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Damped Simple harmonic motion
A particle of mass m is undergoes oscillatory motion if, on displacement from its
equilibrium position, it experiences a restoring force proportional to its displacement
– the well known simple harmonic motion.
The equation of motion is
d 2x
m 2  m 2 x,
dt
where  is the natural frequency of oscillation. If there is also a drag force -mv that
is proportional to the instantaneous speed v the equation of motion becomes
m
where  is a drag coefficient.
d 2x
dx
 m 2 x  m
,
2
dt
dt
The earlier exercise on derivatives showed how to get a numerical approximation to
the first derivative. For x tabulated at intervals of t we have
 dx  xn 1  xn
 dx  xn 1  xn 1
or
.
 
 
t
2 t
 dt 
 dt 
The second form was found to converge more quickly so we will use it here.
The second derivative can be approximated by
 dx   dx 
   
2
d x  dt  n  dt  n 1

dt 2
t
 xn 1  xn   xn  xn 1 

2
d x   t    t  xn 1  2 xn  xn 1


.
2
dt 2
t
 t 
Thus the equation of motion may be approximated by
xn1  2 xn  xn1
x x 
  2 xn    n1 n1  .
2
 2 t 
 t 
Thus we obtain
 2   2  t 2  xn  xn1 1   t / 2 

xn1  
.
1   t / 2 
This equation can be used to generate the positions xn 1 from xn and xn 1 . Thus we
can obtain positions x2 and onwards. However x1 cannot be obtained this way as it
requires a position x1 before the start! The effective initial acceleration, at t = 0, in
the x-direction is  2 x  0   v  0 . We now have a two distinct choices; either the
particle is situated at the origin and given an initial velocity along the x-axis
( x  0  0; v  0  v0 ), or it is displaced from the origin by an amount a (the amplitude)
and then released ( x  0  a; v  0  0 ). We’ll choose the latter case here. Thus using
the equation “ s  ut 

Construct a spreadsheet which contains labelled cells for the quantities
a,  ,  and t. You may find it convenient to calculate the subsidiary quantities
 t 


1 2
ft ” we can approximate
2
2
1

x1  a    2 a   t  .
2

2


, 2   2  t  , 1   t / 2  , 1   t / 2  to avoid repeated calculation of
2
these quantities for each position. If these are positioned near the top-left of
the spreadsheet, then below them under headings n, t and x compute the
position of the particle for integer values of n from 0 to 100. Take t = 0.1 s
and initially set  = 0 s-1, i.e. no drag.
Plot x against t. You should see the familiar cosine curve.
Now set   0.4 . The plot should look similar to that below with only the
oscillatory curve shown.
x-t plot
2.5
2
1.5
1
x
0.5
0
-0.5 0
2
4
6
8
10
-1
-1.5
-2
-2.5
t

Add the amplitude curves with the equations a exp   t / 2 .


Save your spreadsheet as “username-dampedSHM”.
Vary the parameter  through the range 0 to 4 and note the behaviour of the
plotted quantities.