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.
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:
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 m^{2}. 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:
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.
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. |