ON THE DESIGN OF AN ELECTRIC DRIVE FOR TORSIONAL OSCILLATIONS

S. A. Lukovnikova

(Presented by the scientific and technical seminar of the Department of Electrical Equipment, AEMF)

In low-power drives with torsional oscillations of the output shaft, it is expedient to use, as the actuating element, an asynchronous two-phase electric motor with a hollow rotor, since in the oscillatory mode it can provide regulation of the amplitude \(\varphi_m\) and frequency \(f_{\text{osc}}\) of shaft oscillations over wide ranges and according to the most varied laws [1].

In a closed-loop control system, the measuring element will have a substantial influence on accuracy and speed of response. This paper considers the possibility of using oscillatory modulation of the luminous flux for measuring \(\varphi_m\) and \(f_{\text{osc}}\). This type of transducer has a number of significant advantages over contact and electromagnetic transducers and is very convenient for use in optoelectronic systems.

Figure 1 presents the modulation function \(h(t)\) when one of the modulating raster disks oscillates according to a sinusoidal law, for the case in which the center of the circular equal-energy light spot coincides with the centers of the modulating disks. The calculation of \(h(t)\) was performed from the relations given in [2] (the notation for all quantities is the same).

In the curves constructed for various \(\varphi_m^0\) and \(\varphi_{н1}^0\), one can see characteristic extrema 1 and 2, the magnitude and mutual position of which are rigidly connected with \(\varphi_m^0\) and \(\varphi_{н1}^0\). If the line \(h = 0.5\) is taken as the origin, then the distance \(\Delta h_1\) from it to the mean straight line between the characteristic extrema \((0.5 \Delta h_2)\) is proportional to \(\varphi_{н1}^0\), and the distance between the characteristic extrema \(\Delta h_2\) is proportional to \(\varphi_m^0\). It must be borne in mind, however, that what is involved here is the fractional part of the quantities \(\varphi_m^0\) and \(\varphi_{н1}^0\). The integer part of \(\varphi_m^0\) is determined by the number of periods of the segment \(h(t)\) between the characteristic extrema, while the integer part of \(\varphi_{н1}^0\) does not affect the shape of the curves \(h(t)\). To determine the fractional parts of \(\varphi_m^0\) and \(\varphi_{н1}^0\), one may use the equations:

\[ \begin{aligned} \varphi_m^0 &= 0.25\,\Delta h_2,\\ \varphi_{н1}^0 &= 0.25\,(1 - 2\Delta h_1). \end{aligned} \tag{1} \]

Let us note that when \(\varphi_{н1}^0 = 0.25\), \(\Delta h_1 = 0\), and the position of the extrema will depend only on \(\varphi_m^0\). Calculation of the dependence of the amplitude of the \(i\)-th harmonic of the curve \(h(t)\) on the value \(\varphi_m^0\) for various \(\varphi_{н1}^0\) showed,

Fig. 1. Modulation function \(h(t)\) for various values of \(\varphi_m^0\) and \(\varphi_{n1}^0\)

that its form is determined by the Bessel function of the \(i\)-th order, as is seen, for example, from Fig. 2, for the constant component \(h_0\) and the first harmonic \(h_1\).

The use of these dependences for finding \(\varphi_m^0\) is expedient, for \(\varphi_{n1}^0 = 0.25\), at the points of extrema. Since the period of the Bessel functions for small arguments is approximately equal to \(\pi\) and, moreover, because of the presence in the expressions [2] determining \(h_i\) of the factor \((2m-1)^{-2}\), it may be assumed that the roots \(h_i(\varphi_m^0)\) and the points where \(h_0=25\) are determined from the equation \(I_i[2\pi\varphi_m^0]=0\). In this case, on the basis of tabular data, the following approximate relations can be obtained:

\[ [\varphi_m^0]_k^{(0)} = 0.38274 + 0.49921\,(k-1), \]

\[ [\varphi_m^0]_k^{(1)} = 0.60984 + 0.50148\,(k-1), \]

\[ [\varphi_m^0]_k^{(2)} = 0.81736 + 0.50543\,(k-1), \]

\[ [\varphi_m^0]_k^{(i)} = [\varphi_m^0]_3^{(i)} + [0.50762 + 0.0032627\,(i-4)](k-3), \tag{2} \]

a)

b)

Fig. 2. Dependence of the constant component (a) and the first harmonic (b) \(h(t)\) on \(\varphi_m^0\) and \(\varphi_{H1}^0\)

where \([\varphi_m^0]_k^{(i)}\) is the \(k\)-th root of the function \(h_i(\varphi_m^0)\). The latter expression in (2) is valid for \(i, k \ge 3\). The table gives the values of \([\varphi_m^0]_k^{(i)}\) for \(k = 1, 2, 3\) and \(i \ge 3\). The error arising when using relations (2) does not exceed 1%. The indicated relations can be used to estimate the values of \(\varphi_m^0\) at which the \(i\)-th harmonic has maximum amplitude, if it is assumed that the maximu-

the amplitude maxima (the extrema points \(h_i(\varphi_m^0)\)) lie midway between the roots \([\varphi_m^0]_k^{(i)}\).

Analysis of the modulation function \(h(t)\) for the case in which the light spot is located on the periphery of the modulating disks makes it possible to obtain conclusions similar to those presented above. However, in this case there appears a strong additional dependence of \(h(t)\) on the spot parameters \(\varphi_{\mathrm{na}}^0\), \(\varphi_{\mathrm{da}}^0\), which can substantially complicate the regulator \(\varphi_m\) as a whole.

Table

Fig. 3. Oscillograms \(h(t)\), obtained with a modulator based on an oscillatory motor \((a, b)\) and RP-4 \((v, g)\).

For the purpose of experimentally verifying the calculation results, a study was carried out of oscillatory modulation implemented with the aid of an oscillatory electric motor with a hollow shaft and an RP-4 relay. In the first case, modulating disks of diameter \(30\) mm and \(N=24\) were used; in the second, the gratings corresponded to disks of diameter \(200\) mm and \(N=120\).

The frequency and amplitude of the oscillations in the first case were varied by appropriate adjustment of the power-supply parameters of the electric motor, and in the second by using a spring, supporting the grating, made of different materials (steel and bamboo).

Figure 3 presents oscillograms \(h(t)\), obtained with the oscillatory motor for the case in which the light spot is located at the center of the disks (a, b, c) and with the aid of a relay for a spot outside the center of the disks (d, e, f). The oscillograms correspond to the parameters:

a) \(\varphi_m^0=1.375;\ \varphi_{н1}^0=0.25;\)
b) \(\varphi_m^0=2.125;\ \varphi_{н1}^0=0.125;\)
c) \(\varphi_m^0=1,\ \varphi_{н1}^0=0;\)
d) \(\varphi_m^0=1.125,\ \varphi_{н1}^0=0.25.\)

The good agreement of the form of \(h(t)\), obtained by calculation and experimentally, and also the satisfactory agreement (discrepancy no more than \(15\%\)) of the calculated and experimental dependences \(\Delta h_2(\varphi_m^0)\) (Fig. 4, a) and \([\varphi_m^0]_1^{(i)}(i)\) (Fig. 4, b) make it possible to recommend relations (1) and (2) for use.

Fig. 4. Calculated and experimental dependences \(\Delta h^2(\varphi_m^0)\) and \([\varphi_m^0]_1^{(i)}(i)\)

Thus, in designing a low-power automated electric drive for torsional oscillations on the basis of an asynchronous two-phase electric motor with a hollow rotor, the use of oscillatory modulation of the luminous flux is promising. In this case it is advisable to direct the luminous flux to the center of the modulating disks, to keep the parameter \(\varphi_{н1}^0\) constant and equal to \(0.25\), and to vary \(\varphi_m^0\) in accordance with relations (1) and (2). The oscillation frequency \(f_{\text{osc}}\) is determined by the repetition frequency of the characteristic extrema in \(h(t)\).

REFERENCES

1. P. I. Goskov et al. Proceedings of the 6th Scientific-Technical Conference on Problems of Production Automation, vol. 3, Tomsk, 1969.

2. V. I. Lukovnikov. “Estimation of sinusoidal oscillatory modulation of luminous flux.” In the present collection.