Experience in Using an Electrotorsiograph with an Inductive Sensor

V. K. Nechaev, A. T. Boldov, E. N. Nevstruev

Modern techniques for experimental investigation of torsional vibrations of the shafts of power installations with piston engines are characterized by the broad use of electrotorsiographs with various types of sensors \[1, 2, 4, 5, 8, 9, 12\]. However, in the field of electrotorsiography at the present time, a whole series of questions cannot be considered finally resolved: those connected with the choice of the type and structural design of the sensors, the circuitry of the amplifying and recording apparatus, and especially the methodology of electrotorsiography, calibration, analysis of errors arising during torsiographing, and also the methodology for processing experimental data. In addition, in the process of debugging the instrument, individual minor malfunctions always occur; eliminating them takes considerable time, although they can be avoided if their possible sources are known in advance. These circumstances prompted us to share, in the present article, some experience in using an electric torsiograph with an inductive sensor, created and used by us in the laboratory of internal-combustion engines of the Tomsk Polytechnic Institute¹.

Description of the Electrotorsiograph

The block diagram of the electrotorsiograph, with vibration recording by means of cathode-ray and magnetoelectric oscillographs, is shown in Fig. 1.

The design of the torsiograph sensor is shown in Fig. 2. The aluminum body of the sensor consists of two parts, 1 and 2, tightened by three tie bolts 3. A steel shoe 4, together with the coil 5 glued to it, is clamped by its flange between parts 1 and 2 of the body. A cylindrical permanent magnet (seismic mass) 6 is mounted in the body on ball bearings. The magnet is connected to the body by two elastic spiral springs 7 with opposite winding directions. The inner ends of the springs are fastened in clamp 8, connected to the magnet. The outer ends of the springs are fastened to the body by means of brackets 9 and pins with nuts 10.

The sensor is fastened to the end face of the shaft of the engine being torsiographed by means of adapter 11 and mounting sleeve 12 with locking bolt 13.

One lead of the coil is connected to contact ring 14, insulated from the housing; the other lead is connected to the housing. The contact ring is made of brass. Current collection from the ring is carried out by three copper-graphite brushes mounted on a special holder.

The voltage induced in the coil is proportional to the instantaneous angular velocity of the oscillatory motion of the torsiographed section of the shaft. To obtain signals proportional to the angular displacements of this section during torsional oscillations, the output voltage of the sensor is integrated.

Fig. 1. Block diagram of the electrotorsiograph.

In the diagram: inductive sensor; integrating circuit; cathode-oscillograph amplifier; loop-oscillograph amplifier; cathode oscillograph; electromagnetic oscillograph; speed counter; electronic switch; audio generator.

As the integrating circuit, we used the simplest \(RC\) cell.

Fig. 2. Construction of the torsiograph sensor.

The output voltage of the integrating circuit is fed to the input of the direct-current amplifier (Fig. 3).

The amplifier is powered from a separate rectifier with electronic stabilization of the anode voltage.

The output voltage of the amplifier is applied directly to one pair of horizontal plates of a two-beam cathode oscilloscope.

The second beam of the cathode oscilloscope is used to record time marks and engine revolutions. For this purpose we used an electronic commutator in which two 6N8 triodes, operating into a common load, served as the “switch.” The operation of the triodes is controlled by voltage pulses generated by a symmetrical multivibrator assembled with 6P6 tubes.

Fig. 3. Schematic circuit diagram of the direct-current amplifier.

Marks of the revolutions of the engine shaft and signals for synchronizing the sweep generator of the oscilloscope are obtained by means of a revolution marker and an electromagnetic-type synchronizer installed on this shaft, the diagram of which is shown in Fig. 4. Here: 1—permanent magnet, 2—cores, 3—synchronizer coil, 4—revolution-marker coil, 5—steel rod, 6—engine shaft, 7—flanged bushing fastened to shaft 6.

The rod 5 is bolted to flange 7 (as close as possible to the vibration node of the system shafting) in such a way that the moment when rod 5 passes between cores 2 coincides with the moment when the piston of the first cylinder of the engine passes through top dead center.

Time marks were obtained by means of a ZG-2A audio generator, set to a vibration frequency desirable from the standpoint of convenience in subsequent processing of the oscillograms.

To determine the scale of the vibration recording, a millimeter grid made on a transparent base is placed in front of the oscilloscope screen.

Images of the shaft vibration curve (the torsionogram proper) and of the auxiliary curves on the oscilloscope screen, together with the scale millimeter grid, are photographed—

Fig. 4. Diagram of the induction sensor for electrical pulses of synchronization and dead-center marks.

were recorded on narrow film with the aid of a mirror camera installed in front of the screen. The focal-plane shutter of the camera was set to remain open continuously, and the necessary exposure for the shot was provided by an additional central shutter installed in front of the camera lens.

The need for a short exposure when photographing torsiograms and the green glow of the screen of the imported oscilloscope we used made photography difficult. Satisfactory results were obtained when using domestic “Izopanchrom” photographic film with a sensitivity of 250–350 GOST units. The photographic film was processed in a special oscillographic developer.

In addition to frame-by-frame photography from the screen, we also carried out continuous recording of shaft oscillations with the aid of an MPO-2 magnetoelectric oscillograph. For such recording a separate electronic amplifier with a low-resistance output was used, which made it possible in oscillography to employ a type-one loop, having its own natural frequency of oscillation in air of 5000 hertz.

We give a summary of the principal characteristics of the electrotorsiograph described above.

Overall dimensions of the sensor: diameter (outside)—100 mm, length—95 mm, sensor weight—2.5 kg; natural frequency of oscillation of the sensor—about 7 hertz; practical range of measured oscillations 900–6000 cycles per minute (at higher oscillation frequencies the instrument was not used by the authors); practical range of amplitudes (angular) of the measured oscillations 0.067–5.00 degrees; sensor sensitivity

\[ \eta = 4.0 \frac{\text{mV}}{\text{degree/sec}}; \]

type of damping of the movable system of the sensor: friction in the bearings of the magnet armature; number of turns in the sensor coil—2000.

Calibration of the electrotorsiograph

For instruments with an induction sensor, as a rule, dynamic calibration is necessary—in order to determine the scale of recording oscillations at various frequencies, taking into account the characteristics of the entire electrical path from the sensor to the oscilloscope screen.

Several types of calibrators for dynamic calibration have been described in the literature [5, 6, 9, 12].

We used a cam-type calibrator, the main advantage of which is the simplicity of its construction.

The diagram of the calibrator developed and built in the internal-combustion-engine laboratory of TPI is shown in Fig. 5. Here: 1—a cylindrical cam, mounted eccentrically on the shaft axis of a direct-current electric motor; 2—roller; 3—lever; 4—flange, rigidly connected with the lever; 11—spring. During calibration the sensor described above (Fig. 2) is attached to flange 4. When the electric motor rotates and, consequently, the cam rotates, the sensor housing is imparted harmonic oscillatory motion with a specified frequency within the range from 200 to 6000 cycles/min and an amplitude from 10 minutes to 5.0°. The frequency of these oscillations is adjusted smoothly by changing the speed of the electric motor with rheostats.

The required amplitude of oscillation of the housing is set by installing cams with a specified amount of eccentricity.

During calibration, the output voltage of the sensor is fed to the oscilloscope through the same circuit (with a fixed amplifier gain) as during torsiographing.

The number of revolutions of the cam (the oscillation frequency) is determined, when processing the calibration torsiograms, from the revolution and time marks, which are obtained in the same way as in torsiographing. In preliminarily setting the desired number of revolutions of the cam during calibration (i.e., the frequency of the forced harmonic oscillations of the body of the sensor being calibrated), we used a centrifugal-type auxiliary tachometer.

Fig. 5. Diagram of the calibrator of the electrotorsiograph.

The diagram of the device for measuring the angular amplitudes of oscillation of the sensor during calibration is also shown in Fig. 5.

Here: 5—a shaped rod rigidly attached to flange 4 of the calibrator and electrically insulated from it; 6—a contact plate; 7—a contact fork; 8—a micrometer screw; 9—a micrometer vernier; 10—a neon lamp; 12—a direct-current source.

The amplitude was measured as follows. Contact fork 7 is moved first upward (downward) by means of the micrometer screw, and from the vernier of the device a reading \(K_1\) is taken, corresponding to the instant of flashing of the neon lamp (the instant when the leg of the fork contacts the contact plate). The same is done when the contact fork is moved downward (upward)—reading \(K_2\).

The doubled linear amplitude of oscillation of the sensor flange (at the radius) is determined from the expression

\[ 2\Phi_l = H - S - (K_1 - K_2), \text{ mm} \]

where \(S\) is the thickness of contact plate 6, and \(H\) is the distance between the contacts of fork 7.

The quantities \(H\) and \(S\) were measured in each of the experiments. The angular amplitude of oscillation of the sensor will then be equal to\(^1\)

\[ \Phi_T = \frac{\Phi_l}{r}\ \text{radian} = \frac{180}{\pi}\cdot \frac{\Phi_l}{r}\ 60\ \text{minutes} \]

or, in our case, for \(r = 105\) mm,

\[ \Phi_T = 16.4\,[H - S - (K_1 - K_2)]\ \text{minutes}. \tag{a} \]

\(^1\) In the Internal Combustion Engine Laboratory of TPI, an optical calibration was also developed (see the article by V. K. Nechaev and A. T. Bolgov, “On determining hysteresis energy losses in shafts,” Izvestiya TPI, vol. 85, Tomsk, 1956).

As repeated measurements showed, the amplitude \(\Phi_T\) of the sensor housing proved to be the same over the entire interval of oscillation frequencies of interest to us (up to 6000 oscillations per minute), i.e., over the interval of rotational speeds of the cam and the electric-motor shaft. This should be regarded as evidence of the sufficient rigidity of the entire calibration device shown in Fig. 5 and of sufficient tightening of the spring pressing the roller against the cam. Now, being confident that the value of \(\Phi_T\) is independent of the rotational speed of the cam, it was possible to determine the frequency characteristic of the entire measuring channel: sensor—integrating circuit—amplifier—oscillograph. For this purpose it was sufficient to measure the values of the deflections \(B\) (mm) of the beam on the screen of the cathode-ray oscillograph (or of the trace on the film when using a loop oscillograph) with the same cam, i.e., with the unchanged angular amplitude of oscillations of the housing of the sensor being calibrated, as indicated earlier, and at different rotational speeds of this cam. Of course, the setting of the adjustment knobs of the electronic amplifier connected between the sensor and the oscillograph (Fig. 1) was kept unchanged throughout.

Fig. 6 gives, as an example, a calibration graph obtained when recording oscillations by means of a cathode-ray oscillograph.

Fig. 6. Calibration graph of the electrotorsiograph.

To check the linearity of the entire measuring channel from the sensor to the oscillograph screen, we recorded the amplitude characteristics of the electrotorsiograph. One of these characteristics is shown in Fig. 7 (the change in amplitudes \(\Phi\) was provided by changing cams with different eccentricities).

It is evident from Figs. 6 and 7 that the electrotorsiograph described, with a cathode-ray oscillograph, has a good linear characteristic.

A similarly good linear characteristic was obtained by us when operating this electrotorsiograph with a loop oscillograph.

To obtain reliable calibration results, it is desirable that the calibration device impart to the sensor being calibrated pure monoharmonic angular oscillations. We performed an analysis of the distortions of the oscillation form caused by the design of the calibrator by means of

M. E. Garf [3]. The distortion coefficient of the shape of the oscillations of the calibration device \(K_{\phi}\) is determined by the expression

\[ K_{\phi}=\frac{u_{1\max}-u_{2\max}}{u_{2\max}}\cdot 100\%, \]

where \(u_1\) is the actual acceleration obtained by the sensor from the calibration device;
\(u_2\) is the acceleration calculated from the actual amplitude of the sensor housing (measured with a neon indicator), assuming strictly sinusoidal oscillations.

Fig. 7. Graph of amplitude calibration of the electrotorsiograph.

The calibration device described, constructed according to the scheme

Fig. 8. Actual (a) and equivalent (b) diagrams of the calibration device.

of Fig. 8a, shall be represented in the form of a connecting-rod–crank mechanism according to Fig. 8b. For this mechanism we have

\[ K_{\phi}=\left(\frac{\varepsilon_k}{l_p\sqrt{1-\left(\frac{\varepsilon_k}{l_p}\right)^2}\arcsin\frac{\varepsilon_k}{l_p}}-1\right)\cdot 100\%. \]

The values of the coefficient \(K_{\phi}\), calculated from the last expression for the calibration device described here (in our case \(l_p = 80\ \mathrm{mm}\); \((\varepsilon_k)_{\max}=3.5\ \mathrm{mm}\)), do not exceed 0.37%. This indicates that the adopted design of the calibrator, with the dimensions we selected, ensures the production of practically pure monoharmonic oscillations. The small high-frequency “superpositions” that occurred during calibration are explained mainly by errors in the manufacture and assembly of the calibrator mechanism.

Current Collection in Electrotorsiography

Sliding of the brushes of the current-collecting device, as a rule, leads to a certain instability of the contact resistance, which can distort the results of torsiography. In our case, during calibration on the device described above (Fig. 5), the sensor does not rotate. This may lead to an additional error.

We tested the operation of the current-collecting device according to the circuit shown in Fig. 9. During the test, the magnet was removed from the sensor. A second current-collecting ring was mounted on the sensor housing. To this ring was connected the coil lead which, in the operating

Labels in the diagram: rings; brushes; audio generator; oscilloscope; amplifier.

Fig. 9. Circuit for testing the operation of the brushes of the current-collecting device.

state, is connected to ground. The sensor was fastened to the shaft of the electric motor of the calibration device and set into rotation. An audio generator and an amplifier were connected into the circuit of the sensor coil through the brushes. Since the circuit of the audio generator therefore includes brushes, if there are substantial fluctuations of the contact resistance at the point of contact between the brush and the ring, distortions of the waveform or amplitude modulation of the sinusoid will be observed on the oscilloscope screen. When the current-collecting device was tested over the speed range from zero to 2000 rpm, no distortions of the waveform or of the voltage magnitude at the amplifier output were observed.

Selection of Parameters and Verification of the Operation of the Integrating Cell

The parameters of the integrating cell (the values \(R\) and \(C\)) were selected by us according to G. L. Shnirman [10], on the basis of the permissible value of the “scale distortions” \(\Delta V\), equal to

\[ \Delta V=\frac{100}{2\pi fRC}\%, \]

where: \(f\) is the frequency (in hertz) of the process being integrated;
\(R\) is in ohms; \(C\) is in farads.

In order to obtain an objective judgment of the qualities of the adopted integrating cell, its operation was checked by three methods.

1. Signals of Π-shaped form from a special multivibrator are fed to the input of the integrating cell. The output voltage of the integrating circuit is fed through an amplifier to the plates of an oscillograph. With good integration, the Π-shaped voltage should be transformed in the cell into a sawtooth voltage.

2. The frequency characteristic of the integrating circuit is determined experimentally: the dependence of the amplitude of the output (integrated) sinusoidal voltage on its frequency when a voltage of constant amplitude and variable frequency from an audio generator is applied to the input of the cell.

It can be shown that, with exact integration, the points of the indicated characteristic lie on a hyperbola.

1. The phase shift between a sinusoidal voltage and its integral is determined. With exact integration, the phase shift is \(90^\circ\). The actual value of this angle is determined from Lissajous figures, or, on the screen of a two-beam oscillograph, by simultaneous observation of the curves of the voltage supplied to the cell and taken from it.

All these verification methods used by us showed that, at values \(\Delta V = 5 \div 10\%\), a simple integrating cell of the \(RC\) type provides sufficiently high-quality integration over the entire frequency range of interest to us.

It should be noted that, in the torsiograph described, a certain inaccuracy of integration at very low frequencies is compensated by calibration: during calibration and torsiographing, the signals from the sensor are fed to the oscillograph through the same circuit.

Method of Electrotorsiographing a Piston Engine

We adopted the following method of electrotorsiographing (when using a cathode oscillograph).

The sensor was fastened to the end of the shaft of the engine being torsiographed. A specified number of shaft revolutions was set near the calculated critical number of engine revolutions. After synchronization of the oscillograph sweep generator with the recorded oscillatory process (stopping the image on the screen), the necessary amplifier gain was selected so as to make fuller use of the screen area.

The frequency of the output voltage of the audio generator was also synchronized with the sweep frequency. Then photographs were taken of the oscillogram images on the screen with the aid of a narrow-film mirror camera.

At the same time the necessary data were entered into the test log, in particular, the frequency of the audio generator \(f_3\).

At first, at the specified engine speed condition, filming was carried out 2–3 times. The exposure was selected in such a way that only one complete oscillation cycle was recorded in a single frame of the film. Later it proved more expedient to record several oscillation cycles in one frame. In this case there is less consumption of photographic materials and it becomes possible to determine how steadily the specified operating condition of the engine, as well as the mode of the oscillations being studied, is maintained.

Then another desired number of engine revolutions was set and, in a manner analogous to that described above, torsiograms were recorded.

In all, at least 30 torsiograms were taken near each calculated resonance.

After completion of the planned program of torsiographing, the sensor was removed from the engine shaft and, after installation on the calibration device, was subjected to a control calibration according to the method described above.

Then the films (with the principal and calibration torsiograms) were developed, and the frames on them were numbered. Enlarged photocopies of the torsiograms were subjected to further processing.

In processing the calibration torsiograms, the total amplitudes of the oscillations of the beam on screen \(B\) in mm were determined with the aid of a scale grid, and the revolutions of the calibrator cam (the oscillation frequency) were determined; a calibration graph was then constructed.

The number of revolutions of the cam was determined from the expression

Fig. 10. Example of a torsiogram record.

\[ n_k=\frac{60 f_3}{m},\ \text{rpm}. \tag{1} \]

where \(m\) is the number of periods on the time-mark line corresponding to one revolution of the shaft.

On the basis of the calibration graph, the scale of recording of the oscillation amplitudes \(\mu_{\varphi}\) was determined by the formula

\[ \mu_{\varphi}=\frac{2\Phi_T\ \text{min}}{B\ \text{mm}}. \tag{2} \]

The processing of torsiograms taken from the shaft of the installation was carried out as follows.

1. The number of revolutions of the shaft \(n\) was determined from expression (1).

2. Harmonic analysis of the torsiograms was performed by the graphical Fischer-Hinnen method; all auxiliary constructions were made directly on the photocopy of the torsiogram.

3. Resonance curves were constructed for the engine shaft.

4. From the resonance curves, the resonance engine speeds \(n_{\text{res}}\) and the resonance amplitudes \(\Phi_{\text{res}}\) were determined for the torsiographed section of the shaft.

The amplitudes of oscillation of the torsiographed section of the shaft, when processing torsiograms, were determined from the expression

\[ \Phi=\overline{\Phi}\cdot \mu_{\phi}=\overline{\Phi}\frac{2\Phi_T}{B}\quad(\text{minutes}), \tag{3} \]

where \(\overline{\Phi}\) is the segment (in mm) on the torsiogram corresponding to the desired amplitude value. In Fig. 10, as an example, a torsiogram is given that was taken from the shaft of one experimental motor installation in the laboratory of internal-combustion engines of the TPI¹.

Figure 11 presents one of the resonance curves for the same experimental installation at resonances of the second \((h=2)\) and third \((h=3)\) orders.

Torsiographing with the use of a loop oscillograph is less laborious. We performed continuous torsiographing “through resonance” twice: with an increase and a decrease in the rotational speed of the torsiographed shaft. In this case there is no need to make photocopies. The amplitudes of oscillation and the shaft speeds are determined directly from the curves on the film. Processing of the torsiograms is also completed by constructing resonance curves.

Fig. 11. Resonance curve of the shaft of a motor installation.

Analysis of torsiographing errors

As is known, the limiting absolute error of a function of several \((p)\) independent variables of the form

\[ y=F(x_1,\ x_2,\ x_3,\ldots,\ x_p) \tag{4} \]

is determined by the expression

\[ \delta_y = \pm \sum_{i=1}^{i=p} \left| \frac{\partial F(x_1, x_2, x_3, \ldots, x_p)}{\partial x_i}\,\delta x_i \right|, \tag{5} \]

where \(\delta x_i\) is the error in determining the independent variable \(x_i\). The limiting relative error of function (4) is determined as follows:

\[ (\delta_y)_{\mathrm{rel}} = \pm \frac{\delta y}{y}\,100\%. \tag{6} \]

The limiting absolute error \(\delta_n\) of each separate determination of the shaft speed \(n\), on the basis of (1) and (5), is equal to

\[ \delta_n = \pm 60\left(\frac{\delta f_3}{m} + \frac{f_3}{m^2}\,\delta m\right)\ \text{rpm}. \]

Here: \(\delta f_3\) is the limiting absolute error in reading the frequency \(f_3\) on the scales of the ZG-2A audio generator;

\(\delta m\) is the limiting absolute error in estimating the quantity \(m\) when processing torsiograms.

The calibration error of the scale of the ZG-2A generator is equal to \(\pm 2\% \pm 2\) hertz.

The calibration error of the detuning dial is \(\pm 3\% \pm 3\) hertz. The reading on the detuning dial was less than 50 hertz, i.e., less than half of the entire interval of the dial scale (100 hertz). Therefore we take the reading error on the detuning dial to be 4 hertz. Then the total frequency reading error is

\[ \delta f_3 = \pm (0.02 f_3 + 6)\ \text{hertz}. \]

We take the reading error of the quantity \(m\) to be 0.5. Finally we have

\[ \delta m = \pm 60\left(\frac{0.02 f_3 + 6}{m} + \frac{0.5 f_3}{m^2}\right)\ \text{rpm}. \tag{7} \]

The limiting absolute error of each separate determination of the magnitude of the oscillation amplitude \(\Phi\), on the basis of (3) and (5), is equal to

\[ \delta\Phi = \pm 2\left(\frac{\Phi_T}{B}\,\delta\Phi + \frac{\Phi}{B}\,\delta\Phi_T + \frac{\Phi\Phi_T}{B^2}\,\delta B\right)\ \text{minutes}. \]

Here: \(\delta\Phi\), \(\delta\Phi_T\), \(\delta B\) are the limiting absolute errors in determining, respectively, the segment on the torsiogram (in mm), the angular amplitude of the lever oscillations during calibration (in minutes), and the total sweep of the beam oscillations on the screen (on the film) during calibration (in mm).

The errors in measuring the quantities \(\Phi\) and \(B\) are determined mainly by the error in reading these quantities from the torsiograms (the main and the calibration ones). With careful processing of the torsiograms (and sufficient enlargement of the torsiograms when printing photocopies), using a millimeter scale grid recorded on the torsiograms, readings can be taken with an accuracy of up to 0.5 mm. For reliability we assume \(\delta\Phi = \delta B = 1\) mm.

On the basis of \((a)\) and (5) we have

\[ \delta \Phi = \pm 16.4(\delta H+\delta S+2\delta K)\ \text{(minutes)}, \]

where \(\delta H, \delta S\) are the errors in certifying the dimensions \(H\) and \(S\), and \(\delta K\) is the error of reading on the micrometric vernier (see Fig. 10).

The dimension \(H\) during calibration was measured as follows. First, the dimension \(H\) was reproduced (taken off) with the aid of an adjustable gauge, and then the gauge was measured with the aid of a passometer with a division value of \(0.002\ \text{mm}\) (an instrument of the 2nd accuracy class). The limiting absolute error of the passometer readings is \(\pm 0.002\ \text{mm}\) over the entire scale range \((0 \div 15\ \text{mm})\), see [11].

The dimension \(S\) was measured with the aid of a micrometer with conical tips and a division value of \(0.01\ \text{mm}\) (an instrument of the 2nd accuracy class). The limiting absolute error of 2nd-class micrometers is \(\pm 0.008\ \text{mm}\).

The micrometric device for measuring amplitudes was a micrometric pair of a measuring microscope and therefore should be classed as an instrument of the 1st accuracy class. However, taking into account that the reading on the micrometric vernier is taken under conditions of some shaking of the calibrator, we take the reading accuracy \(K=\pm 0.008\ \text{mm}\), equal to the accuracy of instruments of the 2nd class.

Now

\[ \delta \Phi_T = \pm 16.4(0.002 \div 0.008 \div 2\cdot 0.008)=0.427\ \text{minutes}. \]

For reliability we assume

\[ \delta \Phi_T = 0.5\ \text{minutes}. \]

Finally we have

\[ \delta \Phi_T=\pm 2\left(\frac{\Phi_T}{B}+\frac{0.5\overline{\Phi}}{B}+\frac{\overline{\Phi}\cdot \Phi_T}{B^2}\right)\ \text{minutes}. \]

Table 1 gives the values of the limiting relative errors of individual measurements of the numbers of revolutions and amplitudes, determined by the method described above for the most unfavorable torsion-recording conditions encountered in our work¹.

The same table gives the errors of torsion recording under the same conditions with the aid of an ordinary mechanical torsion recorder of the Geiger type. The method of analyzing these errors is given in [13].

Table 1

From Table 1 it is evident that electrotorsiography provides substantially greater measurement accuracy than torsiography with a Geiger instrument.

Prolonged use of the electrotorsiograph described has shown that it is reliable in operation and that its readings are stable. With the aid of the electrotorsiograph it is possible to obtain a torsiogram of large dimensions, convenient for processing. All this gives grounds for recommending torsiographs of the type described above for broad practical use.

REFERENCES

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2. Volchok L. Ya. Electrical Methods of Measurement in Internal-Combustion Engines. NIDI. Book 8, Mashgiz, 1948.

3. Garf M. E. Distortions of the form of oscillations in test installations. Academy of Sciences of the Ukrainian SSR. Scientific Transactions of the Institute of Machine Science and Agricultural Mechanics. Vol. III. Dynamics and Strength of Structures. Kiev, 1951.

4. Measurement of Mechanical Quantities by Electrical Methods. Collection of works published in foreign literature. Edited by Prigorovskii. Mashgiz. Moscow, 1952.

5. Karas’ V. Z. Rubber dampers of torsional oscillations. NAMI, issue 59, Mashgiz. Moscow, 1951.

6. Malkin L. A. Involute gearing for obtaining periodically varying speeds. Vestnik mashinostroeniya, vol. XXVI, No. 7–8, 1946.

7. Menli R. Analysis and Processing of Oscillation Records. Mashgiz. Moscow–Leningrad, 1948.

8. Raevskii N. P. Methods of Experimental Investigation of Mechanical Parameters of Machines. Academy of Sciences of the USSR. Moscow, 1952.

9. Raevskii N. P. Sensors of Mechanical Parameters of Machines. Academy of Sciences of the USSR. Moscow, 1959.

10. Shatalov K. T. Problems of experimental investigations of torsional oscillations of engine shafts. Collection Dynamics and Strength of Crankshafts. Academy of Sciences of the USSR, Moscow–Leningrad, 1948.

11. Shnirman G. L. Electrical methods of differentiation and integration. Academy of Sciences of the USSR. Proceedings of the Seismological Institute, No. 105, Moscow–Leningrad, 1940.

12. Inspection of Means for Measuring Dimensions in Machine Building. Collection of materials and instructions. Committee for Measures and Measuring Instruments under the Council of Ministers of the USSR, Moscow, 1948.

13. Evaluation of Effects of Torsional Vibration. SAE. War Engineering Board, NY, 1945.

14. Bolgov A. T. On certain characteristics of damping torsional oscillations of shafts of installations with piston internal-combustion engines. Candidate dissertation. Tomsk, TPI, 1956.

1. The greatest relative errors are obtained, obviously, when torsion-recording vibrations of small amplitude at low revolutions. In our case we have \(n_{\text{res}}=700\ \text{rev/min}\) and \((\Phi_{\text{res}})=12 \div 15\) minutes. ↩↩↩