HEJ, HU ISSN 1418-7108 Manuscript no.: MET-000123-A

Parts and operation of the ball thread

The ball thread is a part of a gear but it can be unambiguously separated from the gear and consists of the following items:

• pin
• nut
• balls
• ball returning tube
• its fixture

The pin and the nut have the same lead and generally the same profile shape. In this case the angles connecting to the profile of the balls are the same as well that is for these threads the angle of action in the unloaded thread is the same as the connecting angle. In the thread the balls align close and generally contact to each other. The start of the thread is really where the balls enter the thread. This place is the bore made on the nut where the ball returning tube is connected. The other end of this tube is at the end of the nut and it returns the leaving balls from there to the start of the thread. Leaving of the balls are assisted by the so-called ears on the tubes which really inhibit the balls from over-rolling and they take them off. The start and end of the thread can be naturally interchanged depending on the rotation direction of the pin and the nut. For small load and proper geometric form of the thread the stress on the returning tube and its ear is has no account and the tube can properly guide the balls between the ends of the thread.

Increasing the load and in case of certain profile forms the forces affecting the balls can expose the ball returning tube and its ear to a significant load which can lead to the deterioration of the thread.

The profile forming of the thread has an important role in the operation of the thread. The profile of the thread has a circular cross-section, the radius of its manufacturing is larger than that of the ball and the centre of manufacturing is shifted away from the axis of the profile. This shift provides for the appropriate engagement between the thread and the ball for all occurring thread loads. The fit between the ball and the thread profile is shown in the Fig. 1 on which the pin and nut threads are made with different profiles.

 

Figure 1: Perpendicular section of the thread: the thread profile

The work of the thread is to transform the torque to a axial force. This transformation implements in the connection of the thread balls along the nut and pin profiles. Since in the thread, at the contact point of the balls and the profile, the profile tangent includes an acute angle with the central line of the thread at any point of the profile arc. The connection force of the balls and the arc always has a component, which forces the connecting places of the balls toward the edge of the profile. This coercing force increases with loading the thread and due to it both the connecting angle on the pin and the nut as well as the angle of the action lines of the forces on the ball, the action angle decrease.

The elastic deformation developing on the contacting surface of the balls and the profile allows the action angle to change. On the basis of this deformation the change in the action angle can be determined as follows according to Molnár-Dr. Varga[1].

Actual action angle:

$$\alpha=\mbox{arcsin }\left[\frac{\sin\alpha_0+\frac{\delta_a}{A}} {\sqrt{\cos^2\alpha_0+\left(\sin\alpha_0+\frac{\delta_a}{A}\right)^2}}\right]$$ (1)

In load free case the theoretical action angle is as follows:

$$\alpha_0=\mbox{arccos }\left(1-\frac{H_t}{2A}\right)$$ (2)

Distance between the trajectory centres:

$$A=(f_k+f_b-1)d_g$$ (3)

Actual elastic deformation:

$$\delta_a=A^\prime-A$$ (4)

Distance between the centres changed under the effect of deformation:

$$A^\prime=\sqrt{A^2\cos^2\alpha_0+(A\sin\alpha_0+\delta^2_a)}$$ (5)

Relation between the deformation of action direction and the axial deformation:

$$\delta=\sqrt{A^2\cos^2\alpha_0+(A\sin\alpha_0+\delta^2_a)}-A$$ (6)

Osculation of the balls and trajectories:

$$f_k=\frac{r_k}{d_g}$$ (7)

$$f_b=\frac{r_b}{d_g}$$ (8)

Axial load:

$$F_a=zF_g\sin\alpha\cos\varphi$$ (9)

Load of a ball:

$$F_g=K_n\delta^n$$ (10)

Constant depending on the contacting conditions:

$$K_n=\left[\frac{1}{\left(\frac{1}{K_b}\right)^{\frac{1}{n}}+ \left(\frac{1}{K_k}\right)^{\frac{1}{n}}}\right]^n$$ (11)

$n$ is the exponent depending on the contacting points, its value relating on the ball bearings:

$$n=3/2$$ (12)

Contacting constant relating to the pin:

$$K_b=2.14\cdot 10^4\frac{1}{\sqrt{\sum\rho_b\delta^{\ast 3}_b}}$$ (13)

Contacting constant relating to the nut:

$$K_k=2.14\cdot 10^4\frac{1}{\sqrt{\sum\rho_k\delta^{\ast 3}_k}}$$ (14)

Sum of the main incurvations on the pin:

$$\sum\rho_b=\frac{1}{d_g}\left[4+\frac{2\gamma}{1-\gamma}-\frac{1}{f_b}\right]$$ (15)

Sum of the main incurvations on the pin:

$$\sum\rho_k=\frac{1}{d_g}\left[4+\frac{2\gamma}{1-\gamma}-\frac{1}{f_k}\right]$$ (16)

Intermediate constant:

$$\gamma=\frac{d_g\cos\alpha}{d_m}$$ (17)

Curvature relation: on the contacting point between the pin and the ball:

$$F(\rho)_b=\frac{\frac{2\gamma}{1-\gamma}+\frac{1}{f_b}} {4+\frac{2\gamma}{1-\gamma}-\frac{1}{f_b}}$$ (18)

on the contacting point between the nut and the ball:

$$F(\rho)_k=\frac{\frac{2\gamma}{1-\gamma}+\frac{1}{f_k}} {4+\frac{2\gamma}{1-\gamma}-\frac{1}{f_k}}$$ (19)

Elastic deformation at the pin:

$$\delta_b=0.013\cdot\delta^{\ast}\sqrt[3]{F^2_g\rho_b}$$ (20)

at the nut:

$$\delta_b=0.013\cdot\delta^{\ast}\sqrt[3]{F^2_g\rho_k}$$ (21)

The function $\delta^\ast=f(F^2_g\rho_k)$ on the basis of the tests of MGM on Figure 2:[2]

Small semi-axis of the contacting ellipse:

$$b=0.05101\cdot b^{\ast}\sqrt[3]{\frac{F_g}{\sum\rho}}$$ (22)

 

Figure 2: Specific data of the contact between the ball and the ball path

On the basis of the above relations and the diagrams of the Fig. 2 the change in the actual action angle in the function of the load.

The angle of action increases digressively in the function of the load and in case of the well-dimensioned thread profile the edge of the depression ellipse cannot reach the boundary of the profile. However if the load exceeds the dimensioning limit the pitch changes or if the connecting angle given from the manufacturing dimensions increases on the nut the edge of depression can reach the edge of the travelling profile of the nut thread. In this case the surface load both on the ball and the path do not develop according to the Hertz voltage distribution and the proportional elasticity. The laying surface doesn't increase according to the earlier used relations, the load of the ball increases significantly and the edge of the profile depresses in the ball. This large surface load will increase the axial displacement of the thread profiles and causes a bottleneck in the thread.

HEJ, HU ISSN 1418-7108 Manuscript no.: MET-000123-A