Analytic Cycling Logo Performance and Wheels Concepts

How much does a wheel's weight, rotating inertia, and drag affect performance?

Topics on this page:
Magnitudes of the Forces and Measurements

Wheel Aerodynamics
Table of
Coefficients of Wheel Drag

Wheel Rotational Inertia
Table of Wheel Inertia and Mass

Equations of Motion

Forces on Rider

Wheels are a small portion of the forces on a bike and rider. The differences between wheels are small. Ordinarily power can be measured to what is normally a small tolerance, say plus or minus 3%, maybe plus or minus 1% under ideal conditions. As the table below shows, a typical difference between wheels may be 1% or less. Clearly, another approach, other than direct power measurement, is needed.

The table below gives typical values for the forces on two riders. The Standard Rider is on 32 hole standard wheels and the Test Rider is on Specialized tri-spokes. Forces are in grams of force since such forces are often quoted that way.

 

 Standard Rider

 Test Rider
 

 Force (gmf)

%

Force (gmf)

%
 Speed (m/s)

 10.76
 

 10.76
 
Total Force on Rider

 2337

 100

 2287

 100
Wind Resistance

 1811

 77.5

 1811

 79.2
Rolling Resistance

 304

 13.0

 304

 13.3
Gravity Force

 0

 0

 0

 0
Drag on Front Wheel

 127

 5.4

 98

 4.3
Drag on Rear Wheel

 95

 4.1

 74

 3.2

Wheel Aerodynamics

A paper by D. I. Greenwell, et. al., entitled "Aerodynamic Characteristics of Low-Drag Bicycle Wheels", Aeronautical J., Vol. 99, No. 983, Mar. 1995, pp.109-120, has a good discussion of the aerodynamics of bicycle wheels. Conclusions by Greenwell et al:

  • The total drag of the wheels is in the range of 10% to 15% of the total drag on a bike. Drag improvements between wheels can reduce this by 25%, or 2% to 3% of the total drag.
  • Axial drag forces are difficult to measure precisely. Most single valued measurements should be suspect.
  • Deep section aero wheels are better than a conventional 36 spoke wheel and are all about the same within the limits of measurement. Disk wheels are better yet. (Don't run a disk in front if there is any chance of wind.)
  • The rotational drag on a wheel does not change as speed changes or with different wheels.
  • The drag on the rear wheel is reduced by 25% due to the seat tube.
  • The forgoing applies to zero yaw angle. Read the paper if you want to know the results for non-zero yaw angles.

Aerodynamic Formulas

Coefficients of Drag Reported for Various Wheels (1)

 Wheel

 Cxo
Conventional 36-spoke

0.0491
Campagnolo Shamal 16-spoke

0.0377
HED CX 24-spoke

0.0379
Specialized tri-spoke

0.0379
FIR tri-spoke

0.0382
HED disk (lenticular)

0.0361
ZIPP 950 disk (flat sided)

0.0364

Please note that for the coefficients given in the above table, the conventional wheel is significantly different from the deep-section wheels, and deep-section wheels are significantly different from the disk wheels. However, there is no significant difference between the deep-section wheels or between the disk wheels.

Wheel Rotational Inertia

It's easy to calculate a wheel's rotational inertia using a kitchen scale, a stopwatch, and a tape measure.

The general approach is to measure the time period for a wheel swinging at the end of a pendulum. See Figure 1. From the time period of a swing, one can calculate the rotational inertia of the wheel about the point of rotation of the pendulum. The rotational inertia about the point of rotation of the pendulum can be transformed into the rotational inertia about the center of gravity of the wheel.

 

Figure 1, Pedulum Method

Most of the error of the method comes from measuring the period. Timing 100 swings and dividing by 100 gives a good estimate. This minimizes the error of starting and stopping the stopwatch by hand. A pendulum has the property that its period is constant as it slows down. Take care that the wheel swings in the same plane at all times. The method will be invalid if it does not. Go to Calculation of Inertia to calculate rotational inertial for your own wheels.

Data on some wheels is shown in the following table. Wheels were complete, meaning they had tires, tubes, rim strips, rims, spokes, hub, skewers, free wheels, just like they would be ridden. As individual components, rims lend themselves to calculation of rotational inertias; tires and tubes don't. There is a large variation between advertised weights and actual weights as manufactured. More real-world, meaningful results come, in my opinion, from measuring wheels in an "as ridden" state. Hence the values here are for fully rideable wheels, just like the ones handed to you from your support vehicle.

Rotational Inertia and Mass for Various Wheels

 Wheel

 Details

 Ic
(kg m^2)

Mass
(gm)
Wire Spoke Rear, Std Rim, 700, track, 36 spokes, w/o tire, w/ axle, nuts

 0.0528

1177
Wire Spoke Front, Std Rim, 700, 32 spokes, w/ tire, tube, rim strip, axle, skewer

0.0885

1264
Wire Spoke Rear, Std Rim, 700, 32 spokes, w/12-21 cassette, tire, tube, rim strip, axle, skewer

 0.0967

1804
Specialized
tri-spoke
Front, 700, w/ tire, tube, axle, skewer

0.0904

1346
Specialized
tri-spoke
Rear, 700, w/ 12-21 cassette, tire, tube, axle, skewer

 0.1032

1771
Specialized
tri-spoke
Front, 650, w/ tire, tube, axle, skewer

 0.0683

1207
Mavic Front, Std Rim, 650, 28 Bladed Spokes, w/ tire, tube, rim strip, axle, skewer

0.0632

1179
MTB Front, 32 Spokes, w/ tire, tube, rim strip, axle, skewer

0.1504

1847

Equations of Motion

Wheel weight and wheel rotational inertia matter when a rider and bike are accelerating. Drag matters whenever a rider and bike are moving. It is not enough to estimate rider and bike performance under constant conditions. Differential equations are used to describe motion under transient conditions. Such equations let us evaluate the combined effect of wheel weight, rotational inertia, and drag.

The following differential equation, with an appropriate starting point and initial speed, describes the position, speed, and acceleration of a rider over time. Using this equation, a comparison can be made between a "Standard Rider" and a "Test Rider" to see the effect of various alternatives. This is the equations that is evaluated in each of the case studies presented here.

Equations of Motion 

© 1998 Tom Compton