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PRINCIPLES
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PICTURES
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POSSIBILITIES
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ASSUMPTIONS
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Real-worldchallenge
WhatstrapshouldIusetotieacanoetotheroof
ofmycar?
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REAL-WORLD
CHALLENGE
ENGINEERING
PROBLEM
STATEMENTS
Somethingneedstobe
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Textbook-likeproblems
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engineeringtechniques
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Develop:FivePsofProblemDefinition
• Considerations tohelpgetgood
problemstatements
• Applied iteratively,inanyorder
• Keepthefocusonobtaining
appropriateproblemstatements
ProblemStatements
Acanoe istobe strapped on aroof rackonanautomobile. There aretwotiedown straps going from
one side ofthe canoetotheother.Eachtiedown strapisconnected toacanoe support thatisattached
totheroof rack.The strapskeep thecanoe inplaceby means of friction between thegunwales ofthe
canoe andtheroof rack.
A)Ifthe tension inthestrapisT,theweight ofthecanoe isW, theaerodynamic lifton thecanoe isL,
and thestaticcoefficient of friction between thegunwales andtherackisμ,determine themaximum
totalfrictional forcethattherackcanapply tothecanoe.
B)Forawind ofspeed vx inthelongitudinal (x)direction, ifthecross sectional areaof thecanoe isAx
and thecoefficient ofdragisCDx, whatisthelongitudinal forceapplied tothecanoe (Fwx)?
C) Forawind ofspeed vy in thetransverse (y)direction, ifthecross sectional areaofthecanoe isAy and
thecoefficient of dragis CDy,whatis thelateralforceapplied tothecanoe (Fwy)?
D)Foravehicle traveling atamaximum speed of80mph, withamaximum wind speed of40mph, what
straptension isnecessary toensure thatthecanoe staysrigidly attachedtotherack(assume thatthelift
forceis neglible)?
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Begin withasummary oftheproblem,whichshouldincludeabriefwrittenstatementthatclearly
identifiesthedesired answer. Youshouldalsoincluderelevantsketches,suchaspictorialsketchesor
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problemsassociatedwithaparticularsection).Iftheapproachisnotobvious,tryfindingmultiplesolution
approaches,alongwiththeassumptionsnecessaryforeachapproach.Considerthestrengthsandweaknesses
Approach and assumptions
ofeachapproach,andselecttheonethatbestmatchestheproblem.
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equationsthroughoutthesolutionprocess.Waituntilthefinalsteptosubstitutegivenvaluesinthe
equations.Whensubstitutionsaremade,includetheunitsonallnumbersandfollowtheunitsthroughthe
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Find:
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(e.g. laminar flow),doesthesolutionsupporttheseassumptions?Whatlimitations,ifany,haveyouidentified
forthesolution?Clearlycommunicateanyconcernsorlimitationsofthesolutionalongwiththesolutionitself.
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Summary
Theoverallproblemsetisaimedatdetermining thestraptension necessary to
holdthecanoeinplace.Thesubproblems include identifying therelationship
between straptensionandthemaximumfrictional force,therelationship
between lateralandlongitudinal windspeed andlateralanlongitudinal drag
force,andtherelationships betweenlongitudinal andlateralforcesandtotal
frictional force.
Eachoftheseproblemsets isrelativelystraightforward.
ApproachandAssumptions
• Wewillapproach theaerodynamicproblemsusingthedefinitionofdragcoefficient.
• Weapproach thefrictionalproblemusingthedefinitionofstaticfriction.
• Aswedonot knowexactvaluesofμ andCD, wewillselectapproximatevaluesthatarereasonable.Wewillmeasuretheareas
fromadrawingofatypicalcanoe. Wewillestimateweightfromavailabledataon typicalcanoes.
• Fromameasurement:Ax is0.25m 2 and Ay is1.4m 2
•
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•
W is35lb.
•
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Theweightofthestrapsisnegligible.
Thecenterofpressureisatthecenterofmass(onlynecessaryifliftisnotnegligible).
BothstrapshaveequaltensionT.
Thecontactbetweenthestrapsand thecanoeisapproximatelyfrictionless.
Theanglethatthestrapsmakewiththerackisnegligible(i.e.allthetensionisstraightdown).
Assumptions:
Thedensityofairis1.2kg/m 3
Theliftforceisnegligible
Framework
Given:Maximumwindspeedvw of40mph.Maximumvehiclespeedv
of80mph.
Free-bodydiagrams:
Execute
First,weconsidertheeffectsofthestrapsonthefrictionalforcepreventingthe
canoefrommoving.
Fs =2T
Fz=0;-2Fs-W+2Fn+L=0
Fn =(2Fs+W–L)/2
Ffmax=2μFn
=μ(2Fs+W–L)
=μ(4T+W– L)
ForL=0,
T=(Ffmax/μ-W)/4
ThisequationonlyholdsifFfmax/μ isgreaterthanW.Otherwise,theweightofthe
canoealoneissufficienttoholditinplace.
Execute(part2)
Wenowconsider the forceduetotherelative windseen bythecanoe. Thereisnosimple calculation forthewind
forceinanarbitrarydirection. Suchacalculation couldbeperformed usingComputational FluidDynamics (CFD),but
CFDisbeyond thescope ofthis problem. Therefore, wewillassume the worst-case longitudinal forceandtheworstcase transverse force, andaddthese forcestogether vectorially. Werecognize that thiscombination ofworst-case
forcescannot happen simultaneously; nevertheless, thisisasimple calculation thatisconservative inthesense that
thecalculated forceshouldexceed theactual force.Thus, atension based onthisforcewillbegreater thantheactual
tension needed.
Forthelongitudinal force:
Fwx=½ρvwx2 A xCDx
Themaximum forcewilloccur when thewindisaheadwind, so
vwx=v+vw
Fwx=½ρv+vw2 A xCDx
Execute(part3)
Forthe transverse force:
Fwy=½ρvwy2AyCDy
The maximum force will occur when thewind isdirectly fromthe sides, so
vwy=vw
Fwy=½ρvw2AyCDy
Combining these twoworst-case forcesasvectors gives theworst-case windforce:
Fw=(Fwx2+Fwy2) 1/2
Toholdthe canoe securely, Fwmust beless thanor equal toFfmax
T=(Ffmax/μ-W)/4
=(Fw/μ-W)/4
ReflectandReport
Theunitsareconsistent– bothsideshaveunitsofforce.
Theequationmakesphysicalsense– boththetensionandtheweight
ofthecanoehavethesamesign,andtheweighthas¼thecontribution
ofthetension.Thismakessensebecausetherearefourstrap
connectionpoints,eachwithT,andonlyoneweight.
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MECHANICAL ENGINEERING
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Construct
• Modelimplemented
inMATLAB
function [ MinT,fw ] = StrapTension( v, vw, ax, ay, cdx, cdy, w, mu )
%StrapTension: Calculate the minimum tension necessary in a canoe
%
hold down strap to prevent a canoe from moving on a
%
roof rack given a specified aerodynamic load.
%
%
Note: this is a worst-case calculation; more accurate calculations
%
would require CFD simulations
% Input arguments:
%
v: car velocity in miles/hour
%
vw : wind speed (relative to ground) in miles/hour
%
theta: angle between car velocity and wind direction in degrees
%
ax, ay : cross-sectional area of canoe in travel direction and
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lateral direction, respectively. Units are m^2
%
cdx, cdy: coefficient of drag in travel direction and lateral direction
%
w: weight of canoe in lbs
%
mu : coefficient of friction between canoe and rack
% Outputs:
%
MinT: Minimum tension in strap (N)
%
fw: magnitude of wind force (N)
%
phi: angle of wind force with respect to vehicle direction
% Copyright 2016 Carl D. Sorensen
% This code may be freely copied and used for any desired purpose
rho = 1.2; % density of air in kg/m^3
mph2mps = @(v) v*1609.344/3600; % convert from mph to m/s
lb2n = @(w) w * 4.44822; % convert from lbf to N
% get all units to SI
v = mph2mps(v);
vw = mph2mps(vw);
w = lb2n(w);
vwx = v+vw;
vwy = vw;
fwx = rho*vwx.^2.*ax.*cdx/2;
fwy = rho*vwy.^2.*ay.*cdy/2;
fw = sqrt(fwx.^2+fwy.^2);
MinT = (fw./mu-w)/4;
MinT(MinT<0)=0; % Set minimum values to zero
VerifyandValidate
>> W
W =
35
Forv=vw=0,Tshould
be0.
>> mu
mu =
0.4000
>> Ax
Ax =
0.2500
>> Ay
Ay =
1.4000
>> Cdx
Cdx =
0.2500
>> Cdy
Cdy =
0.5000
>> [T,f]=StrapTension(0,0,Ax,Ay,Cdx,Cdy,W,mu)
T =
0
Search
Required strap tension vs. wind speed
400
Wind force vs. vehicle speed
400
350
v = 100
350
300
Required strap tension, N
Wind force, N
250
CDy = 1.00
CDx = 0.25
vw = 60.0
300
vw = 52.5
200
vw = 45.0
150
250
CDy = 0.80
200
CDy = 0.60
150
CDy = 0.40
vw = 37.5
100
100
CDy = 0.20
vw = 30.0
vw = 22.5
50
0
0
10
20
30
vw = 15.0
40
50
60
Vehicle speed, mph
vw = 0.0
vw = 7.5
70
80
90
50
100
0
0
10
20
30
Wind speed, mph
40
50
60
Analyze
Because ofthelarger transverse area andthe larger transverse CD,thewind speed dominates theforce
requirement. Atawindspeed of45miles/hour andavehicle speed of80miles per hour,thevehicle
speed isresponsible forabout15%ofthetotal force.Thus,it’sthewind thatweneed toworry about
themost.
The aerodynamic forces are relatively small inall ofthese situations, however. Even atanunrealistic
transverse CD of1.0,andawindspeed of60mph,the required tension isonly 350N.A1inch polyester
webstrap hasaworking load of3700N,ormore than 10times the required tension. A1inch
polypropylene straphasaworking load of1175N,ormore than3times the required tension.
This calculation hasignored lift force.Ifthere isasignificant lift force,the tension required will increase. Perhaps aCFD,windtunnel test,orreal-world experiment shouldbe performed tomeasure thelift
force.
Translate
A1-inchwidetiedownstrapofeitherpolyesterorpolypropylenewill
carrytherequiredtensiontoholdthecanoe,eveninextreme
conditions.
Communicate
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MECHANICAL ENGINEERING
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REAL-WORLD
ANSWERS
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&$
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IDEALANSWERS
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