MEASUREMENT OF THE DYNAMIC FRICTION TORQUE OF INSTRUMENT BALL BEARINGS BY THE TIME METHOD

V. A. KASPLER, E. F. KOLOTII, I. G. LESHCHENKO

(Presented by the scientific seminar of the Faculty of Automation and Computer Engineering)

A time method developed at the Department of Electrical Measuring Engineering of the Tomsk Polytechnic Institute is proposed for measuring the dynamic friction torque of instrument ball bearings with the bearing axis arranged vertically and with an axial load present. The essence of the method is as follows.

Fig. 1. Diagram of the movable part.

If shaft 2 (Fig. 1) rotates with a constant angular velocity \(\omega\), while the outer ring of the bearing under test 1, together with flywheel 3 mounted on it, remains stationary, then a dynamic friction torque of the bearing acts on the movable part of the device (outer ring—flywheel), entraining the movable part in the direction of rotation. As is known, for a system having one degree of freedom (the movable part), the following equality holds:

\[ \sum M = 0, \tag{1} \]

where \(\sum M\) is the algebraic sum of the driving and resisting torques acting on the system.

In the case under consideration, only forces caused by the presence of a definite moment of inertia \(I\) of the movable part oppose the friction torque \(M_{\mathrm{fr}}\), which is the driving torque. Then equality (1) for our case takes the form

\[ I \frac{d^2 \alpha}{dt^2} - M_{\mathrm{fr}} = 0 \]

or

\[ I \frac{d^2 \alpha}{dt^2} = M_{\mathrm{fr}}, \tag{2} \]

where \(\alpha\) is the angle of deflection of the movable part.

From (2) the angular acceleration of the movable part is determined as

\[ \frac{d^2 \alpha}{dt^2} = \frac{M_{\mathrm{fr}}}{I}. \tag{3} \]

Thus, by measuring the angular acceleration of the moving part and using equality (3), one can determine the friction torque of the bearing being tested.

However, measurement of small angular accelerations presents considerable difficulties; therefore it is proposed to measure the time of rotation of the moving part through a definite angle and to judge the friction torque from it. This conclusion follows from the following reasoning.

Integrating equality (3), we obtain

\[ \frac{d\alpha}{dt}=\frac{M_{\mathrm{tr}}}{I}t+C_1, \tag{4} \]

where \(C_1\) is the constant of integration.

Taking \(C_1=0\) (the initial velocity of motion of the moving part is equal to zero), from equality (4), by integration, we find

\[ \alpha=\frac{M_{\mathrm{tr}}}{I}t^2+C_2, \tag{5} \]

where \(C_2\) is the constant of integration.

Under the condition that \(C_2=0\) (the initial angle of deflection of the moving part is equal to zero), equality (5) takes the form

\[ \alpha=\frac{M_{\mathrm{tr}}}{I}t^2. \tag{6} \]

Hence

\[ M_{\mathrm{tr}}=\frac{\alpha I}{t^2}, \tag{7} \]

where \(\alpha\) is the angle through which the moving part turns from the initial zero position in time \(t\).

Having chosen the angle of rotation \(\alpha\) as constant and measuring the time \(t\) during which the moving part turns through the angle \(\alpha\) under the action of the torque \(M_{\mathrm{tr}}\), one can, using expression (7), determine the dynamic friction torque of the bearing being tested. The smaller the friction torque of the bearing, the longer the time in which the moving part turns through the angle \(\alpha\).

Measurement of the rotation time by electrical methods presents no difficulty and can be carried out with very high accuracy (an ordinary electric vibration stopwatch provides a measurement accuracy of 0.02 sec). Measuring the angle of rotation \(\alpha\) at an unchanged time \(t\), as is done in some existing instruments [1, 2], seems to us less expedient, since in this case the measurement accuracy is sharply reduced.

The graphical dependence of \(M_{\mathrm{tr}}\) on \(t\) at a constant value of the moment of inertia \(I\) of the moving part is presented in Fig. 2 (curves 2, 3, 4). By varying the fixed angle \(\alpha\) of deflection of the moving part, it is possible to select the optimum value of this angle, at which high measurement accuracy and sufficiently high productivity are obtained. From Fig. 2 it is seen that measurement of small bearing friction torques can be carried out with high accuracy, since a slight change in torque corresponds to large changes in the time of deflection of the moving part through a definite angle.

For comparison, curve 1 shows the dependence of \(M_{\mathrm{tr}}\) on \(t\) for a moment of inertia \(I\) ten times smaller.

With an unchanged moment of inertia \(I\) of the moving part, the scale of the device measuring the deflection time \(t\) can be graduated directly in units of the measured friction torque.

Fig. 2. Dependence of \(M_{\mathrm{tr}}\) on \(t\) for various fixed angles of rotation of the moving part.

This method makes it possible to measure the bearing friction torque at any rotational speeds except very low ones, since at low rotational speeds the bearing ring does not have time to make a complete revolution; therefore the moving part will turn through a fixed angle not under the action of the average dynamic friction torque of the bearing, but under the action of a torque that may be either smaller or larger than the average friction torque.

The method described was tested on a mock-up that included the moving part with the ball bearing under test (Fig. 1), a photo relay actuated when the moving part turned through an angle of \(0.5\) rad, and an electric stopwatch switched on and off by the contacts of the photo relay. The time was measured with an accuracy of \(0.02\) sec. The operation of the mock-up was satisfactory at rotational speeds above \(60\) rpm.

Thus, the time method for measuring the friction torque of instrument ball bearings is highly accurate and promising from the standpoint of developing and creating, on its basis, semiautomatic and automatic installations for inspection of manufactured bearings by friction torque.

REFERENCES

1. N. B. Bruevich, Z. M. Breitman, Yu. M. Reznikov. Technical Measurements in the Ball-Bearing Industry. Mashgiz, 1963.

2. R. M. Matveevskii. Investigation of Friction in Instrument Ball Bearings. Mashgiz, 1957.