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June 16, 2017 4 Comments

Let’s talk rotating weight. Rotating weight is a big issue for wheel builders. Why? We make choices that determine wheel weight, its total and location. Builders must understand this topic.

Rotating weight directly affects inertia, so the topic is really inertia. Inertia is the resistance of a mass to acceleration. Moment of inertia (MOI) characterizes this resistance and depends on rotating and non-rotating mass. Builders should be measuring wheel MOI.

I’ll show you how to measure moment of inertia (MOI), plus I’ll share a spreadsheet to shortcut the math supporting MOI measurement and its effect on riding. Plug in numbers and get usable wattage estimates.

**Forces**

Aerodynamic drag, mechanical friction, and inertial energy are the forces we face in cycling.

Aerodynamic drag is the most significant.

Mechanical friction was, historically, a bigger obstacle. Numerous inventions were needed to launch the cycling age.

MOI is less discussed, often passed over as insignificant. Yet all who pedal are aware of the effect of weight, especially wheel weight.

Wheel MOI has two parts:

1/ Rotational energy - required to make a wheel rotate. Rotating takes energy.

2/ Translational energy - needed to accelerate a mass. Mass takes energy to get moving.

**Benefits of reduced MOI**

1/ Sprinting

Lower rotating weight means quicker acceleration and sprinters value it most.

2/ Climbing

Faster climbing is possible with lower weight. Gravity prioritizes weight so its feel is magnified.

Speed varies more on climbs and variations are a larger percentage. Momentum is smaller. A lapse in pedal force makes a larger speed change. Consequently, experts advise that maintaining tempo is an important advantage.

Climbers universally prefer lighter bikes and wheels.

3/ Pedal acceleration

Pedaling is pulse inputs. Micro analyzed, bicycle speed is a sine wave driven by a bumpy, two piston engine. Even smooth pedaling delivers more power on the down stroke. Resulting speed variations include many moments of acceleration. If wheels are lighter, these recoveries are less costly.

While lighter bikes and wheels increase the number and magnitude of speed variations, the competitor senses an advantage. If another climber increases tempo, it is urgent to match the pace, a pure and simple sprinting challenge. Frequent tempo changes in competition biases riders to lighter equipment.

When Campagnolo introduced their extra light Fluid Dynamic disk in 1987, they asserted a lighter wheel is faster because pedal strokes involve small accelerations affected by wheel weight. Download their discussion.

4/ Handling

Low MOI means less force to change angle or direction. Lighter bicycles feel more agile. For touring, fast handling can be distracting, but even sedate tours include moments of lively pedaling. In racing, low MOI is an asset.

Ultimately, so much is subjective. Some worship lightness, others ignore it. It’s often discounted because lightness is associated with weakness, unreliability, and higher maintenance. Yet no riders disagree, lighter weight is generally beneficial. While there are selective benefits to higher weight (stability, for one), they are usually outweighed by the sensation of speed, efficiency, and freedom.

An excellent, approachable discussion of wheel inertia is found at the French website *Roues Artisanales*. Check the inertia section of Adrien’s “Grand Test,” where he offers MOI for nearly 80 wheel sets.

**Measuring MOI**

Who measures MOI? For wheels, the relevant numbers include all components including tire. Wheel builders are often the only people in a position to measure and catalog MOI. Too few builders are checking MOI and using data to guide their choices and advise riders.

Consequently, some place too much emphasis on the advantages of low weight. Others ignore the issue in favor of aerodynamics or bearings. The full picture includes MOI.

Regardless of its relative importance, wheel engineers closely monitor MOI. As Lennard Zinn reported in 2006.

*“Perhaps some of you remember when I did a test in VeloNews seven years ago (in the 6/28/99 issue) of wheel inertia by building a rotational pendulum in my garage. I was just at a Mavic tech seminar in Annecy, France, last week and saw a test machine set up virtually the same in the Mavic test lab.”*

Thankfully, this is simple physics and anyone can build an accurate MOI machine. Here is a trifilar pendulum, the type of machine you should make and use.

From this 1945 paper.

My platform is three, ½” x 3’ dowels, assembled in a triangle. Each corner is connected overhead with 30lb monofilament fishing line. The exact dimensions don’t matter but must be accurately recorded and entered into formulas used to determine MOI.

These are the mechanics:

Inertia = T^2( (W/r)*a*b)/(4*π^2*h)*1000

Inertia = gm^2

T = Oscillation period

W = weight in g

r = gf/N

*a* = diameter of top anchor

*b* = diameter of bottom

*h* = height

To calculate work (accelerate from zero to 30kph) in joules of energy:

Total energy = Rotational energy + Translational energy

Total energy = 1/2(Iw^2) + 1/2(mv^2)

Total energy = 1/2(Tire Inertia + Wheel Inertia)w^2 + 1/2(wheel mass + tire mass)v^2

Wheel total energy = 40% + 60%

w^2 = rotational speed in rad/s (30km/h = 24.54rad/s)

v^2 = wheel velocity in m/s (30km/h = 8.33m/s)

m = weight in kg

I = inertia in kg/m^2

To spare you the math, here is an auto calculating spreadsheet. The yellow cells need your input. Enter dimensions of your own jig, the period of the pendulum motion, and weight and you get an accurate MOI for your wheel, rim, tire, or anything. Pre-existing numbers are from my machine and two recently tested wheels. The other cells calculate based on your entries. Download the sheet (excel) for long-term use.

Watch this movie to see how easy it is to count pendulum cycles. Here is my own unit, whose dimensions are entered in the embedded spreadsheet. Remember to replace those with your own, once you build it.

The results are indisputable. The costs and benefits of MOI can be predicted but real world variables are numerous. The lesson is to understand MOI and establish it for various wheels. Wheel design needs all useful input and, all else equal, lower MOI is almost always preferable.

We want to learn how MOI affects wheel feel and performance. With a dynamic structure in a complex context (rider, bike, terrain), data is our best tool. Besides, if two wheel designs have similar weight, strength, cost, and longevity but one has much better MOI, its design is better. This should be perceived, quantified, and recognized. As we aspire to higher performance standards, all measurable dynamics must be considered!

Have fun building your MOI machine, you won’t be disappointed. Do some tests, take notes, share with customers and others like me. The pace of fundamental learning may seem glacial at times but we can all do a part.

For further insight to the subject, download Jim Martin’s seminal 1998 paper on bicycle power. Thanks also to Adrien Gontier of *Roues Artisanales* and illustrious theoretician, Josh Deetz.

November 02, 2021

This is brilliant, this might be a somewhat effective way to quantify what we qualitatively describe as the “feel” of a wheel. This could very well be what many of us mean when we say “that wheel feels fast.”

I’ve tried for a while to convince people that the weight of the wheel system is less important than the rotational weight. Or rather at least, that the rotational weight is “weighted” more heavily in my thinking as to what makes a nice wheelset.

Sometimes I’d rather ride a given wheelset that’s a moderately heavier (by weight weenie standards), if I know the difference is at the hubs. This might be a useful way to convince people to give slightly heavier, but in my opinion, much more interesting, if not outright nicer hubs, like Onyx Racing hubs a try over something a bit lighter, but maybe a bit finickier in the long run.

November 02, 2021

Ric – I found the formula for MOI (I) very cumbersome to use. I like the 1945 paper version and the formula it uses and that it shows you how to take the mass of the platform in to account. The paper says the three lines need to be equidistant and parallel to use their formula. I would use all English units and then convert to Watts at the end. I enjoyed your article and the references very much. Thanks. Tom

November 02, 2021

I just couldn’t get the units to cancel out correctly on the MOI formula and units like gf are obscure to most in the US. I used lbf and ft in the measurements and applied the formulas in the 1945 paper and everything went smoothly and made sense. I then converted ft-lbf/sec to Watts.

*Comments will be approved before showing up.*

## Doug

November 02, 2021

Dropbox links have died, here’s some that are working as of 2020-01-24:

http://velobase.com/CatalogScans/Campagnolo/Campy1987_FluidDynamics.pdf

http://www.recumbents.com/wisil/MartinDocs/Validation%20of%20a%20mathematical%20model%20for%20road%20cycling.pdf