Re: How can I use the power from a rubber band to it's fullest?

Dear Greg,

Well, I have thought quite a bit about your design, and I hope my following 
info isn't too technical (or boring) and can help you toward your goal. 
First we will start with the math, since this is the basis for all 
engineering (and is too frequently forgotten in the design process).  First 
we will look at the rubber band itself.

A rubber band in tension is, in essence, a spring.  A spring is described 
by the following equation:
F=kD

Where: F = force
k = spring constant
D = distance

The spring constant is a number associated with the individual spring, and 
something you may need to find experimentally.  The distance is how for the 
spring moves and Force is the force the spring pulls when in tension (or 
compression, it works both ways).  For example, say you have a rubber band 
that you want to find the spring constant for.  Attach one end solidly and 
the other end to a spring scale (a fish scale works fine) and get out a 
ruler.  Lets say that at rest (zero pulling) the band is 4 inches long.  
Then you pull on it and at 6 inches the scale reads 1 lb.  (This means that 
the rubber band is pulling with a force of 1 lb.).  You then pull to 8 
inches and get a reading of 2 lbs.  So if we look at our equation and using 
the first set of number we have F = 1 lb, D = 6-4 = 2", so 1 lb = k*2" and 
solving for k we get k=1/2 Lb/in.  To double check ourselves we can use the 
second reading.  For a distance of 4" (8"-4") and k=.5 Lbs/in we get F=.5 
Lb/in*4 in = 2 Lb which correlates to our experiment.  You may need to 
perform this experiment as there are many variables in rubber and not much 
literature to tell you exactly what k value a particular rubber band will 
have.  Why, then, is the spring constant so important?  Once you know the 
spring constant of you rubber band, you can figure out how much force you 
can get from it.

If we have a rubber band that we know the k value for and we know we can 
stretch it 10 times its initial length before we deform it.  If we start 
out with 1 foot of this rubber band and stretch it to 11 feet and it has a 
k value of 2 lbs/in, then at 120 in (10 ft of stretch) your force is 240 
lbs.  Of course, once you start moving and the band is stretched a little 
less (say 119 inches) your force is also less (238 lbs).

So how do we deal with this changing force?  Well, that is what calculus is 
for.  The derivative of location (S) with respect to time gives you 
Velocity (V) (dS/dt = V) and the derivative of Velocity (V) with respect to 
time gives us Acceleration (a) (dV/dt).  By rearranging these and 
eliminating dt we get the following equation (from Engineering Mechanics 
Volume 2: Dynamics 2nd ed. By J.L. Meriam and L.G. Kraige pg. 17)

V*dV = a*dS

Now I don't expect you to have to solve this differential equation 
outright, but if we can solve for our acceleration.  We know F=ma and we 
now know our force F=kD, therefore a=kD/m.  Also D can be described as a 
position S, since we can say that the maximum stretch is where our car 
starts at point zero, and the rubber band just becomes loose at our final 
position, S1.  So our position is the opposite of the stretch of the rubber 
band, or D = S1-S.  Confused yet?  Try and bear with me, it will make sense 
soon.  Anyway, plugging it all in and integrating, assuming that our start 
point is S=0 and initial Velocity V0=0, then we get the following equation:
V^2 = -(kS^2)/2m + (kS0/m)S
Of course that may not make sense to you, but we can solve for the maximum 
Velocity which will occur at the point where the rubber band becomes loose, 
which is point S0.  Solving for that, we get the much simpler equation:

Vmax = sqrt(k/m)*S0

So here is a quick example.  Say we have a rubber band with a k value of 30 
lb/ft and a 150 lb car.  If we can stretch this rubber band 10 ft, what 
will our maximum Velocity be?  Now you may have noticed before that when I 
do all of my calculations I include the units.  This is extremely important 
so that you know the numbers you get are valid.  If I divide a k value that 
is in inches by a mass that includes feet, I am already off by a factor of 
12.  Unit continuity is extremely important and your teacher may be able to 
help you with this if you have further questions.  Anyway, when dealing 
with pounds and mass, we need to get our units right.  Lb. is a measure of 
weight, not mass.  For calculations in English units they need to be in 
slugs, so we divide the weight W by 32.2 ft/sec^2 to get slugs.  So our 150 
lb. car is actually 4.65 slugs (4.65 lb*s^2/ft).  So then plugging into our 
equation we get:

Vmax = sqrt(30 lb/ft / 4.65 lb*s^2/ft) * 10 ft  =  25.4 ft/s  =  17 mph

Pretty cool, huh!  Unfortunately this only would work if you tied the 
rubber band to the front of the car (with light wheels and no friction!  
More later.) and pulled the car back 10 ft from an anchor.  But there is 
some very important information here.  By examining the equation we see 
that we can increase our max Velocity by increasing the k value or 
decreasing mass.  But those numbers are square rooted compared to S0.  
Therefore, increasing your stretch distance is more productive than 
decreasing mass or increasing the k value of the rubber band!  For example, 
using the above equation and same mass, if we double the k value to 60 
lb/ft we get a Vmax = 35.9 ft/s (24.5 mph).  However, if we double the 
stretch length for the 30 lb/ft band we get a Vmax = 50.8 ft/s (34.6 mph!) 
 Therefore, we know for our design that the longer we can stretch it, the 
faster we will go.

Now remember that you just can't by a really long rubber band, it needs to 
be the stretch length, not the overall length.  I would probably recommend 
some kind of surgical tubing like they use for bungee jumping.  Decent k 
values and lots of stretch.

So now if you have waded through all of that, we get to the last part, the 
propulsion to the wheels.  Now I would probably recommend wrapping the band 
around the rear axle.  If you go through any other mechanisms (gearing or 
what not) you will incur losses due to friction.  Wrapping the axle is the 
most direct way to transfer power.  I would recommend using a ratcheting 
hub for each rear wheel so that it runs freely once the rubber band is done 
unwinding, or else you'll end up rewinding the band again and slowing down! 
 One from the rear wheel of a bike might work. 

Also, to have gearing of sorts, you can use a continuously variable 
transmission (some ATV's and snowmobiles have them).  In your case, the 
rear axle should be cone shaped, narrow on one side and thicker on the 
other.  Wind the rubber band beginning at the narrow end and ending at the 
thick end, this way when you start out, you are at the thicker end with 
more torque but less speed for starting out and as you wind down, torque 
decreases but speed increases.  Just like shifting gears.  Of course the 
reduction in force that the unwinding band gives you may make this not 
work.  I tried to figure out a way to prove this mathematically but I'm 
afraid I am stuck.  I might try to amend this once I get it figured out!  
Anyway, it is an idea to keep in mind if you have troubles getting the 
vehicle going from a dead stop.  Also, light wheels will be best, as some 
torque will go into accelerating them in rotation instead of the car.  Of 
course they need to be beefy enough to hold everything up!

And finally the frame.  (Covering is mostly your preference for aesthetics. 
 You won't be going at speeds that aerodynamics is a factor).  I would 
recommend aluminum as it is easy, cheap, and will give you the most 
rigidity for the weight.  Titanium and Carbon Fiber are cool, strong, and 
light, but very difficult to work with.  Aluminum is a little tricky to 
weld, but if you can get someone who knows how, it makes assembly fast and 
lighter than using brackets.  However, be aware that welding can bend the 
frame out of square.

If you really want some great answers to questions that may come up on any 
aspect of the car, or even ideas we may not have even thought of (like how 
to you get the rear wheels to line up with the frame?  Do you have any 
adjustability back there?  Might need it!), you should contact the closest 
University with a Mechanical Engineering department.  Ask if they have an 
SAE Formula One team.  Every year SAE sponsors a contest where teams build 
a small race car.  I am pretty sure the contest is in March, so the teams 
should be in full swing right now.  If you could get a day to visit with 
the team and their professor, it would be an invaluable experience.  The 
cars I have seen built are amazing, and they could help you with many of 
the aspects of your car, such as frame, wheels, brakes, alignment, seats, 
steering, all sorts of things.  A day like that would be worth ten of these 
letters.

Anyway, I know this was long but that isn't an insignificant project you 
have taken on.  I am more than willing to help you individually if you have 
any further questions.  My e-mail address if bradk@jymis.com .  Please feel 
free to write if you get stuck or want some further input as you progress 
on the project.  Best of luck!
BK