PENDULUM WITH A VIBRATING SUPPORT

P. L. Kapitsa

The development of mechanics is undoubtedly closely connected with the study of the pendulum. After Galileo drew attention to the isochronism of its oscillations, it became possible to create a very perfect mechanism for measuring time—the pendulum clock, whose accuracy has only recently been surpassed by quartz clocks. Thanks to the study of the pendulum, methods were found for measuring time as accurately as length and mass were measured, which was necessary in order that the development of mechanics could proceed along a firm path. Naturally, no other mechanical system has received so much attention and such comprehensive theoretical study as all the varieties of the motion of a pendulum. It would seem that in the 300 years that have passed since Galileo’s time this question should have been exhausted, and if anything remained to be studied, it should have had the character of a refinement of previously obtained results. But, apparently, the type of pendulum motion to which this article is devoted has not received sufficient attention, and one very peculiar and interesting variety of pendulum oscillations has remained almost completely unstudied. To draw attention to this type of motion and to the possibilities that open up in its study is the aim of this article.

In Fig. 1 a mathematical pendulum is shown in two positions; it can oscillate at the point of suspension \(l\); the mass \(m\) is concentrated at the end of the rod \(L\).

The position of the pendulum on the left side of the figure \((a)\), when the point of suspension is above the center of gravity, we shall call the normal position. In the drawing on the right \((b)\) the point of suspension of the pendulum is below the center of gravity; we shall call this position the inverted position of the pendulum.

The type of pendulum that we shall consider has the feature that the point of suspension \(l\) moves along the \(y\)-axis near the origin \(O\), while the distance \(Ol\) is a periodic function of time; we shall also assume that the amplitude of the oscillations—

suspension \(a\) is small in comparison with the length of the pendulum \(L\). This is the well-known pendulum with an oscillating suspension.

In the study of this type of pendulum, all attention was concentrated on that kind of motion in which the period of oscillation of the suspension \(T\) differed little from the period of oscillation of the pendulum itself \(\tau\). It was found that in those cases when \(2T\), or a multiple of it, is close to the period \(\tau\), the phenomenon of parametric resonance arises. These investigations amounted to the study of the properties of the solutions of Mathieu’s equation, by which this motion is described for small oscillation amplitudes. Further\(^1\) it was discovered that, for

Fig. 1.

small values of \(T\) in comparison with \(\tau\), the pendulum can acquire a special kind of stability—it can stand, without falling, in the inverted position. The character of the motion of the pendulum in this position and the degree of its stability at high frequencies of oscillation of the suspension apparently remained completely unstudied. Thus the beautiful and instructive phenomenon of the dynamic stability of the inverted pendulum not only did not enter modern textbooks on mechanics, but is even almost unknown to a wide circle of specialists.

One may suppose that such undeserved neglect of this phenomenon was a consequence of the fact that its study is connected with the solution of Mathieu’s equation; it was carried out by infinite determinants (Hill’s method) or by special functions, which led to a solution of a formal character and did not make it possible to describe the motion visually.

Studying this motion, I noticed that, under the condition when the amplitude of oscillations of the suspension \(a\) is small in comparison with the length of the pendulum \(L\), there exists a method of approximate solution of the problem of motion which describes the phenomenon simply and graphically.

In what follows we shall denote the ratio of the amplitude of oscillation of the suspension to the length of the pendulum by \(\alpha\):

\[ \alpha = a/L \ll 1. \tag{1} \]

The quantity \(\alpha\) will play a very important role in the present method, since the accuracy of the results obtained in the types of motion that interest us is mainly determined by it. We shall chiefly study those kinds of motion of the pendulum for which the frequency of oscillation of the suspension is large in comparison with the frequency of oscillation of the pendulum and, moreover, is not at all connected with it by any phase relations; the spectrum of the oscillations of the suspension itself may be a sum of spectra of different frequencies. Therefore, in order to distinguish the motion we are studying from the motion of a pendulum with an oscillating suspension, we shall call it the motion of a pendulum with a vibrating suspension.

The method we have used for solving the problem is based on successive approximations together with the introduction of coordinates averaged over time. A detailed exposition of it and an investigation of the accuracy of the results obtained are given by us elsewhere². Here we shall confine ourselves to a description of the principal results obtained and of the possibility of their practical application.

The method of successive approximations, already in the first step, reduces the problem of the influence of a vibrating suspension point on the motion of the pendulum to a very simple physical picture: it turns out that this influence is equivalent to a moment of forces which behaves exactly like a pair of ordinary forces and tends to set the pendulum so that its mass is always situated in the direction of the vibrations of the suspension. We have called this moment the vibrational moment and denoted it by \(\overline{M}\). As will be seen from what follows, the introduction of the vibrational moment also makes the solution of problems of motion of this type of pendula no more difficult than the solution of problems for ordinary pendula. Further on we shall also describe a simple way to construct a pendulum with a vibrating suspension, on which the theoretical results obtained can be demonstrated.

Let us write the equation for the general case of motion of the type of mathematical pendulum under consideration. If, as shown in Fig. 1, the angle between the rod of the pendulum and the \(y\)-axis is denoted by \(\theta\), then the coordinates of the mass of the pendulum \(x\) and \(y\) will be

\[ x = L \sin \theta; \qquad y = U + L \cos \theta, \tag{2} \]

where \(U\) denotes the distance (along the \(y\)-axis) of the point of suspension of the pendulum \(l\) from the origin \(O\).

We shall denote the forces acting on the mass \(m\) along the axes \(x\) and \(y\) by \(F_x\) and \(F_y\); then we obtain:

\[ \left. \begin{aligned} F_x &= m\ddot{x}=mL(\ddot{\Theta}\cos\Theta-\dot{\Theta}^{2}\sin\Theta),\\ F_y &= m\ddot{y}=m[\ddot{U}-L(\ddot{\Theta}\sin\Theta+\dot{\Theta}^{2}\cos\Theta)]. \end{aligned} \right\} \tag{3} \]

We shall denote the moment of the couple of external forces acting on the mass of the pendulum by \(M_\Theta\); it will be equal to:

\[ M_\Theta=L(F_x\cos\Theta-F_y\sin\Theta). \tag{4} \]

Substituting the values of \(F_x\) and \(F_y\), we obtain:

\[ M_\Theta=mL^{2}\ddot{\Theta}-mL\ddot{U}\sin\Theta. \tag{5} \]

This equation is not difficult to generalize to the case of a physical pendulum. For this, one must regard \(m\) as an elementary mass and integrate the right-hand side of equation (5) over the entire volume of the mass of the pendulum; then, instead of (5), we obtain:

\[ M_\Theta=m(L^{2}+K^{2})\ddot{\Theta}-mL\ddot{U}\sin\Theta, \tag{6} \]

where \(\Theta\) and \(L\) are the coordinates of the center of gravity of the mass of the pendulum, and \(K\) is the radius of inertia of the pendulum.

Let the suspension of the pendulum perform simple harmonic oscillations with amplitude \(a\) and angular frequency \(\omega\); then we have:

\[ U=a\sin\omega t. \tag{7} \]

Differentiating this expression twice with respect to time and substituting the value of \(\ddot{U}\) into (6), we obtain:

\[ M_\Theta=m(L^{2}+K^{2})\ddot{\Theta}+mLa\omega^{2}\sin\omega t\sin\Theta. \tag{8} \]

In the particular case where the moment of the external forces is produced by the force of gravity, it is equal to:

\[ M_\Theta=mgL\sin\Theta, \tag{9} \]

and the equation of motion takes the form:

\[ \ddot{\Theta}=\frac{L}{(L^{2}+K^{2})}(g-a\omega^{2}\sin\omega t)\sin\Theta. \tag{10} \]

This equation is usually subjected to simplification, restricting the problem to the consideration of small values of the angle \(\Theta\) and replacing \(\sin\Theta\) by the quantity \(\Theta\). Under this restriction one obtains Mathieu’s equation, by means of which the problem of the motion of a pendulum with an oscillating suspension has been studied up to the present.

When applying our method of successive approximations, the consideration of the problem is not restricted to small angles \(\Theta\). The basic-

...the idea of this method consists in the assumption that during the period of the rapid oscillation of the suspension the angle \(\Theta\) changes little, remaining close to some value \(\varphi\). We set:

\[ \Theta=\varphi+\beta . \tag{11} \]

The angle \(\beta\) is a periodic quantity, but its value over the period of oscillation \(T\) always remains small. The angle \(\varphi\) may have any value, but over the same time \(T\) it changes little. If we average these quantities with respect to time over the period \(T\), and denote this averaging by a bar, then we have:

\[ \overline{\Theta}\simeq \varphi;\quad \overline{\beta}\simeq \Theta . \tag{12} \]

In studying the motion of a pendulum with a vibrating suspension, we are chiefly interested in the variation of the angle \(\varphi\), which represents the position about which small vibrations occur. Therefore the method of solution is constructed so that, by averaging, the angle \(\beta\) is eliminated from the equation and \(\Theta\) is replaced by the angle \(\varphi\). This, it turns out, can be done if the problem is reduced to a motion in which there participates a vibrational moment equal (for the physical pendulum) to

\[ \overline{M}=-\frac{1}{4}\left(1+K^{2}/L^{2}\right)^{-1}ma^{2}\omega^{2}\sin 2\varphi . \tag{13} \]

Then it can be shown\(^2\) that, for most types of motion of interest to us, in the first approximation (for determining quantities with accuracy of order \(\alpha^{2}\)) the following simple equation of motion holds:

\[ m\left(L^{2}+K^{2}\right)\ddot{\varphi}=M_{\varphi}+\overline{M}, \tag{14} \]

where the moment of the external forces \(M_{\varphi}\) is obtained from \(M_{\Theta}\) by simply replacing the angle \(\Theta\) by \(\varphi\). Thus we obtain the same equations as if the suspension were at rest, but, in addition to the external moment \(M_{\varphi}\), there acts also an additional moment \(\overline{M}\). It is not difficult to see that integrating the equation thus obtained for the angle \(\varphi\) presents no greater difficulties than in the case of the motion of ordinary pendulums with a fixed suspension. This is a consequence of the fact that the vibrational moment \(\overline{M}\), since time does not enter into it, acts in the same way as the moment of ordinary forces. From expression (13) it is seen that the vibrational moment tends to set the pendulum rod along the direction of the \(y\)-axis, i.e. the axis along which the oscillations of the suspension occur. The greatest value of \(\overline{M}\) is attained at \(\varphi=45^\circ\). Further, from (13) it follows that the magnitude of the vibrational moment does not depend on the length of the pendulum and is determined mainly by the kinetic energy imparted to the mass of the pendulum in the process of vibration of the suspension.

With a prescribed and constant vibration of the suspension, the magnitude of the vibrational moment \(\overline{M}\) depends only on the angle \(\varphi\); therefore, as will be seen from what follows, the resulting solutions of the mechanical problems of pendulum oscillation take on a clear form.

The simplification of the solution of the pendulum problem by introducing the vibrational moment is reminiscent of the analogous simplifications of problems of motion of various types of tops and gyroscopes by introducing the concept of gyroscopic moment. In this respect there is a certain analogy between the gyroscopic moment and the vibrational moment.

Let us give a number of examples of solutions of equation (14) that are of practical interest.

We shall first consider problems of “static” equilibrium between an applied moment \(M_\varphi\) and the vibrational moment \(\overline{M}\). The solution of these problems is obtained from equation (14); putting \(\varphi=\mathrm{const}\), we have:

\[ M_\varphi+\overline{M}=0. \tag{15} \]

Suppose that \(M_\varphi\) is produced by the force of gravity, and assume that the axis \(y'\), along which the vibrations occur, makes an angle \(\gamma\) with the plumb line. Then the moment of the force of gravity is equal to:

\[ M_\varphi=mgL\sin(\varphi+\gamma). \tag{16} \]

Substituting this value into (14), and also the value for \(\overline{M}\) from (13), we obtain the following equation:

\[ 4\left(1+\frac{K^2}{L^2}\right)Lg\sin(\varphi_n+\gamma)-a^2\omega^2\sin 2\varphi_n=0. \tag{17} \]

From this expression one can determine those values of the angle \(\varphi_n\) for which an equilibrium position of the pendulum is possible.

A graphical analysis of the equation shows that, depending on the values of the parameters, \(\varphi_n\) may have 4 or 2 values that are roots of it. In the case when there are two roots, only for one of them is the pendulum in stable equilibrium, corresponding to its normal position. In the case when there are four roots, the pendulum is in stable equilibrium for two values of the angle \(\varphi_n\): one of them corresponds to the normal position, and the other to the inverted one. Four roots are possible only when the quantity \(a^2\omega^2\) is sufficiently large, i.e., the vibrations are sufficiently intense.

The two found positions of stability can be demonstrated on the pendulum shown in Fig. 2 (\(a\) and \(b\)). At the end of the vibrating lever two identical pendulums are suspended symmetrically. With sufficient intensity of vibration they assume positions corresponding to each of the two angles \(\varphi_n\) that determine the stability of equilibrium. By giving the pendulum light pushes,

one can verify experimentally that these positions do indeed correspond to stable equilibrium.

In the particular case where the vibration of the pendulum suspension occurs in the vertical direction, i.e. \(\gamma=0\), it is seen that the equation is always satisfied when \(\varphi_1=\pi\) and \(\varphi_2=0\). The value \(\varphi_2=0\), i.e. when the pendulum is in the inverted position, becomes stable only in the case when there are also two values of the angle \(\varphi_n=\varphi_3\) and \(\varphi_n=\varphi_4\) for unstable equilibrium. Equation (17), if in it we set \(\gamma=0\), gives:

\[ \sin\varphi_n=0;\qquad \cos\varphi_n=\frac{2gL}{a^2\omega^2}\left(1+\frac{K^2}{L^2}\right). \tag{18} \]

From this we obtain that \(\varphi_1=\pi\), \(\varphi_2=0\), and \(\varphi_3=2\pi-\varphi_4\); the last angle determines the aperture of that cone from which the pendulum will pass into the stable inverted position at \(\varphi_2=0\). With an initial position of the pendulum at an angle greater than \(\varphi_3\), it will pass into the stable equilibrium with angle \(\varphi_1=\pi\), i.e. into the normal position. The smaller the value of \(\cos\varphi_n\), the wider is that region in the inverted position in which the pendulum is stable. The initial condition necessary for obtaining a stable position of the inverted pendulum is obtained from (18); it has the form:

\[ \frac{1}{2}a^2\omega^2 > gL\left(1+\frac{K^2}{L^2}\right). \tag{19} \]

This condition had already been obtained for the mathematical pendulum and, apparently, this is the only result for characterizing the behavior of the pendulum in the inverted position that has so far been obtained from consideration of Mathieu’s equation. The following physical interpretation was given to this result.^1 For stability in the inverted position, the value of the kinetic energy of the mass of the pendulum, created by oscillation of the suspension, must be greater than the potential energy of the mass of the pendulum above the point of suspension. As is clear from our analysis, this interpretation is valid only for the mathematical pendulum; for the physical pendulum it has no place.

Let us now proceed to the consideration of dynamical problems. Then in equation (14) the angle \(\varphi\) should be regarded as a variable quantity. We shall analyze the simplest case, when the external couple of forces \(M_\varphi\) is absent. Setting it equal to zero, from expression (14) we obtain the following equation of motion:

\[ \left(1+\frac{K^2}{L^2}\right)^2 \ddot{\varphi} = -\frac{1}{4}a^2\omega^2\sin 2\varphi. \tag{20} \]

This equation is easily integrated and leads to elliptic

integrals of the first kind; the pendulum, when the suspension vibrates, even in the absence of external forces, will perform periodic oscillatory motion. If the period of the oscillations with which the angle \(\varphi\) varies is denoted, as before, by \(\tau\), and the period of oscillation of the suspension by \(T\), then from the solved equation (20) we obtain:

\[ \frac{\tau}{T}=\sqrt{2}\,\alpha^{-1}\left(1+\frac{K^2}{L^2}\right)F(k). \tag{21} \]

\(F(k)\) is the complete elliptic integral of the first kind, \(k=\sin\varphi_a\), where \(\varphi_a\) is the angular amplitude of oscillations of the pendulum. For constant or small values of the amplitude \(\varphi_a\), there is a simple proportionality between the period of oscillation of the pendulum and the period of vibration of the suspension. Since \(\alpha\) is a small quantity, the period \(\tau\) will be considerably greater than \(T\). Such oscillations may be reproduced on the pendulum shown in Fig. 3. In these experiments the influence of gravity is excluded if the pendulum is placed horizontally. If the period \(\tau\) is sufficiently large that it can be determined by a simple count of the oscillations, then, knowing from expression (21) the coefficient of proportionality, one can determine the period of vibrations \(T\). The phenomenon described can be used as a kind of simple tachometer.

Let us now consider the oscillations of a conical pendulum in the absence of gravity. We assume that the rotation of the mass of the pendulum takes place about the \(y\)-axis with constant angular velocity \(\Omega\); then for the moment produced by the centrifugal force we obtain the following expression:

\[ M_\varphi=\frac{1}{2}m\Omega^2L^2\sin 2\varphi. \tag{22} \]

The quantity \(M_\varphi\) depends on the angle \(\varphi\) in the same way as the vibrational moment \(M\) (13). Therefore the equilibrium between \(M_\varphi\) and \(M\) does not depend on the value of the angle \(\varphi\), and we obtain the following simple relation:

\[ \frac{\Omega^2}{\omega^2}=\frac{1}{2}\alpha^2\left(1+\frac{K^2}{L^2}\right)^{-1}. \tag{23} \]

It follows from this that the angular velocity \(\Omega\) of rotation of the conical pendulum with a vibrating suspension does not depend on the angle \(\varphi\). It is somewhat difficult to reproduce this type of motion experimentally, since the influence of gravity must be excluded. One can approach it by imparting powerful vibrations to the suspension, so that the vibrational moment considerably exceeds the moment of gravity.

As a more detailed analysis shows,\(^2\) the degree of accuracy obtained for the period of oscillation of a vibrating pendulum,

Fig. 2a. Normal position of the pendulum.

Fig. 2b. Inverted position of the pendulum.

Fig. 3a. Pendulums at rest.

Fig. 3b. Pendulums during vibrations.

altogether be determined by the quantity \(\alpha\), equal to the ratio of the length of the pendulum to the amplitude of vibration of the suspension, and is of order \(\alpha^2\).

The solution of the problem of the oscillations of a pendulum in a gravitational field when the suspension vibrates along the \(y\)-axis, inclined to the vertical at an angle \(\gamma\), is obtained from the solution of equation (14) by substituting into it, for \(M_\varphi\), the value given by expression (16). The resulting equation is integrated, and the solution gives an oscillatory motion in which the amplitude is an elliptic function of time. If the intensity of the vibrations is sufficient for equation (17) to have four roots, then the oscillatory process is possible near two values of the angle \(\varphi\). One corresponds to the inverted position of the pendulum, the other to the normal one. In this case we find that, in the inverted position, the period of oscillation of the pendulum is greater than in the normal position. The period of oscillation of the same pendulum in the absence of oscillations of the suspension has a mean value between these two periods. It follows from the solution of the equation that for any vibrations of the suspension of the pendulum the period of its oscillation in the normal position is always shortened. These phenomena are well demonstrated on the double pendulum shown in Fig. 2. By inclining the apparatus in such a way that the direction of the vibrations makes various angles \(\gamma\) with the vertical, the pendulums can simultaneously be made to swing so that one of them is in the normal position and the other in the inverted one. Then one can visually compare the periods of oscillation and verify the conclusion stated above.

We shall examine in more detail the simplest case of this motion, when the vibrations of the suspension occur in the vertical direction and, consequently, \(\gamma=0\). In addition, we shall take the amplitudes of oscillation to be small, so that \(\sin\varphi\) can be replaced by its argument. Under these conditions equation (14) takes the form:

\[ \left(1+\frac{K^2}{L^2}\right)^3 \ddot{\varphi} = -\alpha^2\omega^2 \left[ \frac{1}{2} \pm \left(1+\frac{K^2}{L^2}\right) \frac{gL}{a^2\omega^2} \right]\varphi . \tag{24} \]

This equation gives, for the variations of the angle \(\varphi\), harmonic oscillations with period \(\tau\), determined by the expression:

\[ \left(\frac{\tau}{T}\right)^2 = \alpha^{-2} \left(1+\frac{K^2}{L^3}\right) \left[ \frac{1}{2} \pm \left(1+\frac{K^2}{L^2}\right) \frac{gL}{a^2\omega^2} \right]^{-1}. \tag{25} \]

In the last two expressions the plus sign corresponds to the position of the center of gravity below the suspension, i.e. to the normal position. The minus sign corresponds to the inverted position; in this case, from (25), we obtain that only when the stability condition (19) is satisfied does the period of oscillation have a real value and, consequently, the oscillatory process in the inverted posi-

is possible. In this problem, by additional analysis one can again show that the periods are determined with accuracy of order \(a^2\).

A further development of the described method of studying a pendulum with a vibrating support is obtained by considering oscillations of the support that have a more complicated character than the simple harmonic motion adopted in expression (7). If the vibration of the support is restricted only by the condition of periodicity, then in the general case the oscillations of the support may be represented in the form of a sum of harmonic frequencies:

\[ U=\sum_n a_n \sin(\omega_n t+\sigma_n). \tag{26} \]

In doing so we introduce a restriction on the magnitudes of the amplitudes of the oscillations:

\[ a^2=L^{-2}\sum_n a_n^2 \ll 1. \tag{27} \]

The former condition (1), restricting the magnitude of the vibrations, is a particular case of this. By the same method as in the preceding case, it can be shown that the motion takes place as though a vibrational moment acted on the pendulum, equal to

\[ \overline{M}=-\frac{1}{4}\left(1+\frac{K^2}{L^2}\right)^{-1} m \sin 2\varphi \sum_n a_n^2 \omega_n^2 . \tag{28} \]

It follows from this that the magnitude of the vibrational moment is still proportional to the mean kinetic energy imparted to the mass of the pendulum by the vibration of the support.

The solution of problems in this case is just as simple as in the preceding one, and usually leads to accuracy of order \(a^2\).

As a further development of this method of solving problems of a pendulum with a vibrating support, one may envisage introducing into the basic equation of motion (14) dissipative forces, for example, those depending on the velocity \(\dot{\varphi}\). This is possible because the forces created by the vibrational moment can perform work which is transformed into the oscillatory energy of the pendulum and, consequently, can be absorbed in motion with friction.

The concept of a vibrational moment may be applied to any body, whether a colloidal particle or a molecule. If the resultant of the forces applied to the body does not pass through its center of gravity, then, under their vibration, a vibrational moment arises, tending to set the body in such a position that its center of gravity would lie on the axis of oscillation. Since the nature of the vibrational moment has so far escaped the notice of theoretical physics, experimentally no orienting action on colloidal and molecular particles has been sought,

which, in the case of their asymmetric form, may be produced, for example, by the application of ultrasonic oscillations or oscillations of an electrical nature. It is interesting to note that anisotropy in the amplitudes of the thermal oscillations of molecules taking place in a crystal lattice will not by itself create a vibrational moment, since, by virtue of the equipartition law for thermal energy, the mean kinetic energy of vibration of the molecules in all directions will be the same; consequently, according to (28), the mean value of the vibrational moment will be zero.

Let us indicate some of the practical possibilities that are opened up by revealing the simple connection existing between vibrations of the suspension and the vibrational moment acting on the body.

A horizontal pendulum, whose motion is described in expression (20), in the case of horizontal vibrations of its suspension, according to expression (26), gives, under the conditions (27), for \(a^2\) the following relation between the kinetic energy of vibration and the angular frequency of oscillations of the pendulum \(\Omega = 2\pi\tau^{-1}\):

\[ \frac{1}{2}\sum_n a_n^2\omega_n^2 = \Omega^2 L^2 \left(1+\frac{K^2}{L^2}\right)^2 . \tag{29} \]

By recording the period \(\tau\) of oscillations of the pendulum, one can determine the energy of horizontal vibrations of the body with which the horizontal pendulum is connected.

Expressions (24) and (25) make it possible to establish the influence of the vertical component of the vibration of the suspension on the period of oscillation of a normal pendulum. Let, in the absence of vibrations, the period of oscillation of this pendulum be \(\tau_0\), and in the presence of vibrations \(\tau=\tau_0+\Delta\tau\); assuming that \(\Delta\tau\) is small in comparison with \(\tau_0\) and neglecting the square \(\Delta\tau^2\), we obtain:

\[ \frac{\Delta\tau}{\tau} = - \frac{a^2\omega^2} {4\left(1+\dfrac{K^2}{L^2}\right)L^2\Omega^2}. \tag{30} \]

We obtain the important result, which we have already mentioned, that vibrations of the suspension always decrease the period of oscillation of the pendulum. The practical interest of this phenomenon lies in the fact that any slight shaking transmitted to the suspension of pendulum clocks, if it has a period shorter than the period of oscillation of the pendulum, will always accelerate the running of the clock.

This is true not only for vibrations of the suspension with a single frequency; using expression (28) for the vibrational moment, one can show that expression (30) can be generalized for sum-

we obtain the kinetic energy at the frequencies of the entire vibration spectrum:

\[ \frac{\Delta \tau}{\tau_0} = -\frac{\sum\limits_n a_n^2 \omega_n^2} {4\left(1+\frac{K^2}{L^2}\right)^2 L^2 g^2}. \tag{31} \]

If there is a pair of identical clocks under identical conditions, but differing in that vibrations of the base are transmitted to the suspension of the pendulum of one of these clocks, while the suspension of the other is isolated from vibrations and is at rest, then from the relative gain of the first clocks one can, with the aid of (31), calculate the mean energy of the vibration spectrum of the base over the period of time during which the difference in the running of the clocks has accumulated.

This opens up the possibility, by measuring the period of the pendulum, of simply studying the mean energy of oscillation of various bases in both the horizontal and the vertical directions. It should be noted that the indicated effect is small, and this may make its practical use difficult for ordinary seismic observations.

A demonstration of the vibrational moment of a pendulum and of the phenomena produced by it does not require especially complicated apparatus and can be carried out with modest laboratory means. Two simple devices are shown in Figs. 2 and 3. As can be seen from the photographs, they consist of a vibrating suspension to which pendulums are freely attached on a hinge. In constructing this device, special attention should be paid to the manufacture of the pendulum itself.

Ordinary demonstration pendulums are made so that, as far as possible, they reproduce the mathematical pendulum. Therefore they consist of a thin rod with a heavy weight at the end. This type of pendulum is completely unsuitable for the present case. If vibrations of the intensity necessary for its stability in the inverted position are imparted to such a pendulum, then from expression (19) it can be shown that an alternating stress will act in the rod, exceeding in magnitude by more than \(L/a\) times the stress caused by the force of gravity of the weight at the end of the rod. Such a force will cause longitudinal bending in the rod, which near the resonance points will create transverse oscillations of the rod with an amplitude exceeding the permissible limits. Therefore one should take rods with such a transverse section that resists longitudinal bending well. We found that, for a pendulum length of 15 to 30 cm, a thin-walled tube of diameter 4 to 8 mm and wall thickness from 0.1 to 0.5 mm is suitable. As for the weight, it is better not to make it at all; but for convenience of demonstration in a large auditorium, a disk of thin sheet metal should be attached to the end of the tube.

PENDULUM WITH A VIBRATING SUSPENSION

(0.1 mm). The hinge at the other end can be made in the form of a fork with a steel pin (1.5 mm), making it possible for the pendulum to oscillate freely at the point of suspension only in one plane.

To demonstrate the phenomenon of stability in the inverted position, with the length of the pendulum rod from 15 to 30 cm and with an amplitude of several millimeters, the frequency of vibration of the suspension should lie within the range from 2 to 7 thousand revolutions per minute. An ordinary small electric motor from a sewing machine is quite suitable for obtaining these frequencies. The motor speed is conveniently regulated by a small variator. We implemented the mechanism for vibrating the suspension point in two ways. Fig. 2 shows the simplest of them. A small ball bearing is eccentrically mounted on the motor shaft; by means of a connecting rod it causes a small cylinder of circular cross-section (5 mm in diameter and 5 cm long) to vibrate, the cylinder sliding in a fixed guide with a cylindrical opening. All these parts must be well fitted and lubricated so as to move with little play. At the outer end of the vibrating cylinder there is a hinge to which the pendulum rod is fastened. The amplitude of the oscillations of the vibrating suspension is set by the degree of eccentricity of the bearing on the motor shaft; in our experiments we usually set it within the range from 2 to 4 mm. To reduce the shaking of the whole apparatus, it is desirable to balance the load on the motor shaft. For damping, it is good to nail small rubber pads made from rubber tubing to the base. With such a pendulum one can easily reproduce all the phenomena described and carry out an approximate quantitative verification of the relations derived.

If it is desired to attach two pendulums in order to compare their behavior in both positions of stability, then it is more convenient to produce the vibration of the suspension point by means of a lever, as shown in Fig. 3. Here the connecting rod is also placed on the motor shaft, but the oscillations are transmitted to a lever, one end of which oscillates about a fixed point, while at the other end two identical pendulums are attached symmetrically on hinges. To avoid vibrations, the lever should be made light and rigid; the best material for it is duralumin. It is important to ensure the rigidity of the oscillating end of the lever in the direction perpendicular to the direction of vibration. This can be achieved with additional braces. In demonstrating the apparatus, the motor can be held in the hand and, by turning it, one can observe the oscillations of both pendulums at once for different directions of vibration.

The demonstration of the phenomenon of oscillation of an inverted pendulum is very effective; the rapid small displacements caused by the vibrations are not visible to the naked eye, and therefore the behavior of the pendulum in the inverted position makes an unexpected impression on the viewer. If

if the apparatus is turned so that the pendulum oscillates in a horizontal plane, the effect of the moment of the force of gravity on the motion is eliminated. If one carefully touches the pendulum rod with a finger and moves it to the side, the finger feels the pressure produced by the vibrational moment, and it is easy to verify that its greatest value corresponds to an angle of rotation of \(45^\circ\). After becoming acquainted experimentally with the dynamic stability of the pendulum in the inverted position, it is difficult not to conclude that it is just as instructive as the dynamic stability of the top, and that it too should occupy an honorable place in lecture demonstrations in mechanics.

References

1. H. and B. S. Jeffereys, Methods of Mathematical Physics, Cambridge, 1951, p. 488.
2. P. L. Kapitza, ZhETF, 21, no. 5 (1951).