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unwinds the coil spring then compresses it again in a
periodic sequence and thereby initiates the oscillation
of the copper wheel. The electromagnetic eddy cur-
rent brake (11) is used for damping. A scale ring (4)
with slots and a scale in 2-mm divisions extends over
the outside of the oscillating system; indicators are
located on the exciter and resonator.
The device can also be used in shadow projection dem-
onstrations.
Natural frequency: 0.5 Hz approx.
Exciter frequency:
0 to 1.3 Hz (continuously adjust-
able)
Terminals:
Motor:
max. 24 V DC, 0.7 A,
via 4-mm safety sockets
Eddy current brake: 0 to 20 V DC, max. 2 A,
via 4-mm safety sockets
Scale ring:
300 mm Ø
Dimensions:
400 mm x 140
Ground:
4 kg
2.1 Scope of supply
1 Torsional pendulum
2 Additional 10 g weights
2 Additional 20 g weights
3. Theoretical Fundamentals
3.1 Symbols used in the equations
D
=
Angular directional variable
J
=
Mass moment of inertia
M
=
Restoring torque
T
=
Period
T
=
Period of an undamped system
0
T
=
Period of the damped system
d
=
Amplitude of the exciter moment
M
E
b
=
Damping torque
n
=
Frequency
t
=
Time
Λ
=
Logarithmic decrement
δ
=
Damping constant
ϕ
=
Angle of deflection
ϕ
=
Amplitude at time t = 0 s
0
ϕ
=
Amplitude after n periods
n
ϕ
=
Exciter amplitude
E
ϕ
=
System amplitude
S
ω
=
Natural frequency of the oscillating system
0
ω
=
Natural frequency of the damped system
d
ω
=
Exciter angular frequency
E
ω
=
Exciter angular frequency for max. amplitude
E res
Ψ
=
System zero phase angle
0S
3.2 Harmonic rotary oscillation
A harmonic oscillation is produced when the restoring
torque is proportional to the deflection. In the case of
mm x 270 mm
7
harmonic rotary oscillations the restoring torque is
proportional to the deflection angle ϕ:
M = D · ϕ
The coefficient of proportionality D (angular direction
variable) can be computed by measuring the deflec-
tion angle and the deflection moment.
If the period duration T is measured, the natural reso-
nant frequency of the system ω
ω
= 2 π /T
0
and the mass moment of inertia J is given by
D
ω
=
2
0
J
3.3 Free damped rotary oscillations
An oscillating system that suffers energy loss due to
friction, without the loss of energy being compensated
for by any additional external source, experiences a
constant drop in amplitude, i.e. the oscillation is
damped.
At the same time the damping torque b is proportional
ϕ
to the deflectional angle
The following motion equation is obtained for the
torque at equilibrium
.
..
⋅ + ⋅ + ⋅ =
ϕ
ϕ
ϕ
J
b
D
b = 0 for undamped oscillation.
If the oscillation begins with maximum amplitude
at t = 0 s the resulting solution to the differential equa-
tion for light damping (δ² < ω
lows
ϕ =
· cos ( ω
ϕ
· e
–δ ·t
0
δ = b/2 J is the damping constant and
ω
=
ω
2
−
δ
2
d
0
the natural frequency of the damped system.
Under heavy damping (δ² > ω
oscillate but moves directly into a state of rest or equi-
librium (non-oscillating case).
The period duration T
of the lightly damped oscillat-
d
ing system varies only slightly from T
oscillating system if the damping is not excessive.
By inserting t = n · T
into the equation
d
ϕ =
· cos ( ω
ϕ
· e
–δ ·t
0
ϕ
and ϕ =
for the amplitude after n periods we ob-
n
tain the following with the relationship ω
ϕ
n
− ⋅
δ
=
⋅
n
e
T
d
ϕ
0
and thus from this the logarithmic decrement Λ:
1
Λ = ⋅
δ
= ⋅
T
In
d
n
is given by
0
.
.
0
ϕ
²) (oscillation) is as fol-
0
· t)
d
²) the system does not
0
of the undamped
0
· t)
d
= 2 π /T
d
d
ϕ
ϕ
n
n
=
In
ϕ
ϕ
0
n+1
0
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