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Oscillation and Decay: a pictorial tour!
-Abhijit Kar Gupta, [email protected]
Oscillatory motion is so common and so fundamental that any beginner in physics should learn
to deal with them with utmost confidence and eagerness.
Any textbook on Mechanics or on Sound describes the motion of a particle or a system where
naturally occurring restoring force and existent damping force in the medium are present. As
the system is set into motion, the restoring force wants to drive the system towards equilibrium
(Hooke’s law), which is proportional to the displacement. Damping force, which also acts
against the motion (basically the friction), is proportional to the velocity. These two forces play
an interesting duel, who will control and how to drive the system to equilibrium and how fast.
Also, there can be the additional force applied to the system which would rescue the system
from decaying!
Any text book would write how a simple second order differential equation can be written out
of the above and then solve them exactly to achieve exact solutions, faithful to the various
scenario that arise. The mathematical solutions can be plotted in order to visualize and one
may have a fair idea of the various possible extents: when the damping force is comparatively
small, medium or large and etc. and so on.
Instead of solving the differential equations mathematically, we here discretize the equations
and use the simple algorithmic steps in computer to obtain results. Thus we will have a nice
tool to play with the equations and can easily plot to see whatever and whenever we like to
see!
Let consider the following damped harmonic oscillation:
(1)
To play with this in computer, we go through the following steps:
Put,
2
In discrete form,
(2)
Now for a fixed value of (damping term) and (restoring term), if we set the time step as
we wish, and start from an initial value of velocity, (say,) and position, (say,), we can find
the trajectory with time and all that.
From (2) we find, (3)
Now,
(4)
This is how the position (4) and velocity (3) evolve as the time increases by .
[I write a simple Fortran program on this and plot the numerical output in ‘Gnuplot’ in Linux.+
In the same way, we can numerically solve any differential equation, adding or manipulating
any suitable term we want. Then we can go for an easy pictorial tour while playing with the
simple computer program. It’s a fun! *Of course, a similar exercise can be done through
software like MATHEMATICA or MATLAB too.]
In the following two panels, the simple harmonic oscillation is shown, where we take zero damping ( ). The R.H.S. is a phase plot (velocity vs. position). In the successive panels we produce the pictures for increasing damping.
𝑑 𝑥
𝑑𝑡 𝑘 𝑥
SHM (No Damping)
3
𝑑 𝑥
𝑑𝑡 𝜆
𝑑𝑥
𝑑𝑡 𝑘 𝑥
Damped Harmonic Motion
[Small damping, under-damped motion, 𝜆 < 𝑘]
Phase plot:
Damped Harmonic Motion
[Larger damping, still under-damped motion,
𝜆 < 𝑘]
Phase plot:
4
Damped Harmonic Motion
[Critical Damping, when, 𝜆 𝑘 ]
Phase Plot: Critical Damping
Overdamped Motion:
The curve with highest peak corresponds to Critical
damping. Other curves are for overdamped
motion. Larger the damping, the slower the decay.
Overdamped Motion:
Same as the curves on the left side pannel. Only
the initial position is not at origin.ping, the slower
the decay. The lower most curve is for critical
damping.
5
We can do similar exercises with lots of other differential equations. For example, we
discretized the non-linear van der Pol oscillator equation which is known to produce Relaxation
oscillation.
Van der Pol Oscillator:
( )
FORTRAN Program: (Damped Harmonic Oscillator)
Open(1,file=’x.dat’)
x=0.0
v=0.5
t=0
dt=0.001
k=5.0
alam=10
dv=-k*x*dt-alam*v*dt
v=v+dv
dx=v*dt
x=x+dx
t=t+dt
write(1,*)t,x
stop
end
𝑑 𝑥
𝑑𝑡 𝜆( 𝑥 )
𝑑𝑥
𝑑𝑡 𝑘 𝑥
Relaxation Oscillation
Van der Pol Oscillator: