Table Of ContentIntroduction to
S
TATICS
and
D
YNAMICS
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Andy Ruina and Rudra Pratap
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Summary of Mechanics
0) Thelawsofmechanicsapplytoanycollectionofmaterialor‘body.’ Thisbodycouldbetheoverallsystemofstudy
oranypartofit. Intheequationsbelow,theforcesandmomentsarethosethatshowonafreebodydiagram. Interacting
bodiescauseequalandoppositeforcesandmomentsoneachother.
I) LinearMomentumBalance(LMB)/Force(cid:2)Balance
EquationofMotion F(cid:2)i =L(cid:2)˙ The total force on a body is equal (I)
to its rate of change of linear
(cid:3) momentum.
I(minpteuglrsaet-imngominetnimtuem) t2(cid:2)F(cid:2)i·dt =(cid:3)L(cid:2) Netimpulseisequaltothechangein (Ia)
t1 momentum.
Con(cid:4)servationofmomentum L(cid:2)˙ =(cid:2)0 ⇒
(if F(cid:2)i = (cid:2)0) (cid:3)L(cid:2)=L(cid:2)2−L(cid:2)1 = (cid:2)0 Wmohmene nthtuemre diso neso nnoett cfohracneg eth.e linear (Ib)
(cid:2)
Stat(cid:2)˙ics F(cid:2)i = (cid:2)0 If the inertial terms are zero the (Ic)
(ifLisnegligible)
net force on system is zero.
II) AngularMomentumBalance(AMB)/Mom(cid:2)entBalance
Equationofmotion M(cid:2) =H(cid:2)˙ The sum of moments is equal to the (II)
C C
rateofchangeofangularmomentum.
(cid:3)
Impulse-momentum(angular) t2(cid:2)M(cid:2) dt =(cid:3)H(cid:2) The net angular impulse is equal to (IIa)
(integratingintime) C C
t1 the change in angular momentum.
C(ifon(cid:4)seMr(cid:2)vati=on(cid:2)0o)fangularmomentum (cid:3)H(cid:2)˙˙HC(cid:2)==(cid:2)0H(cid:2)⇒−H(cid:2) = (cid:2)0 If there is no net moment about point (IIb)
C C C2 C1 C then the angular momentum about
point C does not change.
(cid:2)
S(itfaHt(cid:2)i˙csisnegligible) M(cid:2)C = (cid:2)0 If the inertial terms are zero then the (IIc)
C total moment on the system is zero.
III) PowerBalance(1stlawofthermodynamics)
Equationofmotion Q˙ +P = E˙ +E˙ +E˙ Heat flow plus mechanical power (III)
(cid:5)K (cid:6)P(cid:7) in(cid:8)t
into a system is equal to its change
E˙
in energy (kinetic + potential +
(cid:3) (cid:3) internal).
forfinitetime t2 Q˙dt+ t2 Pdt =(cid:3)E Thenetenergyflowgoinginisequal (IIIa)
t1 t1 tothenetchangeinenergy.
ConservationofEnergy E˙ =0 ⇒
(ifQ˙ = P =0) (cid:3)E = E −E =0 If no energyflows into a system, (IIIb)
2 1 then its energydoesnotchange.
S(itfaEt˙icsisnegligible) Q˙ +P = E˙P+E˙int If there is no change of kinetic energy (IIIc)
K then the change of potential and
internal energy is due to mechanical
work and heat flow.
PureMechanics
(ifheatflowanddissipation P = E˙ +E˙ In a system well modeled as purely (IIId)
K P
arenegligible) mechanical the change of kinetic
and potential energy is due to mechanical
work.
Some Definitions
(Please also look at the tables inside the back cover.)
r(cid:2) or x(cid:2) Position .e.g., r(cid:2)i ≡ r(cid:2)i/Oisthepositionofapoint
irelativetotheorigin,O)
dr(cid:2)
v(cid:2) ≡ Velocity .e.g., v(cid:2)i ≡ v(cid:2)i/Oisthevelocityofapoint
dt
irelativetoO,measuredinanon-rotating
referenceframe)
dv(cid:2) d2r(cid:2)
a(cid:2) ≡ dt = dt2 Acceleration .e.g.,a(cid:2)i ≡ a(cid:2)i/O istheaccelerationofa
pointi relativetoO,measuredinaNew-
tonianframe)
ω(cid:2) Angularvelocity Ameasureofrotationalvelocityofarigid
body.
α(cid:2) ≡ ω(cid:2)˙ Angularacceleration Ameasureofrotationalaccelerationofa
rigidbody.
(cid:4)
L(cid:2) ≡ (cid:12) miv(cid:2)i discrete Linearmomentum Ameasureofasystem’snettranslational
v(cid:2)dm continuous rate(weightedbymass).
= m v(cid:2)
tot cm
(cid:4)
L(cid:2)˙ ≡ (cid:12) mia(cid:2)i discrete Rate of change of linear Theaspectofmotionthatbalancesthenet
a(cid:2)dm continuous momentum forceonasystem.
= m a(cid:2)
tot cm
(cid:4)
H(cid:2) ≡ (cid:12) r(cid:2)i/C× miv(cid:2)i discrete Angular momentum about Ameasureoftherotationalrateofasys-
C r(cid:2)/C ×v(cid:2)dm continuous pointC tem about a point C (weighted by mass
anddistancefromC).
(cid:4)
H(cid:2)˙ ≡ (cid:12) r(cid:2)i/C× mia(cid:2)i discrete Rateofchangeofangularmo- Theaspectofmotionthatbalancesthenet
C r(cid:2)/C ×a(cid:2)dm continuous mentumaboutpointC torqueonasystemaboutapointC.
(cid:4)
1 m v2 discrete
E ≡ 2(cid:12) i i Kineticenergy Ascalarmeasureofnetsystemmotion.
K
1 v2dm continuous
2
E = (heat-liketerms) Internalenergy The non-kinetic non-potential part of a
int
system’stotalenergy.
(cid:4) (cid:4)
P ≡ F(cid:2)·v(cid:2) + M(cid:2) ·ω(cid:2) Powerofforcesandtorques The mechanical energy flow into a sys-
i i i i
tem. Also, P ≡W˙,rateofwork.
Icm Icm Icm
xx xy xz
[Icm]≡ Icm Icm Icm Momentofinertiamatrixabout Ameasureofhowmassisdistributedin
xy yy yz
cm arigidbody.
Icm Icm Icm
xz yz zz
(cid:10)c Rudra Pratap and Andy Ruina, 1994-2002. All rights reserved. No part of this
book may be reproduced, stored in a retrieval system, or transmitted, in any form
orbyanymeans,electronic,mechanical,photocopying,orotherwise,withoutprior
writtenpermissionoftheauthors.
Thisbookisapre-releaseversionofabookinprogressforOxfordUniversityPress.
Acknowledgements. The following are amongst those who have helped with this
book as editors, artists, tex programmers, advisors, critics or suggestors and cre-
atorsofcontent: AlexaBarnes,JosephBurns,JasonCortell,IvanDobrianov,Gabor
Domokos,MaxDonelan,ThuDong,GailFish,MikeFox,JohnGibson,RobertGhrist,
Saptarsi Haldar, Dave Heimstra, Theresa Howley, Herbert Hui, Michael Marder,
Elaina McCartney, Horst Nowacki, Arthur Ogawa, Kalpana Pratap, Richard Rand,
Dane Quinn, Phoebus Rosakis, Les Schaeffer, Ishan Sharma, David Shipman, Jill
Startzell,SaskyavanNouhuys,BillZobrist. MikeColemanworkedextensivelyon
thetext,wrotemanyoftheexamplesandhomeworkproblemsandcreatedmanyof
thefigures. DavidHohasdrawnorimprovedmostofthecomputerartwork. Some
ofthehomeworkproblemsaremodificationsfromtheCornell’sTheoreticalandAp-
pliedMechanicsarchivesandthusareduetoT&AMfacultyortheirlibrariesinways
that we do not know how to give proper attribution. Our editor Peter Gordon has
beenpatientandsupportivefortoomanyyears. Manyunlistedfriends,colleagues,
relatives,students,andanonymousreviewershavealsomadehelpfulsuggestions.
SoftwareusedtopreparethisbookincludesTeXtures,BLUESKY’simplementation
ofLaTeX,AdobeIllustrator,AdobeStreamline,andMATLAB.
MostrecenttextmodificationsonJanuary29,2002.
Introduction to
S
TATICS
and
D
YNAMICS
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Ms
(cid:2)
F
s
ˆ
k
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ıˆ
F1 F2
N N
1 2
Andy Ruina and Rudra Pratap
Pre-printforOxfordUniversityPress,January2002
Contents
Preface iii
Tothestudent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
1 Whatismechanics? 1
2 Vectorsformechanics 7
2.1 Vectornotationandvectoraddition . . . . . . . . . . . . . . . . . 8
2.2 Thedotproductoftwovectors . . . . . . . . . . . . . . . . . . . . 23
2.3 Crossproduct,moment,andmomentaboutanaxis . . . . . . . . . 32
2.4 Solvingvectorequations . . . . . . . . . . . . . . . . . . . . . . . 50
2.5 Equivalentforcesystems . . . . . . . . . . . . . . . . . . . . . . . 69
3 Freebodydiagrams 79
3.1 Freebodydiagrams . . . . . . . . . . . . . . . . . . . . . . . . . 80
4 Statics 107
4.1 Staticequilibriumofonebody . . . . . . . . . . . . . . . . . . . . 109
4.2 Elementarytrussanalysis . . . . . . . . . . . . . . . . . . . . . . 129
4.3 Advancedtrussanalysis: determinacy,rigidity,andredundancy . . . 138
4.4 Internalforces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
4.5 Springs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
4.6 Structuresandmachines . . . . . . . . . . . . . . . . . . . . . . . 179
4.7 Hydrostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
4.8 Advancedstatics . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
5 Dynamicsofparticles 217
5.1 Forceandmotionin1D . . . . . . . . . . . . . . . . . . . . . . . 219
5.2 Energymethodsin1D . . . . . . . . . . . . . . . . . . . . . . . . 233
5.3 Theharmonicoscillator . . . . . . . . . . . . . . . . . . . . . . . 240
5.4 Moreonvibrations: damping . . . . . . . . . . . . . . . . . . . . 257
5.5 Forcedoscillationsandresonance . . . . . . . . . . . . . . . . . . 264
5.6 Coupledmotionsin1D. . . . . . . . . . . . . . . . . . . . . . . . 274
5.7 Timederivativeofavector: position,velocityandacceleration . . . 281
5.8 Spatialdynamicsofaparticle . . . . . . . . . . . . . . . . . . . . 289
5.9 Central-forcemotionandcelestialmechanics . . . . . . . . . . . . 304
5.10 Coupledmotionsofparticlesinspace . . . . . . . . . . . . . . . . 314
6 Constrainedstraightlinemotion 329
6.1 1-Dconstrainedmotionandpulleys . . . . . . . . . . . . . . . . . 330
6.2 2-Dand3-Dforceseventhoughthemotionisstraight . . . . . . . . 343
i
ii CONTENTS
7 Circularmotion 359
7.1 Kinematicsofaparticleinplanarcircularmotion . . . . . . . . . . 360
7.2 Dynamicsofaparticleincircularmotion . . . . . . . . . . . . . . 371
7.3 Kinematicsofarigidbodyinplanarcircularmotion . . . . . . . . . 378
7.4 Dynamicsofarigidbodyinplanarcircularmotion . . . . . . . . . 395
7.5 Polarmomentofinertia: Icmand IO . . . . . . . . . . . . . . . . . 410
zz zz
7.6 Using Icmand IO in2-Dcircularmotiondynamics . . . . . . . . . 420
zz zz
8 Generalplanarmotionofasinglerigidbody 437
8.1 Kinematicsofplanarrigid-bodymotion . . . . . . . . . . . . . . . 438
8.2 Generalplanarmechanicsofarigid-body . . . . . . . . . . . . . . 452
8.3 Kinematicsofrollingandsliding. . . . . . . . . . . . . . . . . . . 467
8.4 Mechanicsofcontactingbodies: rollingandsliding . . . . . . . . . 480
8.5 Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500
9 Kinematicsusingtime-varyingbasevectors 517
9.1 Polarcoordinatesandpathcoordinates . . . . . . . . . . . . . . . . 518
9.2 Rotatingreferenceframesandtheirtime-varyingbasevectors . . . . 532
9.3 Generalexpressionsforvelocityandacceleration . . . . . . . . . . 545
9.4 Kinematicsof2-Dmechanisms . . . . . . . . . . . . . . . . . . . 558
9.5 Advancekinematicsofplanarmotion . . . . . . . . . . . . . . . . 572
10 Mechanicsofconstrainedparticlesandrigidbodies 581
10.1 Mechanics of a constrained particle and of a particle relative to a
movingframe . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584
10.2 Mechanicsofone-degree-of-freedom2-Dmechanisms . . . . . . . 602
10.3 Dynamicsofrigidbodiesinmulti-degree-of-freedom2-Dmechanisms618
11 Introductiontothreedimensionalrigidbodymechanics 637
11.1 3-Ddescriptionofcircularmotion . . . . . . . . . . . . . . . . . . 638
11.2 Dynamicsoffixed-axisrotation . . . . . . . . . . . . . . . . . . . 648
11.3 Momentofinertiamatrices . . . . . . . . . . . . . . . . . . . . . 661
11.4 Mechanicsusingthemomentofinertiamatrix . . . . . . . . . . . . 672
11.5 Dynamicbalance . . . . . . . . . . . . . . . . . . . . . . . . . . . 693
A Unitsanddimensions 701
A.1 Unitsanddimensions . . . . . . . . . . . . . . . . . . . . . . . . 701
B Contact: frictionandcollisions 711
B.1 Contactlawsareallroughapproximations . . . . . . . . . . . . . . 712
B.2 Friction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713
B.3 AshortcritiqueofCoulombfriction . . . . . . . . . . . . . . . . . 716
B.4 Collisionmechanics . . . . . . . . . . . . . . . . . . . . . . . . . 721
Homeworkproblems 722
Answersto*’dproblems 831
Index 837
Preface
Thisisastaticsanddynamicstextforsecondorthirdyearengineeringstudentswith
anemphasisonvectors,freebodydiagrams,thebasicmomentumbalanceprinciples,
and the utility of computation. Students often start a course like this thinking of
mechanics reasoning as being vague and complicated. Our aim is to replace this
loosethinkingwithconcreteandsimplemechanicsproblem-solvingskillsthatlive
harmoniouslywithausefulmechanicalintuition.
Knowledgeoffreshmancalculusisassumed. Althoughmoststudentshaveseen
vectordotandcrossproducts,vectortopicsareintroducedfromscratchinthecontext
of mechanics. The use of matrices (to tidy-up systems of linear equations) and of
differentialequations(fordescribingmotionindynamics)arepresentedtotheextent
needed. The set-up of equations for computer solutions is presented in a pseudo-
languageeasilytranslatedbyastudentintooneoranothercomputationpackagethat
thestudentknows.
Organization
Wehaveaimedheretobetterunifythesubject,inpart,byanimprovedorganization.
Mechanicscanbesubdividedinvariousways: staticsvsdynamics,particlesvsrigid
bodies, and 1 vs 2 vs 3 spatial dimensions. Thus a 12 chapter mechanics table of
contentscouldlooklikethis
I. Statics II. Dynamics complexity
of objects
A. particles C. particles
rigid
1) 1D 7) 1D body
2) 2D 8) 2D particle number of
dimensions
3) 3D 9) 3D
static
B. rigidbodies D. rigidbodies dynamic
4) 1D 10) 1D
1D 2D 3D
5) 2D 11) 2D how much
inertia
6) 3D 12) 3D
However,thesetopicsarefarfromequalintheirdifficultyorinthenumberofsubtopics
theycontain. Further,therearevariousconceptsandskillsthatarecommontomany
ofthe12sub-topics. Dividingmechanicsintothesebitsdistractsfromtheunityofthe
subject. Althoughsomevestigesoftheschemeaboveremain,ourbookhasevolved
toadifferentorganizationthroughtrialanderror,thoughtandrethought,reviewand
revision,andninesemestersofstudenttesting.
The first four chapters cover the basics of statics. Dynamics of particles and
rigidbodies,basedonprogressivelymoredifficultmotions,ispresentedinchapters
five to eleven. Relatively harder topics, that might be skipped in quicker courses,
areidentifiablebychapter, sectionorsubsectiontitlescontainingwordslike“three
dimensional”or“advanced”. Inmoredetail:
iii
iv PREFACE
Chapter1 defines mechanics as a subject which makes predictions about forces and
motionsusingmodelsofmechanicalbehavior,geometry,andthebasicbalance
laws. Thelawsofmechanicsareinformallysummarized.
Chapter2 introduces vector skills in the context of mechanics. Notational clarity is
emphasized because correct calculation is impossible without distinguishing
vectorsfromscalars. Vectoradditionismotivatedbytheneedtoaddforcesand
relativepositions,dotproductsaremotivatedasthetoolwhichreducesvector
equationstoscalarequations,andcrossproductsaremotivatedastheformula
whichcorrectlycalculatestheheuristicallymotivatedconceptofmomentand
momentaboutanaxis.
Chapter3 isaboutfreebodydiagrams. Itisaseparatechapterbecause,inourexperience,
gooduseoffreebodydiagramsisalmostsynonymouswithcorrectmechanics
problem solution. To emphasize this to students we recommend that, to get
anycreditforaproblemthatusesbalancelawsintherestofthecourse,agood
freebodydiagrammustbedrawn.
Chapter4 makesupashortcourseinstaticsincludinganintroductiontotrusses,mecha-
nisms,beamsandhydrostatics. Theemphasisisontwo-dimensionalproblems
untilthelast,moreadvancedsection. Solutionmethodsthatdependonkine-
matics(i.e.,workmethods)aredeferreduntilthedynamicschapters. Butfor
thestretchoflinearsprings,deformationsarenotcovered.
Chapter5 isaboutunconstrainedmotionofoneormoreparticles. Itshowshowfar
youcangousingF(cid:2)=ma(cid:2)andCartesiancoordinatesin1,2and3dimensions
in the absence of kinematic constraints. The first five sections are a thor-
oughintroductiontomotionofoneparticleinonedimension,socalledscalar
physics,namelytheequation F(x,v,t) = ma andspecialcasesthereof. The
chapterincludessomereviewoffreshmancalculusaswellasanintroduction
toenergymethods. Afewspecialcasesareemphasized,namely,constantac-
celeration,forcedependentonposition(thusmotivatingenergymethods),and
theharmonicoscillator. Afteronesectiononcoupledmotionsin1dimension,
sections seven to ten discuss motion in two and three dimensions. The easy
setupforcomputationoftrajectories,withvariousforcelaws,andevenwith
multipleparticles,isemphasized. Thechapterendswithamostlytheoretical
sectiononthecenter-of-masssimplificationsforsystemsofparticles.
Chapter6 isthefirstchapterthatconcernskinematicconstraintinitssimplestcontext,
systems that are constrained to move without rotation in a straight line. In
one dimension pulley problems provide the main example. Two and three
dimensional problems are covered, such as finding structural support forces
inacceleratingvehiclesandtheslowingorincipientcapsizeofabrakingcar.
Angular momentum balance is introduced as a needed tool but without the
usualcomplexitiesofcurvilinearmotion.
Chapter7 treatspurerotationaboutafixedaxisintwodimensions. Polarcoordinates
and base vectors are first used here in their simplest possible context. The
primaryapplicationsarependulums,geartrains,androtationallyaccelerating
motorsorbrakes.
Chapter8 treatsgeneralplanarmotionofa(planar)rigidbodyincludingrolling,sliding
and free flight. Multi-body systems are also considered so long as they do
notinvolveconstraint(i.e.,collisionsandspringconnectionsbutnothingesor
prismaticjoints).
Chapter9 is entirely about kinematics of particle motion. The over-riding theme is the
useofbasevectorswhichchangewithtime. First,thediscussionofpolarcoor-
dinatesstartedinchapter7iscompleted. Thenpathcoordinatesareintroduced.
Thekinematicsofrelativemotion,atopicthatmanystudentsfinddifficult,is
treated carefully but not elaborately in two stages. First using rotating base
PREFACE v
vectors connected to a moving rigid body and then using the more abstract
notationassociatedwiththefamous“fivetermaccelerationformula.”
Chapter10 isaboutthemechanicsofparticlesandrigidbodiesutilizingtherelativemo-
tionkinematicsideasfromchapter9. Thisisthecapstonechapterforatwo-
dimensionaldynamicscourse. Afterthischapteragoodstudentshouldbeable
tonavigatethroughandusemostoftheskillsintheconceptmaponpage582.
Chapter11 isanintroductionto3Drigidbodymotion. Itextendschapter7tofixedaxis
rotation in three dimensions. The key new kinematic tool here is the non-
trivial use of the cross product for calculating velocities and accelerations.
Fixed axis rotation is the simplest motion with which one can introduce the
fullmomentofinertiamatrix, wherethediagonaltermsareanalogoustothe
scalar 2D moment of inertia and the off-diagonal terms have a “centripetal”
interpretation. Themainnewapplicationisdynamicbalance. Inourexperience
goingpastthisistoomuchformostengineeringstudentsinthefirstmechanics
courseafterfreshmanphysics,sothebookendshere.
AppendixA on units and dimensions is for reference. Because students are immune to
preaching about units out of context, such as in an early or late chapter like
thisone,themainmessagesarepresentedbyexamplethroughoutthebook(the
bookmaybeuniqueamongstmechanicstextsinthisregard):
– Allengineeringcalculationsusingdimensionalquantitiesmustbedimen-
sionally‘balanced’.
– Units are ‘carried’ from one line of calculation to the next by the same
rulesasgonumbersandvariables.
AppendixB oncontactlaws(frictionandcollisions)isforreferenceforstudentswhopuzzle
overtheseissues.
Aleisurelyonesemesterstaticscourse,oramorefast-pacedhalfsemesterprelude
to strength of materials should use chapters 1-4. A typical one semester dynamics
courseshouldcovermostofofchapters5-11precededbytopicsfromchapters1-4,
asneeded. Aonesemesterstaticsanddynamicscourseshouldcoverabouttwothirds
ofchapters1-6and8. Afullyearstaticsanddynamicscourseshouldcovermostof
thebook.
Organizationandformatting
Eachsubjectiscoveredinvariousways.
• Everysectionstartswithdescriptivetextandshortexamplesmotivatingand
describingthetheory;
• Moredetailedexplanationsofthetheoryareinboxesinterspersedinthetext.
Forexample,oneboxexplainsthecommonderivationofangularmomentum
balanceformlinearmomentumbalance,oneexplainsthegeniusofthewheel,
andanotherconnectsω(cid:2)basedkinematicstoeˆr andeˆθ basedkinematics;
• Sampleproblems(markedwithagrayborder)attheendofmostsectionsshow
how to do homework-like calculations. These set an example to the student
intheirconsistentuseoffreebodydiagrams, systematicapplicationofbasic
principles,vectornotation,units,andchecksagainstintuitionandspecialcases;
• Homework problems at the end of each chapter give students a chance to
practice mechanics calculations. The first problems for each section build a
student’sconfidencewiththebasicideas. Theproblemsarerankedinapproxi-
mateorderofdifficulty,withtheoreticalquestionslast. Problemsmarkedwith
an*haveanansweratthebackofthebook;
