applied engineering mathematics solution book 2

8/3/2019 Applied Engineering Mathematics Solution Book 2

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ME 427:APPLIED ENGINEERING MATI{EMATICS

BOOK2

Part C: Solution of Boundary Value Problern byVariational Methods of Calculus

Part D: Solution ofBoundary Value Problem by FiniteElement Method


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TitletNo.Table of Contents

PartC: Sohrtion of Bonndaty Veluc hoblcu by Yrrhtbnel Mc{tods of Calculus, Applietion ofEuls-Lagrarlge EquationProblem# 33,34,35, 36q, Determination of Functional for a given hoblem4 Problem # 37,r Minimum Potential Energy Mettrcd for Solving BVPs Problem# 38,39, &,41,42,43,UL Rayleigh-Ritz Method fo'r Solving BVP- Problem#45,46,47,48,49{ Problem# 50,51c Galerkin's Method forSolving BVPo Problem# 52,53,54, 55,56

Part D: Solution of Boundery Yalue Problem by Finite Element Method, Direct Mfhod for krivirry Elment Properfy Mahix^ Problem# 57,58,59, ffi,61,62,63s Element Stiffiress Mafiix using Linwshape Functionse Problem# 64


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Part CSolution of Borrndary Value

Problem byVariational Methods ofCalculus


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Problem # g2 (Dave_:2o/ lt / t t)Find bhe qltabion of bhe cwrue 6oining ?he- bt,ttoPainbs an a- plabe. o bhav vhediekanD bevu@en vhe.nbs is ninimum.For bhe given line,dg:\&ilU,+d":1@"0r.No,rr, Ils: !fr@^*

@,9)^'Jro!0 : j6.:Jr : f , @6,a,) dx

E rAq LagronTe, Eqruafr a n :_oF d 7eF\ _t^q-&\W) :ur o-.+(L\ :odrc\,\Mz)+ L:@K! t*ul'+ U'": K'(l"W")+ b":#: k Leu, k:Ct+ -dg- - T


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Problan #gg (oqp: zl/tl/tl-

Find bho oxhq,miztng funebion b@ uhich pa,sest,bhrs- poin7s (o,o) and (t, t), Ateo a/alrrobe- vhaifwcaional Ig:3olu2ioni',,Thn ?lven runedonal, tq:.f' [t*(H"] oo:Ju',@A,{F:.i'-rho eqyafiwt VnaV gollaxrL b@.,is,OF - d 1of \-ffi-#'/:o216-#e#):o#-a:o76e-saLubion of ahe equavlorL is 8n* bt5b: At+be*av (o,o)ab (,, ,)So(vin& Ael ::t' 6-tDIA:L'/ I-e' t dzd-L

o:Ad+o;1 +6:0 Lr: Ad,*aA, I:' Ao+*:lJq*le6i-'. b: o'-l eu-l


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ItProbtem # g4 (oato: 2l/tl/t)T->-xl_LJut

conerUer a, eimple su-ppornd bam of sport L.The-'bqmis sub6*Ded vo aolf- p@ o,nd or1 end Dl-rrusbTl\attetiat and geonebriql properbies ere- E q.nd L.Theerpr2ssion for bhe VoAal po*erl?lqt entrg?J! of bhe beamis r= * Io' [.r, ffi)"- T (W^ - 2 p @)U ] .b.Find vhe difffare abial e4uabion asgociabed beamdaf lec don using Dhe ene.ru eec-preae'ron.Solubion

7ho

Un B,ven lunc7ionalttd=I:i fz, (Wf - r (M)"ap@a].,.: !' r @ b,t, b')oneqltn)torv bhav Soverns vhe A@L-#W**;(ffi):o+- p@)-#(rW)*#(zu#):o+ ut#+T#-P@-o:> -{gL -* -r !L: P(n\' d*'a ' 4- dcc' EL


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o-%Aeo_&* le@-@ F

a:(#4-(ffi")*#o-(,$+-(%)+&q* ortq su*aAo3 aeq4

neJCuo14vflb "*i

n,affl,n,fr,qr" f[:ftwrf.W)*.ffi)]of f:(m)z'puotqcw(z* ua,np aVL'o

erq +o Q,napunoq otlq t?Ao Vaqlr?eatd 3{ mftwp["F*) *.(#r)Joff:@r'p-Eaur Bu,molot @q4 v>nrr?r4aa 04 &qnolqDupA atlapuadapv Jat uo{


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-

Problem #97 (Dabez zj/illtt)Derivo aho funcbional For vhe foltowing bv p.#nkb-.7,' (sax-*U)

A@-o and #ln=,:tr9ol ubion:Aasume a beav func7ion v @) szrch Vhav 19 (o) : Oi,NoN, "[#*,.U]:vx:+ [,' u #or +1,', ng. x- * I " t* u lo:,- I"t# # 6n*loto,4ud n-lo'tlq-ao


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ktP2k2

Problem #41 (Dabez l/t2/lt)

'Find bhe Mf leenion of nocrtl poinrOonO@ uy bl'te,,applicabion of minimun poVen7ial ene,rg6 madtod

Kt= 40 N/nnRz- 50 N/nmKs- 8O N/nmSolauion:-,-TTe Voful poVen2ial **ffi, tu:U+R z,^o* CL:+ ub|+ IRr6i*!ks6Z: U Kp'+!^"fr*rt\upulLt: - P, u; Pzur?'(4: + Lt", uf * k ti + ks$ + L4G - 2 lvrua4- p11.,r2- p,t)t6n(,.): #,6uft#62)2: o

-P2:O

Pt - 5ONPz:6O N

l.lt : alW:-l ?

( &2") q- k1&tt kzut-Fke?ar I ,^z)u*L"*: koz'L'L- ke14- n:O+{ (ar* kz+,


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Problom #42 (oavo: l/t2/tl)

C f" boblish bhe- s& of da. brole e4/yab\orue fpr' brermining bha delle.cdOn of bho epimga-rralgenenb,*in1 ohe rnin\nun pofunfral 8nor86 me,ftiod.@Varin6 bhe- ratAz bA *aablishing bhe-&na-

eQruabionB bA qpfUina bt@ prinaple of nenharrics.SolubionOThe potenv,a( qq% is giuen bg

4*U+U+ 4, : * kr (ur-rD'* L u;$ + t,.:"lua 41 ; u,tu- nu;E&

Nour, # : ffi tu.t+ *"ot** gfuo% - o' oz - krl'lrk lLz-_ Ft: O + Ft = ktut- R1tz 1::' :k*)z-k11)1+ t'


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t,e,*9*r,

["inA foreo boJanceaD node f :Z F:Oj F1- tq(u,-ul:O-".- + F(: ktllt-ktW

aD node 1"2, & g

ktO

4s6-olaV ncd,e 2:iiF f== O; k u"+ k1(u;u)*4t(u'-24:Or- =) $nu"- kttA* 14y12+t


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'Oerive vho fwrcdonal for ahe obove e\npl6eupporfuAbenm problem I n be,rms of ibs ^qgimllrrL def lendon .Asstrm>, E ard I. are VltO rnazerial alci Beo moVrial'paralnabers o+ ,he barnlsoluDionThe pof/enV"ral energyz-t:Tt+E)v,: I # + ! u6-lq,tCd' - p61o: +l Lut({#,)"- rqbf^q-*|"b"- raNoat, tho eQluation fhaV BoyernA b@ea- o lar \-,o' /,ob ox r og) - oN?aFUd -o

nr. n, n,[i*= i I"Lut (#, "- 2qi$dL+ * y1n)P6W+ -ou** (ut#) +!-kab- P- o-) =t#*!kb: P+Q


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Problam #44 (Davez l/t2/tl)

Find bhe.d.eplaceme,nV of Ahe- 6uncdiol po\nu r,rsingminimurn p ove,r'rA al qqBb m e-Vh od .Qive,n, Enc:7O GPo- , EoD-, eO O G pa-AAt: gOO fiimn, Auo: l2OO mmz

Lt= 2oO f,fr, L2- goOnmP= lqooo N

Solubion

Alo;t, Eae-Aet EclaAs."ou Lat-A9xl0'x7OxO,z lt-t-

IttSxt&LL:lqOOO?)-: 8. g 6xtO6m

r{-P:oCIW -.30xlO x ZOOxlO"

Lea1.2=+

=+=+

)

Alutninum

7" bo2al pot'envial ener%,q'(u): Z.r.+D+ TJ (rr): uat+ Eu; tr u+4,@):++tt+wot- P*

I-AI-

o.3 t&- l0rOOO


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'Proabrn # 45 (Oaro:3/tz/t05olva bho DEg az6 R&dleiBh- tuivz mebhod.

%+t)__#+td:-x- (o


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-

Problem # q6 (oaw: g/tz/11)Solva bho DEg % Ra:6teidh- tuiDz. rnezhod.t*7b+2c


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-

TesU funcDtort., A@):q Q,@: C{ (*-0@E.' . J (d : * 1," @" ex--o)'+ t z rq (n-t) @-4ldac=+r(d: +b{ef+eelaI a. -,l5ft +cr* O:O + C1--*?

eor uvion, b- *E-@-0@4Aga\n, ?esv fuaclton, b@:Q ArUS* cr6t@+b@: q@)@4+c. tg)-@t)- ^ . 4C - + [,"L*t ", @.- t) * Q @- z)* Q" @- t)' * c, (zn-z) (*t-t'+ zxp, t @--t) @-4*c.@- t)- @41] o^

44+I(6): rt-+9'-ec'-r oc1 -ffino*6eq:o*ot9.-3:o-cr + #q-!:o

ro5 g*?qJNoru, EIbE:

AIocraI-oct

4got-,:--t ,79c--- - 231z /79

sotuvion, b:#@-I/\-ffi Wf W)


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Problern # 47 (Dauo: 5/tz/tl@A

For bhe circ4l qr e)qmpeA botr shoaln%ne\der, L: I uni4 f-l tt;rti4 A: I uqivFind oho follouing ^i^B @ one,-turnt o.ndsolaaion o? Ra,bte$ -tuitz me,bhod.

(t)displarerrtenb(tt) surain(tr) euress

@ ar,oo -Vern

I

:TFial

The re4lwrec\ funcDr ort,al ,r (u) : + [:E,4 (*+)' dn- pu(4slufion'- uQ):C( 6,@-C1 x{tz)

+r(1.) = fzn,z (+ -t*4]"_+ PCr+ r(4 : + EAq" + 2 P c1: + S+2trCtNoN, %: +ct+2P=+Fq +


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-' . S olaViort-, u@) : A - % /--@-4sararrv' e:g: -+V.-)gtrress, c- Ee:- T@)

=gs-: -+W-0Again, ArioJ funcDion, U(t4: g ,r\q--4+Cu2r-,-? 4-'. r(t^):+J:EA{Lq+ c/ @-_z)* tc'+zc.xg{'d*+pcr *p?=+r(u) - - t .,' - + cte- E c' + P G*qNor^\ #-#tq *XLoq:o

crJ&tPh#ac':ffo. " . sot uiion' u@) :- gl-nVt)

Ena'rn, ,:#:-+@)Drese, 6: E Tkt=+6:-TWt)


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-

Problem #49 (Oare-: 5/t2/ t)a

%nbilever benm eub6*bed bo a- vdL q/@Solvo *or - O gingle Parame@r

@-fioo PorcnrteberCompare:, deflent\on and bend)ng tttomeslDe uiVhe*.acb sotualon,s of M a_v buo ]oc,o.bion of VttabqmSolu?ionthe reqlirad *uncdr onoJ,

r(d : rl : fn &)"- zq/@bl a".Tilal Solr, b Lu): Qr"- r (uS:+ j: [rrefra-zq@c"*"ldrc=+rq): * Lt, Erq u- ?qW) q /3 lAtoru, tL:o + -# -: BEre4L-

+ Q:# W-l^.F,)^- . ?r : 4/tru'SoluP) on,' A tzEI3 q/@t3: o


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-

Iftrial Sola: A@:eL'+"o#= + I : Ler ee, + Gcz*f- 4 (e,n'+c,*")f M' . r(b) =*, ,r(b)- + Ler (Aef L+ I2qc" t+ 12e"" Le)- i ,_-)n/1n,6, I al4)lzr[/\o \4t_ -4 I /J

Nor4 6r:0+ #: *oQ*#ac2:o#: Er(Be4 L+t2erL") - Z qf =o& -- EL(ze1X+24c2L")- *qt3:O

\-r ' 24 EIr-, q/ L_- --- lzEL-'.Eotufron, A--#, #

Dof leeUion d) fre? end ' L, Ono PararlLerte-r \So(lL: b: i"rtl Tao Pcxranno,J olt : b- #, Exach $olo: b:#


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Pr oblor4 #. 50 (oqve,: 8/ I z/ I l)Frnd bha eolubion of bhe DEg

#+P+N:@Q(o): fr(t):o

asi ng VDO poinb uttocc&ion meiltod uiVh *- t/z orrdx=t/4 eott ocnHton p oin hs.SolrrDionResidurrt funcD)on, |ac): #+@+x-TVial Sola, U (r.) : Ct fr @ + 9+r@:Cr*W)-bCz't@)Na-il, Ool lom Dion qluauions,qL:'/t):o + fze,* cr- + c1- E ."*tl- o

, (x=-t /a): Q + [zq- + a-z- & c,- & cz* ql: oSol v[n4 Ct:-aC.o:-40zt+

:i -'. s oLwvion b:-&x-@-t)-#%'@)


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Problen # 5L (Oate.. gltz/tt)-tilSolve b6 ealetuirlo Mevhod.#*lt:l (o


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''Problem # 54 (Dat;e: 8/ Iz/10lfind aho vwo pararnqD@r Gal qkin'e eolubion Do bhe

oltowing 6V P.d^u '--UL:Tox:t*(o): (Q) - O

iEoluvionRzesrcl ual fwcttiort , tLx): #*l!- l -'ATFiol Sola u@): cr ft@+Qf'($

+ u@): q rc@-z)rCzxQx-


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.iProblem #55 (Oate,'- lO/tz/t)find bho soluhon for displ a.c,e,{nenD orrd sbressfields for Ahe folloruing a%tallt6- toaded bar problemb6 CaJerkln's mezhod

conerder a par2iedar casef=,1 unib: A:l unib, Lt : Lz:l unl D' F:Zrttrio

%mpare ?ho einfle pararneW Galerkin's So(rrDiorruibh hhree obvqined bg dte S/U ryproaeh.5ol ubionQovern'tn$, qpofrron,AIh:F

+# (=n*f):ofueddual funcbion , eq):Tri4 EolL : tL@: CtO@

+ u4): Ax'(Pq#Ge#)


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e@)- EA' 2c1G,aler Rr n' s E4lta-Vi o n{""#(=oS*) o@dac*o

+ [@ @) (ra#)]r'- [eq G,q*b] :- I"(ro#)#* o+ [ o @ G A #ilo: to @ (rn #)J,7r: ["" Go #Y^du-o+ - [' uA#{ a4-4} {{xaal o** [ au), t- Q 0 til:o

+ -[' EA q eb4 @*-4dx+ zA (t) : o+ EA", I' Qx-)"rJ^: 2. r (r -44 rac,[+ B-4 2"4-1 --/+ ct:-s/A

-'. Ois placenenb fieJd, rt@):- + n@4Srress fiet$ o(g): effi:?eU-4Non, for Ga(qkirls mebhod,u(o): O, u(t): O'75, uq:Oo (o): t-5 oQ): q o(z) =8-1"5

h ^ lAA{or S/ Mt4o): O, u():t' 1L(4:Oer (d: l, o(): tl, a(Q:-l


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\r:,:r Part D

Solution of Botrndaay ValueProblems

by Sinite Element Method


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-

Problem # 5A (oqvo: I 5 / tz/ I l)Kr6olb/irtkz:ao0o/inks:1oLb/inlzq- l5ol,b/tnkO- lzolblTx14-l@Utnfi=toolbtr^:9OLbFind Ahe dbplacemena ^?rg FEM.

9olabionGtobql V\ffness Mavrir Qruav\orl

Lnf ru {1u-, : { FJ ,*,k1

*k1Ooo zl:{;l

""J l"

U1lA2l^sl-ltl%

lf u,"ta+t'gt r..t) I -'**4e)L -k1

-ktOOk1+k2+k6t4 -b-tg - lzt-kr* ks kz+t


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-H{r} :{ffvo2go- t20t,tsing Garres E.llnina7iln,

f sqol-,uoI,uo

I @n"){-} :{IE\(oI o.rs

- I o.29I o.uoLo( tLtI t^2lusl:"


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-

Problem # 59 (Oata, tO/tz/ t tl

A= t2.g cm' A- 8.OGcrn"E: G.9xfo'o N/^" E: e.Sxloto ^/^o

f, a 4'46 kNFz-: Z.ZOI


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+ [:::: :;]{:f-{2,::::l1

-' .Q@cVion f orce, 58.86Nt8,G.7KN

{n}'tj:::;^}^{t}:lfx:;+n

Solving,

-4l.l4x@'i--


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-

Problem # Ao (oauoz la/tz/tt)" - Sini lar mct4e-na(s

Ebqt AL E@l4'/ ,;d , 2.tFtlz// lo"

Solue bA reU

Eev=Ooxlf VeiE E.4L:lox8*t

Ft - SOOxtooO(bFz:20Ox(000 Lb{

SolubionQtobal 9viffnes M aPitcc,At _ E,AtLr L1

trqrua2lort-o0EtAt ErAt -* E"At-T Lt ' Lz-

E,At - EzA'L1'LzEtfu__;

O

E"A^--TE"A^ E AyLz'LgEoAsLg

ofuAiLswzoooL2E'A" eAo-Lz'LZ

E-At oEefu

o-fiG

UzL4

EstuLsLs

t6A,\a


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l-0)

c9% --- PFf Ff "FFF F# L'JJ v,lltlE----. ]-P o9 Ro=o d{l \}' NJ f$''B E \L-+\,

'c$o.oilF$rSt

Ico(oe)b ..b-.O 6l--l+FFF#il+ I9e otu$

l-rK$it -{.,| -\9Peil$Na#))lt)


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Problem #62 (DaVe,: t5/t2/t)A @mpoeite wall condoDs of blwe rrctr,eigl lauershnvocDlofl henb Vranofq Dd^es plae,e. av oho\nnuwall afr, Which mdium fuinpdabuqe, T*,:8OO'C andAe 25 N/m"C. DaDqm\ne- pempe-raDttre.dis?rrb yjtion inblte wall btd FEM wi?h rritnL no of ebsnerrD.

ouves NaUp2g"q,

Solubionthe Gtobal Sffifnes lAaVize a4,@iort, titVh gl.1ysla Melettstlt

LKJo*o{TJo., : {gJo.,_bL1

*KlL1oo

Kr KzLr* L"I


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hv,-kn

olOTe.r j

oo- t


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Proulem #ag (Dave: l5/tz/il)An insrlq.S)ng wall oF o lwmogeneo&s la6ers.

Media" near inner wall

RFSONfs'Ckz= AOl.t/ y1'CRs:70N/p1"CLf SAmnLz:95nmLe:25mm

T:tOo'c , 4: lO N/m."Ctu@Aia- nenr ourer utoJ(f:.35? tLo:15 In/rtT

: Find vhe Oernpuatatre. aP \nnqt ou@rbhe inuqfoe- b6 FEM. rual( as udlas ab '

tb

SolubionGlobal SviFfnes f\aVftr Equahon,

L*fa^o{4o*r: {4*,KtL1

_KtL1o

TtT2

Ta

u

n

o

L1n P


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I 0oololZ,J

:ll.s IIL6 l+4oKz

**u -+Kt K+ *k-T T- t-"o-too+

E}

LzKz l'9E* t"_t9Ls

Elinirna7ion[ 47.o 1: I 4l.ss II qa.o6 IIo G.t+ J

I =((

T2Te

T

-t- IOOOt857.t4-857.14o

o-857. t43G5+.14-2800

oo

-28002800

l0to-@oo

OO

ttang GalrcS

@"u)


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-

Problem # 64 (OaVe: 22/ t2/ nSolve Vhe foilowing O\/ p bt6 FEM

#+Rx-:o (o-


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e-q/tlAblor1.

-**l(d* hKg-o-ea8 k#t(r)+ o.tABKNoru, Global Sprffness MArix-

f s -3 o "l ful-. GL: :11/fi1:/lzLo

I U^1 I o.orzt k * t/s f] *l-l a.oeezt

applied engineering mathematics solution book 2

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