Name__________________________________
Class__________________________________
Lab Partners ____________________________
An Euler Project
Purpose
To learn how to solve the damped pendulum equation by numerical integration to predict the
behavior of a pendulum with velocity-dependent friction and constant friction
Procedure for velocity-dependent damping
The pendulum equation
d 2
g
  sin 
2
L
dt
was derived neglecting friction. In the real world a variety of friction (or drag) forces (or torques, in the
case of the simple pendulum) are always present. One important type of drag that is easy to model is
velocity-dependent drag, whose magnitude is proportional to the pendulum's velocity and whose direction
is always opposite that of the velocity. When such a drag effect is included in Newton's second law, the
equation of motion becomes
d 2
g
  sin   
2
L
dt
where  is a "drag coefficient". It is fairly easy to adapt the spreadsheet used for numerical integration of
the pendulum equation to produce a solution to this problem. However, the presence of the velocitydependent term makes this equation even more unstable (numerically speaking) than the drag-free
equation already studied. To avoid numerical problems, we'll use the half-step Euler method.
Step 1: Build a spreadsheet
Construct a spreadsheet that looks like the one used in the first numerical investigation, except
now add a cell to hold a numerical value for . (This number will usually be on the small side.) The first
couple of lines follow this scheme for the half-step method:
g
L


 0   (0)
 1/ 2   0   sin  0   0  dt / 2
 1   0   1/ 2 dt
 3/ 2   1/ 2   sin  1   1/ 2  dt
g
L


etc.
You should already have a copy of the Excel spreadsheet we constructed in class for the half-step
method. Note that you have to change the first two velocity calculations to include the damping term, and
then copy the second one into the cells for the other velocity calculations. You should have at least 250
iterations in your spreadsheet. Set L = .66m, g = 9.8 m/s/s, and set the time step at 1/30 sec (to match
our earlier pendulum observations).
Step 2: Make sure the spreadsheet works with a familiar case
As a starting point for your investigations, use the initial amplitude that you obtained from the
pendulum exercises and set  = 0. Plot your results and scan through the spreadsheet to determine the
period of the vibration. Verify that it's the same as the value you obtained experimentally and from the
numerical calculations with the frictionless model.
Euler Project
 Stephen Luzader
Step 3: Determine how the amplitude changes when damping is present
Keep the initial amplitude at the value used in Step 1. Set the damping parameter at a small
value (.05, for example) and observe the result. Describe how the amplitude changes with time. Try
larger values of the damping parameter and describe its effect. Include printed copies of a few of the
graphs, and be sure to indicate the value of the damping parameter with each graph.
Step 4: Determine how the period is affected by damping
We have already learned how damping affects the amplitude of an oscillation as time passes, and
we have learned experimentally how the period of a simple pendulum depends on the amplitude of its
swing. So your first task is to make a prediction about how you expect the amplitude and the period to
change as time passes. Then set the amplitude at a fairly large value (1 to 1.5 radians) with  = 0 and
determine the undamped period of the oscillation. Then try a few nonzero values of the damping
parameter to see how the period is affected. Describe how you determine the period and tabulate your
results.
Procedure for constant damping torque
It is also easy to model damping due to a constant friction torque that is always in the opposite
direction from the velocity. Mathematically, such a torque might be represented by

  R

where  is a unit vector that points in the direction of  . To create a unit vector in the spreadsheet, we
can use this trick:
 

abs( )
where abs(  ) is the absolute value function of the spreadsheet. This formula always has a numerical

value of 1 and its direction is the same as the direction of  . (Strictly speaking, this equation is incorrect
because it seems to equate a vector with a scalar quantity. However, the context indicates that we are
using algebraic signs to indicate vector directions in a one-dimensional system. Thus this is a useful
functional definition that fits our present application.)
To model this kind of damping, all we have to do is change the formulas in the velocity cells:
g
L
 1/ 2   0   sin  0  
g
L
 3/ 2   1/ 2   sin  1  
 0 
 dt / 2
abs( 0 ) 
 1/ 2 
 dt
abs( 1/ 2 ) 
etc. (Now  doesn’t mean the same thing it did in the first investigation, but for purposes of doing
computations that doesn’t matter.) The computations for  aren’t shown here because they don’t
change.
There is a very important practical problem that arises here. We have been taking the initial
angular velocity to be 0, but this kind of damping requires dividing by the magnitude of the angular
velocity. Setting the initial value equal to 0 will cause the spreadsheet to fill up with error messages, since
we can’t divide by zero. There is a neat numerical trick to solve this problem: Instead of setting the initial
velocity exactly equal to 0, set it equal to some very tiny but finite number like 10-6 rad/s. This doesn’t
affect the numerical results, but the spreadsheet is happy because it isn’t trying to divide by zero. This is a
useful ploy to remember when doing any kind of numerical work. (Note: Here’s how to enter scientific
notation in Excel using 1 x 10-6 as an example: 1e-6.)
To study this kind of damping, repeat Steps 2 through 4 from the velocity-dependent part of the
exercise. Does the amplitude decrease in the same way as it did for velocity-dependent damping, or is it
different?
Euler Project 2
 Stephen Luzader
Format for the report
Your report will consist of two parts, one written and one on disk. The written part must be typed
and must begin with a clear statement of the purpose of the investigation. It must include a brief
theoretical introduction and a description of how you carried out your studies. There must be a discussion
of results, supported by graphs from the spreadsheet. The written report must end with a conclusion
stating whether you satisfied the purpose of the experiment and what the general results were. The other
thing you must turn in is a disk with your spreadsheets on it.
When it’s due...
Each group will submit one report, which is due at the beginning of class on Friday, October 20,
2000. The project is worth 10 points, and will be treated as undroppable when the final grade is
determined. Please organize yourselves and begin working right away, so we can deal with the problems
that are certain to arise!
Euler Project 3
 Stephen Luzader