P2132705

Forced oscillations - Pohl's pendulum

The goal of this experiment is to investigate the oscillation behaviour induced by forced oscillation.

Physics Acoustics Wave Motion

   
Difficulty level Group size Preparation time Execution time

medium - - -

Robert-Bosch-Breite 10 Tel.: 0551 604 - 0 info@phywe.de

37079 Göttingen Fax: 0551 604 - 107 www.phywe.de
P2132705

General information

Application

Pendulum oscillations offer a first understanding of

mechanical systems close to the harmonic oscillator, which
is fundamental in the description of many physical systems
in fields such as particle physics and solid state physics.

This experiment investigates the behaviour of a system,

that is forced into oscillation. It offers insight in
phenomena such as resonance.

Experimental setup

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Other information (1/3)

Prior The behaviour of a singular pendulum shoud be known.

knowledge

If an oscillating system is allowed to swing freely it is observed that the decrease of

Main successive maximum amplitudes is highly dependent on the damping. If the oscillating
system is stimulated to swing by an external periodic torque, we observe that in the
principle steady state the amplitude is a function of the frequency and the amplitude of the
external periodic torque and of the damping. The characteristic frequencies of the free
oscillation as well as the resonance curves of the forced oscillation for different
damping values are to be determined.

Other information (2/3)

Learning The goal of this experiment is to investigate the oscillation behaviour induced by
forced oscillation.
objective

Free oscillation

1. Determine the oscillating period and the characteristic frequency of the undamped
Tasks case.

2. Determine the oscillating periods and the corresponding characteristic frequencies

for different damping values. The corresponding ratios of attenuation, the damping
constants and the logarithmic decrements are to be calculated.

3. Realize the aperiodic and the creeping case.

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Other information (3/3)

Tasks Forced oscillation

1. Determine the resonance curves and represent them graphically using the damping
values of A. Determine the corresponding resonance frequencies and compare
them with the resonance frequency values found beforehand.

2. Observe the phase shifting between the torsion pendulum and the stimulating
external torque for a small damping value for different stimulating frequencies

Theory (1/3)

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 Φ and M2 = −CΦ̇

Φ = angle of rotation. Φ̇ = angular velocity, D0 = torque per unit angle, C = factor of proportionality
depending on the current which supplies the eddy current brake.

The resultant torque M1 = −D0 Φ − CΦ̇

leads us to the following equation of motion:

I Φ̈ + cΦ̇ + D0 Φ = 0 (1)

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Theory (2/3)

Dividing Eq. (1) by and using the abbreviations The solution of the differential equation (2) is

δ= C and ω2
2I 0
=
D
0

I
t
Φ( ) = Φ0 e − δt cos ωt (3)

ω = √ω
−−−−−−
results in Φ̈ + 2δΦ + ω20 Φ = 0 (2) 2
0
− 2 δ (4)

δ is called the “damping constant” and

ω0 = √ DI−
−−0

F = mω r 2

the characteristic frequency of the undamped

system.

Theory (3/3)

Forced oscillation where

If the pendulum is acted on by a periodic torque Φα =

Φ0
(9)
Ma = M0 cos ωa t Eq. (2) changes into √( ωα
1−( ω )
2
)
2
δ ωα
+( 2 ω ω
2
)
0 0 0

Φ̈ + 2 Φ̇ + δ ω20 Φ = F0 cos ωa t (7)

F
and Φα = ω20
M
where F0 = I 0
0

Furthermore:
In the steady state, the solution of this differential
2δωα
equation is tan α= ω20 −ω2a
(10)
t
Φ( ) = Φa cos( a − ωt α (8)

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Equipment
Position Material Item No. Quantity
1 Torsion pendulum after Pohl 11214-00 1
2 Digital Laboratory Power Supply 2 x 0 - 30 V/0 - 5 A DC/5 V/3 A fixed EAK-P-6145 1
3 Digital stopwatch, 24 h, 1/100 s and 1 s 24025-00 1
4 Connecting cord, 32 A, 750 mm, red 07362-01 2
5 Connecting cord, 32 A, 750 mm, blue 07362-04 2

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Setup and Procedure

Setup

The experiment is set up as shown in

Fig. 1 und 2. The DC output of the power
supply unit is connected to the eddy
current brake. The motor also needs DC
voltage. For this reason a rectifier is
Fig. 1: Experimental setup inserted between the AC output (12 V) of
the power supply unit and to the two
right sockets of the DC motor (see Fig.
3). The DC current supplied to the eddy
I
current brake, B , is set with the
Fig. 3: adjusting knob of the power supply and
is indicated by the ammeter.

Connection of the DC Fig. 2: Electrical connection

motor and power supply. of the experiment.

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Procedure (1/5)

A. Free oscillation

1. Determine the oscillating period and the characteristic frequency of the undamped case

To determine the characteristic frequency ω0 of the torsion pendulum without damping (IB = 0),
deflect the pendulum completely to one side,

measure the time for several oscillations.

T
The measurement is to be repeated several times and the mean value of the characteristic period ¯0 is to
be calculated.

Procedure (2/5)

2. Determine the oscillating periods and the corresponding characteristic frequencies for different
damping values.

In the same way the characteristic frequencies for the damped oscillations are found using the following
current intensities for the eddy current brake:

IB ∼ 0.25 A, (U = 4 V), IB ∼ 0.40 A, (U = 6 V), IB ∼ 0.55 A, (U = 8 V), IB ∼ 0.90 A, (U = 12 V)

To determine the damping values for the above mentioned cases measure unidirectional maximum
amplitudes as follows:

deflect the pendulum completely to one side,

observe the magnitude of successive amplitudes on the other side.

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Procedure (3/5)

Initially it has to be ensured that the pendulum pointer at

rest coincides with the zero-position of the scale (see Fig.
4). This can be achieved by turning the eccentric disc of the
motor.

3. Realize the aperiodic and the creeping case

I ∼
To realize the aperiodic case ( B 2.0 A) and the creeping
I ∼
case ( B 2.3 A) the eddy current brake is briefly loaded
with more than 2.0 A. Caution: Do not use current
intensities above 2.0 A for the eddy current brake for
more than a few minutes.

Fig. 4: Pendulum pointer at zero-position.

Procedure (4/5)

B. Forced oscillation

To stimulate the torsion pendulum, the connecting rod of the motor is fixed to the upper third of the
stimulating source. The stimulating frequency ωα of the motor can be found by using a stopwatch and
counting the number of turns (for example: stop the time of 10 turns).

1. Determine the resonance curves and represent them graphically using the damping values of A.

The measurement begins with small stimulating frequencies ωα . ωα is increased by means of the
motorpotentiometer setting “coarse“. In the vicinity of the maximum ωα is changed in small steps using the
potentiometer setting “fine“ (see Fig. 5). In each case, readings should only be taken after a stable pendulum
amplitude has been established. In the absence of damping or for only very small damping values, ωα must
be chosen in such a way that the pendulum does not exceed its scale range.

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Procedure (5/5)

2. Observe the phase shifting between the torsion

pendulum and the stimulating external torque for a
small damping value for different stimulating
frequencies

Chose a small damping value and stimulate the pendulum

in one case with a frequency ωα far below the resonance
frequency and in the other case far above it. Observe the
corresponding phase shifts between the torsion pendulum
and the external torque. In each case, readings should
only be taken after a stable pendulum amplitude has been
established.
Fig. 5: Control knobs to set the motor
potentiometer. Upper knob: “coarse”; lower
knob: “fine”.

Evaluation

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Evaluation (1/6)

The mean value of the period t̄ 0 and the corresponding characteristic frequency ω0 of the free and
undamped swinging torsional pendulum are found to be

ΔT¯0
T¯0 = (1.817 ± 0.017)s; = ±1%
T¯0

and ω̄0 = (3.46 ± 0.03)1/s

Evaluation (2/6)

Plot successive unidirectional maximum amplitudes as a function of time. The respective time is calculated
from the frequency. See Fig 6 for sample results. The corresponding ratios of attenuation, the damping
constants K and the logarithmic decrements Λ are to be calculated as follows:
t
Eq. (3) shows that the amplitude Φ( ) of the damped oscillation has decreased to the e-th part of the initial
t δ
amplitude Φ0 after the time = 1/ has elapsed. Moreover, from Eq. (3) it follows that the ratio of two
successive amplitudes is constant.
Φn
Φn+1
= K = eδT (5)

K is called the “damping ratio” and T = oscillating period and the quantity

Λ = ln K = δT = ln Φn
Φn+1
(6)

is called the “logarithmic decrement“.

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Evaluation (3/6)

Sample results for the characteristic damping values:

−−−−−−
δ δ
I [A]1/ [s] [1/s] ω = √ω20 − δ 2 [1/s] K = ΦΦn+1n Λ
0.25 16.7 0.06 3.46 1.1 0.12
0.4 6.2 0.16 3.45 1.4 0.31
0.55 3.2 0.31 3.44 1.9 0.64
0.9 1.1 0.91 3.34 5.6 1.72

Fig. 6: Maximum values of unidirectional

amplitudes as a function of time for
different dampings.

Evaluation (4/6)

Eq. (4) has a real solution only if ω20 ≥ δ 2 . For ω20 = δ 2 , the pendulum returns in a minimum of time to its
initial position without oscillating (aperiodic case). For ω20 ≤ δ 2 the pendulum returns asymptotically to its
initial position (creeping).

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Evaluation (5/6)

Fig. 7 shows the resonance curves for different dampings.

An analysis of Eq. (9) gives evidence of the following which is
confirmed by the results in Fig 7:

1. The greater F0 , the greater Φα

2. For a fixed value F0 we have: Φ → Φmax for ωα = ω0

3. The greater δ , the smaller Φα

4. For δ = 0 we find Φa → ∞ if ωα = ω
0

Evaluating the curves of Fig 7 leads in this sample result to Fig. 7: Resonance curves for different
medium resonance frequency of ω = 3.41 1/s which comes dampings.
very close to the resonance frequency determined in task
A1.

Evaluation (6/6)

Fig. 8 shows the phase difference of the forced oscillation

as a function of the stimulating frequency according to Eq.
(10). For very small frequencies ωα the phase difference is
approximately zero, i.e. the pendulum and the stimulating
torque are “inphase”.

If ωα is much greater than ω0 , pendulum and stimulating

torque are nearly in opposite phase to each other. The
smaller the damping, the faster the transition from
swinging “inphase” to the “in opposite phase” state can be
achieved.

Fig. 8: Phase shifting of forced oscillation

for different dampings.

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