Statics and Dynamics - Andy Ruina

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Introduction to
S
TATICS
and
D
YNAMICS
Filename:Saskyalaunch3517
Andy Ruina and Rudra Pratap
OxfordUniversityPress(Preprint)
MostrecentmodificationsonAugust21,2013.
Reference Tables: The front and back tables concisely summarize much of the text material.
Summary of Mechanics
0) Thelawsofmechanicsapplytoanycollectionofmaterialor‘body.’ Thisbodycouldbetheoverallsystemofstudy
oranypartofit. Intheequationsbelow,theforcesandmomentsarethosethatshowonafreebodydiagram. Interacting
bodiescauseequalandoppositeforcesandmomentsoneachother.
I) LinearMomentumBalance(LMB)/ForceBalance
EquationofMotion Fi L The total force on a body is equal (I)
to its rate of change of linear
momentum.
Impulse-momentum t2
(integratingintime) Fi·dt L Netimpulseisequaltothechangein (Ia)
t1 momentum.
Conservationofmomentum L=0
When there is no net force the linear (Ib)
(if Fi 0) L=L2 L1 0 momentum does not change.
Statics
Fi 0 If the inertial terms are zero the (Ic)
(ifLisnegligible)
net force on system is zero.
II) AngularMomentumBalance(AMB)/MomentBalance
Equationofmotion M H The sum of moments is equal to the (II)
C C
rateofchangeofangularmomentum.
Impulse-momentum(angular) t2
M dt H The net angular impulse is equal to (IIa)
(integratingintime) C C
t1 the change in angular momentum.
Conservationofangularmomentum H 0
C If there is no net moment about point (IIb)
(if M 0) H H H 0
C C C2 C1 C then the angular momentum about
point C does not change.
Statics
M 0 If the inertial terms are zero then the (IIc)
(ifH isnegligible) C
C total moment on the system is zero.
III) PowerBalance(1stlawofthermodynamics)
Equationofmotion Q P E E E Heat flow plus mechanical power (III)
K P int
into a system is equal to its change
E
in energy (kinetic + potential +
internal).
t2 t2
forfinitetime Qdt Pdt E Thenetenergyflowgoinginisequal (IIIa)
t1 t1 tothenetchangeinenergy.
ConservationofEnergy E 0
If no energyflows into a system, (IIIb)
(ifQ P 0) E E E 0
2 1 then its energydoesnotchange.
Statics
Q P E E If there is no change of kinetic energy (IIIc)
(ifE isnegligible) P int
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 on the system.
Filename:Summaryofmechanics
Some definitions
(Alsoseetheindexandbacktables)
*r or *x Position e.g.,*r *r is the position of a point i
i (cid:17) i=O
relativetotheorigin,O.
d*r
*v Velocity e.g.,*v *v is the velocity of a point i
(cid:17) dt i (cid:17) i=O
relativetoO,measuredinanon-rotatingref-
erenceframe.
d*v d2*r
*a Acceleration e.g.,*a *a istheaccelerationofapointi
(cid:17) dt D dt2 i (cid:17) i=O
relativetoO,measuredinaNewtonianframe.
*
F Force e.g.,theforceonAfromBisF .
AfromB
* * *
M or M M MomentorTorque e.g., the moment of a collection of forces
CD =C
aboutpointC.
*! Angularvelocity Ameasureofrotationalvelocityofarigidob-
ject.*! =angularvelocityofrigidobjectB.
B
*(cid:11) *! Angularacceleration Ameasureofrotationalaccelerationofarigid
(cid:17) P
object.
m*v discrete
* i i
L Linearmomentum Ameasureofasystem’snettranslationalrate
(cid:17) 8 P*vdm continuous (weightedbymass).
<
m R*v
D :tot cm
m*a discrete
* i i
LP Rate of change of linear momen- The aspect of motion that balances the net
(cid:17) 8 P*adm continuous tum forceonasystem.
<
m R*a
D :tot cm
*r m*v discrete
H* i=C(cid:2) i i AngularmomentumaboutpointC Ameasureoftherotationalrateofasystem
=C (cid:17) 8< P*r=C (cid:2)*vdm continuous about a point C (weighted by mass and dis-
tancefromC).
R
:
*r m*a discrete
H*P=C (cid:17) 8< P*r=iC=C(cid:2)(cid:2)*admi i continuous RtuamteaobfocuhtapnogientoCfangularmomen- TtohrequaesopnecatsoyfstmemotiaobnouthtaatpboainlatnCc.es the net
R
:
1 m v2 discrete
E 2 i i Kineticenergy Ascalarmeasureofnetsystemmotion.
K (cid:17) 8 1Pv2dm continuous
< 2
R
:
E (heat-liketerms) Internalenergy The non-kinetic non-potential part of a sys-
int D
tem’stotalenergy.
P F**v M* *! Powerofforcesandtorques The mechanical energy flow into a system.
(cid:17) i(cid:1) i C i(cid:1) i
Also,P W,rateofwork.
P P (cid:17) P
Icm Icm Icm
xx xy xz
(cid:140)Icm(cid:141) 2 Icm Icm Icm 3 Moment of inertia matrix about Ameasureofthemassdistributioninarigid
(cid:17) xy yy yz
centerofmass(cm) object.
66 Ixczm Iyczm Izczm 77
6 7
4 5
iv Chapter0.
(cid:13)c Rudra Pratap and Andy Ruina, 1994-2013. All rights reserved. No part of this book may be reproduced, stored in a
retrievalsystem,ortransmitted,inanyformorbyanymeans,electronic,mechanical,photocopying,orotherwise,without
priorwrittenpermissionoftheauthors.
Thisbookisapre-releaseversionofabookinprogressforOxfordUniversityPress.
Acknowledgements. Thefollowingareamongstthosewhohavehelpedwiththisbookaseditors,artists,texprogrammers,
advisors,criticsorsuggestersandcreatorsofcontent:WilliamAdams,AlexaBarnes,PranavBhounsule,JosephBurns,Jason
Cortell,GaborDomokos,MaxDonelan,ThuDong,GailFish,MikeFox,JohnGibson,RobertGhrist,SaptarsiHaldar,Dave
Heimstra, Theresa Howley, Herbert Hui, Michael Marder, Elaina McCartney, Horst Nowacki, Jim Papadopoulos, Kalpana
Pratap, DaneQuinn, RichardRand, C.V.Radakrishnan, NidhiRathi, PhoebusRosakis, LesSchaffer, IshanSharma, David
Shipman, JillStartzell, SaskyavanNouhuys, TianTang, KimTurnerandBillZobrist. Ouron-againoff-againeditorPeter
Gordonhasbeensupportivethroughout. Manyotherfriends,colleagues,relatives,students,andanonymousreviewershave
alsomadehelpfulsuggestions.
WecertifyArthurOgawa,IvanDobrianov,andStephenHicksasTeXgeniuses.
MikeColemanworkedextensivelyonthetext,wrotemanyoftheexamplesandhomeworkproblemsandmademanyfigures.
DavidHo,R.Manjula,AbhayandMiekeRuinadreworimprovedmostofthedrawings. Creditforsomeofthehomework
problems retrieved from Cornell archives is due to various Theoretical and Applied Mechanics faculty. Harry Soodak and
MartinTierstenprovidedsomeproblemsfromtheirincompletebook.
SoftwarewehaveusedtopreparethisbookincludesTEXshop(forLATEX)withmanycustomfeaturesimplementedbyStephen
Hicks,AdobeIllustrator,GraphicsConverterandMATLAB.
IntroductiontoStaticsandDynamics,(cid:13)c AndyRuinaandRudraPratap1994-2013.
IntroductiontoStaticsandDynamics,(cid:13)c AndyRuinaandRudraPratap1994-2013. 1
2 Chapter0.
Brief Contents
Fronttables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
BriefContents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
DetailedContents . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Preface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Tothestudent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Part I: Basics for Mechanics
24
1 Whatismechanics? . . . . . . . . . . . . . . . . . . . . . . . . . 24
2 Vectors: position,forceandmoment . . . . . . . . . . . . . . . . 38
3 FBDs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
Part II: Statics
186
4 Staticsofoneobject. . . . . . . . . . . . . . . . . . . . . . . . . 186
5 Trussesandframes . . . . . . . . . . . . . . . . . . . . . . . . . 258
6 Transmissionsandmechanisms. . . . . . . . . . . . . . . . . . . 324
7 Tension,shearandbendingmoment . . . . . . . . . . . . . . . . 378
8 Hydrostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402
Part III: Dynamics
418
9 Dynamicsin1D . . . . . . . . . . . . . . . . . . . . . . . . . . . 418
10 Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498
11 Particlesinspace . . . . . . . . . . . . . . . . . . . . . . . . . . 544
12 Manyparticlesinspace . . . . . . . . . . . . . . . . . . . . . . . 592
13 Straightlinemotion . . . . . . . . . . . . . . . . . . . . . . . . . 618
14 Circularmotionofaparticle . . . . . . . . . . . . . . . . . . . . 658
15 Circularmotionofarigidobject . . . . . . . . . . . . . . . . . . 690
16 Planarmotionofanobject . . . . . . . . . . . . . . . . . . . . . 770
17 Time-varyingbasisvectors . . . . . . . . . . . . . . . . . . . . . 856
18 Constrainedparticlesandrigidobjects . . . . . . . . . . . . . . . 926
Appendices
994
A Unitsanddimensions . . . . . . . . . . . . . . . . . . . . . . . . 994
B Friction: perspectivesonfrictionlaws . . . . . . . . . . . . . . .1006
C ThesimplestODEsandtheirsolutions . . . . . . . . . . . . . . .1016
D TheoremsforSystems . . . . . . . . . . . . . . . . . . . . . . .1020
Answerstosomehomeworkproblems . . . . . . . . . . . . . . . . .1030
Backtables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1039
IntroductiontoStaticsandDynamics,(cid:13)c AndyRuinaandRudraPratap1994-2013.
Detailed Contents
Fronttables i
Summaryofmechanics . . . . . . . . . . . . . . . . . . . i
Somebasicdefinitions . . . . . . . . . . . . . . . . . . . . ii
BriefContents 2
DetailedContents 3
Preface 12
General issues about content, level, organization, style and motivation.
Studyadvicestartsonpage ??.
Tothestudent 16
Howtostudy. Theuseofcomputers.
0.1 Anoteoncomputation . . . . . . . . . . . . . . . . . . . . . 20
Box: Informalcomputercommands . . . . . . . . . . . . . 23
Part I: Basics for Mechanics
24
1 Whatismechanics? 24
Mechanicscanpredictforcesandmotionsbyusingthethreepillarsofthe
subject: I. models of physical behavior, II. geometry, and III. the basic
mechanicsbalancelaws. Thelawsofmechanicsareinformallysumma-
rized in this introductory chapter. The extreme accuracy of Newtonian
mechanicsisemphasized,despiterelativityandquantummechanicssup-
posedly having ‘overthrown’ seventeenth century physics. Various uses
oftheword‘model’aredescribed.
1.1 Thethreepillars . . . . . . . . . . . . . . . . . . . . . . . . 25
1.2 Mechanicsiswrong,whystudyit? . . . . . . . . . . . . . . 31
1.3 Thehierarchyofmodels . . . . . . . . . . . . . . . . . . . . 33
2 Vectors: position,forceandmoment 38
The key vectors for statics, namely relative position, force, and mo-
ment, are used to develop vector skills. Notational clarity is empha-
sized because good vector calculation demands distinguishing vectors
fromscalars. Vectoradditionismotivatedbytheneedtoaddforcesand
relativepositions. Dotproductsaremotivatedasthetoolwhichreduces
vectorequationstoscalarequations.Andcrossproductsaremotivatedas
IntroductiontoStaticsandDynamics,(cid:13)c AndyRuinaandRudraPratap1994-2013. 3
4 Chapter0.DetailedContents DetailedContents
theformulawhichcorrectlycalculatestheheuristicallymotivatedquan-
titiesofmomentandmomentaboutanaxis.
2.1 Notationandaddition . . . . . . . . . . . . . . . . . . . . . 40
Box2.1Thescalarsinmechanics . . . . . . . . . . . . . . 41
Box2.2TheVectorsinMechanics . . . . . . . . . . . . . 42
2.2 Thedotproductoftwovectors . . . . . . . . . . . . . . . . 58
Box2.3Basicfeaturesofthevectordotproduct. . . . . . . 58
Box2.4abcos(cid:18) a b a b a b . . . . . . . 63
x x y y z z
) C C
2.3 Vectorcrossproduct . . . . . . . . . . . . . . . . . . . . . . 67
Box2.5Usesofthecrossproduct . . . . . . . . . . . . . . 68
Box2.6Crossproductasamatrixmultiply . . . . . . . . . 74
Box2.7Thecrossproduct: fromgeometrytocomponents . 75
2.4 Moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
2.5 Solvingvectorequations . . . . . . . . . . . . . . . . . . . . 93
Box2.8Therulesofvectoralgebra. . . . . . . . . . . . . 94
Box2.9Vectortrianglesandthelawsofsinesandcosines . 96
Box2.10Existence,uniqueness,andgeometry . . . . . . . 108
ProblemsforChapter2 . . . . . . . . . . . . . . . . . . . . . . . 113
3 FBDs 122
Afree-bodydiagramisasketchofthesystemtowhichyouwillapplythe
lawsofmechanics. Thediagramshowsallofthenon-negligibleexternal
forcesandcoupleswhichactonthesystem. Thediagramtellswhatma-
terial is in the system and also what is known, and what is not known,
about the forces. Mechanics reasoning depends on free body diagrams
so we give tips about how to avoid common mistakes. On a free body
diagram systems of forces are often replaced with ‘equivalent’ forces, a
specialcaseofwhichisaweightforceatthecenterofgravity.
3.1 Equivalentforcesystems . . . . . . . . . . . . . . . . . . . . 125
Box3.1 meansadd . . . . . . . . . . . . . . . . . . . . 127
Box3.2Equivalentatonepoint equivalentatallpoints 128
P )
Box3.3A“wrench”canrepresentanyforcesystem . . . . 129
3.2 Centerofmassandgravity. . . . . . . . . . . . . . . . . . . 134
Box3.4Like ,thesymbol alsomeansadd . . . . . . . 135
Box3.5Eachsubsystemislikeaparticle . . . . . . . . . . 139
P R
Box3.6TheCOMofatriangleisath=3 . . . . . . . . . . 143
3.3 Interactions,forces&partialFBDs . . . . . . . . . . . . . . 149
VectornotationforFBDs . . . . . . . . . . . . . . . . . . 151
Box3.7Freebodydiagramfirst,mechanicsreasoningafter 159
Box3.8ActionandreactiononpartialFBD’s . . . . . . . 161
3.4 Contact: Sliding,friction,androlling . . . . . . . . . . . . . 164
ProblemsforChapter3 . . . . . . . . . . . . . . . . . . . . . . . 178
IntroductiontoStaticsandDynamics,(cid:13)c AndyRuinaandRudraPratap1994-2013.
Chapter0.DetailedContents DetailedContents 5
Part II: Statics
186
4 Staticsofoneobject 186
One object is in equilibrium if the forces and moments balance. For a
particle,forcebalancetellsall. Butforanextendedobject,momentbal-
ance is also essential. There are special shortcuts for an objects that
hasexactlytwoorexactlythreeforcesactingonit. Iffrictionforcesare
relevant the possibility of motion needs to be taken into account. Many
real-world problems are not statically determinate and thus yield either
only partial solutions, or yield full solutions after you have made extra
assumptions.
4.1 Staticequilibriumofaparticle . . . . . . . . . . . . . . . . . 188
Box4.1Existenceanduniqueness . . . . . . . . . . . . . 192
Box4.2Thesimplificationofdynamicstostatics . . . . . . 195
4.2 Equilibriumofoneobject . . . . . . . . . . . . . . . . . . . 202
Box4.3Two-forcebodies . . . . . . . . . . . . . . . . . . 207
Box4.4Three-forcebodies . . . . . . . . . . . . . . . . . 208
Box4.5Momentbalanceabout3pointsissufficientin2D . 209
4.3 Equilibriumwithfrictionalcontact . . . . . . . . . . . . . . 214
Box4.6Undrivenwheelsandtwoforcebodies . . . . . . 218
4.4 Internalforces . . . . . . . . . . . . . . . . . . . . . . . . . 228
4.5 3Dstaticsofonepart . . . . . . . . . . . . . . . . . . . . . 234
Box4.7Staticallydeterminatewaystoholdanobjectin3D 239
ProblemsforChapter4 . . . . . . . . . . . . . . . . . . . . . . . 243
5 Trussesandframes 258
Hereweconsidercollectionsofpartsassembledsoastoholdsomething
up or hold something in place. Emphasis is on trusses, assemblies of
bars connected by pins at their ends. Trusses are analyzed by drawing
free body diagrams of the pins or of bigger parts of the truss (method
ofsections). Frameworksbuiltwithotherthantwo-forcebodiesarealso
analyzedbydrawingfreebodydiagramsofparts. Structurescanberigid
or not and redundant or not, as can be determined by the collection of
equilibriumequations.
5.1 Methodofjoints . . . . . . . . . . . . . . . . . . . . . . . . 260
5.2 Themethodofsections . . . . . . . . . . . . . . . . . . . . 277
5.3 Solvingtrussesonacomputer . . . . . . . . . . . . . . . . . 284
5.4 Framesandstructures . . . . . . . . . . . . . . . . . . . . . 295
Box5.1The‘methodofbarsandpins’fortrusses . . . . . 298
5.5 Advancedtrussconcepts: determinacy . . . . . . . . . . . . 305
Box5.2Structuralrigidityandgeometriccongruence . . . 310
Box5.3Rigidity,redundancy,linearalgebraandmaps . . 311
ProblemsforChapter5 . . . . . . . . . . . . . . . . . . . . . . . 316
6 Transmissionsandmechanisms 324
Some collections of solid parts are assembled so as to cause force or
torque in one place given a different force or torque in another. These
include levers, gear boxes, presses, pliers, clippers, chain drives, and
crank-drives. Besides solid parts connected by pins, a few special-
IntroductiontoStaticsandDynamics,(cid:13)c AndyRuinaandRudraPratap1994-2013.
6 Chapter0.DetailedContents DetailedContents
purpose parts are commonly used, including springs and gears. Tricks
for amplifying force are usually based on principals idealized by pul-
leys, levers, wedges and toggles. Force-analysis of transmissions and
mechanisms is done by drawing free body diagrams of the parts, writ-
ingequilibriumequationsforthese,andsolvingtheequationsfordesired
unknowns.
6.1 Springs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326
Box6.1‘Zero-length’springs . . . . . . . . . . . . . . . . 327
Box6.2Howstiffaspringisasolidrod . . . . . . . . . . 334
Box6.3Stifferbutweaker . . . . . . . . . . . . . . . . . . 334
Box6.4Apuzzlewithtwospringsandthreeropes. . . . . . 335
Box6.52Dgeometryofspringstretch . . . . . . . . . . . 338
6.2 Forceamplification . . . . . . . . . . . . . . . . . . . . . . 347
6.3 Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . 358
Box6.6Shearswithgears . . . . . . . . . . . . . . . . . . 362
ProblemsforChapter6 . . . . . . . . . . . . . . . . . . . . . . . 369
7 Tension,shearandbendingmoment 378
The ‘internal forces’ tension, shear and bending moment can vary from
point to point in long narrow objects. Here we introduce the notion of
graphingthisvariationandnotingthefeaturesofthesegraphs.
7.1 Arbitrarycuts . . . . . . . . . . . . . . . . . . . . . . . . . 379
7.2 Singularityfunctions . . . . . . . . . . . . . . . . . . . . . . 395
ProblemsforChapter7 . . . . . . . . . . . . . . . . . . . . . . . 400
8 Hydrostatics 402
Hydrostaticsconcernstheequivalentforceandmomentduetodistributed
pressure on a surface from a still fluid. Pressure increases with depth.
With constant pressure the equivalent force has magnitude = pressure
times area, acting at the centroid. For linearly-varying pressure on a
rectangular plate the equivalent force is the average pressure times the
area acting 2/3 of the way down. The net force acting on a totally sub-
merged object in a constant density fluid is the displaced fluid’s weight
actingatthecentroid.
8.1 Fluidpressure . . . . . . . . . . . . . . . . . . . . . . . . . 403
Box8.1AddingforcestoderiveArchimedes’principle . . . 406
Box8.2Pressuredependsonpositionbutnotonorientation 407
ProblemsforChapter8 . . . . . . . . . . . . . . . . . . . . . . . 415
Part III: Dynamics
418
9 Dynamicsin1D 418
ThescalarequationF Dmaintroducestheconceptsofmotionandtime
derivatives to mechanics. In particular the equations of dynamics are
seen to reduce to ordinary differential equations, the simplest of which
have memorable analytic solutions. The harder differential equations
needbesolvedonacomputer. Weexplorevariousconceptsandapplica-
IntroductiontoStaticsandDynamics,(cid:13)c AndyRuinaandRudraPratap1994-2013.