Influence of Fixing Devices on the Accuracy of Measuring the Static Friction Torque of Instrument Ball Bearings

E. F. Kolotii

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

As is known [1], to measure the static friction torque of instrument ball bearings one may use an electrical (rotary) transducer whose output quantity is a force or a torque. The general structural diagram of an instrument with an electrical transducer for measuring the static friction torque of instrument ball bearings is shown in Fig. 1. The starting circuit actuates a generator, at the output of which a gradually increasing electrical quantity \(y\) (current or voltage) appears. With the aid of the electrical transducer this quantity is converted into the torque \(M_{\beta}\), applied to the outer ring of the bearing under test. The torque \(M_{\beta}\) also gradually increases with time. When the condition \(M_{\beta} > M_{\mathrm{tr}}\) is satisfied, the outer ring of the bearing acquires a certain acceleration. This circumstance is used to record that value of the generator output electrical quantity at which acceleration of the moving part appeared (the outer ring of the bearing and the flywheel mounted on it, producing a specified axial load on the bearing). The function of converting the acceleration of the moving part into an action on the generator is performed by the fixing device.

Fig. 1

Labels in the diagram: starting circuit; generator; electrical transducer; indicating instrument; fixing device; \(y\); \(M_{\beta}\); \(M_{\mathrm{tr}}\); \(\Delta M\).

As shown in [1], the value \(\Delta M = M_{\beta} - M_{\mathrm{tr}}\) can be found if the moment of inertia \(J\) of the moving part and the smallest (threshold) value of the acceleration of the moving part are known,

\[ \left(\frac{d^{2}\alpha}{dt^{2}}\right)_{\mathrm{thr}}, \]

at which the fixing device is actuated:

\[ \Delta M = J \left(\frac{d^{2}x}{dt^{2}}\right)_{\text{th}} . \]

Knowing \(\Delta M\) and using the expression \(M_{\text{fr}} = M_{\beta} - \Delta M\), one can determine the static friction torque of the bearing.

However, it is difficult to create a fixing device that responds to the acceleration of the moving part, since the magnitude of this acceleration is usually small. For this reason, we propose making fixing devices that respond not to the acceleration of the moving part, but to the angle \(\alpha\) of rotation of the moving part from the initial (zero) position. In this case, obviously, it makes sense to speak of the smallest (threshold) value of the angle \(\alpha\) at which the fixing device is actuated.

Let the torque \(M_{\beta}\) increase linearly with time, i.e., \(M_{\beta}=\kappa t\) (where \(\kappa\) is the proportionality coefficient). In this case the quantity \(\Delta M\) will likewise increase linearly, and the difference \(\Delta M = M_{\beta} - M_{\text{fr}}\) will appear at the instant \(t_1\) (Fig. 2), and the rate of increase of \(\Delta M\) will be equal to the rate of increase of \(M_{\beta}\), that is,

\[ \frac{d(\Delta M)}{dt} = \frac{dM_{\beta}}{dt} = \kappa . \]

It is obvious that \(\Delta M = \kappa (t-t_1)\). Under the action of the torque \(\Delta M\), the moving part of the instrument acquires the angular acceleration \(\frac{d^{2}\alpha}{dt^{2}}\). Neglecting air resistance and taking into account that there is no opposing torque, we arrive at the following equation of motion of the moving part:

\[ J \frac{d^{2}x}{dt^{2}} - \kappa (t-t_1) = 0 . \tag{1} \]

Taking the instant \(t_1\) as the origin of time, we simplify expression (1) as follows:

\[ J \frac{d^{2}x}{dt^{2}} - \kappa t = 0 . \tag{2} \]

Integrating expression (2), we obtain

\[ J \frac{d\alpha}{dt} = \frac{1}{2}\kappa t^{2} + C_1 . \tag{3} \]

Taking \(C_1=0\) (the initial velocity of motion of the moving part is equal to zero), from equality (3) by integration we find:

\[ J\alpha = \frac{\kappa t^{3}}{6} + C_2 . \tag{4} \]

Under the condition that \(C_2=0\) (the initial angle of deflection of the moving part is equal to zero), equality (4) takes the form

\[ \alpha = \frac{\kappa}{6J}t^{3}. \tag{5} \]

Expression (5) makes it possible to find the time \(t_2\) during which the moving

part will turn through an angle \(\alpha_{\mathrm{th}}\), sufficient for actuation of the locking device:

\[ t_2=\sqrt[3]{\frac{6J\alpha_{\mathrm{th}}}{\kappa}} . \tag{6} \]

Using expression (6), we determine the value of the moment \(\Delta M\) by which the moment \(M_\beta\) will differ from the measured moment \(M_{\mathrm{fr}}\):

\[ \Delta M=\kappa t_2=\sqrt[3]{6\kappa^2 J\alpha_{\mathrm{th}}}. \tag{7} \]

Fig. 2.

The true value of the measured torque \(M_{\mathrm{tr}}\) can be found as

\[ M_{\mathrm{tr}}=M_{\beta}-\Delta M = M_{\beta}-\sqrt[3]{6\kappa^{2}J\alpha_{\mathrm{por}}}, \tag{8} \]

where \(M_{\beta}\) is the value of the compensating torque at the instant when the locking device is actuated;

\[ \kappa=\frac{dM_{\beta}}{dt}=\mathrm{const} \]

is the rate of increase of the compensating torque,

\(J\) is the moment of inertia of the moving part,

\(\alpha_{\mathrm{por}}\) is the threshold (smallest) value of the deflection angle of the moving part at which the locking device is actuated.

Expression (8) can be used both in processing the measurement results (if the indicating instrument is calibrated in values of the torque \(M_{\beta}\)) and in calibrating the indicating instrument directly in values of \(M_{\mathrm{tr}}\).

As expression (6) shows, when \(\alpha_{\mathrm{por}}\) and \(\kappa\) change, the time \(t_2\) required for the moving part to rotate through the angle necessary to actuate the locking device changes; this, in turn, leads to changes in the quantities \(M_{\beta}\) and \(\Delta M\). A change in the quantity \(M_{\beta}\) at the instant when the locking device is actuated will not cause a significant measurement error (the indicating instrument is calibrated in values of \(M_{\beta}\)); however, a change in the quantity \(\Delta M\) can affect the measurement result, i.e., can cause a measurement error.

Let the instabilities of the angle \(\alpha_{\mathrm{por}}\) and of the rate of increase \(\kappa\) be respectively

\[ \gamma_{\alpha}=\frac{\Delta\alpha_{\mathrm{por}}}{\alpha_{\mathrm{por}}} \quad \text{and} \quad \gamma_{\kappa}=\frac{\Delta\kappa}{\kappa}. \]

Then, using expression (7), we determine the instability of the quantity \(\Delta M\). For this purpose, taking the logarithm of expression (7), we obtain

\[ \ln(\Delta M)=\frac{1}{3}\left(\ln 6+2\ln\kappa+\ln J+\ln\alpha_{\mathrm{por}}\right). \tag{9} \]

Taking the derivative of the left- and right-hand sides of expression (9), we find

\[ \frac{d(\Delta M)}{\Delta M} = \frac{1}{3}\left(2\frac{d\kappa}{\kappa} + \frac{d\alpha_{\mathrm{por}}}{\alpha_{\mathrm{por}}}\right) \tag{10} \]

or, passing to finite increments, we obtain

\[ \gamma_{\Delta M}=\frac{2}{3}\gamma_{\kappa}+\frac{1}{3}\gamma_{\alpha}. \tag{11} \]

Thus, the true value of the difference \(\Delta M\) can vary within the limits

\[ \Delta M(1\pm\gamma_{\Delta M}), \]

where \(\Delta M\) is the value of the difference torque calculated from expression (7).

Using expression (8), we find that

\[ \delta(M_{\mathrm{tr}})=\delta(M_{\beta})+\delta(\Delta M), \tag{12} \]

where \(\delta(M_{\mathrm{tr}})\) is the absolute error in measuring the torque \(M_{\mathrm{tr}}\),

\(\delta(M_{\beta})\) is the absolute error of the torque \(M_{\beta}\),

\(\delta(\Delta M)\) is the absolute error of the difference torque \(\Delta M\).

Taking \(\delta(M_\beta)=0\) (we assume that the readings of the pointer instrument correspond exactly to the output quantity of the electrical transducer—the moment \(M_\beta\)), we obtain

\[ \delta(M_{\mathrm{tr}})=\delta(\Delta M)=\Delta M\cdot \gamma_{\Delta M}. \tag{13} \]

Hence the relative error in measuring the static friction torque \(M_{\mathrm{tr}}\) of the bearing under investigation is found as

\[ \gamma_{M_{\mathrm{tr}}} = \frac{\delta(M_{\mathrm{tr}})}{M_{\mathrm{tr}}} = \frac{\gamma_{\Delta M}\cdot \Delta M}{M_\beta-\Delta M} = \]

\[ = \frac{ \left(\dfrac{2}{3}\gamma_k+\dfrac{1}{3}\gamma_\alpha\right) \sqrt[3]{6k^2J\alpha_{\mathrm{por}}} }{ M_\beta-\sqrt[3]{6k^2J\alpha_{\mathrm{por}}} }. \tag{14} \]

Expression (14) makes it possible to assert that the relative instabilities of the rate of increase of the compensating moment \(\gamma_k\) and of the threshold value of the actuation angle of the fixing device \(\gamma_\alpha\) have an insignificant effect on the total error in measuring the static friction torque of bearings. This is illustrated by the following example.

Let, at the moment of actuation of the fixing device, the indicating instrument give a reading of \(2.5\ \text{g}\cdot\text{cm}\). The magnitude of the moment is \(\Delta M=0.5\ \text{g}\cdot\text{cm}\). The values of \(\gamma_k\) and \(\gamma_\alpha\) are, respectively, \(3\%\) and \(6\%\). Then

\[ \gamma_{M_{\mathrm{tr}}} = \frac{ \left(\dfrac{2}{3}\cdot 3\%+\dfrac{1}{3}\cdot 6\%\right)\cdot 0.5\,\text{gcm} }{ (2.5-0.5)\ \text{g}\cdot\text{cm} } =1\%. \]

Thus, even not overly stringent requirements on the stability of the quantities \(k\) and \(\alpha\) make it possible to create a sufficiently accurate instrument for measuring the static friction torque of instrument ball bearings. In connection with the foregoing, it should be noted that the principal role in the error of measuring the friction torque by the indicated method will evidently be played by the instability of the conversion coefficient of the electrical transducer and by the error of the indicating instrument. However, discussion of this question lies beyond the scope of the present work.

REFERENCES

1. E. F. Kolotii, I. G. Leshenko, On the question of measuring the friction torque of instrument ball bearings, Izvestiya TPI, vol. 141, 1965.