Download How do we determine the speed and load

How do we determine the speed and load-carrying capability of our
vehicle?
The goal is to move the greatest mass the fastest. We know that there is a trade-off
between the load we can carry and the speed at which we can carry that load. This tradeoff is expressed as the mechanical power P = force x velocity = torque x angular velocity.
For the same power, if we increase the load (force or torque) then the linear or angular
velocity is decreased. With our motor-gear system we determined that, for each gear
ratio, there is a maximum mechanical power. That is, there is a combination of force and
velocity at which the product of force and velocity is optimized. That product tells the
force and velocity at which we will move the greatest mass the fastest. It is the
combination that provides the maximum power output.
You determined, for two gear ratios, the combinations of torque and angular velocity that
provide maximum power output. Pick one of those combinations. The objective is then to
choose the wheel size and the load such that the motor-gear combination ends up
operating at its optimum torque and velocity.
For purposes of this discussion, suppose that the vehicle driving axle is the axle for the
final gear and for the vehicle drive wheel, as shown in the figure.
Torque
r
Wheel
Axle
Gear
Wheel
F
We can start by saying we want the axle to be rotating at the same angular velocity as it
does at the point of maximum power. If we reproduce the load that produced the torque
that resulted in optimum power output, we will create the condition for optimum angular
velocity as well.We need to find the load that results in the optimum torque in the axle.
What causes the vehicle to go up the ramp is the force between the wheels and the ramp.
That force depends on the torque in the axle and wheel and the radius of the wheel.
F = T/r
Pick a wheel size of radius, r. Then the force we want equals the optimum torque divided
by the radius of the wheel.
What load results in that force? In our lab experiments we lifted loads vertically, directly
against the force of gravity: F = mg. The force to move the load up a ramp is less; the
vehicle moves a longer distance to reach the same height. The force that we are looking
for, calculated above as F = T/r, is shown in the diagram below. This is the force that
must be applied by the wheels to the ramp in order to move the vehicle up the ramp at a
constant velocity. Notice that we need the elevation angle of the ramp to make the
calculation: F = mg sin where m is the total of the vehicle mass and the payload mass.
mgSin F
mg
q

Since we know F, g, and sin , we can calculate the total mass (vehicle plus payload) that
will theoretically make the motor gear system operate at maximum power output.
Actually, the mass will be somewhat smaller, since the above analysis assumes that there
are no additional efficiency losses in the vehicle itself.
Now we can predict the velocity of the vehicle. First, the vehicle velocity is the same as
the circumferential velocity of the wheel, as shown in the diagram below.
Axle travel
C
Suppose the circumference of the wheel “unrolls” on the surface of the ground. When a
complete circumference has “unrolled” the axle of the wheel has traveled the same
distance. Since both have occurred in the same time, the forward velocity of the axle (i.e.
vehicle) is the same as the circumferential velocity of the wheel. That velocity can be
calculated as v = r, where  and r are the angular velocity and radius of the wheel.
Picking a larger wheel radius increases the vehicle velocity but reduces the load that can
be carried.
These calculations can be used to predict load and velocity to provide maximum
performance. Testing of the vehicle on the ramp is necessary to determine what the actual
load is to provide maximum performance.