Efficiency of Bicycle Pedalling

Question

Consider a bicycle with multiple gears. Suppose that you are in a starting position with someone holding your bike upright (so when you start there's no issue with clipping in etc). It's well-known (and easily testable) that if you start in a low gear, then you accelerate away faster than in a high gear. (Obviously, if you start in the lowest gear and have a climbing mtb, then you'll have to change gear which can slow you etc - to ignore this, suppose you're on a road bike and using a not too low gear etc.)

Now, if the bike (and your body) is (are) perfectly efficient, then applying the same amount of force over a given distance gives the same work done. However, this could be done at a different rate (power). (Does the human body output at a certain power or is it the work done?) However, this outcome clearly isn't realised, so there must be some inefficiencies. Here are a few that I can think of (mainly the first one then the next two):

1. Torque. Am I correct in thinking that changing gears is very similar to the situation, say, of undoing a nut and bolt with a spanner and changing the length of the spanner. (Consider also opening a door by pushing in different places (horizontally) on the door.)

2. The body is far more efficient using the optimal cadence than a very bad cadence. Eg, pushing really hard and slowly is inefficient: the extra effort doesn't translate into extra power; similarly, if the cadence is too high, then you cannot move your legs up and then back down fast enough to give the required speed.

3. A very high strain on the chain is inefficient ("uses up" a lot of the power).

4. Very fast/slow turning of the legs can cause loss of balance, so effort can be spent on trying to maintain this.

Is there something key that I'm missing, or are these just the main things?

Any insight would be appreciated! :) - thanks!

Answer 1

The question is quite complex, but there are several fairly simple things that can be observed:

First off, the standard bicycle, as a machine, is quite efficient. Very little energy is lost in the "drive train", with the vast majority of "lost" energy (mechanical energy input at the crank that is not converted to momentum) being expended as either air friction or friction between tire and roadway.

The human body, on the other hand, is often an incredibly inefficient machine. Not only are there simple concerns of "energy efficiency" -- how many calories of food, say, it takes to produce an erg of "work" -- but there are also major issues of "durability" and "endurance", both in the short term and long-term.

The average human body tends to have a "sweet spot" for cycling where the cadence is (depending on the individual and the circumstances) somewhere between maybe 60 and 90 RPM. Cycling within the "sweet" range for the individual produces a large amount of energy (though perhaps not the "peak" energy) and, more importantly, minimizes fatigue and optimizes endurance (as measured, say, in total energy produced in a given 24-hour period, including rest, eating, sleeping, etc).

In terms of gear ratio, in addition to determining cadence on relatively level ground, it also, of course, affects climbing. An individual is limited as to the total torque they can place on the bike crank arms, and hence what degree of incline they can climb at a given gear ratio. Lowering gear ratio (obviously) reduces the torque required to turn the crank arms and hence enables climbing a steeper incline. Here the "sweet spot" (for a relatively short climb) is below the level ground "sweet spot", but there still is one.

When considering cadence both on level ground and climbing it needs to be considered that muscles are more efficient when in "aerobic" mode -- burning "fuel" using oxygen supplied from the lungs via the bloodstream. Aerobic mode is perhaps twice as efficient as anaerobic mode (though don't quote me on that number), and, of major importance, it produces far fewer metabolic byproducts which can accumulate in the body and eventually become toxic. Although there are several factors that determine whether exercise is aerobic or anaerobic, a major one is, in fact, cadence, with lower cadences being more likely to be anaerobic.

Answer 2

The power inserted into a mechanical system equates to the power dissipated by the system plus the rate of gain of energy that stays in the system. As a formula it would look like:

$$P_{in} = P_{out} + \dot E_{internal}$$

$P_{in}$ is the power the cyclist is introducing into the system by moving his legs.

$P_{out}$ is the energy being lost from the system. This is what you're looking for here in your question.

$\dot E_{internal}$ is the rate of gain of internal energy, and, for a bike, mainly deals with rates of gain of kinetic and gravitational potential energy. Elastic energy might also be including, in the metal frame of the bike, but I doubt it will have too much of an impact over time.

Some of the sources of $P_{out}$ that I can think of:

When you pedal, you will be combating a frictional torque due to the rubbing together of the mechanical parts. The work done against this frictional torque leads to some heat loss.

When you are moving, you will be working against air resistance. The air particles collide with the moving bicycle, and the collision of the air transfers some of the energy away from the bicycle. The faster the bicycle, the faster air resistance dissipates energy. This also occurs on a smaller scale when the pedals go around very fast: the pedals collide with the air particles when rotating, and hence lose energy.

The tyres on the road do a good job at not slipping, but when they do (which I would suspect must happen regularly, if not constantly to some extent), the tyres are doing work against the friction from the road. This results in heat dissipation.

And, as an intentional design of the bicycle, the brakes do a wonderful job at contributing to $P_{out}$, because, when you use your brakes, you really want to drain as much of the kinetic energy (stored as part of $E_{internal}$) as possible. This works by constantly applying a strong frictional force against each wheel, dissipating a lot of heat energy.

That is what I could think up of for now, but I'm sure there are additional means of losing energy from the system that I missed.