SOME REMARKS ON THE PROBLEM OF THE EMERGENCE OF SUBHARMONIC OSCILLATIONS IN SYSTEMS WITH DAMPING

V. I. KOPYTOV

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

It is known that the response of a nonlinear system even to an external harmonic action is of a very complex character. In particular, at frequencies exceeding the “fundamental” frequency of the system, in addition to forced oscillations with the frequency of the external action, stable oscillations may arise with a frequency an integer number of times smaller than the frequency of the external action—subharmonic oscillations [1—9]. As experiments and investigations on electronic modeling installations show [1—5], the emergence of stable subharmonic oscillations is characteristic of conservative systems or of systems with small damping. Obviously, in practical cases it is very important to be able, at least approximately, to obtain limiting values of damping at which subharmonic oscillations may occur. The relevance of this problem is determined by the fact that quite many modern instruments and complexes, representing certain nonlinear systems, are subjected in operation to the action of high-frequency vibration. The emergence of stable subharmonic oscillations of relatively large amplitudes in this case is an extremely undesirable phenomenon, since it reduces the accuracy of readings, the reliability of operation of instruments, etc.

In works [7, 8, 9], using the Duffing equation as an example and in the presence of viscous damping, subharmonic oscillations of order \(\frac{1}{2}\) and \(\frac{1}{3}\) were determined. It should be noted that an analytical investigation of subharmonics makes sense only in the presence of viscous damping (friction).

In the present article this problem is considered using the example of an “elastic link—mass—rigid limiter” in the presence of viscous damping. The solution is carried out by means of the method of Acad. B. G. Galerkin.

Thus, let us consider an oscillatory system of the form (Fig. 1), where

\(x\) — the coordinate of the mass \(m\) from the position of static equilibrium of the system;

\(a_0\) — the distance (“clearance”) from the position of static equilibrium to the limiter;

\(c\) — the stiffness coefficient of the elastic link;

\(\alpha\) — the coefficient of viscous damping forces \((R = \alpha \cdot \dot{x})\);

\(Q(t) = H\sin\omega t\) — the external periodic force of frequency \(\omega\), acting on the mass \(m\).

The differential equation describing the behavior of the system will be represented in the form

\[ m\ddot{x}+\alpha\dot{x}+cx=H\sin \omega t \]

or

\[ \ddot{x}+2\beta\dot{x}+\omega_0^2 x-h\sin\omega t=0, \tag{1} \]

where

\[ \omega_0=\sqrt{\frac{c}{m}};\qquad 2\beta=\frac{\alpha}{m};\qquad h=\frac{H}{m}. \tag{2} \]

We use the condition of contact of the mass \(m\) with the stop after choosing the form of the solution. Choosing the solution of equation (1) in the form (the first approximation of B. G. Galerkin’s method)

\[ x=x_0+a\sin\omega t+a_n\sin\frac{1}{n}(\omega t-\gamma), \tag{3} \]

where \(x_0\) characterizes the equilibrium position of the system “elastic connection—mass—stop”;
\(a\sin\omega t\) are the oscillations of the mass \(m\) with the frequency of the external action;
\(a_n\sin\dfrac{1}{n}(\omega t-\gamma)\) are the subharmonic oscillations of the mass \(m\);
\(\gamma\) is the phase-shift angle between the subharmonics and the external action.

Fig. 1. Mechanical oscillatory system with a rigid stop

The phase-shift angle of the oscillations of the mass \(m\) with the frequency of the external action may be taken equal to zero. From the condition of contact during subharmonic oscillations of the mass \(m\) with the stop we have

\[ x=a_0,\qquad \frac{1}{n}(\omega t_1-\gamma)=\frac{\pi}{2}, \]

whence

\[ \omega t_1=n\left(\frac{\pi}{2}+\gamma\right). \tag{4} \]

Then solution (3), taking (4) into account, is written in the form

\[ a_0=x_0+a\sin n\left(\frac{\pi}{2}+\gamma\right)+a_n, \qquad \text{whence } x_0=a_0-a\sin n\left(\frac{\pi}{2}+\gamma\right)-a_n. \]

And finally solution (3) takes the form

\[ x=a_0+\left[\sin\omega t-\sin n\left(\frac{\pi}{2}+\gamma\right)\right]a +\left[\sin\frac{1}{n}(\omega t-\gamma)-1\right]a_n. \tag{5} \]

In accordance with the method of Acad. B. G. Galerkin, we substitute this solution into equation (1), multiply by the variation \(\delta x\), integrate over the limits from \(0\) to \(\dfrac{2\pi n}{\omega}\), and set the result equal to zero.

Owing to the independence of the variations \(\delta a\), \(\delta a_n\), and \(\delta\gamma\), after integration we obtain a system of three equations in the form:

\[ \left. \begin{aligned} &\left[1+2\sin^2 n\left(\frac{\pi}{2}+\gamma\right)-\frac{\omega^2}{\omega_0^2}\right]a +2\sin n\left(\frac{\pi}{2}+\gamma\right)\cdot a_n=\\ &\qquad =x_{\mathrm{st}}+2\sin n\left(\frac{\pi}{2}+\gamma\right)\cdot a_0;\\[6pt] &2\sin n\left(\frac{\pi}{2}+\gamma\right)\cdot a+ \left(3-\frac{1}{n^2}\cdot\frac{\omega^2}{\omega_0^2}\right)\cdot a_n=2a_0;\\[6pt] &\frac{1}{n}\cdot \nu\,\frac{\omega}{\omega_0}\cdot a_n^2 -n\cdot\cos n\left(\frac{\pi}{2}+\gamma\right)\cdot a\cdot a_n +n\cdot\cos n\left(\frac{\pi}{2}+\gamma\right)\cdot\\ &\qquad \cdot a_0\cdot a -n\cdot\cos n\left(\frac{\pi}{2}+\gamma\right)\cdot \sin n\left(\frac{\pi}{2}+\gamma\right)\cdot a^2=0, \end{aligned} \right\} \tag{6} \]

where

\[ x_{\mathrm{st}}=\frac{h}{\omega_0^2}=\frac{H}{c};\qquad \nu=\frac{\beta}{\omega_0}. \]

The system obtained is a system of nonlinear equations with respect to \(a\), \(a_n\), and \(\gamma\). Its exact solution is extremely difficult. Therefore we shall restrict ourselves to a particular problem, namely, we shall find the conditions determining the upper bound of the values of the resistance forces for which subharmonic oscillations are possible.

Thus, we shall assume that \(n=2;\ 4;\ 6;\ \ldots\), that is, we consider subharmonics of even order. In addition, as experiments show, subharmonic oscillations in phase “lag” behind the external action by the angle \(\frac{\pi}{2}\). Consequently,

\[ \left. \begin{aligned} &\sin n\left(\frac{\pi}{2}+\gamma\right)=\sin n\pi=0;\\ &\cos n\left(\frac{\pi}{2}+\gamma\right)=\cos n\pi=1. \end{aligned} \right\} \quad \text{for } n=2;\ 4;\ 6;\ldots . \tag{7} \]

and the system of equations (6) will take the form

\[ \left. \begin{aligned} &\left(1-\frac{\omega^2}{\omega_0^2}\right)a=x_{\mathrm{st}},\\[6pt] &\left(3-\frac{1}{n^2}\cdot\frac{\omega^2}{\omega_0^2}\right)a_n=2a_0;\\[6pt] &\frac{1}{n}\nu\cdot\frac{\omega}{\omega_0}\cdot a_n^2 -n\cdot a\cdot a_n+n\cdot a_0\cdot a=0. \end{aligned} \right\} \tag{8} \]

From the first two equations we obtain dimensionless expressions for the amplitudes of the oscillations with the frequency of the external force and of the subharmonics:

\[ \frac{a}{x_{\mathrm{st}}}= \frac{1}{\frac{\omega^2}{\omega_0^2}-1}; \qquad \frac{a_n}{x_{\mathrm{st}}}= \frac{2n^2\cdot\frac{a_0}{x_{\mathrm{st}}}} {3n^2-\frac{\omega^2}{\omega_0^2}}. \tag{9} \]

From the third equation we determine the boundary of the values of the damping coefficient at which subharmonic oscillations are possible. We have

\[ a_n= \frac{ na+\sqrt{n^2a^2-4\nu\,\frac{\omega}{\omega_0}\cdot a_0\cdot a} }{ \frac{2}{n}\nu\cdot\frac{\omega}{\omega_0} }. \]

It follows from this that \(a_n\) can have real values only when the expression under the radical is positive, that is,

\[ n^2 a^2 - 4 \nu \frac{\omega}{\omega_0}\, a_0 \cdot a > 0, \]

which is satisfied if

\[ \nu < \frac{n^2 a}{4a_0 \cdot \dfrac{\omega}{\omega_0}} . \tag{10} \]

Or, in dimensionless quantities, taking (9) into account,

\[ \nu < \frac{n^2}{4\,\dfrac{a_0}{x_{\mathrm{st}}}\cdot \dfrac{\omega}{\omega_0}\left(\dfrac{\omega^2}{\omega_0^2}-1\right)} . \tag{11} \]

Expressions (10) or (11) give the upper bound of the values of the resistance coefficient \(\nu\) for which subharmonic oscillations are possible. Example:

\[ \text{let } \frac{a_0}{x_{\mathrm{st}}}=1;\quad n=2,\ \text{ then for } \frac{\omega}{\omega_0}=2,\ \nu < \frac{1}{9}\simeq 0.11; \]

\[ \text{for } \frac{\omega}{\omega_0}=2.5,\quad \nu < 0.051. \]

Conclusions

In the oscillatory system “elastic connection—mass—limiter,” for a given parameter \(\nu\) of the coefficient of viscous damping, the possibility of the existence of stable subharmonic oscillations of order higher than 2, 3, 4, ... sharply decreases.

Inequalities (10) or (11) make it possible to give a quantitative estimate of the parameter \(\nu\), which is very important in studying the dynamics of systems on electronic modeling installations.

REFERENCES

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