Parametric Excitation of Vibrations
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Formal Metadata
Title 
Parametric Excitation of Vibrations

Alternative Title 
Parametrische Anregung von Schwingungen

Author 

Contributors 

License 
CC Attribution  NonCommercial  NoDerivatives 3.0 Germany:
You are free to use, copy, distribute and transmit the work or content in unchanged form for any legal and noncommercial purpose as long as the work is attributed to the author in the manner specified by the author or licensor. 
DOI  
IWF Signature 
C 1627

Publisher 
IWF (Göttingen)

Release Date 
1987

Language 
English

Producer 

Production Year 
1985

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

Content Metadata
Subject Area  
Abstract 
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

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A fairground. Sometimes one can meet the good old swingboat.
00:56
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 socalled 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
01:24
a certain Eigenfrequency f0, which is given by the acceleration of free fall, g, and the length of the thread, l. l and g are the socalled frequency determining parameters.
01:36
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.
02:28
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
02:51
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.
03:05
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 hereacting 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 hereacting force is smaller than at the center point. By this means energy is pumped into the oscillating system.
04:17
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
04:39
is the double of the Eigenfrequency f0, and if secondly the feeding in of the energy always is done at the center point.
04:48
In comparison if we shift the
04:50
phase of the excitation for half a period, now, we observe that, still with constant
04:56
pump frequency, the amplitudes are decreasing. Being at rest at last, no variation of the
05:08
thread length is able to produce new oscillations, because in this state the energy supplied per period equals the energy taken out.
05:18
Here the model of a swingboat. In this model the distance between the suspension point and the center of gravity of the little man is the frequencydetermining parameter. A control circuit provides the
05:31
correct pump frequency and correct phase. It does so by means of this excenter.
05:40
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
05:55
double of the Eigenfrequency. With optimum adaptation very soon our model achieves large amplitudes. What happens, if the pumping occurs with wrong phase adaptation?
06:10
Switching over the control circuit has the effect that now the
06:13
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.
06:37
Now we understand the role of the swingboat attendant. He has to make the swingboat swing before any energy supply is possible. This Foucaultpendulum 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.
07:15
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
07:39
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 electromagnetic oscillatory circuit, consisting of an inductance L and a capacitance C.
08:15
These parameters are responsible for the Eigenfrequency f0 according to the formula of the mechanical pendulum. Let us observe a period of oscillation.
08:28
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?
09:05
Let's do an experiment of thought.
09:08
An iron cord is pushed into the coil and is now pulled out.
09:14
Pushing in is possible without any acting force, when no current flows
09:20
and therefore no magnetic field exists.
09:23
On the other hand work is supplied to the coil by pulling the core out of the magnetic field
09:28
while the current is flowing. In this way the oscillating circuit
09:32
gains energy. It functions parametrically because one of the parameters, namely the inductance L, is
09:37
varied by moving the iron core in and out. In practice, of course, the inductance is not varied mechanically.
09:48
Here on the righthand 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 Eigenfrequency 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.
10:26
On the lefthand side the control winding, on the right the oscillatory circuit.
10:33
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,
10:45
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.
11:27
So one pulls the plates apart when they are charged and puts them together again in the discharged state.
11:35
That means energy is supplied at the charged state,
11:39
but no energy is removed at the discharged state. Obviously, again the optimum pump frequency
11:48
is two times the Eigenfrequency
11:50
of the circuit. In practice, the capacitance is not varied mechanically but electrically, for example by means of this semiconductor diode.
11:59
It is much smaller than a pfennigcoin. The value of its capacitance can be varied by the applied voltage.
12:09
In this circuit, this socalled 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
12:26
the varactor diode is incorporated (on the righthandside.) Such amplifiers are used for receiving very weak signals from radio satellites.
12:41
Remarkably, for this modern hightech purpose, the same principle of parametric excitation is used as for the good old swingboat at the fairground.