**3. The calculation of radial stiffness of nodal points**

### **3.1. Assumptions of solution**

According to the Hertz assumptions [13], [14], there is a dependence between load "*P*" and deformation "*δ*" at the contact point of the ball with the plane, given by the relationship

$$\mathbf{P} = \mathbf{k}\_{\delta}. \delta^{3/2} \tag{11}$$


### **3.2. Stiffness of nodal points with directionally-arranged bearings**

The calculation of the stiffness of a nodal point is based on the stiffness of the bearing itself [15], which is defined as:

$$K\_{r1} = \frac{d}{d} \frac{F\_{r1}}{\mathcal{S}\_{r0}} \tag{12}$$

As radial displacement r0 is a function of contact deformation 0 of the ball with the highest load [13], the equation for calculating the stiffness of bevelled radial bearings will have the form:

$$K\_{r1} = \frac{d}{d} \frac{F\_{r1}}{\delta\_0} \frac{d}{d} \frac{\delta\_0}{\delta\_{r0}} \tag{13}$$

When calculating stiffness, the distribution of load among the rollers must be determined, and the dependence between the load on the top ball and external load must be found. The distribution of load in the bearing can be derived from the balance under static conditions [14],

$$F\_{r1} = \frac{F\_{r1}}{i} = \sum\_{j=0}^{z} P\_j \cdot \cos\left(\alpha\_j\right) \cdot \cos\left(j\cdot\gamma\right) \tag{14}$$

where <sup>360</sup> *z* is the spacing angle of the balls.

The values of contact deformations *<sup>j</sup>* and angles *<sup>j</sup>* differ from each other around the circumference of the bearing and can be expressed as follows, (Figure 8).

$$\mathcal{S}\_{j} = l\_{r\dot{j}} - l\_p = \sqrt{\left[l\_z \sin\left(\alpha\_z\right) + \delta\_p\right]^2 + \left[l\_z \cos\left(\alpha\_z\right) + \delta\_{r0} \cos\left(\dot{j}\gamma\right)\right]^2} - l\_p \tag{15}$$

$$\cos\left(\alpha\_{j}\right) = \frac{l\_{z}\cos\left(\alpha\_{z}\right) + \delta\_{r0}\cos\left(j.\gamma\right)}{\sqrt{\left[l\_{z}\sin\left(\alpha\_{z}\right) + \delta\_{p}\right]^{2} + \left[l\_{z}\cos\left(\alpha\_{z}\right) + \delta\_{r0}\cos\left(j.\gamma\right)\right]^{2}}}\tag{16}$$

By loading the pre-stressed bearing with a radial force, the distance, OAOip, between the centre of the balls is constant, (Figure 8 b, c).

$$l\_p.\sin\left(\alpha\_p\right) = l\_{r\dot{\jmath}}.\sin\left(\alpha\_{r\dot{\jmath}}\right) = konst.\tag{17}$$

The dependence between the deformation of the jth ball and the top ball can be determined by the relation

$$\mathcal{S}\_{\dot{j}} = \mathcal{S}\_0 \cos \left( \mathbf{j}.\mathcal{Y} \right) \tag{18}$$

Radial Ball Bearings with Angular Contact in Machine Tools 63

**Figure 8.** Detailed bearing scheme, a – unloaded, b – pre-stressed, c – radial loaded

 

<sup>0</sup>

2/3 1/3 2 2

 = .cos .cos . cos cos j. *j j*

After inserting equations (25) and (19) into equation (13) we get the resulting relation for the

 

*z z ro z zp <sup>d</sup> j j <sup>j</sup> <sup>d</sup> <sup>j</sup>*

 

1 cos cos . 1 sin

*<sup>d</sup> <sup>j</sup> <sup>d</sup>*

sin 3 = . . . .cos . .cos .

*<sup>z</sup> <sup>j</sup> j jj j rj Ki k P <sup>P</sup> <sup>j</sup> <sup>l</sup>*

stiffness of a pre-stressed nodal point with directionally-arranged bearings.

1

 <sup>2</sup> <sup>2</sup> <sup>1</sup> 2 1 cos cos . cos . . cos cos .

 

 

 2

(26)

(25)

*j*

  (24)

 

is calculated from equation (15)

*j z z ro*

by inserting equations (24) and (22) into equation (23)

0

*r*

0

2

Where

0 *j r*

2

r

*ro*

*d d*

By derivation of equation (14) we get

$$\frac{d}{d\ \delta\_0} \stackrel{F\_{r1}}{=} \stackrel{z}{\text{i}} \sum\_{j=0}^{z} \left[ \frac{d}{d\ \delta\_j} \text{.} \cos(\alpha\_j) - P\_j \sin(\alpha\_j) \frac{d}{d\ \delta\_j} \frac{\alpha\_j}{\delta\_j} \right] \frac{d}{d\ \delta\_0} \stackrel{a}{\text{.}} \cos(j\ \varphi) \tag{19}$$

The unknown derivatives in equation (19) can be calculated by changing the relations (11), (17), (18).

$$\frac{dP\_{\dot{j}}}{d\delta\dot{\delta}} = \frac{3}{2} k\_{\delta}^{2/3} P\_{\dot{j}}^{1/3} \tag{20}$$

$$\frac{d\alpha\_j}{d\delta\_j} = -\frac{\text{tg}\left(a\_j\right)}{l\_{rj}}\tag{21}$$

$$\frac{d\delta\_j}{d\delta\_0} = \cos(j\omega\gamma) \tag{22}$$

The interdependence of the contact deformation and radial displacement, Figure 8, can be determined from the relation

$$\frac{d}{d\,} \frac{\delta\_0}{\delta\_{r0}} = \left(\frac{d\,\,\delta\_j}{d\,\,\delta\_0}\right)^{-1} \cdot \frac{d\,\,\delta\_j}{d\,\,\,\delta\_{r0}}\tag{23}$$

**Figure 8.** Detailed bearing scheme, a – unloaded, b – pre-stressed, c – radial loaded

Where 0 *j r d d* is calculated from equation (15)

62 Performance Evaluation of Bearings

by the relation

(17), (18).

The values of contact deformations *<sup>j</sup>*

cos

By derivation of equation (14) we get

determined from the relation

*j*

centre of the balls is constant, (Figure 8 b, c).

circumference of the bearing and can be expressed as follows, (Figure 8).

 *ll l* 

*l l* 

> 

and angles *<sup>j</sup>*

<sup>0</sup> .sin .cos .cos . *j rj p z z p z z r <sup>p</sup>*

*z zr*

*z zp z zr*

By loading the pre-stressed bearing with a radial force, the distance, OAOip, between the

.sin .sin . *p p rj rj*

The dependence between the deformation of the jth ball and the top ball can be determined

<sup>0</sup>.cos . *<sup>j</sup>*

<sup>1</sup>

3 2/3 1/3

= . .cos .sin . . .cos .

0 0 0

*iP j d d d d*

 

2 *j*

*d tg d l*

cos . *<sup>j</sup> <sup>d</sup> <sup>j</sup> <sup>d</sup>* 

The interdependence of the contact deformation and radial displacement, Figure 8, can be

*d d d d dd*

 

*<sup>j</sup> <sup>j</sup> dP k P*

*<sup>j</sup> <sup>j</sup> r j j*

1

 

00 0 =. *<sup>j</sup> <sup>j</sup> r r*

The unknown derivatives in equation (19) can be calculated by changing the relations (11),

*<sup>z</sup> <sup>j</sup> j j <sup>r</sup> jj j j j j*

*d F d P d d*

*d*

0

0

   

(19)

 

 

(20)

(21)

(22)

(23)

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

 0

 

.cos .cos .

*l j*

.sin .cos .cos .

*ll j*

 

*<sup>l</sup> j l* (15)

2 2 0

 

differ from each other around the

(16)

 

> 

*konst* (17)

*<sup>j</sup>* (18)

 

$$\frac{d\delta\_j}{d\delta\_{vo}} = \frac{1}{2} \cdot \frac{2\left(1\_z \cos a\_z + \delta\_{vo} \cos(j.\gamma)\right) \cos(j.\gamma)}{\sqrt{\left(1\_z \cos a\_z + \delta\_{vo} \cos(j.\gamma)\right)^2 + \left(1\_z \sin a\_z + \delta\_p\right)^2}} = \cos a\_j \cos(j.\gamma) \tag{24}$$

by inserting equations (24) and (22) into equation (23)

$$\frac{d}{d\,} \frac{\delta\_0}{\delta\_{r0}} = \frac{1}{\cos(\text{j.}\,\text{\textdegree})} . \cos(\text{a}\_{\text{j}}) . \cos(\text{j.}\,\text{\textdegree}) = \cos(\text{a}\_{\text{j}}) \tag{25}$$

After inserting equations (25) and (19) into equation (13) we get the resulting relation for the stiffness of a pre-stressed nodal point with directionally-arranged bearings.

$$K\_{\mathbf{r}} = \mathbf{i}\_r \sum\_{j=0}^{z} \left[ \frac{3}{2} \boldsymbol{k}\_{\boldsymbol{s}}^{2/3} \cdot \boldsymbol{P}\_{\boldsymbol{j}}^{1/3} \cdot \cos^2 \left( \alpha\_{\boldsymbol{j}} \right) + \boldsymbol{P}\_{\boldsymbol{j}} \cdot \frac{\sin^2 \left( \alpha\_{\boldsymbol{j}} \right)}{\boldsymbol{l}\_{\boldsymbol{r}\boldsymbol{j}}} \right] \cdot \cos^2 \left( \boldsymbol{j} \cdot \boldsymbol{\gamma} \right) \tag{26}$$

#### **3.3. Stiffness of nodal point with bearings arranged according to shape**

When calculating the nodal point with bearings arranged according to shape, we divide the nodal point into part "1" and part "2" (Table 1), with the *same* orientation of contact angles in nodes with directionally-arranged bearings, and the stiffness of the parts is calculated as follows:

$$\begin{aligned} \mathop{K}\_{\text{r}1} &= \overset{z}{\underset{j=0}{\rightleftharpoons}} \left[ \frac{3}{2} \, k\_{\,\,s}^{2/3} \, P\_{j}^{1/3} \cdot \cos^{2} \left( a\_{1j} \right) + P\_{j} \cdot \frac{\sin^{2} \left( a\_{1j} \right)}{l\_{r1j}} \right] \cdot \cos^{2} \left( j \, \gamma \right) & \quad \text{(a)}\\ \mathop{K}\_{\text{r}2} &= \overset{z}{\underset{j=0}{\rightleftharpoons}} \left[ \frac{3}{2} \, k\_{\,\,s}^{2/3} \, P\_{j}^{1/3} \cdot \cos^{2} \left( a\_{2j} \right) + P\_{j} \cdot \frac{\sin^{2} \left( a\_{2j} \right)}{l\_{r2j}} \right] \cdot \cos^{2} \left( j \, \gamma \right) & \quad \text{(b)} \end{aligned} \tag{27}$$

For example in Figure 9 the total numbers of bearings in the front node SBS is 5: *i1* = 3, *i2* = 2, *contact angles α1* = *α2* = 25.

We determine the total stiffness of the nodal point by the addition of both parts of the node with the equation:

$$K\_{\rm r} = K\_{\rm r1} + K\_{\rm r2} \tag{28}$$

Radial Ball Bearings with Angular Contact in Machine Tools 65

**Figure 9.** Horizontal machining centre, Thyssen-Hüller Hille GmbH , Germany; Work nodal – 3x71914

If the magnitude of the spindle bearing contact angles is not greater than 26 degrees, then

Taking these assumptions into consideration, we obtain the relationship for the approximate calculation of the radial stiffness of a bearing angle with directionally placed bearings in the

> -3 2 2/3 2/3 2/3 1/3 r 1/3 3.10 cos = . . . .F . 4 sin

> >

r 1 1/3 2/3 2 1/3

4 sin .cos .sin *<sup>p</sup>*

<sup>5</sup> 10 . 1,25 . *<sup>w</sup> c d*

The pre-stressing value "Fp" can be calculated according to the standard, STN 02 46 15. Some foreign manufacturers (for example, SKF, FAG, SNFA ...) publish this value in their catalogues. The number of balls "z" and their diameters "dw" of some types of bearings are

> 2 22 1 5 3 3 3 3 33 3 3 2 1 1 1 2 5

 

3.10 sin sin 1

3.10 cos .cos .sin


*i*

*i*

 

11 1 2

 

(33)

2 5

3 3 1 2

*i*

sin

 

(31)

(32)

(34)

the value of the second expression in equations (27a) and (27b) is negligible.

*K zki <sup>p</sup>* 

and with bearings arranged according to shape in the form:

where the approximate value of the deformation constant is

2

*a p*

*K zki*

dw – is the diameter of the balls.

quoted in the literature, e.g. [16].

= . . . .F . . 1


*<sup>i</sup> K zkiF*

ACGB/P4 - 2x71914 ACGB/P4, Opposite side– 6011-2Z

form:

In order to optimize the stiffness and load-bearing capacity for specified technological conditions, the manufacturers of machine tools have come out with a new, non-traditional solution for nodal points. By diminishing the contact angle of the bearing in Part 2, the axial stiffness of the nodal point is partially decreased, but at the same time, the value of the radial stiffness and boundary axial load is increased.

#### **3.4. Approximate calculation of stiffness**

When evaluating the overall stiffness of a spindle, the designer must take into account the approximate calculation of the stiffness of the nodal points.

If all the balls are loaded, and there are more than 2 per bearing [14], the following equation can be applied:

$$\sum\_{j=0}^{z} \cos^2 \left( j \lrcorner \gamma \right) = \frac{z}{2} \tag{29}$$

If the bearing angle is loaded only in an axial direction by the pre-stressing force, then the load on the rollers is constant around the whole circumference and can be expressed, for the particular parts of the nodal point [11] in the form

$$P\_{1j} = \frac{F\_p}{i\_1 z. \sin\left(\alpha\_{p1}\right)}; \qquad \qquad \qquad P\_{2j} = \frac{F\_p}{i\_2 z. \sin\left(\alpha\_{p2}\right)}\tag{30}$$

**Figure 9.** Horizontal machining centre, Thyssen-Hüller Hille GmbH , Germany; Work nodal – 3x71914 ACGB/P4 - 2x71914 ACGB/P4, Opposite side– 6011-2Z

If the magnitude of the spindle bearing contact angles is not greater than 26 degrees, then the value of the second expression in equations (27a) and (27b) is negligible.

Taking these assumptions into consideration, we obtain the relationship for the approximate calculation of the radial stiffness of a bearing angle with directionally placed bearings in the form:

$$K\_r = \frac{3.10^{-3}}{4} . z^{2/3} . k\_{\phantom{\alpha}}^{2/3} . i^{2/3} . F\_p^{1/3} . \frac{\cos^2 \left(\alpha\right)}{\sin^{1/3} \left(\alpha\right)}\tag{31}$$

and with bearings arranged according to shape in the form:

$$K\_r = \frac{3.10^{-3}}{4} z^{2/3} \, k\_s^{2/3} \, i\_1^{2/3} \, \mathrm{F}\_p^{1/3} \cdot \frac{\cos^2 \left(a\_1\right)}{\sin^{1/3} \left(a\_1\right)} \left[1 + \frac{i\_2^{2/3} \cdot \cos^2 \left(a\_2\right) \cdot \sin^{1/3} \left(a\_1\right)}{i\_1^{2/3} \cdot \cos^2 \left(a\_1\right) \cdot \sin^{1/3} \left(a\_2\right)}\right] \tag{32}$$

where the approximate value of the deformation constant is

$$c\_{\mathcal{S}\_{\mathcal{S}}} = 10^5 \sqrt{1 \, 25 \, . \, d\_w} \tag{33}$$

dw – is the diameter of the balls.

64 Performance Evaluation of Bearings

follows:

**3.3. Stiffness of nodal point with bearings arranged according to shape** 

When calculating the nodal point with bearings arranged according to shape, we divide the nodal point into part "1" and part "2" (Table 1), with the *same* orientation of contact angles in nodes with directionally-arranged bearings, and the stiffness of the parts is calculated as

1 2/3 1/3 2 2

sin <sup>3</sup> = . . . .cos . .cos . (a) <sup>2</sup>

sin <sup>3</sup> = . . . .cos . .cos . (b) <sup>2</sup>

For example in Figure 9 the total numbers of bearings in the front node SBS is 5: *i1* = 3, *i2* = 2,

We determine the total stiffness of the nodal point by the addition of both parts of the node

In order to optimize the stiffness and load-bearing capacity for specified technological conditions, the manufacturers of machine tools have come out with a new, non-traditional solution for nodal points. By diminishing the contact angle of the bearing in Part 2, the axial stiffness of the nodal point is partially decreased, but at the same time, the value of the

When evaluating the overall stiffness of a spindle, the designer must take into account the

If all the balls are loaded, and there are more than 2 per bearing [14], the following equation

<sup>2</sup>

If the bearing angle is loaded only in an axial direction by the pre-stressing force, then the load on the rollers is constant around the whole circumference and can be expressed, for the

> 1 2 1 1 2 2

. .sin . .sin *p p*

*F F*

*<sup>z</sup> <sup>j</sup>* 

2

*p p*

cos .

0

;

*j j*

*P P*

*j*

*z*

2 2/3 1/3 2 2

0 1

*<sup>z</sup> <sup>j</sup> j jj j r j*

*Ki k P <sup>P</sup> <sup>j</sup> <sup>l</sup>*

 

0 2

*<sup>z</sup> <sup>j</sup> j jj j r j*

*Ki k P <sup>P</sup> <sup>j</sup> <sup>l</sup>*

r1 1 1

*contact angles α1* = *α2* = 25.

with the equation:

can be applied:

r2 2 2

radial stiffness and boundary axial load is increased.

approximate calculation of the stiffness of the nodal points.

**3.4. Approximate calculation of stiffness** 

particular parts of the nodal point [11] in the form

*i z*

 

> 

*KK K* r r1 r2 (28)

(29)

 

*i z* (30)

(27)

2

2

The pre-stressing value "Fp" can be calculated according to the standard, STN 02 46 15. Some foreign manufacturers (for example, SKF, FAG, SNFA ...) publish this value in their catalogues. The number of balls "z" and their diameters "dw" of some types of bearings are quoted in the literature, e.g. [16].


$$K\_a = \frac{3.10^{-3}}{2} z^{\frac{2}{3}} \cdot k\_{\delta}^{\frac{2}{3}} \cdot i\_1^{\frac{2}{3}} \cdot F\_p^{\frac{1}{3}} \cdot \sin^3 a\_1 \left| 1 + \frac{i\_2^{\frac{2}{3}} \cdot \sin^{\frac{5}{3}} a\_1}{i\_1^{\frac{2}{3}} \cdot \sin^3 a\_2} \right| \tag{34}$$

and substituting the equation in brackets

$$T\_1 = 1 + \frac{i\_2^{\frac{2}{3}} \cos^2 \alpha\_2 \sin^{1/3} \alpha\_1}{i\_1^{\frac{2}{3}} \cos^2 \alpha\_1 \sin^{1/3} \alpha\_2} \text{in equation} \tag{35}$$

Radial Ball Bearings with Angular Contact in Machine Tools 67

weight will also be decreased, and this fact will allow an increase in the maximum

 The number of bearings in bearing arrangement "i" is the significant factor which can favourably influence stiffness. But the increased number of bearings will reduce the maximum revolving frequency and therefore it is possible to use this solution only for

 The preload has a relatively small effect on the stiffness of bearing arrangements. The preload real value also depends on the type of flange used. When fixed flanges are used the preload value can exceed the nominal value by several times. This will cause excessive preload values which produce heat and the bearing arrangement will break

 The contact angle *"α"* has a significant influence on the variation of the stiffness of the bearing arrangement. When the value of the contact angle is increased, the radial stiffness and maximum revolving speed of the bearing arrangement is also decreased. On the other hand, the axial stiffness of the bearing arrangement will be significantly

In addition to the bearing arrangements, the temperature properties of the bearing supporting node have an increasingly greater significance on the high-speed capability of the bearing. The main goal of this section is to show the SHS design under real operating conditions, taking into consideration the temperature-related behaviour of the spindle and

The value of the changes in SHS temperature depends on the temperature gradient, the type of bearing arrangement (DB, DF, DT, …), the contact angle of the bearing, and the distance

The stiffness of the given example was analysed using the application software "*Spindle* 

The analysis identified the optimal stiffness, which was then applied to the headstock of the

Bearings: FAG B 7016 C.TPA.P4.UL in O arrangement

= 1...1,5 m

DB 24 fy. Ex-Cell-O GmbH., Eislinger precision boring machine, Figure 11, [18]

The headstock used for analysis had the following parameters:

Maximum speed: nc max = 5500 min-1

Output power: P = 3 kW

Shape inaccuracy at boring Es 1,5 m

Surface roughness: Et

Bearing lubrication: grease

**4. Optimization of the spindle-bearing system in relation to** 

revolving speed.

increased.

**temperature** 

bearing nodes.

low speed spindle-bearing systems.

down much sooner than expected.

between the bearings arranged in the node.

*Headstock"* [3], developed in our department.

and

$$T\_2 = 1 + \frac{i^{\frac{2}{3}} \sin^{\frac{5}{3}} a\_2}{i^{\frac{2}{3}} \sin^{\frac{5}{3}} a\_1} \text{in equation} \tag{36}$$

the dependence between the axial and radial stiffness can be expressed by the relation

$$K\_r = \frac{K\_a}{2} \cdot \frac{1}{\text{tg}\,2\alpha\text{1}} \cdot \frac{T\_2}{T\_1} \tag{37}$$

When 1 2 in a nodal point with bearings arranged according to shape, or i = 0 in nodal points with bearings arranged according to direction, the quotient of the constants T1, T2 will be equal to 1 and the relation (37) will be simplified. Thus

$$K\_r = \frac{K\_a}{2} \cdot \frac{1}{\text{tg}\,2\alpha} \tag{38}$$

Taking equations (32) and (34) into consideration, it is evident that the stiffness of the bearing arrangement depends on the number of bearings (*i1* and *i2*) in the arrangement, the dimensions of the bearings (*z1, dw1* and *z2, dw2*), the contact angle (*α1* and *α2*) and the preload value *Fp*.
