008618266359149
Abstract: The calculation method is introduced for clearance of four point contact ball bearing under negative clearance． Taking the thin － walled four point contact ball bearing QJ1830 under no － load as an example，the influence of rings deformation on contact load between steel ball and raceway under negative clearance is considered，and the main factors affecting friction torque of bearing are analyzed． The friction torque of bearing under no － load is calculated according to conservation law of energy and compared with the measured value． The results show that the less the clearance is，the less the error between the theoretical calculating value and the measured value of friction torque is． The error is less than 30% when the clearance is less than － 7 μm．
Key words: four point contact ball bearing; thin wall; negative clearance; friction torque
Negative clearance four point contact ball bearings have the function of equivalent paired pre tensioned angular contact ball bearings, which can eliminate radial and axial clearance, reduce load deformation, obtain greater stiffness, make the bearings more compact, weigh less, and can withstand combined radial, axial, and torque loads. Therefore, this type of bearing is applied in some aerospace components, and can achieve miniaturization and lightweight of the host. In order to meet the requirements of aerospace environment, this type of bearing requires solid lubrication, so it is necessary to calculate the Friction torque of this type of bearing.
1. Calculation of clearance
The negative clearance of a four point contact ball bearing refers to the radial clearance, which is the pre tightening amount in the diameter direction of the bearing after assembling a large diameter steel ball. The negative clearance value cannot be measured by instruments, and can only be measured by assembling a small gauge steel ball to measure the radial clearance in a non preloaded state. Then, the diameter difference between the actual assembled steel ball and the testing process steel ball can be calculated and obtained. Therefore, the accuracy of the calculation will affect the Friction torque and stiffness of the sleeve assembled bearing.
Assuming that the radial clearance of the process steel ball used for bearing assembly testing is Gr, and the gasket angles of both the inner and outer rings are β. The diameter variation between the actual assembled steel ball and the process steel ball used for testing is Δ Dw, then the radial clearance change caused by assembling these two types of steel balls Δ The estimated relationship of Gr is (Figure 1)
Δ Gr=2 Δ Dw cos β ; (1)
So, after replacing the actual steel ball with the process steel ball used for testing, the theoretical calculation value of the radial clearance of the bearing is
Grt=Gr - Δ Gr. (2)
Figure 1 Cross section diagram of negative clearance four point contact ball bearing
2. Stress and deformation
The negative clearance four point contact ball bearing has a four point contact with the groove under no-load condition. If it bears certain axial and radial loads and different rotational speeds, the number of contact points may be reduced to 2 or 3, so the Kinematics characteristics of the steel ball are very complex.
Taking the negative clearance thin-walled four point contact ball bearing QJ1830 as an example, the contact stress and deformation under no-load state are analyzed. The structural parameters of QJ1830 bearing are shown in Table 1. Due to the thin wall of the bearing ring and significant deformation under internal stress, it is necessary to consider the impact of changes in the inner and outer diameters of the bearing on internal stress.
Table 1 Structural parameters of QJ1830 bearing
2.1 Deformation of bearing inner and outer diameters
Both the inner and outer rings of the bearing are thin-walled circular rings. Under negative clearance, there are contact stresses and deformations at all four contact points between the steel ball and the groove. At the same time, the outer ring expands while the inner ring shrinks. The relationship between clearance, contact deformation, bearing inner diameter compression, and outer diameter expansion is
Δ D+ Δ D+2 δ ICOS β I+2 δ E cos β E-Grt=0, (1)
In the equation: Δ D is the reduction in the inner diameter of the bearing; Δ D is the increase in the outer diameter of the bearing; δ I, δ E is the contact deformation between the steel ball and the inner and outer rings, respectively.
When there is a negative clearance, the inner and outer rings are subjected to equally distributed loads in the circumferential direction. As the inner and outer rings are thin-walled flexible parts, the classical energy method of bending thin-walled rings can be used to calculate the overall deformation of the inner and outer rings. Excessive expansion of bearing outer diameter under negative clearance Δ D is
Compression of bearing inner diameter under negative clearance Δ D is
In the formula, Qri and Qre are the radial components of the normal load between the steel ball and the inner and outer channels, respectively, Qri=2Qicos β， Qre=2Qe cos β; Ri and Re are the curvature radii of the neutral layer in the inner and outer rings of the bearing, respectively; E is the elastic modulus of the ferrule material; Ii, Ie are the moments of inertia of the inner and outer rings, respectively (see reference [5] for the calculation method of Ri, Re, Ii, Ie); Δψ = 180 °/Z.
2.2 Normal contact load between steel ball and channel
According to Hertz contact theory, the relationship between normal contact load and deformation between a steel ball and a channel is
In the formula, Qi and Qe are the normal contact loads between the steel ball and the inner and outer channels, respectively; Ki and Ke are the contact deformation constants, respectively. The contact angle between the steel ball and the inner and outer rings when the negative clearance four point contact ball bearing is in an unloaded state（ α I, α E) Equal to its gasket angle（ β I, β E), that is, the contact angles of the inner and outer rings are equal. Meanwhile, the gravity of the ferrule is very small compared to the normal load at the contact point and can be ignored. Therefore, the normal load and friction force at the left and right contact points of the inner and outer channels are equal, and the steel ball's rotation axis is parallel to the bearing axis. The steel ball is in equilibrium under the normal load at the contact point.
From equations (1) to (6), it can be determined that Δ D, Δ D, δ I, δ e. Qi, Qe.
3. Calculation of Friction torque under no-load condition
QJ1830 bearing adopts solid lubrication with low speed. Its friction mainly comes from rolling friction and Sliding friction between steel ball and channel, which are mainly caused by elastic hysteresis, differential sliding, spin sliding, steel ball slipping and gyro rotation. Due to the low working speed of the bearing, the preload between the steel ball and the channel forms a four point contact, and there is significant spin sliding between the steel ball and each channel (Figure 2). The rotation of the steel ball relative to the channel can be decomposed into angular velocity ω R= ω Cos β Rotation and angular velocity around the tangent direction (MM axis) of the contact point ω S= ω Sin β Rotation around the normal direction (NN axis) of the contact point, where ω The angular velocity of the steel ball relative to the ring can be determined according to the law of angular velocity vector synthesis [4]. ω R is the rolling component of the steel ball relative to the channel, and the Friction torque generated by this component is mainly caused by differential sliding, which accounts for a small proportion of the total Friction torque; And the spin component of the steel ball relative to the channel ω The Friction torque caused by s is the main source of Friction torque when the bearing is in motion.
Figure 2 Analytical diagram of rolling and spin sliding at the contact between the steel ball and the channel
Assume that the Sliding friction factor of each point in the contact area is μ (constant), then the total spin Sliding friction torque on the contact ellipse is
In the formula: Q is the normal load at the contact point; L (K) is the second kind of complete Elliptic integral related to the shape of the contact surface; K is the eccentricity of the ellipse, K=b/a; a. B represents the long and short half axes of the contact ellipse, respectively.
Ignoring the remaining friction with less internal influence of the bearing, according to the Conservation of energy, the sum of friction power consumption caused by spin sliding at all ball contact points is the total friction power consumption of the bearing.
Where: Msil, Msir, Msel, Mser are the spin Friction torque between each steel ball and each channel respectively; ω Sil, ω Sir, ω Sel, ω Ser is the spin angular velocity component between each steel ball and each channel. M is the total Friction torque of the bearing; Ni is the inner race speed.
The total Friction torque M of the bearing can be calculated by simultaneous equations (1) to (9). See Table 2 for the comparison between the calculated value and the measured value. It can be seen from Table 2 that the measured value of the total Friction torque is larger than the calculated value. This is because only the friction caused by spin sliding is considered in this paper. With the decrease of the clearance value, the proportion of the Friction torque generated by spin sliding in the total Friction torque gradually increases. Therefore, the error between the theoretical calculated value and the measured value also decreases. Because there are many factors affecting Friction torque, the fluctuation of Friction torque test value of a single set of solid lubrication bearings is generally about 30%, so the calculation error of this model is acceptable.
Table 2 Comparison of Friction torque under different clearances
4. Conclusion
When calculating the internal contact load of thin-walled bearings such as QJ1830 in the negative clearance state, it is usually not possible to assume that the ring is a rigid body, otherwise the negative clearance amount will be fully converted into contact deformation, which will cause the calculated contact load to be larger than the actual value. Therefore, the influence of the deformation of the inner and outer rings on the internal contact stress should be considered. When installing this type of bearing, special attention should be paid to the impact of changes in the inner and outer diameters of the bearing under negative clearance conditions on the fit between the bearing, shaft, and bearing seat. Especially when the rigidity of the shaft and bearing seat is large, the interference amount will be converted into contact deformation inside the bearing, significantly affecting the contact stress state inside the bearing.
By analyzing the movement of the steel ball, it can be seen that the Friction torque of the four point contact ball bearing under negative clearance is mainly caused by the spin sliding of the steel ball; The error between the theoretical calculation value and the measured value of the total Friction torque is small when the clearance value is small.
More about KYOCM Spherical Roller Bearing：
A spherical roller bearing is a rolling-element bearing that permits rotation with low friction, and permits angular misalignment. Typically these bearings support a rotating shaft in the bore of the inner ring that may be misaligned in respect to the outer ring. The misalignment is possible due to the spherical internal shape of the outer ring and spherical rollers.[1] Despite what their name may imply, spherical roller bearings are not truly spherical in shape. The rolling elements of spherical roller bearings are mainly cylindrical in shape, but have a (barrel like) profile that makes them appear like cylinders that have been slightly over-inflated.
http://www.kyocm.com/products/Spherical-Roller-Bearing/739.html
2023-07-24