Analytic Cycling Logo Glossary

Air density, kilograms per cubic meter, depends on temperature, barometric pressure and altitude and to some extent on water vapor (humidity). Air density is calculated here as a function of temperature, barometric pressure, and altitude, neglecting the effect of water vapor which is small.   

Average force on the pedals, Fav, during a revolution is related to power and the speed of the pedal, Vp. 

Cadence, Revolutions per minute of the pedals. 

Coefficient of Rolling Resistance, Crr, Dimensionless parameter describing the retarding force of rolling divided by the weight of a rider.

Coefficient of Wheel Drag, Cx0, Dimensionless parameter describing the drag on a wheel as speed and wheel diameter vary.

Coefficient of Wind Resistance, Cw, dimensionless parameter describing the retarding force exerted by the air as a rider moves.

A Differential Equation defines a rider's speed and acceleration as a function of time, starting point, and initial speed. Equations are solved using state-of-the-art numerical methods.

Equations of Motion

Fit Points, Thigh and Shin Extensor and Flexor Fit Points are lists of points that the Pedal Model uses to define the Thigh and Shin Extensor and Flexor strength functions. Such lists should be values spaced evenly over a range of motion and are in units of N m. 

Fit Points Example: 

          {0., 88., 200., 232., 240., 232., 200., 88., 0.} 

  • The list begins and ends with "curly braces, {}"
  • Each value must be separated by a comma.
  • The model requires at least four points.
  • Values can be zero; a "zero valued moment function." is {0,0,0,0}. 
    A number in front of the list has the effect of multiplying all the values in the list by this number, i.e., 2 {0., 88., 200., 232., 240., 232., 200., 88., 0.} would give the same result as {0.,166.,400.,464.,480.,464.,400.,164.,0.}. 

Example of Fit Points used to create Strength Functions:

    Plot of Thigh Strength Functions 

    Plot of Shin Strength Functions


Effective Pedaling Force, Feff, gives the force in each of two legs that is required to give the same average force, Fav, while pedaling in only a portion, Eff, of a full rotation of the pedals.

Effective Portion of Pedal Stroke, Eff, the portion of the pedal stroke where most of the power from one leg is exerted, degrees.

Frontal Area, A, area in square meters presented by a bike and rider. It's hard to measure.  Often it's calculated from some form of a coasedown test or from speed verses power data.  Typical values are around .5 m2.  A large rider may have a larger frontal area.

Gear Chart, a gear chart which gives a relation between the forward motion of a bike and one revolution of the pedals.  It is derived from the time of ordinary bikes (the old ones with the very large front wheels that were driven directly by the pedals without the benefit of gears).  The values in the table are equivalent to the diameter of the ordinary's front wheel.  The values in the table are (Chainring  Teeth\Cog  Teeth)\ Wheel Diameter and the units are inches. 

Grade, GradHill, slope of hill, positive if up hill, negative if down. Expressed as a decimal fraction, rise divided by run.

Gravity Forces, Fsl, pull the rider and bike down the slope.

Hip's Horizontal Distance, horizontal distance from vertical line through bottom bracket to point of rotation of hip. Rear direction, positive, forward negative.  See Figure 1.

Hip's Vertical Distance, vertical distance from line through bottom bracket to horizontal line though point of rotation of hip. See Figure 1.

Pedal Model Points is the number of divisions of a complete rotation of the pedals used in the pedaling model.  A value of 12 gives good results for most cases.  However, if the torque at the bottom bracket is all in a small portion of the pedal rotation, a larger value will give better results.  See Figure 1.

Plot Position, a plot of the pedaling geometric relationships at a point in the rotation of the pedals, one of the Pedal Model Points, a positive integer less than or equal to Pedal Model Points above. See Figure 1.

Power is the work required per unit of time to overcome the net forces acting on the rider and bike. 

Power Profile Function, power as a function of a short period of time. 

When a rider commits to a sprint, power increases rapidly, reaches a peak and then trails off.  The shape of this power curve is reported to be about the same for most riders.  Magnitudes change, but the basic shape stays the same. 

Data for power curves can be taken in a Wingate Test where a load is suddenly applied to a bicycle ergometer and power is measured over 30 seconds.  Or data could be taken on a bike using a power measurement device such as an SRM.

A Power Profile Function puts such measurements in the form of a function that can be used in quantitative calculations. Where a Power Profile is used on an input form Maximum Power must be greater than Average power and all input values must be greater than zero. An example plot:

Plot of Power Profile

Rear Shelter— The rear wheel is sheltered by the frame and rider and therefore creates less aerodynamic drag than the front wheel. The estimate of this effect is called "Rear Shelter".

Rolling Resistance, Frl, is the force in newtons on the rider and bike caused by the rolling friction on the road. Variables affecting rolling resistance are the coefficient of rolling resistance, Crr, and the weight of the rider and bike, Wkg.

Rotational Drag, the drag on a wheel from simple rotation. It does not change by any appreciable amount as speed changes and is almost the same for all wheels.

Rotational Inertia, the tendency to rotate as a force is applied. The higher the rotational inertia, the more work and hence power is required to get the wheels up to speed.

Shin Length, from the point of rotation of the knee to center of the pedal spindle. See Figure 1.

Slope Force, Force exerted by gravity pulling a rider down a hill.

Speed of the pedal, Vp, depends on the cadence, Cd, and the crank length, Cl.

Strength Functions, supply torque to the pedaling model. Muscles rotate the thighs about the hip and rotate the shin about the knee.  The strength and speed of the motions of the thighs and shins  power the pedals.  Thighs and shins can move in two directions.  Muscles that bend the thigh and shin relative to the torso are called thigh and shin flexors.  Muscles that move the thighs and shins in the opposite direction are called thigh and shin extensors. The torque produced by the thigh or shin can be different at each point in the range of motion and depends on the speed of motion.  The torque at each point is described by a function here called a strength function.  The strength and direction of the thigh and shin extensors are represented by green arcs on the thigh and shin in Figure 1.   (Some readers may think of these strength functions as strength curves or moment functions or couples.) 

Thigh Length, from point of rotation at hip to point of rotation of knee. See Figure 1.

Torque, see Strength Functions.

Weight, Wkg, weight of rider and bike. 

Wind Resistance, Fw, is the force in newtons on the rider and bike caused by wind drag. Variables affecting drag are effective frontal area of bike and rider, A, drag coefficient, Cw, air density, Rho, and speed, Vmps.

Work of Revolution, the work performed by one pedal in one revolution of the pedals. (The area under the curve in Figure 1.) Work of Revolution at a given cadence is related to output power. The area under the Work-of-Revolution curve shows where in the pedal stroke work is done.

 Pedal Model Nomenclature

Yaw Angle, the direction relative to the direction in which a rider is riding from which the wind is blowing. Wind is assumed to be zero in speed and direction for analysis here.

©1997 Tom Compton