On the Measurement of the Starting Torque of Instrument Ball Bearings
Abstract Generated abstract
This paper examines measurement of the starting torque of instrument ball bearings in devices where shafts undergo small angular displacements under axial load. It develops a general method in which a gradually increasing torque is applied to the bearing’s moving part and the onset of acceleration is used to register the torque required to overcome friction, with correction for the moving part’s moment of inertia and the threshold sensitivity of the recording device. The authors compare possible electrical transducers for generating the test torque, including magnetoelectric, electromagnetic, and induction mechanisms, and discuss their calibration characteristics and principal error sources. The contribution is a framework for designing instruments that measure bearing starting torque by relating an easily measured electrical input to the applied mechanical torque.
Full Text
On the Measurement of the Starting Torque of Instrument Ball Bearings
E. F. Kolotii, I. G. Leshchenko
(Presented by the scientific seminar of the Faculty of Automation and Computer Engineering)
There are a large number of units and devices in which shafts undergo small angular displacements in their supports, with stops (the supports of gyroscope gimbal suspensions, supports of selsyn transmissions, and so on). It is obvious that, in order to obtain high operating accuracy in such devices, the starting torque of the bearings is of decisive importance. In this connection, it becomes necessary to measure the starting torque of instrument ball bearings.
This article considers methods for measuring the starting torque of instrument ball bearings when the bearing axis is vertical and an axial load is present, developed at the Department of Electrical Measuring Engineering of the Tomsk Polytechnic Institute.
General approach to the solution of the problem
If a gradually increasing torque \(M_\beta\) is applied to the moving part of the bearing under test 1 (Fig. 1), consisting of the outer ring and the flywheel 3, which creates an axial load, then, when the torque \(M_\beta\) becomes equal to the starting torque \(M_{\text{tr}}\) of the bearing, the moving part will start from rest. The fact that acceleration of the moving part appears is used to record the value \(M_\beta\) at which acceleration of the moving part appeared. To create a gradually increasing torque, various types of transducers may be used, whose output quantity is a force or a torque. The type of input quantity of such transducers is not of fundamental importance; however, preference should be given to electrical transducers, to whose input electrical signals are applied (current, voltage, power), since in this case it is extremely convenient to measure the input quantity of the transducer and, from it, to judge the value of the output quantity.
Fig. 1. Diagram of the moving part.
Strictly speaking, the moving part can start from rest only when \(M_\beta > M_{\text{tr}}\), that is, it rotates under the action of the difference of the torques \(\Delta M = M_\beta - M_{\text{tr}}\).
The acceleration of the moving part will be the greater, the smaller its moment of inertia \(I\), which follows from the equality
\[ I \frac{d^{2}a}{dt^{2}}=\Delta M, \]
where \(a\) is the angle of deflection of the moving part.
As already indicated, the recording device must respond to the appearance of acceleration of the moving part; however, the creation of devices that respond to arbitrarily small accelerations is extremely difficult, and therefore it is meaningful to speak of the smallest (threshold) value of the acceleration of the moving part \(\left(\dfrac{d^{2}a}{dt^{2}}\right)_{\text{thr}}\), at which the recording device is actuated. In this case the magnitude of the difference torque \(\Delta M\) will be the smaller, the smaller the moment of inertia \(I\) of the moving part. With an unchanged moment of inertia \(I\) of the moving part, the quantity \(\Delta M\) will decrease as the sensitivity of the recording device is increased (as the threshold value of the acceleration \(\left(\dfrac{d^{2}a}{dt^{2}}\right)_{\text{thr}}\) is decreased).
The error of the method of measuring the friction torque of a ball bearing is defined as
\[ \gamma_{\text{method}}=\frac{\Delta M}{M_{\text{fr}}}, \]
however, this error can always be taken into account when processing the results, provided that \(I\) and \(\left(\dfrac{d^{2}a}{dt^{2}}\right)_{\text{thr}}\) are known. In this case
\[ M_{\text{fr}}=M_{\beta}-\Delta M=M_{\beta}-I\left(\frac{d^{2}a}{dt^{2}}\right)_{\text{thr}} . \tag{1} \]
The value of the torque \(M_{\beta}\) is determined from the known value of the input quantity of the electrical transducer and its known conversion coefficient.
The general block diagram of the device for measuring the friction torque of instrument bearings by the proposed method is shown in Fig. 2. The starting circuit starts the generator, at the output of which there appears a gradually increasing electrical quantity \(y_{\text{in}}\), supplied to the input of the electrical transducer. The out-
Fig. 2. General block diagram of the instrument circuit for measuring friction torque
Diagram labels: starting circuit; generator; indicating instrument; \(y_{\text{in}}\); electrical transducer; indication-recording circuit; \(\Delta M\); \(M_{\beta}\); \(M_{\text{fr}}\).
the input quantity of the transducer is the torque \(M_\beta\) applied to the moving part of the device. When a difference in torques \(\Delta M\) appears, the locking device is actuated, recording the value of the quantity \(y_{\text{in}}\) at which acceleration of the moving part appears,
\[ \left(\frac{d^2\alpha}{dt^2}\right)_{\text{thr}} . \]
The quantity \(y_{\text{in}}\) is measured by an indicating instrument calibrated directly in units of the measured torque.
Application of Certain Transducers for Producing the Torque \(M_\beta\)
a) Magnetoelectric Transducer
Any magnetoelectric measuring mechanism may be regarded as a transducer of an input electrical quantity (current \(I_{\text{in}}\)) into an output, non-electrical quantity (torque \(M_\beta\)). As is known [1, 2], the relation between the input and output quantities of such a transducer has the form
\[ M_\beta = BSwI_{\text{in}}, \tag{2} \]
where \(B\) is the induction in the gap of the magnetoelectric mechanism;
\(S\) is the area of the frame;
\(w\) is the number of turns of the frame.
Hence the conversion coefficient of the magnetoelectric mechanism is
\[ k = \frac{M_\beta}{I_{\text{in}}} = BSw . \tag{3} \]
The error of the conversion coefficient
\[ \gamma_k = \frac{\Delta k}{k} = \frac{\Delta B}{B} \]
is determined only by the instability of the induction value in the gap and amounts to no more than \(0.02\%\) [3].
Thus, with a gradual increase of the current \(I_{\text{in}}\) from zero, the output torque \(M_\beta\) applied to the moving part of the mechanical assembly (Fig. 1) will also gradually increase. The ammeter measuring the input current of the transducer \(y_{\text{in}}\) is calibrated directly in units of torque, taking into account the conversion coefficient (3). With an unchanged value of the moment of inertia of the moving part and of the threshold value of acceleration at which the locking device is actuated, the ammeter may also be calibrated using equality (1). The error in measuring the starting torque in this case is determined mainly by the error of the indicating instrument (ammeter).
b) Electromagnetic Transducer
The natural output quantity of an electromagnetic transducer is the attractive force applied to the moving part through a core made of ferromagnetic material. This force creates a torque acting on the moving part owing to the presence of a certain arm,
\[ M_\beta = F_\text{e}l, \]
where \(F_\text{e}\) is the tractive force of the electromagnet;
\(l\) is the length of the arm.
The tractive force of the electromagnet is some function of the current \(I\) flowing through the winding and of the magnitude of the air gap \(\delta\) between the electromagnet core and the ferromagnetic projection of the moving part, that is,
\[ F_{\mathrm{e}}=f(I;\delta). \]
If the material of the cores is in a region far from saturation, then, as the current \(I\) in the winding increases, the attractive force of the electromagnet increases, which leads to an increase in the torque \(M_{\mathrm{e}}\).
For constant values of \(l\) and \(\delta\), the torque developed by the electromagnet is uniquely related to the current flowing through the winding. An instrument measuring the current can be graduated directly in units of torque [4].
The measurement error in this case will be determined by the instability of the magnetic characteristics of the core material.
The total measurement error also depends on the instability of the lever-arm length \(l\) and the gap \(\delta\), as well as on the error of the indicating instrument measuring the current in the electromagnet.
Temperature fluctuations cause changes in the magnetic properties of the core material, which also leads to the appearance of an additional temperature error in the measurement. It should be noted that, owing to the nonlinear dependence of the magnetic flux on the current flowing through the winding of an electromagnet with a core, and also owing to the presence of magnetic hysteresis in the core material, the calibration characteristic of the electromagnetic transducer (the dependence of the torque \(M_{\mathrm{e}}\) on the current \(I\)) becomes extremely complicated.
When an electromagnet without a core is used, the dependence of \(M_{\mathrm{e}}\) on \(I\) is simplified; however, in this case, in order to obtain the same torque, it is necessary to increase the magnetic-field intensity considerably (to increase the number of ampere-turns of the electromagnet winding).
c) Induction transducer
As is known [5], an induction measuring mechanism develops a torque applied to the moving part, the magnitude of which is proportional to the currents \(I_1\) and \(I_2\) flowing through the transducer windings, and to the sine of the phase-shift angle between the magnetic fluxes \(\Phi_1\) and \(\Phi_2\) produced by these currents, that is,
\[ M_{\mathrm{e}}=k I_1 I_2 \sin \widehat{(\Phi_1,\Phi_2)}. \]
With a gradual increase of the currents \(I_1\) and \(I_2\), the torque \(M_{\mathrm{e}}\) will also increase gradually. In this case, two variants are possible for obtaining an increasing torque \(M_{\mathrm{e}}\):
1) to increase the current in one of the windings while maintaining the current of the other winding unchanged;
2) to increase the currents simultaneously in both windings.
The first variant presupposes the presence of a stabilizing device for supplying one of the windings with a current that is constant in time, and the stabilization error is then fully included in the total measurement error. The calibration characteristic of the transducer in this case is linear,
\[ M_{\beta}=k_1 I, \]
where \(I\) is the varying current in one of the windings.
The second variant is simpler to implement; there is no need for current stabilization, but the calibration characteristic becomes nonlinear.
The question of using an induction transducer to measure the starting torque of instrument ball bearings is considered in greater detail in [6].
References
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M. M. Fetisov. New Automatic Compensation Instruments for Measuring Nonelectrical Quantities. Part I, LDNTI, Leningrad, 1960.
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I. G. Leshchenko, E. F. Kolotii, Yu. A. Bannov. Application of an electromagnetic transducer for measuring the starting torque of instrument ball bearings. (Article included in this collection.)
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P. N. Goriunov, S. M. Pigin, N. N. Shumilovskii. Electrical Counters. Gosenergoizdat, Moscow—Leningrad, 1951.
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E. F. Kolotii. Induction method for measuring the starting torque of instrument ball bearings. (Article included in this collection.)