Coupled pendulums

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Example of a coupled pendulum

When coupled pendulums are pendant designated between which an exchange of energy (for example by a coil spring can) take place, so that as a Paired harmonic oscillators act. The vibrations carried out are also called coupling vibrations . A directional moment acts in every pendulum , which is caused by gravity and which endeavors to pull the pendulum back into the rest position. In addition, the existing coupling becomes noticeable in the form of an additional straightening torque, which acts in such a way that the spring is relaxed as much as possible.

Several equal pendulums arranged in a row and interacting with their immediate neighbors are known as oscillating chains.

Historical observations

The Dutch astronomer and physicist Christiaan Huygens observed coupled pendulum oscillations as early as the 17th century when he discovered that two identical pendulum clocks , which were installed on board a ship in a common housing, oscillated synchronously after half an hour , regardless of the starting position Pendulum at the beginning. The pendulum weights transferred energy to the watch case and influenced each other. (see: lock-in effect )

Physical-mathematical consideration

coupled pendulum in rest position by the moment of the spring

As a model, let us consider the case of two identical pendulums connected by a spring. Then due to the torque caused by gravity and the opposing moment of the spring, the two pendulums are deflected into a new equilibrium position.

If you now deflect pendulum 2 by the angle to the right, assuming that the length of the spring in the relaxed state is equal to the distance between the suspension points, for small deflections you get an approximate total moment of:

where is the spring constant of the coupling spring.

If you also deflect pendulum 1 by a small angle to the right, an approximate total moment of:

One can proceed in the same way for pendulum 1 and obtain the two differential equations :

is the moment of inertia of a pendulum. If it is a thread pendulum, then applies .

There are three characteristic oscillation forms of the pendulum system:

Example: coupled pendulum as an eigenvalue problem - normal vibration analysis

The equations of motion of the coupled pendulum can be calculated with the Lagrange formalism . For this, the Lagrange function of the system is set up:

where , the kinetic energy of the system is given by:

and , the potential energy of the system, is given by:

From which follows:

The small-angle approximation can be used for small deflections . If only terms up to the 2nd order are taken into account, then as well . With the Euler-Lagrange equation

we get two coupled equations of motion of the form:

.

With the approach that every fundamental oscillation of the normal oscillation described here has the form

you get the matrix representation

.

With the approach:

one determines the eigenvalues (= squares of the eigencircle frequencies, i.e. ). The point behind the approach is that the matrix has full rank - that is, its column vectors are linearly independent . In this case there would only be the trivial solution and thus no oscillation, but the pendulums would simply remain in place.

For the special case one obtains the eigenvalues:

The eigenvectors must now be determined for the eigenvalues . For this example:

and

The solution of the normal vibration analysis ( eigenvalue problem ) for the special case is thus :

Case distinctions

The variables and can be discussed with the graphic below. Figure 1 shows the case that ; Figure 2 shows the case that and Figure 3 shows the case that .

Alternating oscillation: If two similar pendulums are suspended from the same cord and only one is deflected, the energy of the oscillation is periodically transferred from one pendulum to the other.

Oscillation in the same direction

The two pendulums swing with the same amplitude and phase.
oscillation of a coupled pendulum in the same direction

Opposite vibration

The two pendulums swing with the same amplitude but in opposite phase.
opposite oscillation of a coupled pendulum

Beat case

If only one of the two pendulums is deflected from its starting position at the beginning, the oscillation energy slowly moves back and forth between the two pendulums.
Beat case of a coupled pendulum
With the Wilberforce pendulum , a mass suspended from a helical spring can perform both a vertical translation and a rotational movement, which interact with one another via the helical spring. At a certain mass of the vibrating element, the two movements alternate.