Hi this is Pam and Gilbert of 2 Bikes 4 Adventure. In reaction to our Brompton versus road
bike video, several subscribers commented on the superior acceleration of Brompton's small wheels.
They wondered how the Brompton would fare against a full-size bike in a
stop-and-go urban environment as opposed to the continuous 60 kilometer ride example we used.
Even brompton's own website mentions that the small wheels allow quick acceleration.
But we need to take a closer look at some basic principles of physics to see if we
can confirm this assertion. We'll focus on three different aspects: mechanical advantage,
rotational inertia and total acceleration. Then we'll analyze the results using a software model.
Forward force on the contact patch is proportional to the force applied to the
bicycle crank by the rider. The contact patch is the area of the tire directly touching the ground.
While pedaling at 20 km/h or so, this force is equal to about 15% of the
rider's weight. In contrast, during a sprint or a quick acceleration, the force can reach three
times the rider's weight. The relationship between this applied force and the tire contact patch force
is called "mechanical advantage". Like a lever on a pivot, the principle is to trade displacement
for force by moving the pivot closer or further away along the lever. Similarly, in 1st gear,
a cyclist would generate a large output force, but a small displacement while in 6th gear, she
would create a small output force and a large displacement. The question is, does the Brompton
smaller wheel provide an enhanced mechanical advantage over a regular full-size bicycle wheel.
As the chainring rotates, the chain drives the rear sprocket to make the bicycle go forward.
The distance traveled for each rotation of the chain ring will be proportional to the
ratio between the chain ring and the number of sprocket teeth as well as the wheel circumference.
The term "Meter of Development" refers to this relationship.
In some parts of the world, the term "Gear Inch" is preferred. It's basically the same equation,
but using the wheel diameter in inches instead of its circumference in meters.
A regular Brompton would range from 33 to 100 gear- inches or about 2.6 to 8 meters of development.
A hybrid bike such as the Trek FX3 10 speed would provide a wider range, from 24 to 99 gear inches
or 1.8 to 8 meters. As we explained in a previous video, the Brompton 2-minus or 3rd gear is its
most energy efficient. This 52 gear-inch (or 4.1 meters of development) corresponds to the Trek's
6th gear. From now on, we'll assume the Brompton will use its 3rd gear and the Trek hybrid bike
at 6th gear for the acceleration test. As seen earlier, changing gears is similar to moving
the pivot along a lever. It affects the distance traveled and inversely the output leveraged force.
So the mechanical advantage is basically the ratio between the sprocket and chainring
multiplied by the ratio between the pedal crank length and the wheel radius. A closer look at the
equation shows we can replace most of it by the gear-inch equation above multiplying the crank
length by 2 and converting to inches gives the mechanical advantage value. Assuming both bikes
have a 170 mm crank and a 52 gear-inch setting, they will both have the same propulsion
force as measured on the tire contact patch, in this case 26% of the pressure applied on the pedals.
This means that the smaller wheel does not provide a force advantage over the full-size bike.
Most of us would expect a smaller wheel to be easier to accelerate than a larger one if
the same force is applied. An experiment that can easily be performed at home to confirm this theory
is to attach a weight, for example an energy drink can, on the front wheel of an upside down Brompton
and measure the distance traveled in one second. Then repeat with a hybrid bike's larger wheel.
The can attached to the larger wheel would travel 50 % less distance than the Brompton in one
second, which seems to confirm that larger wheels are slower. However, an additional test with the
similarly sized road bike wheel would give a distance traveled somewhere in between the two.
Clearly, radius is not the only factor. This phenomenon is explained by the difference
in rotational inertia. The linear acceleration, or the acceleration of the energy drink falling down,
is a function of the weight of the wheel, not its radius. So assuming the same force is applied to
the tire, the wheel linear acceleration will be ONLY a function of its weight and NOT its
radius. Since most Brompton wheels are lighter than full-sized bike wheels, the Brompton should
have an advantage. For a hypothetical acceleration test, we'll take a 1Kg Brompton wheel and
a 2Kg hybrid wheel plus two weights, one of 1Kg and one of 2Kg.
Assuming no friction under the two weights nor rolling resistance under the wheels,
we apply the same force to the four objects for one second and measure the distance they travel.
As expected, the 2Kg weight travels half the distance of the 1Kg weight.
Interestingly enough, the 1Kg wheel would have traveled the same distance as a 2Kg
weight. That's because half the energy would have been used to accelerate the wheel rotation speed,
like our can of energy drink, and the other half to accelerate the mass of the wheel forward.
As discussed earlier, the radius of the wheel has no direct effect, only the wheel mass. Similarly,
the 2Kg wheel would have covered a similar distance to a 4Kg weight.
Now let's assemble the two bikes. Both are 12 Kg total, the hybrid bike's heavier wheels
being compensated by its lighter frame. At rest, we can replace the wheels by their corresponding
weights. During acceleration, we can multiply the equivalent wheel weight by two to represent
rotational acceleration. We can see the hybrid bike would feel 2Kg heavier than the Brompton,
thus giving the Brompton what seems to be a 14% acceleration advantage.
However, we need to add the weight of the rider to both bikes and while the
hybrid bike would still feel 2Kg heavier than the Brompton during acceleration,
that would only translate to 2% once a total weight by the bike and rider is factored in.
There are other factors to take into consideration. So far, we assume no rolling resistance nor
aerodynamic drag. In real life, the bicycle will accelerate until the sum of the rolling resistance
(shown in yellow) and the aerodynamic drag (in blue) equals the forward force on the tire contact patch,
at which point the bicycle speed will stabilize. The hybrid bike with its
bigger tires has less rolling resistance, thus it stabilizes at a higher speed.
Let's see if the Brompton small wheel 2% advantage holds up to the Trek's superior rolling resistance.
All the physics properties discussed so far can be represented in a mathematical equation.
In a more usable form in a software model, we can predict bicycle performance at any
phase of the acceleration process. Assuming a 200 watt input and a 95%
power transfer efficiency on both bikes, the Brompton will take an early lead, but this will
be short-lived. At one second after the start, it will be 1/2 cm ahead of the Trek.
Four seconds after that, it'll be 15 cm behind the Trek, and the gap will keep widening.
However, the Brompton looks good in comparison to a 16 kilogram Gazelle city bike,
which would be 65 cm behind the hybrid bike and behind the Brompton too.
To sum up, bicycle acceleration assuming equal power setting and gearing,
comes down to total weight and not wheel radius. A lighter wheel will accelerate faster. But
the Brompton's 2% advantage will be quickly negated by its poor rolling resistance,
so as much as we would wish or feel that the Brompton has superior acceleration,
the truth is it will be quickly overtaken by most full-size bikes of similar weight.
We hope this won't discourage you from kicking into high gear on your bicycle adventure.