# bike design1

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Cycling Bio-Mechanics

n Basic Terminology (fill in the details as a class) Work

Energy

Power

Force

Torque

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Things Ive always

wondered about1. Why do we shift gears on a bicycle?

2. What determines how fast our bike goes for a

given power input?3. Are toeclips worth the trouble?

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Newtons Second Law

F = ma = m dv/dt

F1

F2

F3

F4

m

aC.G. A Rigid Body

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External Forces acting on BikeRIDER WEIGHT WIND RESISTANCE

HANDLEBAR FORCE

BIKE WEIGHT

GROUND REACTION FORCES

PEDALFORCE

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Force Transmission

Purpose of bike transmissionis to convert the high force, low

velocity at the pedal to ahigher velocity (and necessarilylower force) at the wheel. Thepower at pedal (F1 x V1) equalsthe power at the wheel (F4 x V4)(assuming no friction losses)

L1

L2

L3F1

F2F3=F2

F4

L4

F4 = F1 x ?

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Pedal Forces

(ref 3, pg 105)

A clock diagram showing thetotal foot force for a group ofelite pursuit riders using toe

clips, at 100 rpm and 400 W.Note the orientation of the forcevector during the first half of therevolution and the absence ofpull-up forces in the secondhalf.

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Pedal Force Components

Fr = Total Foot Force

Fe=Effective Force

(causes useful Torque)

The total foot force can be resolvedinto vector components

PEDAL

CRANK

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Effective Pedal Force

(ref 3, pg 106)

EFFECTIVE FORCE

RESULTANT FORCE

UNUSED FORCE

NEGATIVE EFFECTIVE FORCE

CRANK ANGLE (Degrees)

FORCE(N)

0 180 360

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HorizontalForce between

(ref 3, pg 107)

A plot of the horizontal force between the rear wheel and the road,due to each leg. The total force is shown as the bold solid line.Note that this force is not constant, due to the fact that the forceapplied at the pedal is only partly effective.

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Pedal Speed

Optimum speed for mostpeople is 55-85 rpm.This yields the mostuseful power output for a

given caloric usage (ref 3, pg 79)

MOST EFFICIENTPEDALLING SPEED

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Human Power Output

n Most adults can deliver .1 HP (75 watts)continuously while pedaling which results in atypical speed of 12 mph

n Well-trained cyclists can produce .25 to.40 HPcontinuously resulting in 20 to 24 mph

n World champion cyclists can produce almost .6HP (450 watts) for periods of one hour or more -resulting in 27 to 30 mph

Why do the champion cyclists only goabout twice as fast if they can producenearly 6 times as much power?

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(ref 3. pg 112)

Human Power Output

The maximum power output that can be sustained forvarious time durations for champion cyclists. Averagepower output over long distances is less than 400 W.

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The Forces Working Against Us

Drag Force due to air resistance: Fdrag =CdragV2 A

Cdrag = drag coefficient (a function of the shape of the body and thedensity of the fluid)

A = frontal area of body

V = velocity

and since: Power = Force x Velocity

This means that, to double your speed requires 8 times

as much power just to overcome air drag (sincepower ~ velocity3)

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Some Empirical

Data

(ref 3, pg 126)

Drag force on a cycle versus speed

showing the effect of rider position. Thewind tunnel measurements are lessthan the coast-down data because thewheels were stationary and rollingresistance was absent.

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Forces - continued

n Rolling Resistance Frr=Crr x Weighttypical values for Crr:

knobby tires .014

n Mechanical Friction (bearings, gear train)absorbs typically only 3-5% of power input if well maintained

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Other Energy Absorbers

n Hills (energy storage or potential energy)Change in Potential Energy = Weight x Change in elevation (Dh)

Dh Here, the rider has stored up

energy equal to the combinedweight of rider and bike timesthe vertical distance climbed.

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The First Law of

Thermodynamicsn Conservation of Energy, for any system:

Energyin = Energyout +Change in Stored Energy

SYSTEM

Energy input

Energy Output

Internal Energyof System

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Now Put it All Together:

Velocity = f[ power input (pedal rpm, pedal force), road slope,rider weight, bike weight, frontal area, rider position, gearratio, tire type and inflation, maintenance ...]

Your task:(as homework, due in one week, use computer(spreadsheet program like EXCEL) for analysis and presentation ofresults)

1. Using first law of thermodynamics, derive the relation between therelevant factors to calculate V (bike velocity). Clearly state allassumptions.

2. Generate a graph relating speed to hill grade (from 0% to 20%) for

riders weighing 120, 140, 160, 180 and 200 pounds who areexerting a continuous power of 0.1 HP.

3. Determine the terminal velocity of the 160 lb rider coasting goingdown a 10% grade.