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Performance Simulation - Part 3

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Fumdomemil
The importance ofachieving a clear understanding f engineering un-damentals hould not be underestimated. s a design analyst who can-notr{ait to get starled eith FEA,Iou might flnd this chapter o be a ittleheary on theory. f so, do not ger bogged down: you can proceed olater chapters nd refer o rhi: one as nccessary. n facr, ou will findmanv eferences o specific eccions n this chapter hrougirout he estof rhe book, You should always ake the time to understand he con-cepts being discussed. n lhe end, fiis chapter will likeb become a verepowerful engineering ompanion n vour naly.is challinges.
First Printiples
Fr t ,, ,
Dody under Externol oading
llhen .performing enginearing analysis, ou are virtually always on.cemed with how a body will behave under external oading. Nervlon,slaws, or the laws hat will mosr generally govern his behavior, are istedbelow.. Firn Lau: A,body,Nill emain at rest or will continue ls straighLline motion with conslanr velocity f there is no unbalancedforce acting on it.. Second alx: The accelemtion of a body wilt be propo ional tothe resultant ofall forces acring on it and n the direcrion of rhe
r ciullanr.
. Third, au: Acnon and reaction orces between nleracdng bod-ies will be equal n magnitude, collinear, and opposite n direc-tion.'fhe mosl impo{ant engin€ering equation adsing rom these aws ol-lows:
ES. .1
where I is the resultant force vector, ?,? s the mass of the body underconsideration, and a is its acceleration vecror.Becanse accelemtion is the time deivative of veloci.r.l dn/dt), arrd, G =rp constitutes the Inaar mommturLlecLot of a body, the above equationcan also be rvdtten as ollows.
Eq.2.2 F=^+=c
In other words, Newton's second aw may also be interpreted ar statingthat the time rate of a body's change of momentum will be propor-tional to the resultant forcq actinE on it and in the sarne direction.
Fig. 2.1 General ree bodydiagram a). Resultant orcesand noments (b). SecondFo(c)
b)
The most usefirl ool for undersranding rd implementing the loadsT9: r:Tr.:. * ,ydary onditiorc rhat govem a body.s behavior, sue ttre 1od) dtagram.'Ihe general ree bod', diagTam bove (a) repre-senre dle body n space emoved rom ie opera-ring ys,.*. aif .J, r-nauy applied loadt and reaction forces are represented with vectorc on
the body, f-the body s n equitiMtnn, ll theseio... u ro., rnuriula ii
[o zero. orh n magnitude nd direcdor.,]l-:l: T?r,,,8.1.d *nse, extemally pplied oading n a rhreedimen_slonar gtcl ody annor nlyaher 6 Eanslation, uiits rotation s well.Referring o resulbnt orces and moments b) ;d the ,. orra t.*equi €Ie-nt c) in Fig, 2.1, the .coresponding puti f q; ti;i.-;iotion or a dgid body ollowrEqs. .1
la=x
where EF and:M are he orce and momenr ecrcr ums, esDectivelv.of all externally pptied oading, ncluding . .,i ;4, ;; ii ;'u*; ;;#rdr nonzntum ector of the body. Borh M and i musr be ca.lculaidaoour ne same oinr on the body.Fig. 2.1 shovs.free-body otion where his point cofiesponds o q the
Xi .I:f ryly f t]r. body..Forconsuainel ^ ,i . ii.;;;;;;i;
u. Ine nxed pohr about which he body orates. n Eqs. 2.3, he dmederi rive of H is a complex quanliw ro deal with * ,i.;u;;iu.-b.u.rsutcce r to sav hat H ii a runction of both il;- ;gr;; .ffi;';;angular cceleration f the body. ts neroa component s oot the massof the body but its rllalf hzoment f inai.a tensor l),l{hich is a 3 x 3matrix omprised.of ass onmts y nmia g;5, and nass ,oduas f i)-.4 \riJ oenveo lrlti respecr o rhe bodv oordinate xes.-These uanti,
l,:i::::::.I,*: *s ora isid ;dy s.atdb,;J;;';il;';
urr cnosen xes. he general quations or these uan ities olJow:Eqs. .4 t t,= \j,-kr)dar r Iiid,iLXl.. ,; rl -O I -. u y combination of rhe rhiee coordinate axes cho_
,MG
Fig. 2-2. Unianal sprlng andanper systen (a). Planar \ ]body notian (b).
. (b)
Constaining the body ro uniaxial motion and allowing for an extemal
spring nd damper n the s)6rem, s shown n Fig.2,2(a), q.2,1expands o the following:
Eq, 2,5 Fx= di+ cr+ k,
with denoting he spring tifness, the damper oefiicient, rld 4, x,J and x *re body's esultant pplied orce, position, elocity, nd accel-€ration long he x axis.Consuaining he body o planar morion see Fig. 2,2(b)], Eqs. .3 sim-pliry o the ollowing xpression,Eq. 2.6 >F = mocluo = oaIn the above quation, G s the vectorial cceleration f the center ofgravity c.9,) of the body, .[15 s t]re sum of all moments bout he -same oint. 16 s the mass momett ofinertia of the body about an axirnormal o lhe plane of modon rhrough he c.g., and q is rhe bodFsangular ccelenrion,Barring dlrramic anal,Ees, EA r{ill always eal with bodies n equilib-dum, By definition, body n such a state must have ero accelerationiso har he esuk ofall exrcmally pplied orces musr be zero. This )?eof analpis s called slota Although this condition sounds ery imitinb,consider hat many imes a well-underenecdvely educed o a quasi-sratic o,ltll1-df *tt rystem may be
appllng fr..;fr;.;;;iJ;#'?;:,::#:T?:'rTfi:' *
xtemal orce.
S : T ::,, Oues ovemeit f a bocly n space ith irde or no
etecEon, bending, or, more genetallv.
ar on rr ch eate.s ;;;;::ffi *llT..Jl,1lil1',j,[il,Hl,?l;
els s a case f rigid body modon as well, piv.,iig fr .fy U.u, in*a:,i. i.considered igid body motion. BoundarT conditions n an FEA modelmust remove all possibiliry of risid body *.6.^ ;;;; ,p.;;;dl.namic soludon s requested whic-h an resolve onstatic equilibrium.Kr€rcr^-oody onon is represeDred n a modal *ari uy -r- *.i?l-uencv ot zero.
Areo Momenls l Inertio
i,ffi :rim:::if J 1'::*,TiiH::i::::ilT'tl
nil:'i'i:ilHT::i ; :i :'rfl -T* uto ' - *i'' itr' , ili
,r,*ai ,r, .5r .io,i; -i;J'ilriiilr) T#fof, f o*'*nof o'
For example, Fig. 2.3(a) shows a submerged eftical wall subject o adistributed pressure (1) that is proportional to the depth ()) below thehorizontal surface line. The total moment experienced bv the wallabout he surface ine is kbzdA, where is dre co;sran( ofproporrional-itt. In the same manner, as ryill be discussed ater in dris chapter, anelastic beam under pure bending [FiS.2.3(b)] will develop n iti crosssection a linear distribution of normal force intensity (or suess, o) thatis proportional to the vertical distance ()), fiom a neutral axis. Hence,the total moment on this cross secrion will once again be nb2dA. A firlalexample coocems an elastic bar under torsion fsee Fig. 2.3(c)]. Thisto$ional moment will cause a distdbution of tangential shear stless t,that s proportional o the radial distance r), from the shaft center, nthis caie, the total mom ent is hlfdA.Fig. 2.3. Subnerged al (a)Beam h PUe bondlng b)Bar n Pure arslan c)
Definitions
Flg. 2,4. Rectengutat nopolat momants f nertla G),Pafttlsl exis heoran (b)
ffi'@
(c)
C Lr
(b)
T] fi r<r rwo examDles n Fig. 2 3 utilize what is knor'n as he rartangulard in Fig 24(a) These are areamoments ofineftia ofa secion' unstrate-o-ent of irrertia about a rectangular or Cart'ria'? coordinate systefi ';;;;;;i;t,h ,ection of inteiest' ar'd are given bv the followingequation.Eq. 2.7 t, - lY2dA
t,= l,-a
On the other hand, the last examPle makes use of a 'otat momerlt ori ;;;;;;; ;;;' normat to thi plane of the section Taken aboutthe oriqrn ofthe same ectangular coordinate system'
F'q.2.s = JldA
It is useful o note tha ,beQanse 2 + 12 = 72 'Eq.2.9 t.= r,+ t,

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