FORCED OSCILLATIONS OF A TRACKING SYSTEM WITH CLEARANCE

Yu. M. Nefedov

(Presented by the scientific-technical seminar of the Department of Gyroscopic Instruments and Devices)

For a mechanical transmission of any type, the presence of clearance between the transmission elements is characteristic. The influence of clearance on the operating accuracy of a system proves to be substantial when the displacements of the driving element of the mechanical transmission are small (of the same order as the magnitude of the clearance) [1]. Knowledge of the characteristics of forced oscillations of a system, taking into account a nonlinear characteristic of the clearance type, makes it possible to assess both qualitatively and quantitatively the dynamic accuracy of such a system.

A tracking system with clearance is a first-class nonlinear system, containing in the general case two nonlinear characteristics, one of which is a complicated function of the amplitude and frequency of the forced oscillations [1]. The block diagram of the tracking system under consideration is shown in Fig. 1, a.

The nonlinear characteristic of the clearance \(x_3 = F_1(x)\) may be described as follows [1]:

\[ \begin{aligned} F_1(x) &= x - x_0 && \text{for } p x_3 > 0;\\ F_1(x) &= x + x_0 && \text{for } p x_3 < 0;\\ F_1(x) &= \mathrm{const} && \text{for } |x - x_3| < x_0, \end{aligned} \tag{1} \]

where

\[ x_0 \]

is one half of the clearance of the mechanical transmission, concentrated in one place.

It should be noted that formulas (1) correspond to the condition of immobility of the driven element (the controlled object) during the taking up of the clearance. This condition is realistic for most tracking systems.

The motion of the motor shaft of the tracking system, taking into account only the moment of inertia of the controlled object, is described by the equation (Fig. 1, a):

\[ F_2(p^2 x,\; px) = k_1 x_2 \tag{2} \]

or

\[ \begin{aligned} (T_1 p + 1)px &= k_1 x_2 && \text{for } p x_3 = 0,\\ \bigl[(T_1 + T_2)p + 1\bigr]px &= k_1 x_2 && \text{for } p x_3 \ne 0, \end{aligned} \tag{3} \]

where

\[ T_1 \]

is the time constant of the motor of the tracking system;

\[ T_2 \]

is the time constant proportional to the moment of inertia of the controlled object.

It is assumed that the control signal \(x_1\), which represents the forced oscillations of the system, varies according to the harmonic law

\[ x_1 = B\sin \omega t . \]

Fig. 1

The equivalent complex gain of the nonlinear element \(F_1(x)\), as a result of harmonic linearization, takes the following form [2]:

\[ N(ja)=\kappa_{\mathrm{н}}(a)\cdot e^{j\varphi_{\mathrm{н}}(a)}; \]

where

\[ \begin{aligned} \kappa_{\mathrm{н}}(a)&=\sqrt{q^2(a)+q_1^2(a)},\\ \varphi_{\mathrm{н}}(a)&=\arg \frac{q_1(a)}{q(a)} . \end{aligned} \tag{4} \]

The coefficients of harmonic linearization \(q(a)\) and \(q_1(a)\) for a clearance-type characteristic have the form

\[ \left. \begin{aligned} q(a)&=\frac{1}{\pi}\left[ \frac{\pi}{2} +\arcsin\left(1-\frac{2}{a}\right) +2\left(1-\frac{2}{a}\right) \sqrt{\frac{1}{a}\left(1-\frac{1}{a}\right)} \right],\\[4pt] q_1(a)&=-\frac{4}{\pi a}\left(1-\frac{1}{a}\right) \end{aligned} \quad \text{for } a\ge 1, \right\} \tag{5} \]

where

\[ a=\frac{A}{x_0} \]

is the relative amplitude of the sinusoidal signal at the input of the nonlinear element \(F_1(x)\).

Accordingly, the following notation is introduced:

\[ b=\frac{B}{x_0} \]

is the relative amplitude of the control signal;

\[ c=\frac{C}{x_0} \]

is the relative amplitude of the sinusoidal output signal of the tracking system.

For harmonic linearization of the nonlinear function \(F_2(p^2x, px)\), one should use the following relation [1]:

\[ F_2(p^2 x,\; px)= \left[ \frac{q_3(A,\omega)}{-\omega^2}p^2+ \frac{q_2(A,\omega)}{\omega}p \right]x, \tag{6} \]

where

\[ \left. \begin{aligned} q_2(A,\omega) &=\frac{1}{\pi A}\int_0^{2\pi} F(-A\omega\sin\omega t,\; A\omega\cos\omega t)\sin\omega t\,d\omega t,\\[6pt] q_2(A,\omega) &=\frac{1}{\pi A}\int_0^{2\pi} F(-A\omega\sin\omega t,\; A\omega\cos\omega t)\cos\omega t\,d\omega t, \end{aligned} \right\} \tag{7} \]

for \(x=A\sin\omega t\).

The phase shift of the forced oscillations at the input of the nonlinear element relative to the forcing oscillations of the system is not taken into account in determining the expression of the harmonically linearized nonlinear function.

Thus, on the basis of equations (3), (6), and (7), after harmonic linearization the nonlinear function (2) can be written in the following form:

\[ \{[T_1+q_4(a)T_2]p+1+q_1(a)T_2\omega\}\,px=k_1x, \tag{8} \]

where

\[ q_4(a)=\frac{1}{\pi} \left[ \frac{\pi}{2}+\arcsin\left(1-\frac{2}{a}\right) -2\left(1-\frac{2}{a}\right) \sqrt{\frac{1}{a}\left(1-\frac{1}{a}\right)} \right], \]

for \(a>1\).

Expressions (4) and (8) make it possible to represent the structural diagram of the tracking system with clearance in the form of a two-loop system (Fig. 1, b). Here the internal feedback is positive, since the harmonic linearization coefficient \(q_1(a)>0\) for \(a>1\). In addition, the gain coefficient of the internal feedback depends on the frequency of the forcing oscillations.

In the tracking system under consideration, the frequency characteristics of the internal loop constitute a nonlinear dependence on the amplitude of the sinusoidal signal at the output of the nonlinear element \(N(ja)\), that is, both nonlinear dependences are determined by one and the same coordinate \(x\). Thus, for subsequent construction of the characteristics of forced oscillations of the tracking system, one can use the procedure given in [3], based on the method of harmonic linearization and the method of logarithmic frequency characteristics.

For convenience in performing all constructions, the equivalent complex gain coefficient of the nonlinear element \(N(ja)\), according to equation (4), is represented graphically (Fig. 2, a).

The second nonlinear dependence, on the basis of equation (8) and Fig. 1, b, can be represented in the form of the following frequency transfer function:

\[ W_{\mathrm{in}}(a,j\omega)= \frac{x_4(a,j\omega)}{x_2(a,j\omega)} \frac{1}{[T_1+q_4(a)T_2]j\omega+1+q_1(a)T_2\omega}. \tag{9} \]

The logarithmic frequency characteristics corresponding to equation (9), for several fixed values of the amplitude \(a\), are shown in Fig. 2, b. The adopted procedure presupposes the determination

characteristics of the forced oscillations at the input of the nonlinear element as a function of the forcing oscillations of the system, in order, relying on this dependence, to determine the characteristics of the forced

Fig. 2

oscillations of the output coordinate of the system for various fixed values of the amplitude of the forcing oscillations.

To this end, first of all, a family of logarithmic amplitude-frequency and phase-frequency characteristics of the forcing oscillations of the system is constructed for various fixed values of the amplitude \(a\). The indicated constructions are carried out on the basis of the following expressions (Fig. 1, c):

\[ b= \left| \frac{N(ja)} {\dfrac{W_{\mathrm{n}}(a,j\omega)\,W(j\omega)\,N(ja)} {1+W_{\mathrm{n}}(a,j\omega)\,W(j\omega)\,N(ja)}} \right|a, \tag{10} \]

\[ \varphi=\arg\left[ \frac{N(ja)} {\dfrac{W_{\mathrm{n}}(a,j\omega)W(j\omega)N(ja)} {1+W_{\mathrm{n}}(a,j\omega)W(j\omega)N(ja)}} \right], \tag{10a} \]

where \(\varphi\) is the phase shift of the forcing oscillations relative to the forced oscillations at the input of the nonlinear element;

\[ W(j\omega)=\frac{\kappa_1}{j\omega} \]

is the frequency transfer function of the linear part of the tracking system.

The dependences (10) are constructed with the aid of the closure nomogram [4] and the characteristics in Figs. 2, a and b. In the plane of the parameters \(b(a)\) and \(\varphi(a)\), as functions of \(\omega\), on the basis of the frequency criterion for the stability of a periodic solution [5], a region is singled out in which the nonlinear tracking system operates in the capture mode and in which the construction of frequency characteristics is meaningful.

Next, the inverse dependences \(a(b,\omega)\) and \(\varphi(b,\omega)\) are determined. Figure 3 presents the characteristics of the amplitude of the forced oscillations \(a(\omega)\) for several fixed values of the amplitude \(b\).

Fig. 3

The hatched region is the region of complex oscillations, in which there is no capture mode. On the boundary of the region of complex oscillations a point \((b=-\infty\,\partial b)\) is marked, which determines the amplitude and frequency of the system’s self-oscillations.

By means of simple graphical constructions the following dependences are determined:

1. The dependence \(\dfrac{a}{b}(\omega)\) for several fixed values of \(b\); here the dependence \(\varphi(\omega)\) for the same values of \(b\) remains unchanged.

2. The dependence \(N(b,j\omega)\). This dependence is determined in the form of the characteristics \(\kappa_{\mathrm{n}}(j\omega)\) and \(\varphi_{\mathrm{n}}(j\omega)\) for various fixed values of \(b\), on the basis of the characteristics \(a(b,\omega)\) and \(N(ja)\).

Finally, a family of characteristics of the forced oscillations of the output coordinate of the system is determined for selected values of the amplitude \(b\). In these constructions the obtained dependences

\[ \frac{x}{x_1}(b,j\omega) \quad \text{and} \quad N(b,j\omega)=\frac{x_3}{x}(b,j\omega) \]

are used; their product gives the characteristics of the forced oscillations of the closed-loop system

\[ \frac{x_3}{x_1}(b,j\omega). \]

in the form of a family of amplitude-frequency characteristics \(A_m(\omega)\) and phase-frequency characteristics \(\varphi_m(\omega)\) for various values of \(b\). The results of the constructions are shown in Fig. 4. The constructed characteristics correspond to the region of the capture mode; the points indicate the boundaries of this region. From a comparison of the curves in Fig. 4 it is seen that, with a decrease

Fig. 4

in the amplitude \(b\), the amplitude of the forced oscillations increases, that is, the ratio \(\dfrac{c}{b}(b,j\omega)\) increases, while the frequency range of the possible capture mode decreases.

Conclusions

1. A tracking system with clearance may have regions of a capture mode and a mode of complex oscillations.

2. For certain values of the amplitude of the forcing oscillations, jump-like resonance is possible in the system.

3. With a decrease in the amplitude of the forcing oscillations, the resonant amplitude of the forced oscillations of the system increases.

REFERENCES

1. V. A. Besekerskii, E. P. Popov. Theory of Automatic Control Systems. “Nauka,” Moscow, 1966.

2. E. P. Popov, I. P. Pal’tov. Approximate Methods for the Study of Nonlinear Automatic Systems. Fizmatgiz, 1960.

3. Yu. M. Nefedov. Influence of play in the actuator element of an aircraft stabilization system on the amplitude of forced oscillations. Tr. LIAP, issue 49, 1966.

4. N. T. Kuzovkov. Theory of Automatic Control. Oborongiz, 1957.

5. J. Teler, M. Pestel. Analysis and Design of Nonlinear Automatic Control Systems. “Energiya,” Moscow, 1964.