*2.4.2. Criteria for selecting the arrangement of bearings*

The number of spindle bearing systems supported on ball bearings with angular contact increases proportionally with increasing demand on the machine tool. By varying the bearings and their arrangement in the bearing nodes (DB, DF, DT, TBT,TTF, QBC, ..), the value of the contact angle, magnitude of preload, and type of flanges can be optimized to suit the required, resulting stiffness and speed-capability of the spindle-bearing system.

Radial Ball Bearings with Angular Contact in Machine Tools 59

therefore they are used only for the mounting of spindles with the lowest values of coefficient N. Developments in the field of increased speed capability is focused on bearings

GROUP 3: Spindles mounted in bevelled radial bearings with optimized structure (design) and using new composite materials enabling high-speed operation, N = (0.8 – 2.5) x 106,

Spindle mountings using only radial bevelled bearings, (table 1), [11] can be divided into 2



A typical feature of the nodes of spindle bearings is the application of pre-stressing, which provides the stiffness of the nodal point and reduces any skidding of the rollers at high

a. Sprung flange: thermal expansion (dilatation) is eliminated by changing the length of the elastic materials positioned between the flange and the bearings, which ensures

b. Stiff (Rigid) flange: provided by a fixing nut or casing. This design provides better stiffness characteristics. The pre-stress value is changed due to the influence of thermal

c. Controllable flange: axially adjustable (by means of hydraulics), which ensures the

The highest values of the coefficient N can be achieved by using spindles mounted on nodes with a "directional" arrangement of bearings, 1, 2 and 3. When used in conjunction with the controllable flange, the correct types of lubrication and cooling, speeds which are comparable with the maximum revolutions of the bearings themselves can be achieved. Thus they can be applied in high-speed machining [11]. These mounting types, in

For difficult technological operations requiring considerably higher stiffness in the radial and axial directions, nodal points with bearings arranged according to shape, together with

There is negligible use of hybrids of the basic types of mounting (mounting 5), as shown in table 1. In such cases one nodal point has bearings arranged according to shape, while the other has directionally arranged bearings, (Figure 2). The pre-stressing in the front nodal

point is ensured by a stiff flange, and in the rear nodal point by a sprung flange.

orientation of contact angles in each nodal point 1, 2, 3, and 7, table 1.

are arranged in "O" or (X) shape, in combination with "T".

Pre-stressing can be achieved through three flange design principles:

required pre-stressing for different operational conditions.

combination with the sprung support, are mostly used for grinding.

minimum change in the pre-stress value.

with point contacts, as these have better friction characteristics.

which is typical for high-speed machining.

basic types:

revolutions.

dilatation.

fixed supports are typical.

In order to assess the maximum permissible speed of different types of spindle rotations, a parameter for the so-called high-speed characteristics has been introduced: *N = nmax.dmid*, where "*nmax"* denotes the maximum spindle revolutions and "*dmid"* the medial diameter of the bearings. Following this parameter, roller bearings of machine tool spindles can be divided into 3 basic groups, [10]:

**Figure 7.** Arrangements in nodal points

GROUP 1: N = (0,1 – 0,5) .106: Headstocks of heavy duty machines for turning, milling and drilling operations. In these machines the spindles are predominantly mounted using double-row roller bearings in combination with axial ball bearings, or tapered bearings. We can assume that the linearization of the deformation curve in roller bearings is sufficiently accurate, which simplifies the calculation of the radial stiffness of the nodal point, [2]. These *mountings* offer high stiffness and load-bearing capacity and quiet operation.

GROUP 2: N = (0.4 – 1) .106 is characterized by bearings of medium size and are found in smaller NC and CNC turning, milling, drilling and grinding machine tools. The maximum possible speed in bearings with linear joints is limited by the heat produced in the head, and therefore they are used only for the mounting of spindles with the lowest values of coefficient N. Developments in the field of increased speed capability is focused on bearings with point contacts, as these have better friction characteristics.

GROUP 3: Spindles mounted in bevelled radial bearings with optimized structure (design) and using new composite materials enabling high-speed operation, N = (0.8 – 2.5) x 106, which is typical for high-speed machining.

Spindle mountings using only radial bevelled bearings, (table 1), [11] can be divided into 2 basic types:


A typical feature of the nodes of spindle bearings is the application of pre-stressing, which provides the stiffness of the nodal point and reduces any skidding of the rollers at high revolutions.

Pre-stressing can be achieved through three flange design principles:

58 Performance Evaluation of Bearings

divided into 3 basic groups, [10]:

**Figure 7.** Arrangements in nodal points

*2.4.2. Criteria for selecting the arrangement of bearings* 

The number of spindle bearing systems supported on ball bearings with angular contact increases proportionally with increasing demand on the machine tool. By varying the bearings and their arrangement in the bearing nodes (DB, DF, DT, TBT,TTF, QBC, ..), the value of the contact angle, magnitude of preload, and type of flanges can be optimized to suit the required, resulting stiffness and speed-capability of the spindle-bearing system.

In order to assess the maximum permissible speed of different types of spindle rotations, a parameter for the so-called high-speed characteristics has been introduced: *N = nmax.dmid*, where "*nmax"* denotes the maximum spindle revolutions and "*dmid"* the medial diameter of the bearings. Following this parameter, roller bearings of machine tool spindles can be

GROUP 1: N = (0,1 – 0,5) .106: Headstocks of heavy duty machines for turning, milling and drilling operations. In these machines the spindles are predominantly mounted using double-row roller bearings in combination with axial ball bearings, or tapered bearings. We can assume that the linearization of the deformation curve in roller bearings is sufficiently accurate, which simplifies the calculation of the radial stiffness of the nodal point, [2]. These

GROUP 2: N = (0.4 – 1) .106 is characterized by bearings of medium size and are found in smaller NC and CNC turning, milling, drilling and grinding machine tools. The maximum possible speed in bearings with linear joints is limited by the heat produced in the head, and

*mountings* offer high stiffness and load-bearing capacity and quiet operation.


The highest values of the coefficient N can be achieved by using spindles mounted on nodes with a "directional" arrangement of bearings, 1, 2 and 3. When used in conjunction with the controllable flange, the correct types of lubrication and cooling, speeds which are comparable with the maximum revolutions of the bearings themselves can be achieved. Thus they can be applied in high-speed machining [11]. These mounting types, in combination with the sprung support, are mostly used for grinding.

For difficult technological operations requiring considerably higher stiffness in the radial and axial directions, nodal points with bearings arranged according to shape, together with fixed supports are typical.

There is negligible use of hybrids of the basic types of mounting (mounting 5), as shown in table 1. In such cases one nodal point has bearings arranged according to shape, while the other has directionally arranged bearings, (Figure 2). The pre-stressing in the front nodal point is ensured by a stiff flange, and in the rear nodal point by a sprung flange.


Radial Ball Bearings with Angular Contact in Machine Tools 61

(11)

(12)

(13)

The stiffness of the bearing arrangement *(KA, KB)* is the specific parameter which influences the consequent spindle distortion. We have developed a simplified mathematical model for

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

> 3/2 P k .

a. the bearings in the nodal points are of the same type and dimensions, with precise

b. the value of the contact angle is the same for all directionally-arranged bearings in the

The calculation of the stiffness of a nodal point is based on the stiffness of the bearing itself

*d* 

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

*dF d*

*d d*

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],

0

1

*r*

*K*

1

*r*

*<sup>z</sup> <sup>r</sup> r jj j <sup>F</sup> FP j <sup>i</sup>*

*K*

1

is the spacing angle of the balls.

1

0 *r*

*r d F*

1 0

0 0 . *<sup>r</sup>*

<sup>1</sup>

.cos .cos .

(14)

 

*r*

nodal point, which delivers equal distribution of strain on these bearings c. radial load is equally distributed onto all the bearings in the nodal point

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

*2.4.3. Stiffness* 

calculating radial and axial stiffness, [11], [12].

**3.1. Assumptions of solution** 

geometric dimensions

[15], which is defined as:

form:

where <sup>360</sup>

*z*

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

**Table 1.** Type of SBS using radial ball bearings with angular contact [11]

### *2.4.3. Stiffness*

60 Performance Evaluation of Bearings

CONFIGURATION *N= nmax.dmid. 106*

**node Forward bearing node** 

t1=1, t2=0 t1=0, t2=1

t1=1, t2=0 t1=0, t2=2

t1=2, t2=0 t1=0, t2=2

t1=1, t2=1 t1=1, t2=1

t1=1, t2=0 t1=1, t2=2

t1=1, t2=1 t1=1, t2=2

t1=2, t2=0 t1=3, t2=0

**Table 1.** Type of SBS using radial ball bearings with angular contact [11]

*[mm.min-1]* 

1,2 -2,5

0,8 – 1,6

0,8 -1,4

0,6 – 1

0,4 – 0,9

0,3 – 0,6


> - suitable for extremely short spindles - medium axial loads



loads



0,5 – 0,9 - medium axial



> - grinding internal holes - milling - drilling




> - milling - boring

Characteristic Use **Rear bearing** 

Seq. No.

1.

2.

3.

4.

5.

6.

7.

The stiffness of the bearing arrangement *(KA, KB)* is the specific parameter which influences the consequent spindle distortion. We have developed a simplified mathematical model for calculating radial and axial stiffness, [11], [12].
