The damped driven pendulum and
applications
Presentation by,
Bhargava Kanchibotla,
Department of Physics,
Texas Tech University.
Overview
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The Dynamics of a simple pendulum.
Dynamic steady state.
Cases of dynamic steady state.
Applications
Conclusions
The Dynamics of Simple Pendulum
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The angular acceleration of the pendulum is produced
by a gravitational torque mgR sin φ Corresponding to
the equation of motion
2
d
φ
mR 2 2 + mgR sin φ = 0
dt
Continued….
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Small angular displacement linearizes the
problem by making the torque proportional to
the displacement and the motion is simple
harmonic with characteristic frequency
⎛ g⎞
⎟
ω0 = ⎜⎜
⎟
R
⎝
⎠
φ
Continued…..
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If a torque is applied to
a stationary pendulum,
it swings through an
angle φ and the
restoring force restores
the pendulum as
equilibrium position
N = mgR sin φ
Continued….
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Greater the torque, the
larger the angle φ and
there is a critical torque N
c
The critical torque
assumes a value 90
degrees. If N exceeds
the critical value, then
the applied torque
becomes larger then
the restoring torque
Continued….
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Assuming the damping force, the equation
motion of the pendulum is given by the
following equation adding the restoring and
the damping torque as
2
d
φ
dφ
2
N = mR
+η
+ mgR sin φ
2
dt
dt
Dynamic Steady State
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When a constant torque is applied to the
pendulum at rest, there will be a initial
transient behavior that eventually settles
down to a dynamic steady state after the
transients die out. There are several cases of
this dynamic steady state.
Static Steady state
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1. For low applied torque, N ≤ N there is a
steady state
C
N = N
C
sin φ
in which all the time derivatives vanish after
the initial oscillations have dies out.
Dynamics steady state
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For under damped motion, with a constant
applied torque, we have the following
equation as below
2
d
φ
torque = N − mgR sin φ = mR 2 2
dt
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The torque has specific values at four
particular angles
Continued…
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If the applied torque
exceeds the critical torque,
the motion will be
continuously accelerated
rotation and the pendulum
increases its energy as the
time goes. The angular
speed also increases, but
with fluctuations that repeat
every cycle as indicated in
the figure
Continued…..
When damping is present with ω << ω
and N > N the angular speed increases untill
the damping term approaches the applied
torque.The acceleration fluctuates around an
average that is zero and the pendulum
undergoes a quasi static motion
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c
C
0
Quasi Static motion
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This has the following equation
N
1 dφ
=
+ sin φ
ω C dt
N C
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and the solutions are as follows
ω = 0 forN < N C
ω = ωC
⎡⎛ N
⎢⎜⎜
⎢⎣⎝ N C
⎛ N ⎞
2
⎤
⎞
⎟⎟ − 1⎥ forN > N c
⎥⎦
⎠
⎟⎟ωC forN >> N C
ω = ⎜⎜
⎝ NC ⎠
Quasi Static Motion
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The solutions of the
quasi static motion can
be plotted as shown in
the figure
Quasi Static Motion
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The cyclic variations in
angular speed for
points A and B are in
this plot.
At Point A, the applied
torque has a value N = 1.2 N
and the net torque
varies between 0.2 N C and
2 .2 N C
C
Continued….
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For point B, we have
N = 2 N C so the net torque
varies between N C
and 3 N C producing
more regular variations
in angular speed.
Continued…
For a negligible damping case η → 0 and ω >> ω
We have the following solutions for all valuses
of N
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c
ω = 0 forN ≤ NC
⎛N⎞
ω = ⎜⎜ ⎟⎟ωC for0 ≤ N
⎝ NC ⎠
0
Continued……
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These solutions are
plotted as a graph.It is
seen that the system
exhibits hyteresis
When the torque is
increased for N < N C the
pendulum stabilized at
the angle (pi) satisfying
the relation N = N C sin φ
Continued….
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When N reaches the
critical torque the
angular speed jumps to
the value ωC and then
rises linearly with
further increase in N
For decreasing torque, ω
remains proportional to
N all the way to the
origin as shown.
Continued….
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For the particular case
we have the
ω C = 2ω 0
following plot. For
increasing torques
there is the usual initial
rise in then at zero
frequency until the
critical value N is
reached.
C
Continued….
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For decreasing torques. There is a hyteresis
with the zero average frequency reached at a
torque N C' which is less than N C .
Application
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The damped driven pendulum equation has
a particularly important applications in solid
state physics When 2superconductorsin
close proximity with a thin layer of insulating
material between them, the arrangement
constitutes a Joseph son junction,
Continued….
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Current exceeding the value is accompanied by the
presence of a voltage, and the plots of current I
versus voltage V for the junction exhibit
hysteresis.The Joseph son junction satisfies the
same differential equation as the damped oscillator
with the current playing the role of the torque, the
voltage playing the role of the average angular
speed, the capacitance acting like a moment of
inertia and the electrical conductance serving as the
viscosity.
Conclusion
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The damped driven pendulum is a very
important system that has a very significant
application in the field of solid state physics.
Also the system is very important to be
understood as it has a lot of physics involved
in