LEP
1.3.27
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Forced Oscillations – Pohl’s pendulum
Determination of resonance frequencies by Fourier analysis
Related topics
Angular frequency, characteristic frequency, resonance frequency, torsion pendulum, torsional vibration, torque, restoring torque, damped/undamped free oscillation, forced oscillation, ratio of attenuation/decrement, damping constant, logarithmic decrement, aperiodic case, creeping.
Principle
If an oscillating system is allowed to swing freely it is observed
that the decrease of successive maximum amplitudes is highly dependent on the damping. If the oscillating system is stimulated to swing under the influence of an external periodic
torque, we observe that in the steady state the amplitude is a
function of the frequency and the amplitude of the external
periodic torque and of the damping. Characteristic parameters such as frequencies of the free oscillation can be determined and the curves corresponding to the different cases of
oscillation as well as the resonance curves describing the
forced oscillation for different damping values are to be displayed. Therefore, the oscillations are recorded with the
Cobra3 system in connection with the movement sensor.
Equipment
Torsion pendulum after Pohl
Power supply, universal
Bridge rectifier, 30 V AC/1 A DC
Stopwatch, digital, 1/100 sec.
Digital multimeter
11214.00
13500.93
06031.10
03071.01
07134.00
Connecting cord, l = 250 mm, yellow
Connecting cord, l = 750 mm, red
Connecting cord, l = 750 mm, blue
Cobra3 Basic Unit
Power supply, 12 VRS 232 data cable
Cobra3 Translation/Rotation Software
Movement sensor with cable
Adapter, BNC-socket/4mm plug pair
Adapter, BNC-socket-plug, 4 mm
Silk thread, l = 100 m
Tripod base ”PASS”
Support rod ”PASS”, l = 250 mm
Right angle clamp ”PASS”
PC, Windows® 95 or higher
07360.02
07362.01
07362.04
12150.00
12151.99
14602.00
14512.61
12004.10
07542.27
07542.20
02412.00
02005.55
02025.55
02040.55
2
2
3
1
1
1
1
1
1
1
1
1
1
2
Tasks
A. Free oscillation
1
1
1
1
1
1. To determine the oscillating period and the characteristic frequency v0 of the undamped case.
2. To determine the oscillating periods and the corresponding characteristic frequencies for different damping values. Successive, unidirectional maximum amplitudes are to be plotted as a function of time. The corresponding ratios of attenuation, the damping constants
and the logarithmic decrements are to be calculated.
3. To realize the aperiodic case and the creeping.
Fig. 1a. Experimental set-up for free and forced torsional vibration
PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen
P2132711
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LEP
1.3.27
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Forced Oscillations – Pohl’s pendulum
Determination of resonance frequencies by Fourier analysis
B. Forced oscillation
1. The resonance curves are to be determined and represented graphically using the damping values of A.
2. The resonance frequencies are to be determined and
compared with the resonance frequency values found
beforehand.
Use a paper-fastener or the weight holder and hang it at the
end of the thread to tighten it. So, the thread runs parallel and
always leaves the pendulum perpendicular.
Fig. 2. Connection of the movement sensor to the Cobra3
Basic Unit
3. The phase shift between the amplitudes of the torsion
pendulum and of the stimulating external torque is to be
observed for a small damping value with the stimulating
frequency far below the resonance frequency in one
case, and far above it in the other.
Fig. 1b: Electrical connection of the experiment.
red
black
yellow
BNC1
BNC2
Procedure
Set-up
The experiment is set up as shown in Fig. 1a and 1b. The DC
output U_ of the power supply unit is connected to the sockets on the right side of the DC motor. The eddy current brake
also needs DC voltage. For this reason a rectifier is inserted
between the AC output U_ of the power supply unit and the
entrance to the eddy current brake. The value of the DC current supplied to the eddy current brake, IB, is indicated by the
ammeter.
Perform the electrical connection of the movement sensor to
the Cobra3 Basic Unit according to Fig. 2.
To obtain a connection between the movement sensor and the
pendulum do as follows:
Place the movement sensor at the end of the table to ensure
that the used thread can easily swing.
Take about 100 cm of thread (dependent on the distance
between pendulum and sensor), tie a knot on it to form a loop
and place it carefully in the groove of the copper-pendulum
starting from the bottom. rotate the pendulum half and fix one
end with some adhesive tape.
Wind the other side of the closed thread around the larger of
the two cord grooves on the movement sensor once, in a way
similar to a bicycle chain. To reduce friction, let the trread run
parallel to the plane of the copper-disk of the pendulum and
adjust the length of the thread. Too much of a tension increases friction, though there must be some friction at the grooves
of the pendulum and of the movement sensor in order to be
able to record the oscillation.
Note: Make sure to place the knot between the pendulum and
the movement sensor in a way that it will not cause extra friction at the grooves.
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FREE OSCILLATION – UNDAMPED CASE
To determine the characteristic circular frequency v0 = 2π/T0
of the torsion pendulum without damping (IB = 0), the time for
several oscillations—is measured repeatedly and the mean
value of the period T0 calculated.
Set the measuring parameters in measure according to Fig. 3.
Select 12 mm for the ”axle diameter” of the movement sensor.
The axis diameter in the ”Rotation” menu item is twice the distance from the pivot point of the pendulum to the groove
around which the silk thread is wound, i.e. here about
185 mm.
Fig. 3. Measuring parameters
PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen
Forced Oscillations – Pohl’s pendulum
Determination of resonance frequencies by Fourier analysis
Press the button ”continue” to enter the measurement window.
Here, the actual value of the movement sensor is displayed.
Set the pendulum in motion (oscillation amplitude up to 15
scale divisions) and click on ”Start measurement”.
After approximately 5-10 oscillations click on the ”Stop measurement” icon.
Fig.4 shows the recorded curve of this undamped oscillation.
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1.3.27
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If the values (50 ms) in the ”Get value every (50) ms” (Fig. 3)
dialog box are too high or too low, noisy or non-uniform measurements can occur. In this case adjust the measurement
sampling rate appropriately.
There are different opportunities to determine the oscillation
period and the characteristic frequency of the undamped
case.
The duration of the period can either be calculated with the aid
of the cursor lines, which can be freely moved and shifted
onto the adjacent maxima or minima of the oscillation curve
(Fig. 5).
Alternatively, use the item ”curve analysis” in the analysis
menu, where the extrema of the chosen curve can be calculated.
Or use ”Fourier analysis”, where the peak displays directly the
period of the oscillation.
FREE OSCILLATION – DAMPED CASE
Characteristic frequencies for the damped oscillations are
found in the same way as for the undamped case.
Rebuild the experimental set-up as described above, now
using the following current intensities for the eddy current
brake (controlled by the ammeter):
Table for the current values for the eddy current brake:
Remarks
Initially, it has to be ensured that the pendulum pointer at rest
coincides with the zero-position of the scale. This can be
achieved by turning the eccentric disc of the motor.
IB
IB
IB
IB
IB
~
~
~
~
~
0.16
0.34
0.52
0.70
0.88
A,
A,
A,
A,
A,
(U~
(U~
(U~
(U~
(U~
=
=
=
=
=
2 V)
4 V)
6 V)
8 V)
12 V)
Fig. 6 shows the curve of the damped case with IB ~ 0.25 A.
Try to rotate the pendulum carefully and release it quickly without friction of your hand. This avoids disturbances of the
thread and damping of the pendulum, which can lead to measurement errors.
To realize the aperiodic case (delta ~ v0, IB ~ 1.5 A) and the
creeping Case (delta > v0, IB ~ 1.7 A) the eddy current brake
is briefly connected directly to the DC output U_ of the power
supply unit.
Fig. 5: Calculation of the period duration with cursor lines
Fig. 6: Recorded curve of the damped oscillation
PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen
P2132711
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LEP
1.3.27
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Forced Oscillations – Pohl’s pendulum
Determination of resonance frequencies by Fourier analysis
The calculation of the oscillating periods is the same as
described above. To evaluate the characteristic damping values, determine the magnitude of successive amplitudes on
the same side (i.e. either minima or maxima). In order to do
that, use the cursor lines again or chose the curve-analysis
function in the analysis menu. Here, you can calculate the
extrema of the curve and use them directly for the determination of the damping ratio (Fig. 7). Alternatively, use ”line fitting”, where the coefficients of a damped cosinus curve
(ae-dt · cos (bx+c) which fits best the obvservation, correspond
to the values of the amplitude a, frequency b, phase c and
attenuation d.
Forced oscillation
To stimulate the torsion pendulum, the rim in the motor is fixed
to the upper third of the rod in the lever arm at the back side
of the pundulum in order to obtain sufficiently large amplitudes. The DC voltage U_ of the power supply unit must be
set to maximum.
The stimulating frequency va of the motor can be found by
using a stopwatch and counting the number of turns.
The amplitudes of the forced oscillation are recorded in the
same way as for the free oscillations. The measurement
begins with small frequencies. va is increased by means of the
motor-potentiometer setting ”coarse”. In the vicinity of the
maximum amplitude in the resonance case va is changed in
small steps using the potentiometer setting ”fine”. In each
case, values should be taken into account only after a stable
pendulum amplitude has been established. The higher the
damping values are the faster this steady state is reached.
Remarks
— In extremely short oscillation periods, signal transients or
deformations can occur. These can be reduced if the sampling rate is changed. In any case, error-free recorded
intervals can be selected from the measuring signal after
completion of the measurements.
— Sickle-shaped deformation of the oscillations are due to
slippage of the thread across the cord groove on the
movement sensor. This is avoided if the thread is wound
around the cord groove once.
— Since the movement recording is not performed without
contact, slight damping of the measured oscillations does
occur, but the difference between ”nearly undamped” and
damped oscillations is very significant.
Theory and evaluation
A. Undamped and damped free oscillation In case of free and
damped torsional vibration torques M1 (spiral spring) and M2
(eddy current brake) act on the pendulum. We have
M1 = – D0 f and M2 = – Cf
f
·
f
D0
C
=
=
=
=
angle of rotation
angular velocity
torque per unit angle
factor of proportionality depending on the current
which supplies the eddy current brake
The resultant torque
M = – D0f – Cf
Fig. 8 shows the measured curve near the resonance frequency for a small damping value (IB ~ 0.16 A).
leads us to the following equation of motion:
..
If + Cf + D0f = 0
In the absence of damping or for only very small damping values, va must be chosen in such a way that the pendulum does
not exceed its scale range (resonance catastrophe).
Therefore, in the resonance regime the measurement has to
be stopped before the maximum values are reached.
Again, the maximum oscillation amplitudes can be calculated
with the aid of the cursor lines or with the curve analysis.
The resonance frequencies can be determined by Fourier
analysis.
(1)
I.. = pendulum’s moment of inertia
f = angular acceleration
Fig. 8: Forced oscillations for a small damping value
Fig. 7: Curve analysis parameters
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P2132711
PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen
Forced Oscillations – Pohl’s pendulum
Determination of resonance frequencies by Fourier analysis
where
Dividing Eq. (1) by I and using the abbreviations
d=
C
D0
and v20 =
2I
I
fa results in
..
f + 2 df + v20f = 0
f0
va 2 2
d va 2
d
e1 c d f c2
v
v
A
0
0 v0
(9)
(2)
and f0 =
d is called the “damping constant” and
v0 =
LEP
1.3.27
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D0
B I
F0
v20
Furthermore:
the characteristic frequency of the undamped system.
tan a =
2 dva
v20 – v2a
The solution of the differential equation (2) is
respectively
f(t) = f0e-dt cos vt
(3)
a = arc tan
with
v = 2v20 d2
(4)
Eq. (3) shows that the amplitude f(t) of the damped oscillation
has decreased to the e-th part of the initial amplitude f0 after
the time t = 1/d has elapsed. Moreover, from Eq. (3) it follows
that the ratio of two successive amplitudes is constant.
fn
= K = edT
fn1
(5)
2 dva
(10)
v20 – v2a
An analysis of Eq. (9) gives evidence of the following:
1. The greater F0 , the greater fa
2. For a fixed value F0 we have:
f S fmax for va v0
3. The greater d, the smaller fa
4. For d = 0 we find:
fa S ∞ if va = v0
K is called the “damping ratio” and the quantity
= ln K = dT = ln
fn
fn1
(6)
is called the “logarithmic decrement”. Eq. (4) has a real solution only if
v20 d2.
For v20 = d2, the pendulum returns in a minimum of time to its
initial position without oscillating (aperiodic case). For v20 < d2,
the pendulum returns asymptotically to its initial position
(creeping).
Fig. 9: Phase shift between the amplitudes of the pendulum
and of the motor in the case of the forced oscillation for
different damping values
B. Forced oscillation
If the pendulum is acted on by a periodic torque Ma =
M0 cos vat Eq. (2) changes into
..
f + 2 df + v20f = F0 cos vat
where F0 =
(7)
M0
I
In the steady state, the solution of this differential equation is
f(t) fa cos (vat – a)
(8)
PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen
P2132711
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LEP
1.3.27
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Forced Oscillations – Pohl’s pendulum
Determination of resonance frequencies by Fourier analysis
Results
Typical values for this experimental set-up are:
For very small frequencies va the phase difference is approximately zero, i.e. the pendulum and the stimulating torque are
”in phase”. If va is much greater than v0, pendulum and stimulating torque are nearly in opposite phase to each other.
—
T0 = (1.75 ± 0.05) sec;
— = (3.59 ± 0.10) sec–1
v
0
I
1/d
d
v
K
0.16
11.1
0.09
3.58
1.17
0.16
0.34
3.8
0.26
3.57
1.58
0.46
0.52
2.0
0.51
3.54
2.47
0.91
0.70
1.2
0.85
3.50
4.60
1.53
0.88
0.8
1.30
3.42
10.89
2.39
Fig. 9 shows the phase difference of the forced oscillation as
a function of the stimulating frequency according to Eq. (10).
Fig. 10 illustrates the decrease in the unidirectional amplitude
values as a function of time for free oscillation and different
damping values.
Fig. 11 shows the resonance curves for different damping values.
Evaluating the curves leads to medium resonance frequency
v = 3.59 s–1.
Furthermore, the software allows to calculate the Fourierspectra of the recorded oscillations. With this opportunity it
can easily be shown that for increasing damping values the
frequency of maximum amplitude shifts to smaller values (see
Fig.12).
Fig.10: Maximum values of unidirectional amplitudes as a function of time for different damping values.
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PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen
Forced Oscillations – Pohl’s pendulum
Determination of resonance frequencies by Fourier analysis
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Fig. 11: Resonance curves for different damping values.
Fig. 12: Fourier-spectra for different damping values, increasing left to right.
PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen
P2132711
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LEP
1.3.27
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Forced Oscillations – Pohl’s pendulum
Determination of resonance frequencies by Fourier analysis
P2132711
PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen