1BADNSX said:
Yes, your definitions are correct, but please show me what this proves other than r^2/r^2=1! How did you get 2:1 or 1.5:1?
The 2:1 is, I hope, obvious:
total energy to accelerate mass at edge of wheel =
energy to accelerate forward + energy to spin the wheel =
1/2 *
m *
v^2 + 1/2 *
m *
v^2
energy to accelerate non-rotating mass =
1/2 *
m *
v^2
For any given
m and
v, those energies are in a 2:1 ratio.
I can only estimate the ratio for a wheel; as the
wheel's mass is distributed at a range of distances
from the axis, it's a pain to calculate the wheel's
exact moment of inertia.
I modeled a 1995 OEM 16" front wheel as if all of its
mass were at a 7.5" distance from the axis. That's about
where the bottom of the wheel well is. There's some
metal farther than 7.5" out (bead seat and wheel lip) and
also a goodly amount of metal inboard (spokes and hub).
The radius of the full wheel (axis to tread surface) is 11.8".
Do the same calculations as I did last time, except
that the moment of inertia is lower by a factor of
(7.5 / 11.8) ^ 2, which is about 0.4. That gives an
energy ratio of (1 + 0.4) : 1.
I'd like to ask you a similar question. How did
you estimate the moment of inertia of the tire?
2:1 makes sense if you model a tire as having
all of its mass is at the surface of the tread,
but tires also have mass closer to the axis in
their sidewalls and beads.
I computed power, which as you know is energy per time, therefore by definition it is dependent on time. Because two actual vehicles with different acceleration rates are also required to shift at different points, this makes the computation dependent on gearing. I don’t have my original solution in front of me, but if I remember correctly I plotted equivalent horsepower gain versus speed and it wasn’t constant versus speed because of variables like the power curve and shift points. Then I believe I averaged the data over time to generate the generic values of 2:1 and 1.5:1.
That's certainly a more complete analysis, but when
all is said and done it should give an answer very
close to that of the energy ratio as I calculated.
I stuck with the energy ratio because it's easy to
understand and calculate, and it's a fair way to
contrast the effect on acceleration of rotating and
non-rotating mass. I think there's value in offering
a method that can be followed rather than just
giving the end result of a large complicated
calculation. Anyone who understands basic physics
can follow my reasoning and see for themself.