The stiffness of machine tool spindle bearings is an important performance indicator. The stiffness is not only related to the load and speed, but also to the frictional heat and pre tightening method. Stiffness calculation is also the basis for analyzing the dynamic characteristics of spindle units.

1、 The influence of pre tightening method and rotational speed

Under constant pressure preloading, the radial stiffness of the bearing slightly increases with the increase of rotational speed, while the axial and angular stiffness rapidly decrease. Under pre tensioning, the radial, axial, and angular stiffness of the bearing increase rapidly with the increase of rotational speed, but the increase in axial and angular stiffness is relatively gentle. The stiffness variation law of ceramic ball bearings is similar to that of all steel bearings, but the variation is relatively gentle. Under pre tensioning, the centrifugal force between the inner ring and the ball, as well as the frictional heat, increase the contact load between the inner and outer rings. At the same time, the contact angle between the outer ring decreases and the inner ring increases, resulting in an increase in contact stiffness. However, the decrease in the contact angle of the outer ring slows down the increase in axial and angular stiffness.

Under constant pressure preloading, the increase in centrifugal force of the ball increases the contact load on the outer ring, while the contact angle decreases. Due to the axial displacement allowed by the inner and outer rings, the contact load of the inner ring remains basically unchanged, but the contact angle increases. Thermal displacement and centrifugal displacement have almost no effect on the contact load and contact angle between the inner and outer rings. Although the normal contact stiffness of the outer ring increases, the normal contact stiffness of the inner ring remains basically unchanged. The result of the series effect is that the radial stiffness increases slightly, but not significantly, while the decrease in the contact angle of the outer ring significantly reduces the axial and angular stiffness. Under positioning pre tension, the stiffness of ceramic ball bearings is lower than that of all steel bearings, while under constant pressure pre tension, the stiffness of ceramic ball bearings is higher than that of all steel bearings. Under pre tensioning, the contact load of all steel bearings is more than twice that of ceramic ball bearings. Despite the high elastic modulus of ceramic balls, the stiffness of all steel bearings is greater than that of ceramic ball bearings. Under constant pressure pre tightening, the contact load of the inner ring does not change much, and the high elastic modulus of the ceramic ball makes the stiffness of the ceramic ball bearing greater than that of the all steel bearing.

1. The influence of preload load

As the preload increases, the radial, axial, and angular stiffness of the bearing slightly increases, but the impact is minimal. Compared with positioning pre tightening, this effect is more significant on constant pressure pre tightening. This is due to the increase in preload, which increases the contact angle between the inner and outer rings, as well as the contact load, resulting in an increase in radial, axial, and angular stiffness. However, the changes in contact load and contact angle caused by preloading are relatively small compared to the changes caused by rotational speed and part displacement, therefore, their impact on bearing stiffness is limited. This is also the reason why the change under positioning preload is smaller than that under constant pressure preload.

2. The influence of channel curvature radius

As the curvature radius of the inner and outer ring channels increases, the radial, axial, and angular stiffness decrease. However, this effect is minimal, with only a slight change in stiffness under positioning preload. This is because the increase in channel curvature radius leads to an increase in contact deformation. Therefore, when selecting the curvature radius of a channel, its impact on stiffness can generally be ignored.

3. The Influence of Ball Count

Under pre tensioning, an increase in the number of balls slightly increases the radial, axial, and angular stiffness. An increase in the number of balls increases the stiffness, but under the same preload, an increase in the number of balls will reduce the contact load. Although their combined effect can increase the stiffness of the bearing, it is relatively small.

Under constant pressure preloading, the increase in the number of balls leads to a significant increase in radial stiffness, while when the rotational speed increases to a certain value, the axial and angular stiffness actually decrease, but the change is small. This is because under constant pressure preloading, although the increase in the number of balls reduces the contact load of the inner ring, it also reduces the contact angle of the inner ring. Their combined effect significantly increases the radial stiffness of the bearing, while the axial and angular stiffness slightly decrease.

Therefore, when the number of balls increases, the preload should be correspondingly increased. Only when the contact load is the same, increasing the number of balls can increase the stiffness of the bearing.

4. The impact of ball diameter

Under pre tensioning, the ball diameter increases, and the radial, axial, and angular stiffness slightly increase accordingly. The increase in ball diameter increases the centrifugal force of the ball, reduces the contact angle of the outer ring, and increases the contact angle of the inner ring, but at the same time increases the contact load between the inner and outer rings. The combined effect of these factors increases the stiffness of the bearing. The effect of centrifugal force variation on contact load is relatively small under positioning preload, so the effect of ball diameter variation on stiffness is minimal.

Under constant pressure preloading, as the ball diameter increases, the radial stiffness also increases, while the axial and angular stiffness decrease, but the impact is relatively small. This is because the increase in ball diameter increases the centrifugal force of the ball, reduces the contact angle between the inner and outer rings, increases the contact load on the outer ring, while the contact load on the inner ring remains basically unchanged, resulting in an increase in radial stiffness and a slight decrease in axial and angular stiffness. Therefore, reducing the ball diameter not only improves speed performance, but also does not reduce stiffness performance. This also proves theoretically that reducing the diameter of the ball is one of the current trends in the development of spindle bearings.

5. The impact of initial contact angle

Under pre tensioning, an increase in initial contact angle leads to a significant decrease in radial stiffness, while axial and angular stiffness increase significantly. This is because as the initial contact angle increases, the radial component of the contact stiffness decreases, the axial component increases, and at the same time, the contact load decreases under the same preload.

Under constant pressure preloading, an increase in initial contact angle leads to a significant decrease in radial stiffness; At low speeds, the axial and angular stiffness increase, while at high speeds, there is basically no change. This is because under constant pressure preloading, the inner and outer rings allow axial displacement. In order to maintain force balance, the contact angle of the outer ring is almost zero, and the initial contact angle size has little effect on the contact angle of the outer ring. Similarly, as the initial contact angle increases, the contact load decreases under the same preload.

Therefore, increasing the initial contact angle of the bearing under positioning preload can improve axial and angular stiffness, while increasing the initial contact angle under constant pressure preload not only fails to improve axial and angular stiffness, but also reduces radial stiffness.