Parametric Excitation of Vibrations

Video in TIB AV-Portal: Parametric Excitation of Vibrations

Formal Metadata

Parametric Excitation of Vibrations
Alternative Title
Parametrische Anregung von Schwingungen
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IWF Signature
C 1627
Release Date
IWF (Göttingen)
Production Year

Technical Metadata

IWF Technical Data
Film, 16 mm, LT, 151 m ; F, 14 min

Content Metadata

Subject Area
The film illustrates essential characteristics of parametric oscillations and demonstrates some important applications.
Keywords parametric oscillations vibration, electrical vibration, mechanical amplitude pendulum frequency pumping frequency rocker model foucault / pendulum phase relationship
Computer animation
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Computer animation Fahrgeschwindigkeit Ballpoint pen Thrust reversal Spare part Force Pendulum Centrifugal force Negativer Widerstand Orbit Angular velocity
Audio frequency H-alpha Cosmic distance ladder Bus Weight Ballpoint pen Pump (skateboarding) Electronic component Force Pendulum FACTS (newspaper) Computer animation Fahrgeschwindigkeit Thrust reversal Orbital period Centrifugal force Angeregter Zustand
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Audio frequency Computer animation Pump (skateboarding)
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A fairground. Sometimes one can meet the good old swing-boat.
Swinging up and down, one can achieve surprisingly high amplitudes without the need to push oneself off the ground. This swinging is a special case of so-called parametric oscillations. How is such a parametric excitation possible? Let us watch, for simplicity, firstly the oscillation of a thread pendulum having the constant length l between its suspension point and the center of mass of the ball. It oscillates with
a certain Eigen-frequency f0, which is given by the acceleration of free fall, g, and the length of the thread, l. l and g are the so-called frequency determining parameters.
Without stimulating the pendulum to oscillations, at first the length l is varied periodically by means of the excenter, seen in the left part of the picture. Now the same periodic variation of l, but this time while the pendulum is oscillating. Increasing amplitudes are observed without any force acting in the direction of the oscillation. This means that this is not a forced oscillation. The length of the pendulum, l, which in this case is varied, is the frequency determining parameter. This explains the name: parametric oscillation. How are these increasing amplitudes of the oscillating pendulum produced? We can understand this by observing the forces working at different phases of the oscillation.
Here a pendulum of length l. During its oscillation the ball is swinging on a part of an orbit. At the reversal points the momentary velocity is equal to zero. Passing the center position the ball has its maximum velocity. At this point the force drawing at the thread is not only the full weight of
the ball Fs = mg but also the highest possible centrifugal force Fz = m w2 • 1. ù is the maximum angular velocity at the center position.
At the reversal points the centrifugal force equals zero because the velocity is equal to zero. And besides this fact along the thread only the component of the weight mg • cos alpha is acting. That means: at the reversal points in total a smaller force is drawing than at the center point. What happens when exciting the oscillations? While the pendulum is passing the center its length is shortened; while going through the reversal points it is lengthened. To recapitulate: what about the balance of energy? At the center point we raise the ball against the here-acting force for the distance 2 Al, thereby putting in a certain amount of energy. In comparison at the reversal points we take out energy. Bus this energy is lower because the here-acting force is smaller than at the center point. By this means energy is pumped into the oscillating system.
Let us watch once more the whole period. Obviously in every period there are two maxima and two minima of the acting force. Thus we are able to supply energy to the oscillating pendulum two times per period. The maximum excitation is achieved, if firstly the pump frequency fp
is the double of the Eigen-frequency f0, and if secondly the feeding in of the energy always is done at the center point.
In comparison if we shift the
phase of the excitation for half a period, now, we observe that, still with constant
pump frequency, the amplitudes are decreasing. Being at rest at last, no variation of the
thread length is able to produce new oscillations, because in this state the energy supplied per period equals the energy taken out.
Here the model of a swing-boat. In this model the distance between the suspension point and the center of gravity of the little man is the frequency-determining parameter. A control circuit provides the
correct pump frequency and correct phase. It does so by means of this excenter.
For controlling the correct operating, namely the man straightening at this phase, suitable for supplying energy, the state of the oscillation is measured by a magnetic sensor. Pumping occurs with the
double of the Eigen-frequency. With optimum adaptation very soon our model achieves large amplitudes. What happens, if the pumping occurs with wrong phase adaptation?
Switching over the control circuit has the effect that now the
man straightens at the reversal points and crouches at the center point. As expected, now energy is removed from the system. The amplitude decreases. And when the swing is at rest even prolonged pumping produces no observable amplitudes.
Now we understand the role of the swing-boat attendant. He has to make the swing-boat swing before any energy supply is possible. This Foucault-pendulum is oscillating in a fixed plane. Without external interference it would shortly be standing still. By parametric excitation it gets the proper energy for oscillating with constant amplitudes.
This is the mechanism on the upper story. It raises and lowers the body of the pendulum with the correct pump frequency and the correct phase. The fact that parametric excitation is without influence on the plane of the oscillation is of special significance for
the Foucault pendulum. The constant site of the plane of oscillation is known to be a presupposition for proving the rotation of the earth, which can be observed by the fact that after 4 hours we can look at the plane of oscillation from another direction of view. Not only mechanical oscillations can be excited parametrically. Here an electro-magnetic oscillatory circuit, consisting of an inductance L and a capacitance C.
These parameters are responsible for the Eigen-frequency f0 according to the formula of the mechanical pendulum. Let us observe a period of oscillation.
For the starting state we assume a maximum electrical charge on the capacitance. The capacitance discharges, and now it is empty. In turn it is charged again, but with the opposite sign of the charge. Let us have a look at the inductance. A current is flowing, which now is highest and then decreases to zero. With the current a growing and vanishing magnetic field is combined. How can this system be supplied with energy?
Let's do an experiment of thought.
An iron cord is pushed into the coil and is now pulled out.
Pushing in is possible without any acting force, when no current flows
and therefore no magnetic field exists.
On the other hand work is supplied to the coil by pulling the core out of the magnetic field
while the current is flowing. In this way the oscillating circuit
gains energy. It functions parametrically because one of the parameters, namely the inductance L, is
varied by moving the iron core in and out. In practice, of course, the inductance is not varied mechanically.
Here on the right-hand side an oscillatory circuit with a capacitor and a coil on the blue core. A control winding, left, on the yellow core serves for exciting with the pump frequency fp, the circuit having the Eigen-frequency f0. That's because a suitable current through the control winding varies the magnetic resistance and with it the inductance L of the circuit. Again for the optimum case is valid: fp = 2 f0. According to this scheme of electrical excitation this so called paraformer is constructed.
On the left-hand side the control winding, on the right the oscillatory circuit.
As an example for a suitable application here a constant voltage circuit, an important part of which is the paraformer. Back to our oscillatory circuit,
consisting of capacitance C and inductance L. We saw that, by a suitable variation of L at the right moment, energy can be supplied. Equally, it should be possible to supply energy by variation of the capacitance C. What is the most feasible moment for inducing energy? Without any charges on the plates of the capacitor, no force is acting between them. When charged, the plates attract each other.
So one pulls the plates apart when they are charged and puts them together again in the discharged state.
That means energy is supplied at the charged state,
but no energy is removed at the discharged state. Obviously, again the optimum pump frequency
is two times the Eigen-frequency
of the circuit. In practice, the capacitance is not varied mechanically but electrically, for example by means of this semiconductor diode.
It is much smaller than a pfennig-coin. The value of its capacitance can be varied by the applied voltage.
In this circuit, this so-called varactor diode is the most important part of a parametric highfrequency amplifier to which a pump voltage up is applied. The oscillatory circuit is a resonant waveguide, into which
the varactor diode is incorporated (on the right-hand-side.) Such amplifiers are used for receiving very weak signals from radio satellites.
Remarkably, for this modern high-tech purpose, the same principle of parametric excitation is used as for the good old swing-boat at the fairground.