Math in Motion :: The Bicycle

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Math in Motion The Bicycle

• Jason Achilich
• GED 613
• Math Notebook

2
Lets Start with the Frame

• There are many examples of math in bicycle
  frames.
• The most basic is the size,
• Size is measured from the center of the cranks to
  the top of the seat tube.

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What about the reach of the bike?

• Now that we see how the height of the bike is
  measured.
• What about the reach?
• The top tube is another important number to
  fitting bike frames.
• This number is proportional to the seat tube, as
  one gets larger, the other measurement grows as
  well.

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These Two Tubes Are Proportional!
5
Angles in a Frame

• Yes there are angles in bicycle frames.
• Both the seat tube (ST) and the head tube (HT)
  are angles.
• These angles affect how the rider is positioned
  on the bike and how the bicycle handles.

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Math in Bike Wheels

• Sometimes different wheel sizes are used for
  performance, other times to go faster, sometimes
  to fit a smaller person on a bike.
• Generally a very short person will have smaller
  wheels on their bike to make positioning easier.

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How does wheel size vary with height?

• You can see in this picture that a shorter person
  rides a bike with proportionally smaller wheels
  than a larger person.

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A larger mountain bike wheel

• Over the past few years, some mountain bikes have
  started to have larger wheels.
• Why?

• The larger size wheel makes objects in the trail
  feel smaller, since the wheel hitting the object
  is proportionally larger.

• This makes rolling over objects easier since the
  object is hitting lower down on the wheel.

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Spokes in the wheel

• Wheels for different styles of cycling sometimes
  have different numbers of spokes.
• The more spokes that a wheel has the stronger it
  is.

Why?

• All wheels are built to a certain tension on the
  spokes to hold them together. When a wheel has
  more spokes the tension is distributed across
  them all, resulting in lower tension on each
  spoke.

• Many bike also come with wheels that have very
  few spokes. Yes you guessed right, less number
  of spokes equals more tension or force on each
  spoke, since there is less distribution of force.

Does it matter?

• Yes it does, it helps to create a stiffer wheel,
  plus there are fewer spokes to hit the wind. But
  also the greater force on each spoke requires
  more work to be done by each spoke to keep the
  system neutral. If one of them break, the system
  is greatly out of balance.

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Can a Wheel not be Round?

• A wheel will not be oval shaped, but it is
  possible for a wheel to not be perfectly round.
• When viewed closely, there can be high and low
  spots.
• These hops and dips are measured in millimeters,
  and though these small variations do not affect
  the ride, they can lead to imbalances in the
  system.

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Penny Farthings

• Penny what? Penny Farthing, a term based upon two
  British coins of the time, a Penny, and a
  Farthing which is a quarter Penny. The two coins
  resemble the bicycle.
• These were the first bicycles, dating back to the
  1860s. At this point there were no gears to
  make a rider go faster or slower.
• James Starley found that with a direct drive
  bicycle, a larger wheel could travel further each
  rotation because of the larger circumference of
  the wheel.

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Examples of Penny Farthings

• The diameter of the drive wheel on this bike is
  40 inches.

• While this drive wheel is 58 inches in diameter.

Which bike will go faster when pedaled at the
same cadence?
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Making the Wheels Turn

• Unlike the Penny Farthings, modern bicycles have
  an assortment of gear ratios. This range of
  ratios allows the rider to work more or less,
  depending on how fast they would like to travel.

Lets look at the gears

• This would be a hard gear for every one rotation
  the front chainwheel makes, the rear makes four
  rotations.

• This gear is very easy it would take many
  rotations of the front chain wheel to move the
  rear gear once. Very slow, but very easy to
  climb big hills with.

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Using the Gear Ratios

• Track racers use ratios to their advantage.
• Sprinters need quick acceleration, by the use of
  a lower gear, to get a jump on the competitor.
• Putting their work into a lower gear allows the
  whole system to increase in speed quickly.

• Compared to racers against the clock use a
  higher gear to keep a constant speed.

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Pedaling the bike

• The length of the crank arm also is affected by
  math.
• A longer crank are will give the rider more
  leverage to push the gear, but the circle their
  feet must travel is larger, causing a slower
  rotational time.
• While some riders will use a shorter crank
  length, giving them less leverage, but a quicker
  cadence.
• The eternal question, which is more
• efficient?

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Numbers where bikes are ridden.

• Track racers compete on an elliptical course
  called a velodrome.
• These velodromes can range in length from 250
  meter to 400 meters.
• A 250 meter long track can have a banking of 50
  degrees in the two turns.
• A 400 meter track has banking of 30 degrees
• Longer turn equals less need for banking.

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Riders using the banking

• Track racers will use the banking to their
  advantage.
• To gain an advantage a racer will get above their
  competitor on the banking.
• Once it is time to sprint, they will use the
  slope of banking to help them accelerate.
• Here a rider gets up on the banking to accelerate
  out of the turn.

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Traveling around the Velodrome

• In track racing much of the winning and losing
  depends on your line around the track.
• The lines on the track represent lanes.
• Riders cant ride below the black line near the
  bottom, so riders will hug it, enabling them to
  ride a shorter distance than the rider trying to
  come around them.

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How many Kilometers in an Hour?

• The hour record on the track is the purest form
  of cycling ability.
• It is the race against the clock.
• Competitors attempting to break the hour record
  will know the average Km/Hr that they will need
  to ride.
• The rider shown here averaged 49.431 Km/Hr in
  1972 to set the new standard.

• How many kilometers did he ride?

• The current hour record holder rode an average
  speed of 49.7 Km/Hr in 2005.

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Meters, whats that in miles?

• Cycling roots are very European, so most all
  events are measured in metric units.
• Many cycling fans that are not using the metric
  system are familiar with the kilometers to miles
  conversion.
• It is common for a cycling fan to know some key
  conversions.
• One kilometer is about a six tenth of a mile and
  one mile is about 1.6 kilometers.
• Which allows a rough estimate of a large European
  road race of 200km, to about 125 miles.

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Percent Grades

• Since cyclists are powering themselves, they
  worry about the steepness of the hills they are
  riding.
• This steepness is generally represented in
  percent grades.
• To find the grade, divide the rise, by the run,
  and multiple it all by 100.
• A mellow grade is below 5 percent.
• 22 percent is the steepest grades found in Vermont

Percent Grade
Rise
Run
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Route Planning

• This spring I am participating in the New England
  Fleche.
• A Fleche is a ride where there is an ending
  point, but the starting point can vary teams
  build their own route to the finish around
  specific rules regarding distance and time.
• The main rule requires teams to ride at least
  380km in a 24 hour period. What will the average
  speed be for this criteria?
• My team has put together a spread sheet adding up
  our mileage and times. Each segment leg is added
  to our total, so we know how far we will traveled
  in 24 hours.
• Our time is also added up by segments to form the
  whole. Thus ensuring we follow the rules.

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Our Trip to Westfield, MA

• You can see how we have added the mileage in each
  leg to form the total mileage we will be covering.

Running time 12 hour time Location Total Mileage Leg Mileage
000 800AM Start, Burlington, VT 0 0
45 845AM Control 1 Essex Junction, VT 7.9 7.9
315 1115AM Control 2 Middlesex, VT 37.4 29.5
500 100PM Control 3 Waitsfield, VT 55.9 18.5
1030 630PM Control 4 Ludlow, VT 116.4 60.5
1415 1015PM Control 5 West Hill Shop, Putney, VT 153.7 37.3
1800 200AM Control 6 Northfield, MA 176.1 22.4
2200 600AM Control 7 Amherst, MA 200.5 24.4
2400 800AM Finish, Westfield, MA 224.4 23.9
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Rider position on the Bicycle

• A rider can benefit from proper positioning on a
  bicycle. A right angle, is also the strongest.
• In this photo you can see all the different
  angles in play.
• It is important to notice the
• angles on the torso.
• She is using structure not
• muscle to hold herself up.

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Handlebar Measurements

• Comfort on a bike can even be affected by your
  handlebar width.
• Too wide and you are using strength to hold
  yourself up.
• The right width you are able to use
• a 90 degree angle to help support
• your weight.

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Closing

• In my searches for information concerning math in
  cycling, many websites devoted to the topic were
  uncovered.
• The information presented here barely scratches
  the surface of the topic, but what is presented
  are some of my favorites that I use and think
  about most often.
• Math is everywhere in cycling, looking at the
  sport it is staggering how in-depth you could
  take many of these examples!

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OnwardA final thought on Pi.

• The meaning of Pi now has meaning to me, as I
  almost daily use it to determine the
  circumference of a wheel while installing a
  cyclo-computer.
• I am looking forward to exploring more cycling
  and math relationships down the road.