00:17

Here we see a beam that is subjected at its ends to compressive forces along its axis, so that its stable equilibrium is in the bent state. Apart from the depicted lower position the upper position is

00:38

stable as well. The two equilibria are indicated by the marks an both sides. We are going to study oscillations of the beam. The movement of the beam is excited by the small box in the middle. This box serves the symbolizing a harmonic excitation. For instance, this symbol simply stands for an unbalanced engine. The position of such an "unbalance" is marked by the pointer. The various possible oscillations can be calculated with the help of this so called Duffing equation and fixed constants as here in the table. F is the frequency. Now the engine rotates with a frequency of 3 Hz. The pointer rotates once in about two seconds, or more precisely, in 2 pi/F seconds. We watch the response of the beam: it oscillates with a small amplitude about the point of equilibrium. Later, this oscillation will be shown once more. Depending on both exciter frequency and the initial position of the beam a variety of different types of vibrations are possible. The following 6 scenes will show 6 different oscillations, all of which have been calculated with the help of the same Duffing equation. Before looking at the first oscillation with the exciter frequency of 1 Hz, we should glance at its corresponding phase-portrait.

02:59

The velocity x-point of motion is plotted as ordinate. Both stable equilibria are marked by crosses; the right-band cross matches the upper equilibrium of the beam. The left hand one the lower equilibrium. The first phase-portrait is symmetrical with respect to the origin. The phase curve is closed; this indicates a periodic movement. The large-amplitude oscillation passes both equilibria. This picture depicts the maximum displacement of the beam versus time. The beam-oscillation is roughly in phase with the exciter cosine-oscillation. That is to say, excitation and beam are approximately synchronized. That is: the pointer will be directed upward when the beam is in the upper position as well. This is the reason for the large amplitude. Now we are going to see the oscillation, which is an harmonic oscillation, that is, the period of excitation is equal to that of the oscillating beam. Here we see the phase-portrait of the second scene. The frequency is the same as in the previous scene, the oscillations however are quite different. The movement is by no means symmetrical with respect to the origin. The phase difference causes opposite displacements of the beam and the excitation. This is the reason for the small amplitude.

06:02

In the 3rd scene the exciter frequency has been increased. With the frequency 3 Hz the beam has no time for a large-amplitude oscillation. As this phase-portrait shows, the beam oscillates with a small amplitude about only one of the two equilibria. So far, the exciter frequency of 3 Hz has been kept constant. In the fourth scene, however, the frequency varies. By slowly lowering the frequency, the amplitude increases.

07:01

Now we shall first see scene No. 3 and, immediately afterwards, scene 4. Starting with frequency 3 Hz, the excitation will be continuously slowed down. In the 5th scene we see a subharmonic oscillation, here it is the second subharmonic. The exciter frequency is again kept constant, F is equal to 0.4 Hz. With a subharmonic oscillation, we distinguish between the oscillation periods of the beam and that of the excitation. As always, the period of the excitation is 2 pi divided by the frequency, which in this case is about 15 seconds. The response of the beam in the second subharmonic has a period of about 30 seconds, which is about twice as long as the excitation period. The last scene shows a quite different kind of oscillation, again with constant exciter frequency. The oscillation however is not periodic at all. The phase-portrait is consequently not a closed curve as before. Instead of this, certain areas of the phase plane are covered in a virtually irregular manner. Finally we show an aperiodic oscillation. This oscillation is called "chaotic" or sometimes "strange attractor".