# Drivetrain Power Transfer Efficiency

### Pedaling Power into Cycling Power

We know that Marco Pantini was producing 308 Watts of cycling power during his record climb of the Alpe d’Huze. What we do not know from that number was how much pedaling power was he producing.

What we are asking is really a question about the cycle drivetrain. How efficient is it in transforming Pedaling Power into Forward Power? The actual physical calculation is a bit involved, but let’s jump to the answer for those who may not wish to wade through the details.

The transfer coefficient is 1 except for some small losses due to mechanical friction. The drivetrain converts all of the Pedaling Power into Forward Power. If you need 308 Watts of Forward Power to accomplish a scenario, then you need to produce  slightly more than 308 Watts of Pedaling Power.

### Connecting Cycling to Pedal Power

We are looking for an equation expressing Cycling Power in terms of Pedaling Power. The approach relates each of the Cycling Power components to its match in the Pedaling Power. First we connect each of the three component pairs:

TireForce = φ * PedalForce

TireRotation = σ * PedalRotation = σ * Cadence

TireCircumference = μ * PedalCircumference

Then we plug these into the Cycling Power equation, and collect terms to get:

CyclePower = TireForce * TireRotation * TireCircumference /60

= (φ * PedalForce) *  (σ * PedalRotation ) * (μ * PedalCircumference)/60

= (μ * σ * φ) * (PedalForce * PedalRotation * PedalCircumference)/60

CyclePower = β * PedalPower

where β = μ * σ * φ

Therefore, our approach for analyzing the cycle drivetrain will be to compute the three μ, σ, φ relationship factors, and multiply them together to get β. We have already derived the φ coefficient connecting force transfer. Remeber it is expressed as a combination of the four radii, two of which are related to the current gearing.

### Computing σ: Connecting PedalRotation to TireRotation

σ is the ratio of how fast the rear tire is rotating as compared to the Pedal Rotation. But PedalRotation is just the Cadence and we know based on the current gearing configuration that the rear wheel rotates at a rate of the GearRatio * Cadence. This is simply a result of the two being connected by a chain. We therefore have our second parameter with much less pain than with the first.

σ = PedalRotation/ TireRotation

TireRotation = σ * PedalRotation = σ * Cadence

σ = PedalRotation/ TireRotation = CurrentGearRatio

σ = CurrentGearRatio = R2 /R3

### Computing μ: Connecting TireCircumference to PedalCircumference

This ratio is important in the power equation because it helps us relate the distance covered by rear tire for a given distance covered by the pedals around the pedal circle. Its computation is  the easiest of the three.

TireCircumference = μ * PedalCircumference

μ = TireCircumference/ PedalCircumference

μ = R4/R1

### Computing β: Power Transfer Coefficient

We now have computed the factors needed to determine how effectively the drivetrain transfers power. But before we jump in and starting inserting numbers, we need to finish up with some algebra as we will find our answer simplifies down significantly.

We know β is the product of our three factors. So let’s compute that and find out what surprise lurks behind all of the products.

β = (φ  * σ  * μ)

β = ((R1/R4) * (R3//R2)) * ( R2 /R3) *  (R4/R1)

β = 1

How did that happen? After all our calculations we end up with 1. Its good news because it verifies what we had hoped for. The drivetrain translates all of the PedalPower directly into CyclePower. Notice also that this result is valid for all gearing configurations.

CyclePower = β * PedalPower

CyclePower = PedalPower

### Computing δ: Accounting for Drivetrain Frictional Loss

There is one caveat remaining. The drivetrain is a mechanical device using forces and torques and therefore, dissipating some PedalPower as frictional heat. So we should be adding in some factor to account for a loss of pedal force. We are using δ to represent this.

CyclePower = δ  * PedalPower

Estimating δ is a bit of an art and is dependent on a number of factors. In general, frictional loss is in the 2 – 5% range, and depends on the cyclist riding style. Elite riders tend to lose less to friction than others because they ride in a way that produces the least amount of friction.

Next Topic:   Drivetrain Efficiency Summary