Application of an Inductive Inverse Transducer for Measuring the Dynamic Friction Torque of Instrument Ball Bearings
Abstract Generated abstract
The paper proposes using an induction mechanism with a traveling magnetic field as a compensating torque source for measuring the dynamic friction torque of instrument ball bearings by the compensation method. It describes a device based on a tangential induction mechanism from a reactive energy meter, with automatic current regulation through an undercompensation photoelectric transducer, so that equilibrium between bearing friction torque and induction torque permits torque to be inferred from winding current. The instrument is analyzed as a closed-loop automatic control system, yielding expressions for dynamic sensitivity and measurement error. The analysis indicates that steady-state error is governed mainly by the induction feedback mechanism and indicator accuracy, potentially not exceeding about 1 percent, while temperature and supply-frequency variations require stabilization for high-accuracy measurements.
Full Text
APPLICATION OF AN INDUCTION REVERSED TRANSDUCER FOR MEASURING THE DYNAMIC FRICTION TORQUE OF INSTRUMENT BALL BEARINGS
E. F. KOLOTII
(Presented by the scientific seminar of the Faculty of Automation and Computer Engineering)
When measuring the dynamic friction torque of instrument ball bearings by the compensation method, it is necessary to create a compensating torque. It is proposed to use for this purpose an induction transducer with a traveling magnetic field. As is known [1, 3], the rotating torque in an induction mechanism arises only in the presence of two or more alternating magnetic fluxes, shifted in phase relative to one another and not coinciding in space. Most modern induction mechanisms have three alternating working magnetic fluxes acting on the moving part and creating a rotating torque. The magnitude of this torque is determined as
\[ M_{\beta}=c\,\frac{f}{\rho}\,\Phi_1\Phi_2\sin\psi, \tag{1} \]
where \(f\) is the frequency of the alternating current;
\(\rho\) is the specific electrical resistance of the disk material;
\(\psi\) is the phase-shift angle between the magnetic fluxes \(\Phi_1\) and \(\Phi_2\);
\(c\) is a proportionality coefficient.
It may be assumed approximately that
\[ \Phi_1 \equiv I_1 \quad \text{and} \quad \Phi_2 \equiv I_2, \]
where \(I_1\) and \(I_2\) are the currents in the first and second windings of the induction mechanism.
If the condition
\[ I_1=\kappa I_2, \]
is satisfied, that is, the currents in the windings are proportional, then expression (1) can be represented in the form
\[ M_{\beta}\simeq c_1\,\frac{f}{\rho}\,I_1^2\sin\psi. \tag{2} \]
Thus, in this case the torque created by the induction mechanism can be determined from the readings of an ammeter measuring the current in one of the windings.
If the scale of the ammeter is linear for the current being measured, then when this ammeter is graduated in units of torque the scale will be quadratic (expanded at the beginning and compressed at the end).
For the purpose of measuring the dynamic friction torque of ball bearings by the compensation method, with the bearing axis in a vertical position and with an axial load present, we used a tangential induction mechanism from a reactive-energy meter, rated for a voltage of up to 100 V and a current of up to 5 A.
Fig. 1. Diagram of an instrument with an induction reversed transducer for measuring dynamic friction torque
The layout of the device with an induction transducer is shown in Fig. 1. When shaft 1 rotates at a constant angular velocity \(\omega\), the outer ring of the bearing under test 2, together with flywheel 3 and disk 4, is carried along by the friction torque of the bearing in the direction of rotation of shaft 1. In addition to the friction torque \(M_{\text{in}}\), a compensating torque \(M_{\beta}\), produced by induction mechanism 5, acts on the moving part of the device. The magnitude of the current through the windings of the induction mechanism is automatically set by means of an undercompensation transducer [2], consisting of illumination device 6 and photoresistor 7, and of an amplifying device that changes the current in the windings of the induction transducer when the illumination of the photoresistor changes. In the established equilibrium position \(M_{\text{in}} = M_{\beta}\), and from the instrument measuring the current in one of the windings of the induction mechanism, one can judge the friction torque of the bearing.
Fig. 2. Structural-functional diagram of the instrument
The structural-functional diagram of the instrument (Fig. 2) is a typical closed single-loop automatic-control system.
For analysis of the dynamic properties, questions of stability, dynamic errors, speed of response, etc., of this device, it is necessary to—
allows the mathematical apparatus of automatic control theory to be applied.
It is convenient to analyze the errors of such a device using the simplified structural-functional diagram (Fig. 3).
For the compensation-conversion circuits, the transfer function or, in other words, the dynamic sensitivity has the form [2]:
\[ S(p)=\frac{I_{\text{out}}(p)}{M_{\text{in}}(p)} =\frac{\kappa(p)}{1+\kappa(p)\beta(p)}, \tag{3} \]
where
\[ \kappa(p)=\frac{I_{\text{out}}(p)}{\Delta M(p)} \]
is the transfer function of the direct-conversion circuit;
\[ \beta(p)=\frac{M_{\beta}(p)}{I_{\text{out}}(p)} \]
is the transfer function of the feedback circuit.
Fig. 3. Simplified structural-functional diagram of the instrument
From expression (3) we find that the static error of the compensation circuit is
\[ \gamma'=\frac{\Delta S}{S}=\gamma_{\kappa}a+\gamma_{\beta}(1-a), \]
where
\[ \gamma_{\kappa}=\frac{\Delta\kappa}{\kappa} \]
is the error of the direct-conversion circuit;
\[ \gamma_{\beta}=\frac{\Delta\beta}{\beta} \]
is the error of the feedback circuit;
\[ a=\frac{\Delta M}{M_{\text{in}}} \]
is the relative undercompensation.
In the total error of the instrument, besides the error of the compensation circuit, the error of the indicating instrument should also be included. Then
\[ \gamma=\gamma' + \gamma_{\text{ind}}, \]
where \(\gamma\) is the total error of the instrument;
\(\gamma_{\text{ind}}\) is the error of the indicating instrument.
In the steady-state mode the value \(\Delta M=0\); therefore the error of the instrument as a whole does not depend on the error of the \(\kappa\) circuit and is determined as
\[ \gamma=\gamma_{\beta}+\gamma_{\text{ind}}, \]
where \(\gamma\) is the error of the instrument in the steady-state mode. Taking into account the fact that the basic error of the induction mechanism does not exceed 0.5% [2], the overall error of the instrument, when a sufficiently accurate indicating instrument is used, may be no higher than 1%.
In measurement it should be taken into account that the temperature error of the induction measuring mechanism, due to variation of the properties of the magnetic core and to variation of the specific electrical resistance of the disk material [see expression (1)] with temperature changes, may reach up to 1% per \(10^\circ\text{C}\).
Instability of the frequency of the supply voltage will also be fully included in the overall error of the induction mechanism.
Thus, in order to obtain high accuracy of readings, it is necessary to employ temperature and frequency stabilization of the device.
References
-
Edited by V. T. Prytkov and A. V. Talitskii. Course in Electrical Measurements. Part I, Gosenergoizdat, Moscow–Leningrad, 1960.
-
M. M. Fetisov. New Automatic Compensation Instruments for Measuring Nonelectrical Quantities. Part I, LDNTI, Leningrad, 1960.
-
P. N. Goryunov, S. M. Pigin, N. N. Shumilovskii. Electric Meters. Gosenergoizdat, Moscow–Leningrad, 1951.