and determine the direction which would lead to the optimal SBS design, [6].

The main methodological advantage of computer analysis is the possibility of repeating single calculating algorithms in a matrix shape. To this end a special software package, "*Spindle Headstock"* was developed at the Department of Production Engineering in the Faculty of Mechanical Engineering at STU, Bratislava, [7].

The resulting radial deformation, *yrc*, of the front spindle end is shown in Figure 4.

Resulting static distortion of the front-end spindle equals

$$\mathbf{y}\_{rc} = \mathbf{y}\_0 + \mathbf{y}\_1 + \mathbf{y}\_t + \mathbf{y}\_a + \mathbf{y}\_v + \mathbf{y}\_{sb} + \mathbf{y}\_h \tag{2}$$

Radial Ball Bearings with Angular Contact in Machine Tools 55

(7)

4 4 4 4 and

 

(6)

<sup>64</sup> <sup>64</sup> *a aa L LL J Dd J Dd*

The definition of the quantities is shown in Figure 5. The individual headstock parts (spindle, bearing arrangement,) create a serial spring arrangement, and it is evident that the

At the same time, parameters *"Fr"*, *"a"*, *"L"* influence the value of both deflections. The spindle deflection caused by bending moments can be decreased by the following methods:


The resulting static distortion of the spindle front-end can be explicitly described by a multi-

<sup>F</sup> <sup>r</sup> a a a L L L A B, y f [E, F , a, L, J ,, D , d J D , d , K , K ]

There remains one significant problem with the calculation of the bearing nodes, and that is

resulting stiffness *Krc* is limited by the stiffness of the weakest part, [1] [2] and [9].



**Figure 5.** Cross section scheme of the spindle-bearing system



*"da", "dL"*.

parametrical equation in the form of:


that they are statically indeterminate systems.

Our experience has shown that whatever mathematical method and software is used, the spindle distortion caused by *bending moments y0* and by *bearing compliances yl* have the greatest influence on the resulting front end spindle distortion, [6].

**Figure 4.** Factors influencing the resulting deflection

Then

$$\mathbf{y\_{rc}} = \mathbf{y\_0} + \mathbf{y\_1} \tag{3}$$

where the distortion caused by bending moments is as follows:

$$y\_o = \frac{F\_r a^2}{3E} \left[ \frac{a}{J\_a} + \frac{L}{J\_L} \right] \tag{4}$$

and the deflection caused by bearing compliance is as follows:

$$y\_I = \frac{F\_r}{L^2} \left[ \frac{a^2}{K\_B} + \frac{\left(L + a\right)^2}{K\_A} \right] \tag{5}$$

Increasing moments of inertia *"Ja"*, *"JL"* were calculated as follows:

$$J\_a = \frac{\pi}{64} \left[ D\_a^4 - d\_a^4 \right] \quad \text{and} \quad J\_L = \frac{\pi}{64} \left[ D\_L^4 - d\_L^4 \right] \tag{6}$$

The definition of the quantities is shown in Figure 5. The individual headstock parts (spindle, bearing arrangement,) create a serial spring arrangement, and it is evident that the resulting stiffness *Krc* is limited by the stiffness of the weakest part, [1] [2] and [9].

At the same time, parameters *"Fr"*, *"a"*, *"L"* influence the value of both deflections. The spindle deflection caused by bending moments can be decreased by the following methods:


The resulting static distortion of the spindle front-end can be explicitly described by a multiparametrical equation in the form of:

$$\mathbf{y}\_{\text{F}} = \mathbf{f} \begin{bmatrix} \mathbf{E} \ \mathbf{F}\_{\mathbf{r}} \ \mathbf{a} \ \mathbf{L} \ \mathbf{J}\_{\text{a}} \ \mathbf{J}\_{\text{a}} \ \mathbf{J}\_{\text{a}} \end{bmatrix} \begin{Bmatrix} \mathbf{D}\_{\text{a}} \ \mathbf{d}\_{\text{a}} \end{Bmatrix} \mathbf{J}\_{\text{L}} \begin{Bmatrix} \mathbf{D}\_{\text{L}} \ \mathbf{d}\_{\text{L}} \end{Bmatrix} , \begin{Bmatrix} \mathbf{K}\_{\text{A}} \ \mathbf{K}\_{\text{B}} \ \mathbf{J}\_{\text{B}} \end{Bmatrix} \tag{7}$$


54 Performance Evaluation of Bearings

Faculty of Mechanical Engineering at STU, Bratislava, [7].

Resulting static distortion of the front-end spindle equals

**Figure 4.** Factors influencing the resulting deflection

**"ya"**

**Deflection of the spindle from bending moments "yo"** 

where the distortion caused by bending moments is as follows:

and the deflection caused by bearing compliance is as follows:

Increasing moments of inertia *"Ja"*, *"JL"* were calculated as follows:

2

*F a a L <sup>y</sup> EJ J*

Axial force **"yh"**

**Resulting deformation of spindle front end is influenced by "yrc"**

*a L*

<sup>2</sup> <sup>2</sup>

 

*B A*

*F a L a <sup>y</sup> <sup>L</sup> K K*

 

3 *r*

2 *r l*

*o*

Deflection of headstock box **"yv"**

Then

greatest influence on the resulting front end spindle distortion, [6].

The main methodological advantage of computer analysis is the possibility of repeating single calculating algorithms in a matrix shape. To this end a special software package, "*Spindle Headstock"* was developed at the Department of Production Engineering in the

Our experience has shown that whatever mathematical method and software is used, the spindle distortion caused by *bending moments y0* and by *bearing compliances yl* have the

> **Bearing compliance "yl"**

0 l t a v sb h y y y y y y y *rc y* (2)

rc 0 l y y y (3)

Stiffening effect of bearings **"ysb"**

(4)

Drive force

Spindle deflection by transversal forces **"yt"**

(5)

The resulting radial deformation, *yrc*, of the front spindle end is shown in Figure 4.

There remains one significant problem with the calculation of the bearing nodes, and that is that they are statically indeterminate systems.

**Figure 5.** Cross section scheme of the spindle-bearing system
