**4.1. The optimisation of SHS with regard to temperature**

The temperature dilatation of the spindle can be described by the equation:

$$
\Delta L = \mathcal{A}\_t.L.\Delta t\tag{39}
$$

Radial Ball Bearings with Angular Contact in Machine Tools 71

**Figure 13.** Temperature deformation of bearing arrangement B 7016 C TPAP4UL in "DB"

If the distance between the bearings in the DB arrangement is short (Figure 13a), the dilatations in a radial direction is greater, [18]. The temperature gradient causes the dilatation of the inner bearing rings to be greater than that of the outer rings. Consequently, the original preload increase in temperature will be higher in the bearing node. The elevated temperature will influence the temperature gradient, and the preload value could cause bearing node failure.

The preload change was defined by the change in the distance between the centres of the radii of the rolling raceways:

$$d\_0 = r\_A + r\_I - d\_w \tag{40}$$

The distance *<sup>t</sup> l* at given temperature gradient in accordance with Figure 13 is

70 Performance Evaluation of Bearings

bearing node failure.

The distance *<sup>t</sup>*

radii of the rolling raceways:

**4.1. The optimisation of SHS with regard to temperature** 

The temperature dilatation of the spindle can be described by the equation:

. . *<sup>t</sup> L Lt* 

If the distance between the bearings in the DB arrangement is short (Figure 13a), the dilatations in a radial direction is greater, [18]. The temperature gradient causes the dilatation of the inner bearing rings to be greater than that of the outer rings. Consequently, the original preload increase in temperature will be higher in the bearing node. The elevated temperature will influence the temperature gradient, and the preload value could cause

The preload change was defined by the change in the distance between the centres of the

*l* at given temperature gradient in accordance with Figure 13 is

(39)

<sup>0</sup> *<sup>A</sup> I w l r rd* (40)

$$l\_t = \sqrt{a^2 + b^2} \tag{41}$$

Radial Ball Bearings with Angular Contact in Machine Tools 73

**0**

**2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20**

**Fp =f ( t), Bm =27mm**

**Bm (mm) Fp (N)**

**Bm =f ( t )**

tI = 70 stC, to= 20 stC

Bearings B 7016 CTB t

Bearings arrang.: O

change of temperature in the bearings arrangement system.

**4.2. Recommendation for improvements in construction** 

tA = tI -

**t ( st C)**

**Figure 14.** The inter - dependence of bearing preload change, ideal distance between bearings and

The recommendations from the point of view of temperature optimisation for the DB 24 SHS boring machine are based on the results of the analysis undertaken. From the perspective of temperature, it can be seen that a change in bearing node arrangement to

individual spindle supports from DB to DT would be advantageous, Figure 15.

**Figure 15.** Model of the spindle

No driving force

*Working conditions* 

Results for spindle with arrangement DB - DB.

Lubrication: Plastic grease

Cooling: Good cooling

 Radial load: Fr =1000 N Axial load: Fa = 200 N Desired spindle speed: nOt = 5500 min-1

where

$$a = l\_0.\sin\alpha + \frac{\dot{\lambda}\_l}{2} \cdot \left[ B\_m.\left(t\_I - t\_A\right) + l\_0.\sin\alpha \cdot \left(t\_I + t\_A - 2.t\_0\right) \right] \tag{42}$$

$$b = l\_0 \cos \alpha + \frac{\dot{\lambda}\_l}{2} \left[ D\_m \left( t\_A - t\_I \right) + l\_0 \cos \alpha \right. \left( t\_I + t\_A - 2.t\_0 \right) \right] \tag{43}$$

The magnitude of deformation will be

$$
\Delta \mathcal{S}\_- = l\_0 - l\_t \tag{44}
$$

and preload change in accordance with [18] will be

$$
\Delta F = \Delta \delta \text{ .} \, z^{2/3} \, x\_{\delta}^{2/3} \, . \sin^{8/3} a \tag{45}
$$

where

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

In the twin bearings FAG B 7016 C.TPA.P4.UL with DB arrangement, at a temperature gradient of 10 C, and with bearing distance Bm = 29 mm, the preload will increase by 13,32 N.

Conversely, if the distance of the bearing in O arrangement is long (Figure 13b), the dilatations in the axial direction prevail and cause a decrease in the value of the preload.

In the twin bearings FAG B 7016 C.TPA.P4.UL with DB arrangement, at a temperature gradient of 10C , the preload will be decreased by 5,88 N.

In DB arrangement, the main goal of temperature optimization is dependant on the determination of the optimum distance between the bearings at which a change in the preload at the given temperature gradient would be minimal.

In accordance with Figure 13 the condition

$$I\_0 = I\_t \tag{46}$$

must be satisfied.

By substituting equations (42) and (43) into (46), the optimal bearing separation distance from the point of view of temperature can be deduced from:

$$B\_{mpl} = D\_m \cdot \frac{\cos \alpha}{\sin \alpha} \quad - \frac{l\_0 \cdot \left(t\_I + t\_A - 2.t\_0\right)}{t\_I - t\_A} \cdot \left(\frac{1}{\sin \alpha}\right) \tag{47}$$

Figure 14 shows the change of optimal bearing distance at various values of the temperature gradient for the analysed SBS, Figure 11.

**Figure 14.** The inter - dependence of bearing preload change, ideal distance between bearings and change of temperature in the bearings arrangement system.
