On the Fundamentals of the Theory of Single-Stroke Electromagnetic Vibrators with a Half-Wave Rectifier

Yu. Ya. Kovylin

In our works [1] and [2] an approximate theory is set forth for an electromagnetic vibrator operating in a near-resonance mode. Recently electromagnetic vibrators have been finding ever wider application in instrument and machine building. At the same time, depending on the specific requirements imposed on the operation of the vibration device, the elastic system of the mechanical circuit often must be detuned sufficiently far from resonance. In this connection there arises the need to develop a method suitable for calculating such vibrators.

The electromagnetic vibrator under consideration is shown in Fig. 1. The alternating force developed by electromagnet 1 excites oscillations of armature 2, fastened on springs 3.

Fig. 1.

It is assumed that

a) the magnetization characteristic of the electromagnet is linear (or linearized);

b) the leakage magnetic flux is negligibly small in comparison with the working magnetic flux;

c) the magnetic field of the eddy currents and the energy losses associated with the excitation of these currents are so small that they may be disregarded;

d) the energy source has unlimited power.

Under these assumptions the initial system of equations may be written in the following form:

\[ M \frac{d^{2}Y}{dt^{2}} + H \frac{dY}{dt} + F(Y) = \frac{I^{2}}{2}\,\frac{dL(Y)}{dY}; \tag{1} \]

\[ \frac{d}{dt}\,[L(Y)\cdot I] + [R + R_{\mathrm{d}}(I)]\cdot I = U(t); \tag{2} \]

\[ L = \frac{L_{0}}{1 - Y/S_{0}};\qquad R_{\mathrm{d}} = \begin{cases} 0, & \text{if } I \geq 0,\\ \infty, & \text{if } I < 0. \end{cases} \tag{3} \]

Here \(M\) is the mass of the moving parts of the vibrator, reduced to the armature;

\(Y\) is the instantaneous value of the armature displacement, measured from the position of its static equilibrium;

\(H\) is a coefficient integrally accounting for all inelastic resistances in the mechanical circuit;

\(F(Y)\) is the elastic force of the springs;

\(L\) is the instantaneous value of the inductance of the electromagnet winding;

\(R, R_{\mathrm{d}}\) are the active (ohmic) resistances of the magnet winding and the rectifier (diode);

\(U(t)\) is the voltage at the terminals of the power source;

\(I\) is the instantaneous value of the current in the magnet winding;

\(S_0\) is the reduced gap, which includes the entire air gap \(S_{\mathrm{v}}\) and the additional air gap \(S_{\mathrm{m}}\), equivalent to the magnetic circuit with respect to the resistance to the working magnetic flux.

If \(F(Y)=CY\) and \(U(t)=\sqrt{2}\,U_{\mathrm{e}}\sin\omega t\), then the solution of equations (1) and (2), valid for any whole period of variation of the voltage \(U(t)\), can be approximately represented in the following form:

\[ Y=A_0+\sum_{n=1}^{\infty} A_n \cos (n\tau+\alpha_n-\delta_n)\quad (\tau=\omega t); \tag{4} \]

\[ I=\frac{\sqrt{2}\,U_{\mathrm{e}}(1-Y/S_0)} {\omega L_0(1+r_{\mathrm{pr}}^2+\kappa)} \begin{cases} r_{\mathrm{pr}}\sin\tau-\cos\tau+e^{-r_{\mathrm{pr}}\tau}, & \text{if } 0\le \tau \le \tau_{\mathrm{d}},\\ 0, & \text{if } \tau_{\mathrm{d}}<\tau<2\pi \end{cases}; \tag{5} \]

\[ r_{\mathrm{pr}}\sin\tau_{\mathrm{d}}-\cos\tau_{\mathrm{d}}+e^{-r_{\mathrm{pr}}\tau_{\mathrm{d}}}=0 \quad (\pi<\tau_{\mathrm{d}}<2\pi); \tag{6} \]

\[ \left. \begin{aligned} A_0&=\frac{L_0}{S_0C}\left(\frac{U_{\mathrm{e}}}{\omega L_0}\right)^2 \frac{q_0}{(1+r_{\mathrm{pr}}^2+\kappa)^2};\\[4pt] A_n&=\frac{L_0}{S_0C}\left(\frac{U_{\mathrm{e}}}{\omega L_0}\right)^2 \frac{q_n}{(1+r_{\mathrm{pr}}^2+\kappa)^2 \sqrt{[1-(n\xi)^2]^2+\varepsilon^2(n\xi)^2}};\\[4pt] \tan\delta_n&=\frac{n\varepsilon\xi}{1-(n\xi)^2}; \qquad \xi=\frac{\omega}{\omega_0}; \qquad \omega_0=\frac{C}{M}; \qquad \varepsilon=\frac{H}{M\omega_0}. \end{aligned} \right\} \tag{7} \]

The coefficients \(r_{\mathrm{pr}}\) and \(\kappa\) take into account the effect of armature vibration on the change in current in the electrical circuit.

The value of \(r_{\mathrm{pr}}\) is determined from the condition of balance of the powers dissipated in the equivalent constant resistance \(R_{\mathrm{pr}}\) and the variable resistance \(R(1-Y/S_0)\):

\[ r_{\mathrm{pr}}=\frac{R}{\omega L_0} \left( 1-\frac{A_0}{S_0} -\sum_{n=1}^{\infty}\frac{A_n}{S_0}\rho_n\cos\delta_n \right). \tag{8} \]

The correction \(\kappa\) is found from the condition that solution (5) satisfy equation (2) with a minimum root-mean-square deviation on the interval \((0;\tau_{\mathrm{d}})\):

\[ \kappa=\frac{R}{\omega L_0} \sum_{n=1}^{\infty}\frac{A_n}{S_0}\chi_n\sin\delta_n. \tag{9} \]

The quantities \(q_0, q_n, \alpha_n, \rho_n\), and \(\chi_n\) depend uniquely on \(r_{\mathrm{pr}}\). The plotted graphs (Fig. 2 and Fig. 3) make it as easy as possible to determine them (for \(n=1\) and \(2\)).

The calculation according to (7)–(9) is carried out by the method of successive approximations. In the zeroth approximation one takes \(r_{\mathrm{pr}}=\dfrac{R}{\omega L_0}\), \(\kappa=0\). The presented

The method makes it possible to establish that, when powered from a sinusoidal voltage source, the amplitude–frequency curves in the zone of the vibrator’s operating regimes have practically the same form as in an ordinary linear oscillator.

Fig. 2

In resonant regimes, the mutual exclusion of the components of motion, characteristic of nonlinear systems, is weakly manifested.

Fig. 3

With appreciable detuning from resonance, the vibrator generates oscillations that, with sufficient accuracy, can be approximated by no fewer than the first three terms of a Fourier series. In a number of practically important cases this is a substantial drawback of the vibrator.

Experimental verification, carried out on a special test bench in the laboratory of mechanism dynamics of the Department of Technical Mechanics of TPI, showed the very high accuracy of the method for detunings from resonance satisfying the inequality $\omega_0 < 3\omega$, which covers all operating modes of electromagnetic vibrators of practical significance.

The calculations necessary for plotting the graphs shown in Figs. 2 and 3 were performed by engineer S. A. Basov.

REFERENCES

1. Yu. Ya. Kovylin, S. I. Basov. Interaction of an electromagnetic vibrator with a source of sinusoidal voltage. Physicotechnical Problems of Mineral Processing. Novosibirsk, Publishing House of the Siberian Branch of the USSR Academy of Sciences, No. 10, 1965.

2. Yu. Ya. Kovylin, S. I. Basov. Study of a vibrator with a single-stroke electromagnetic drive with a half-wave rectifier. In: Problems of Vibration Technology. Kiev, “Naukova Dumka,” 1968.