PROCEEDINGS
OF THE TOMSK ORDER OF THE RED BANNER OF LABOR POLYTECHNIC
INSTITUTE named after S. M. KIROV

Vol. 139　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　1965

ON THE NATURE OF THE AMPLITUDE–FREQUENCY CURVES OF AN ELECTROMAGNETIC VIBRATOR OPERATING FROM A PULSATING CURRENT SOURCE

Yu. Ya. Kovyilin

(Presented by the scientific seminar of the departments of the Faculty of Automatic Systems)

Under certain assumptions, the force applied to the armature 1 from the side of the magnet 2 of an electromagnetic vibrator (Fig. 1) is determined by the well-known formula

\[ P=\frac{L_0 S}{(S-Y)^2}\cdot \frac{i^2}{2}. \tag{1} \]

Here \(i\) is the strength of the current flowing in the magnet winding;
\(S\) is the reduced length of the air gap between the electromagnet and the armature;
\(Y\) is the instantaneous value of the displacement of the armature, measured from the position of static equilibrium;
\(L_0\) is the inductance of the electromagnet winding at \(Y=0\).

If a pulsating current flows through the electromagnet winding,

\[ i=i_0[1+\xi\cos(\omega t+\delta)], \tag{2} \]

then the motion of the armature will be described by the following nonlinear differential equation:

\[ M\frac{d^2Y}{dt^2}+H\frac{dY}{dt}+CY = P_0 \frac{[1+\xi\cos(\omega t+\delta)]^2} {\left(1+\dfrac{Y}{S}\right)^2}. \tag{3} \]

Fig. 1.

Here \(M\) is the mass of the moving parts of the vibrator, reduced to the armature;
\(H\) is the coefficient accounting for energy absorption in the mechanical circuit of the vibrator;
\(C\) is the spring stiffness coefficient;
\(P_0\) is the force that the electromagnet will develop at \(Y=0\), if a constant current \(i_0\) flows through its winding;
\(\xi\) is a coefficient equal to the ratio of the amplitude of the harmonic component of the current to the magnitude of the constant component of the current \((0\le \xi \le 1)\);

\(\omega\) — the circular frequency of current variation;
\(t\) — time.

An exact solution of equation (3) is unknown. Attempts to find approximate solutions have been made repeatedly. In most known works the solutions are constructed under the assumption

\[ \left|\frac{Y}{S}\right|_{\max} \ll 1. \]

In this case it is considered possible to restrict oneself to the first one [1], two [2], or, at most, three [3] terms of the expansion

\[ \frac{1}{\left(1-\dfrac{Y}{S}\right)^2} \sim 1+2\frac{Y}{S}+3\left(\frac{Y}{S}\right)^2+\ldots \tag{3′} \]

But since vibrators are usually operated at values

\[ \left|\frac{Y}{S}\right|_{\max}=0.5 \div 0.7 \]

in order to make the greatest possible use of the gap (for reasons of reducing the dimensions and weight of the vibrators and saving electric power), it becomes obvious that the results obtained on the basis of (3′) do not cover the cases of greatest practical importance.

In [4] the method of direct linearization [5] was applied to equation (3), without using the expansion (3′). However, the great mathematical difficulties arising in the study of equation (3) by this method over a wide frequency range forced the author of [4] to confine himself to an approximate consideration of only three special cases: \(\delta=0\) (far before resonance); \(\delta=-\dfrac{\pi}{2}\) (resonance); \(\delta=\pi\) (far beyond resonance).

This, of course, is insufficient for judging the amplitude-frequency characteristics of an electromagnetic vibrator, since, for example, most vibration-conveying machines with electromagnetic excitation of oscillations are tuned to a near-resonance regime. The latter, as is known, helps to reduce the influence of instability of the coefficient \(H\) on the amplitude of oscillation and at the same time makes it possible to provide the required vibration amplitude of the armature with a comparatively small magnitude of the disturbing force.

The purpose of this article is an approximate study of the general character of the amplitude-frequency curves of a single-stroke electromagnetic vibrator with bias magnetization in the range of near-resonance regimes.

Denoting

\[ \frac{Y}{S}=y,\qquad \frac{C}{M}=\omega_0^2,\qquad \frac{P_0}{CS}=y_c \]

and, in addition, in accordance with the known hypothesis of Bock, taking

\[ \frac{H}{M}=h_0\frac{\omega_0}{\omega}, \]

we give equation (3) a more convenient form:

\[ \frac{1}{\omega_0^2}\frac{d^2 y}{dt^2} +\frac{h_0}{\omega_0\omega}\frac{dy}{dt} +y = y_c \frac{\left[1+\xi\cos(\omega t+\delta)\right]^2}{(1-y)^2}. \tag{4} \]

We shall seek an approximate periodic solution of this equation in the form

\[ y=a_0+a\cos\omega t, \tag{5} \]

where \(a_0\) is the dimensionless displacement of the mean line of oscillation, arising as a consequence of the “asymmetric” nonlinearity and of the presence of a constant component of the disturbing force;

\(a\) is the dimensionless amplitude of the fundamental tone of the armature oscillation.

Let us require that (4) be satisfied by the solution (5), at least at those instants of time when \(t=0;\ \dfrac{\pi}{2\omega};\ \dfrac{\pi}{\omega}; \ldots\) Then, substituting (5) into (4) and omitting in the denominator of the right-hand side the quantity \(a_0\), which is small in comparison with unity, we obtain a system of equations for determining \(a\) and \(a_0\):

for \(t=0\)

\[ a_0+a\left[1-\left(\frac{\omega}{\omega_0}\right)^2\right] = yc\,\frac{\left[1\pm \xi\sqrt{1-\left(\frac{a}{a_m}\right)^2}\right]^2}{(1-a)^2}, \tag{6} \]

for \(t=\dfrac{\pi}{\omega}\)

\[ a_0-a\left[1-\left(\frac{\omega}{\omega_0}\right)^2\right] = yc\,\frac{\left[1\mp \xi\sqrt{1-\left(\frac{a}{a_m}\right)^2}\right]^2}{(1+a)^2}. \tag{7} \]

Here

\[ a_m=\frac{2yc\,\xi\omega_0}{h_0}. \]

The system obtained is easily solved graphically. For this purpose, in the \(x\)—\(z\) coordinates (Fig. 2), an auxiliary contour is constructed, formed by the curves \(z_1(x)\) and \(z_2(x)\) according to the equations

\[ z_1=yc\,\frac{\left[1+\xi\sqrt{1-\left(\frac{x}{a_m}\right)^2}\right]^2}{(1-x)^2}, \tag{8} \]

\[ z_2=yc\,\frac{\left[1-\xi\sqrt{1-\left(\frac{x}{a_m}\right)^2}\right]^2}{(1-x)^2}, \tag{9} \]

where \(x\) may take both positive and negative values, \(-a_m\leq x\leq a_m\). It is not difficult to show that the curves \(z_1(x)\) and \(z_2(x)\) are smoothly conjugate to one another at the points with abscissae \(x=\pm a_m\) (see Fig. 2).

Fig. 2.

Now, assigning some value to the oscillation amplitude \(a\leq a_m\) (moreover \(a<1\)), we find the points of intersection of the constructed contour with the ordinates \(x=a\) and \(x=-a\). Next, through the obtained points \(1, 2, 2', 1'\) we draw the straight lines \(1—2\) and \(1'—2'\), whose angular coefficients

of which, as is clear from (6) and (7), are equal to \(1-\left(\dfrac{\omega}{\omega_0}\right)^2\). Thus, from the angular coefficients of the constructed straight lines, two values of \(\dfrac{\omega}{\omega_0}\) are determined which correspond to the specified oscillation amplitude \(a\).

The segments on the ordinate axis, bounded on one side by the origin and on the other by the points of intersection of the straight lines \(1—2\) and \(1'—2'\) with the ordinate axis, are equal (in the scale of the drawing) to the displacements \(a_0\) for the specified oscillation amplitude, corresponding to the two values of \(\dfrac{\omega}{\omega_0}\) just found.

Following the method set forth, it is not difficult to outline a procedure for the numerical determination of the quantities \(\dfrac{\omega}{\omega_0}\) and \(a_0\) corresponding to a specified oscillation amplitude \(a\). It consists of the following. Substituting \(x=\pm a\) into (8) and (9), we find the values \(z_1(a)\), \(z_1(-a)\) and \(z_2(a)\), \(z_2(-a)\). Further, those frequency ratios at which the specified amplitude can occur are determined by the formula

\[ \frac{\omega}{\omega_0} = \sqrt{ 1 \mp \frac{z_1(\pm a)-z_2(\mp a)}{2a} }, \tag{10} \]

where the upper signs must be taken first, and then the lower signs.

The approximate values of \(a_0\), respectively, are

\[ a_0= \frac{z_1(\pm a)+z_2(\mp a)}{2}, \tag{11} \]

and here also, as in the preceding case, the upper signs must be taken first, and then the lower signs.

Figure 3 shows the amplitude-frequency curves of an electromagnetic vibrator, constructed by the approximate method described, for

\[ \frac{h_0}{\omega}=0.05;\quad \xi=1;\quad y_c=0.01;\quad y_c=0.015;\quad y_c=0.02. \]

Fig. 3.

As we see, with an increase in the exciting force there occurs not only a shift of the maxima of the amplitude curves to the left, which is in agreement with [4], but at the same time the very shape of the amplitude curves changes substantially. If, for small forcing forces, they have practically the same form as in an ordinary linear system, then for sufficiently strong disturbances the left branch of the amplitude curve takes on a form characteristic of a nonlinear system with a soft spring characteristic, whereas the form of its right branch, within the physically realizable region, is close to that of a linear system. In connection with this, during operation “in the resonance region” unstable regimes should occur. This conclusion is confirmed-

is confirmed by the solution of equation (4) on the MN-7 electronic model and by direct experiments with an electromagnetic vibrator.

REFERENCES

1. E. U. Zharikov. On the problem of calculating the mechanical part of single-tube vibratory conveyors with electromagnetic and mechanical exciters. Transactions of the Novocherkassk Polytechnic Institute, vol. 108, Novocherkassk, 1960.

2. O. Taniguti, M. Sakata, M. Sudzuki. On the steady-state motion of an electromagnetic vibrator. Mekhanika, Collection of Translations of Foreign Literature, 6 (76), Moscow, 1962.

3. L. G. Etkin. On the excitation of vibrations of elastic systems by electromagnetic exciters. Priborostroenie, No. 2, 1960.

4. R. A. Mozniker. Investigation of Vibration Test Installations with Electromagnetic Exciters. Publishing House of the Academy of Sciences of the Ukrainian SSR, Kiev, 1960 (in Ukrainian).

5. Ya. G. Panovko. Fundamentals of the Applied Theory of Elastic Vibrations. Mashgiz, Moscow, 1957.