3B SCIENTIFIC PHYSICS 1002956 Gebruiksaanwijzing - Pagina 3

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By inserting δ = Λ / T
, ω
d
0
into the equation
ω
=
ω
2
−
δ
2
d
0
we obtain:
Λ
2
=
⋅
+
T
T
1
d
0
π
2
4
whereby the period T
can be calculated precisely pro-
d
vided that T
is known.
0
3.4 Forced oscillations
In the case of forced oscillations a rotating motion with
sinusoidally varying torque is externally applied to the
system. This exciter torque can be incorporated into
the motion equation as follows:
.
..
⋅ + ⋅ + ⋅ =
ϕ
ϕ
ϕ
J
b
D
After a transient or settling period the torsion pendu-
lum oscillates in a steady state with the same angular
frequency as the exciter, at the same time ω
be phase displaced with respect to ω
tem's zero-phase angle, the phase displacement be-
tween the oscillating system and the exciter.
ϕ =
· sin ( ω
· t – Ψ
ϕ
S
E
The following holds true for the system amplitude
M
E
J
ϕ
=
2 2
2
ω
ω
−
) +
(
0
E
The following holds true for the ratio of system ampli-
tude to the exciter amplitude
ϕ
S
=
ϕ
E
ω
−
E
1
ω
0
In the case of undamped oscillations, theoretically
speaking the amplitude for resonance (ω
increases infinitely and can lead to "catastrophic reso-
nance".
In the case of damped oscillations with light damping
the system amplitude reaches a maximum where the
exciter's angular frequency ω
tem's natural frequency. This frequency is given by
2
ω
=
ω
⋅
−
1
Eres
0
= 2 π / T
and ω
= 2 π / T
0
d
⋅
ω
⋅
(
)
M
sin
t
E
E
can still
E
. Ψ
is the sys-
0
0S
)
0S
ϕ
S
2
2
δ
ω
⋅
4
E
M
E
J
2 2
2
2
δ
ω
+
 ⋅
E
4
ω
ω
0
0
equal to ω
E
0
is lower than the sys-
E res
δ
2
ω
2
0
Stronger damping does not result in excessive ampli-
d
tude.
For the system's zero phase angle Ψ
true:
Ψ
=
arctan
0S
For ω
= ω
(resonance case) the system's zero-phase
E
0
angle is Ψ
= 90°. This is also true for δ = 0 and the
0S
oscillation passes its limit at this value.
In the case of damped oscillations (δ > 0) where
ω
< ω
, we find that 0° ≤ Ψ
E
0
it is found that 90° ≤ Ψ
In the case of undamped oscillations (δ = 0), Ψ
for ω
< ω
and Ψ
E
0
4.1 Free damped rotary oscillations
• Connect the eddy current brake to the variable volt-
age output of the DC power supply for torsion pen-
dulum.
• Connect the ammeter into the circuit.
• Determine the damping constant as a function of
the current.
4.2 Forced oscillations
• Connect the fixed voltage output of the DC power
supply for the torsion pendulum to the sockets (16)
of the exciter motor.
• Connect the voltmeter to the sockets (15) of the
exciter motor.
• Determine the oscillation amplitude as a function
of the exciter frequency and of the supply voltage.
• If needed connect the eddy current brake to the
variable voltage output of the DC power supply for
the torsion pendulum.
4.3 Chaotic oscillations
• To generate chaotic oscillations there are 4 supple-
mentary weights at your disposal which alter the
torsion pendulum's linear restoring torque.
• To do this screw the supplementary weight to the
body of the pendulum (5).
)
8
the following is
0S
2 δ ω
ω
ω
2
−
2
ω
0
≤ 90° and when ω
0S
≤ 180°.
0S
= 180° for ω
> ω
.
0S
E
0
4. Operation
> ω
E
0
= 0°
0S