applied engineering mathematics solution book 1
TRANSCRIPT
BAIYGLAI}ESH TJNTVERSITY OF ENGINEERING & TECHNOLOGY
ME 427: APPLIED ENGINEERING MATHEMATICS
#sstgmff,p#f?* *'f f
Date of Submission
Name: Aashique AIqm BenuanStd. No.:06 10 AI'2Sec:'A'
Dept.: Mechanical F.ngineering
ME 427:
APPLIED ENGINEERING MATHEMATICS
BOOK 1
Part A: Solution of Initial Value Problem
Part B: Solution of Boundary Value Problem by FiniteDifference Method
$: ritte
Table of Contents
Part A: Solution of Initial Value Problem
Simultaneous Solution of a Set of First Order Ordinary DifferentialEquationsProblem# 1,2,3
Solution of Higher Order Ordinary Differential EquaticnsProblem # 4, 5, 6,7,8,9, l0
Solution of Non-Linear ProblemProblem # llSolution of Higher Order Partial Dffirential EqtntionsProblem# 12,13
Solution of Non-Linear Algebraic EquationsProblem # 14,15,16
Part B: Solution of Boundary Value Problem by Finite DifferenceMethod
Boundary Value Problem with Dirichlet Typ Boundary ConditionsProblem # 17, 18, 19\Boundary Value Problem withNumann & Mixed Type BoundaryConditionsProblem # 20,21,22
Finite Difference Solution of Eigen Value ProblemProblem # 23
Transformation Relations for Polar & Cylindrical Co-ordinateProblem # 74,25
Finite Diflerence Solution for Polar & Cylindrical Co-ordinateProblem # 26, 27, 28, 29, 30, 3 I
Page Na.
1-8
9 -?8
29 -32
33-39
40-47
48-56
57-81
82*86
87 -9{)
9t -ttz
s/No.1
23
List of Figures
Figure Title
Lorenz Strange Attractor {using ODE45)Lorenz Strange AtFactor (using RK 4r' order system)Position of Satellite
4 Beam Deflection (Cantilever)5 Temperatrne Profile of a Rod for insulation case
6 Temperature Profile of a Rod exposed to Room Temperatrue7 Deflection of a Variable Cross Section Short Cantilever Beam8 Deflection of Beam (Comparison of Two Deflection Theo.y)I Deflection of a Simply Supportd Beam10 Deflection of a Popped Cantilever B€am11 Axial Stress (max) along the axis of the Beam12 Shear Stress (avg) along the axis of the Beam13 Flexural Stress along the Beam Depth14 Horimntal Shear Stress along the Beam Depth15 Deflection of a Beam with linearly distributed loading16 Deflection of a Beam supported by an elastic element17 Radial Temprature Distribution of a Hollow Cylinder18 Radial Temperature Distribution of a Hollow Sphere19 Radial Temperature Distribution of a Solid Cylinder20 Radial Temperature Uistrilutiln of a Solid Sphere
\
List of .m Files
ffr. .m Fite Titte
1 RK4_sys
2 NewtonSys
3 GaussPivot
Ftmction
Runge Kutta 4e Order Method
Newton R-ryhson Method forSolving a Set of Equations
Gauss Elimination with rowPivoting
Prablem Page
373871581792192410 28
11 32
18 55
2A 61
?fi 63zfi 65
20 6720 692L 7522 81
27 97?8 101
30 108
31 112
Problem Page
387t4818921I11
16
17,18,n,21,n,27,28,30,31
?+
31
47
51
Part A
Solution of Initial Value Problems
1:P!9blom # L (Oaue: t9/ og/tt)
{: A -l rc(o)- ol
W=-"J u(o):r J
Find bho ls> sAep solubion uiv11 sbop size O.l
SoluVion
Given, x':b Inibta| C;ondiDion, x-(O):O
U':-* b(o): t
-TFlu$, #:Adu-u --KdP
, For SimUlfuneOLLS eolubiort Of & eeD of lsD Order ADE,
dcc6p-: f'(v'n'b)-b
dad,
ectfrt: ri+ *(t,* zk2+2 ke+lxt)
%i*t: Vio* t^t+Lnzt \rns+m)
nhoro, kt: g'-F, (st'i,*i'bi)lfrt: 0"f ,(Oirrcc,bi,)
kz: 1tf, (Yio k/t, ,Li+ Rr /2, bi + n t /2)
n2: h+ ,(aA + k/2, reif Ktlz; Aromt/z)
2:kO: /"f t (Airklt,Li+ t4/2, Ui*^,/r)frg : 0t f ,(bi +/r/2, Ki* K1z , Uio rU/z)
l<q: htf t(ai*0', Ki+kO, Ua *ryy)
[/]4: htr(a;-r2t, x/,'f kb, bi*nQ
No4, Por, gvAp () bir,: bi+ h: OrO"l : O.t
kt: O.l x P, (O, O,@)- O./xl -o.lrr:rr : o. lx N, (o, o,l@) : o
kz:O. lxt,(o O5,O.O5, l): O. lxl: o'I
tnz: o.lx f, (o,o5, o.o5, t) : o.tx co.o') - -o- co5
ks: o.l x f, (o 05, o.o5, 0.975) - o.lxo.$v1:o.@v5
ffie: o.tx rr(a,oe,o,o7,O.g7O) * O.I^F o 05) ':-Q. OO5
4: o.l x +r (o.(,O. ogT 5,, o.g5) * o.lxo.95 - Q.Og5
tf14:o.lx f2-(o.l ,o.og+5-,o.g5) - Q.l^(- o.og75):-o.o0915
ffiu's, Li*t- Q, t (o.r *z^o.l + 2xo'og 5 + o'o95)
: Q.Og Be
aiot: l+ t(o _ 2xo.oo7_ 2xo'oo7- O.oog75)
: O. gg5
x(o.i) - o.os83Ans:
A@ r): o'sgo
3:Pnoblom #2 (Daue: I9/o9/tt)
9olve, bho Pollowing Oob of ?4ua"?ions
o(-rL+d+rt rc(o):1] o< b<r 1sr.x:)
V'= bb*x J b(o) : -t J
lleo ebep sir<e of O.5
Solubion
6iven, x'-rtdtb Inrbial Condi bicn, 40):lU':aL)n A@--l
Thu,s, # -Lyrrt : f, (u,x, 0
t# - Db+*- rz(b''L'$
For simullzane.Ctlrs golu/ton- Ol aseb o+ lsv orde-r ODE,
%i+r-Ki * * Q<1+2kz+ Zksr k4)
Ua*, :
b i* * Qn,+ z tnzt- 2Tne*Yna)
nhorQ., kr: Ltf t @ r'i' xi'bi){Ttr: 0"fz(vi, rci,bi)
kz: 0,'f t (annh/2,tui tt*/z'bi"ru+/t)
hz: 0"lz (ui+ htzrtui+Arlz,bt* mP/4)
kg: ht",(ai * h/2, Ki * k'/t, bi a nz lz)
(ilg : 0'l r-(Vt+h'/ z t Xi + k'/2, b i+ n'/z)
k4: h+1 (Aa +ht Li+ k-b, Ai, *^u)
714: A"b (ai * AL, Ni a kg t bt *frs)
4:Alout, for sUep I , biut: br-+ft-: OtO.5 - O.5
kt: O.O*{t(o, tr- t) : o-bxftx(-l) rOJ : - o.5tTlt : O.5 x f"(O,, t,-t)- O.5x fO^er)+ l] : O.3
k z : O. 5 x F t (o. z O, O .7 b,- o.7 O) : O. O x fo: sx e0 75) +o 25J
: - O.lOG25
trt z : o' 5 x sz (o . 25, o.7 5, - o 7 5) : Q 5 [ o. z 5x (- o -7 Q+ o'7 s]
o'28t25ks: O.5xf, (o. 25,o.gzt7r>.,- O. aOggz;)
: o->x I o.9ztg45x(- o. 85gga5)+ o' zQ
o-2v I I tB
trte: o. 5 ^ f. (o.25, o. gzt 31 5,- o.A5gg1 5)
: o.6x [o.rsx(o. f,59e.75)+ o' 9zr8+5f
- O.35 eO l6
4- o"5*f,( o.5) o.72ABB2,- o,646.48.4)
- o. 5" I o.72a}Bzx(- o. G4c.484) + o.5f: e. Ol45go
tnl: O.5*fr(O.g, O.vzEbZz )-O' G46'484)
- o-5xfo 5^?o-a.4l4e.4)+ o;2e,8;e27
: Q.zO 282
Ki*: x(o.e) : l+ *( o-5 - zx ol 5625 -zx o' 27 tt tB
- o.v+f,60s +o' q&sq
5:NOtr,, for of,Ap 2 , bd+t: bi+ 0u: O. 5 +O.5 : I
kt: O.$*f, ( O"5,O.7+6A,-o.6+t3): - O.Ol O6G'6
fh t : o .5x f? (o. 6, 0.7+66,- o.6+ t 3): O. 2 20 4+5
kz: o.5><fr (o.ts, o.771 267,- 0.5610G'25)- o. 11?8,636
ffi z : O.5 x f. (o.7 5, O.7 7 t 2c.7, -o.66to625):O. 175235
ks: o.5-r, (o.t5, o.Bo7gtB,. o.57g68g)- Q. Pe2oB
) ng: o.5 xfz(o.+5,o.Bo 5.9^18., - o.O\e6gg) = o.4 tgl 56k4: o.O * fr U, o.go r gog,- o. 2 o Ot44) - o. egOAO]rr14: o.5^Fz (t, o.gOt Aog- o.2oot{a): o.O2 Oez
-'.%t+t- K(t) : o.geev 4A
Ait,: x|U) o'oB27o2
Ans'.
b K ao I -l
o,5 0.7+6,609 -0:6+t275
I 0.933746 -0.98270)
6:Pr obtom # 3 (Dabez I9/a9/tt)
ThreO pero-mobers of a- ph6siu-l s6sDem is governeA
A6 bho f otloning setu or e4P4s
K':- grX*Old
A': )rc-b-Kz<': -ax+x6
lrse iniyial condi bion for all bho varia'ble's a-s 5 ttttit'obbain bho sgttrDions f or O-{ r< lO e@ rksin8 a- s'ePsize ol leec. Shoru Pke solubions on & sin$e graph.
5ol*aon
Given, cr: lO): za
b: B/g
6iven, {:-crct-obA': )r-A-x=-/-- bx+xbdrThus,dD
-3f : ) r-A-x.=. - tz (v,tU,=)
#For sOlving A- seD of hhree lg,b order ODE uJa need
bhrea iniviat vatue. Flere, *(O): g(O): z(O):5Tho MAT LAO f uncb)on's for So{ving O- seb o} ql7z
rcsing bho 764rah order ntun6>- Kubao- meahod and
A1O irrpub and oublonb a-re given bolou .
ulhexe, o:tO)):2ob_ 8/3
7:Problem # 3 {Date: 19/09/11}(Using b'IATLAB Built in Ftmction ODE45)
MATfuIB Functianfunction xprime : Lorenz(t,x);sig:1 6 'beta:B /3;rho:28;xprime- [-sig*x { 1) +sig*x (2) ; rho*x ( 1 i -x (2 ) -x {1 i *x {3},' -beta*x ( 3) +x ( 1 ) *x (2) ] ;
InWt Command>>x0:t5551>> tspan : [0,10]
>> plot (t. x)>> legend{ tx', 'y', 'z')>> xlabel {'Ti-me, t {sec} '}>> y1abe1- ( 'x, y, z' )
il'IATLAB Outtrut
'i,4lF ;t
i 11,lli,r
/ .. ,",
I
I
I.,a I
I
The Lorenz Strange AttractorLo!"*r slrtr"te .n,|lracltt
45
&l
I
:1ll
n 1t'
:IJ
t5
10
j.. i-.. iii, i: j ... ! \, li;1:ai! .i',.j rl , i \j- r.; i.,, i iiJ
t' -/..-.. ti(r/ ')l ,'/ tI ,
tl :it i,l tll i
t'n' 't,,r'/ !
5L
B:Px'*h},es$ # 3 i**ie: 13.r$9ll il
MATIAB Functionfunction [x, u] : RK4_sys{f, tspan, u0, n)a:tspail(1i; b:tspan{2i,- h: {b-ai/n; x: ia:h:b}'; ut1, :): u0;for i : 1:n
k1 : h*fevalif. x{i}, u{i,:}} r;k2 : b*fewal- (f, x(i) +h/2, u(i,:)+k1/2J' ;k3 : h*fewaf (f, x(1)+h./2, uti, z|+k212)';k4 = h*feva1 {f , x {i) +h, u (i, : } +k3) ',-u(i+1,:) : u (i,:) + k!/6 + k2/3 + k3/3 + k4l6;
end
Input Commsnd>> f : inline{' i {-u{1) *10+u{2}'10},' {28*u{1) -ui2}-u{3) *u(1} },'
{-{8/3)*u{1)+u{1).u(2}) j','r','u');>> [t, u] = RK4_sys(f, t0,101, f5, 5, 51, 10]t>> plot(t,u)>> legend { 'x' , 'y' , ' z' }
>> xlabel ('Time, t (sec) ')>> ylabel('x, y, z')
MATI-ILB {}utout
E
! tl;6 F\t*t;;gs n* !:t.d i te:l
r--T--iI
iilj
Ii
L
I
i,l.'t-,'iil't,f-iriii
t,
liill.iI
II
Ir*-,-! !--_-l--- -L- ,.L,-01lli5r:
lII
I
-f1i
Li--.-78!
qd&i I i:r!e:€i ;{e d i!i':€ 1 e.cl
9:Probtom #4 (D at'e'- tg/og/ tt)
Conver7 frho Pollouing probletrt ir'rpo an 4Jlivale1DiniDial valu,e problom .
A"'* a{ - b{- zA: fu tA@): {/(o):o; {(o)= t
Solubion
6iv8n, a", *ba"- bu'- 2g-@b b(o):{'(o):ou'(o):t
rhuca, #*r#,-r$ -rb:tu>
+ #--b#,*r#+zb+bTo atve & *tird order or=iTIS';?)U ao re-dt-L@ byle
probtem in bo M.a- sbs ou, of bhree fir'sb ordet
P4yPvDi on,
LoV b: =,
-dg-- d,f- : 22 ', zt@): A@):odP dr/ ^
{A: d,1? - zo ) Zz(o): U'(o):tdu' dtz v
!3- * bze+Dzz+2ztD ) zo(o):{(o):OaF- d" L
An ' Eqtuivalanb tniaral VoJUe Problen'
dz, r(o) : Od':22 )
#-.Ze i Z"(o):l
# Dzs-rbzz+221r b; 'zs(O):O
10:Problem # 5 (oauo: A/Ogl tt)
ConvorA fhe- f olloning 2-D o+ vibra-binB sbsvq^ invo an
eqwvalenb inibid value Problem.
SolsYisn
Governtng eq,ua7i orts
For marss, ml
Acrelerabing lorce- m,#*J Z f : - Fot- Fez--K1Kf kt@t-x-)
' d'xt+ mrffi:- ktxt- kzxttkzXz
+V&*i", ;e-- ?)
frr ma,sg, ffi,AcelerQft'tng ForCQ-:
^"*dD2
+{ZF-- Fez:-k2(r'-xS
) ^rffi Rzxo+ k"x-4
i d'x,_ -l+ Vff-+^zKz-*':" i
Lnibol C,ondibion( *r:O
aD D:o 1-,*r:o
11_:To solve a- stdsrem of bno se"c.ond ordor ODE
nuneri*JlA, HQ nee.d ao reduco vha Vrobtem bo ae}€rern of Four tirsU order eq/uo.VionS
Leb Kt: zt , h_:zo-@- -
dzt - z - tz,/n\:1,/o):Odb da : 22 , zt(o): KtLo):
# : # : - #, [{u,*a4 Z 1- k'=eJ', =,(o):4(o)=(
dr" dzg*-: d, : 2.4 \ ze(O) : %'(A)-O
#: *# : - #. [- o'zo - t<'=,f ; z.t@) -x"(o) 4
Ans: EqwvatenD inibial vatuQ problarn
dz,-Zo ) zt(O):O
d, 1
dz' -- ( [k,*k)zt-u-z=u7 ; 'z'(o):o
ola fl1
rlzoY--.z- 7. ; z.s(O):Oda <'1
9a: - * Su,4- k,zt7 ; za(o) : od, tYTz
12:)oblom # 6 (oqve: 24/ oO / t t)
Converb bho fottoruing 2-D of vibra\inB s?6sbern intoQn e4luivAtenD in.ibial vatu.e probtem.
A
f"sintfult m
L2
Cr Lt1tJ3;
la
-IT k1
Tr11
"lI-r 7u"
fTl 2
31Ir ?uu
Solubion
Gove,rn'tn\ ltuabiansFor mclss., lhrt ^,ffirq# *C.r(#-ffi+k61+t+@rp)
: Fo ei nbDFor rna*. . ff12"" ,"ffi+cz(#- #S +cg# r R,Qr;xf u<sx'=c
To solve a- sbssem of bu)o second order oDEnurnor)uttg, b@ l1erztJ Uo reduee Pho Problem bo a-
3?ds1ern of f our f irs? order eq/ua"biom s
Letz, X-1: z. 1 , x2:Z-3
#:#- 22 1 z-r:xt(o)
Ye: t7 *L.,Zztc,G"-za)+Q^ru')<y :*u- f"si nutu]dx"-: -d-? : 24 ) -s- xr(o)d3 df/
# : # hL"'?q-=-) + cg z4 * k2z t +(k*xeYfl
13:Problem #7 (Oatez z4/o9/ tt)
If same simplifging o-ssr,r-rnPuions qro rnode., tthe eTt-nbions
of mobion of & sabDeibo o-round a- @bral bodu qre
A:-4rs x(o)= o'4 , x(o) - o
A:-Wrs b(o) -o, b@):z
whers., r= [*rgtlt/zEvaludve x- qrtd b For fz: o'5 (o'e) to'o
Plov bho solubions on a-graph.
Solubion
6i ven, Qover ni ng Eqiuafr)ons,
t Z r9):o-4,x'(o):odb'- 7^3
d?di" - -rt
nhare, y: fc&+A1'r'
To solve a seb ol bwo second order OOE- nLLmericallt4,
tlJO nwd Ao redUce Vhe probtem bo A-sbsaem olFour firs A order eQluadon.
Le-tt, X: z( , b: z.gdx -dzi -? -) -2,(o):x-(o):a- ola dr - 1z
d'^ - A= =, ..=. ; zr(a): K(o):AM - -dt- fz1'+z-$)ut"
qg : d? * z4, zb(o) - td(o) - oo,t dv4:e: d,t. : - = =3 ==, ) zq(o): b(9:2AF
: do - l=,'* zrtJe/z
L4:Probl,en # 7 fl)ate 24f|9/ll)
MATI-AB Fmctionfunclion [x, u] : RK4_sys if, tspan, u0, n)a: tspan(1); b: tspan{2); h: (b-a)/n; x: (a:h:b)'; u(1, :} : u0;for i : l:n
k1: h*feval(f, x(i), uti,:))';k2 : h*feva1 {f, x(i)+h12. u(i,:)+k7/21 ,;k3: h*fevaL(,f , xli)+h/2, u{ir:)+k2 lZ},;k4 : h*feval (f, x{i)+h, u{i,:)+k3) ';u{i+1,:} : u {i,:) + k1/6 + k2/3 + k3l3 + k4/6;
end
Iw* Command>> f : inline(' i (u (2)) ; (-u{1,)/(sqrt(u(l)^2+u(3t ^2) )^3) ;
{ut4) ); {-u{3) /{sqrt(u(1}^2+u{3)^2i )^3i l','t','u');>> [t, u] : RK4*sys{f, [0, 10], [0.4, 0, 0, 2f, 50];>> plot (t,u(. rL),t,u{ :,3} )
>> legend(txtr'y')>> xlabel ('Time, t (sec)')>> ylabel { I Position (unj_t) ' )
>> title { 'Position of the Satal-.1-ite' )
MATLAB Otrtput
t x Y
0.5000 -0.06s9 0. 61591 - 0000 -o -6292 0.7984I .5000 -1.0487 0 7 r272.0000 -1.3391 0. 534 92.5000 -1.5173 0 - 31113.0000 -1.5929 0.06583.5000 -1-5697 -0. 18374.0000 -7 - 4468 -o - 42124.5000 -4.2L16 -0.621Ls. 0000 -a -8692 -0 - 769s5,5000 -o -3824 -0.?8066. 0000 a.22AO -0. 45806 - 5000 0. 2550 a - 44427 . 0000 -0.34?3 0.77197 .5000 -0.8436 0.7718I .0000 -1. 1995 0. 63348.5000 -J. .4341 0 .42949. 0000 -L.5625 0 . 19169.5000 -1.5891 -0. 0583
10. 0000 -1.5178 -0.3045
15:
0.5
Position of the SatalliteProbbm # 7
5Time, t {sec}
gc5
c -nqo€tt,o{L
-1
-1.5
-2
I
10
16:Problem # I @aW: 24/og/ I t)
Find ahe. dof ter-bion of bhe given @ndtover bq'm dua 6
ibs se-/f weighD ontt4- The Soverni,^B e4pa'vion of ivs
de:ftec,vion is,
tr#:PB[,- (#)'fu" [*(*- *) * #lGivon, L: 2m, P: to kg/mu , IE = /4o o t<glne/s'
Ueo a. s@p size oF a.4 n.thow bho defreevion of ahe otausvic glffvo on Q' graph '
Solr-ltzion
Givgyt, Gover dmA qlua?i or1,
# : # [," (#)z1e/z [n(*- i) " * I7ni Dial Condi ai ort
'
&D, x-:o t orilu
To oolue & s,aond order aDE, Ne ne-bo\ tu raduco ohe
qlUabian Uo O- s6sf/em al A)ro firsD otder qluavlons'
Le-b, A: =,9b: + :22 ) zt(o): U(o):OdN d/-e : *- : # [t *=t]e/z [x@-i) ** ]'d/: de- LE L
zr(o): A,p): o
17:
Beam DeflectionProblem # 8
-0_5
-0.4
{.3
-02
-0"1E
coo(l,
(uG 0.'t
0-2
0.3
4.4
0.5o.2 0.60"4
1B:Problem # 8 fl)ate: 24lt)9/11)
MATLAB Ftmctionfunction [x, u] : RK4*sys{f, tspan, u0, n)a: tspan(1); b: tspan{2}; h: (b-ai/n; x: (a:h:b)'; u(1, :} : uOifor i : 1:n
k1 - h*feva]{f, x(i). u(i,:))',.k2 : h*feva1 {f, x(ii+h/2, u(i,:}+k1/2}';k3 : h*feval-if. xliJ+b/2,,:{1, :)+k2/2)' ;k4 : h*feval(f, x{i}+h, u(i,:)+k3}';u(i+1,;) : u (i,:) + ki/6 + k2/3 + k3/3 + k4/6;
end
Input Command>> f : inline('[(u{21); { {(10*9.81) /24OO}*{ (1+u12}^2}^{3/2) }*{t*it-.5)+4 j)l', 't','u');
>> plot (t, u (:, 1) )
>> xlabel ('Length, (m) ')>> ylabel ('Deflection, (m) ')>> title { 'Beam Def lect.ion' )
>> axis ('ij ')>> hold on>> stem{t, u(:,1).'k')
19:Problom #9 (oate.: z4/o9/t0
gteal Rod(O: zc.m)
T = zO"C, room
O Find tzho bemperab*re distzri bubian in ?horod nhen
Ahe onbire rod sirr*ace is Vhe,rmalh6 insalat/ed'
@ Find bho 1omperqbure dis2ri bubion in bho rod whon
uhe rod is a--xfosed bo room air.
T=ZOO'1T:40'C
Solwuion
Governing Aq/U"AA\on oF bernperaVurC. disDri bu7iort For Q"
rod u,th\ch is bher m atlg insula Aed,
boundar b Condivlon,Ab %:O, T:4O"Cab X-: lO,T: \OO"C
t2++=[cJx'
To SotUe a S Uond order ODE n?Lrre-ricatlr6, ao need Vo
rOdtrcO Ahe, probem Ao a s$sbam of Aato firsD o-rder
eqtr aDion ,
Lor/, T: z(dT dztdr dtcd'r dz'
22 ; <r(o) :T(o):40
odza^ dx
20:This is a boundarg valuo problern, vo eoll/e aha problem
wibh bho inivial ualuo boc.hniqpn Ne, need ,o Bzress bhe
eecond eondiVion and @mPa.re vhe boundar\ vaLue
aiht ahe glven boundarU c,ozrdi b)on
bt4 lin*r i rlturPolaDo
Loh, 2r(0): lO A gain, LeD, z,(o)=zo zr(o): tGLoh, zr(o): lo
K To 4oI 502 60o 70
4 BO
6 9oG l@o7 lto3 12Cl
I teoto 14c
x To 40
, 602 803 (oo
4 (20
5 14c,
6 t607 too
a 2(o-C^
g 220
lo 24c,
T(to): 24O"c
K To 40
I 562 72
e a84 l04
b 12c.
6 te67 152
a l6ag t8'4lo 200
T ((O) : 2OO'e'' ' T(ro) : I4O'c T(lo)
Thus, ="(O): # I*:o: t6
21:Problem # 9 fa) fi)ate: 24l$9/11)
MATfuIB Functionfr:nctlcn lx, ul : RK4_sys{f, tspan, u0, n}a: tspan(1); b: tspan(2i; h: {b-a}/n; x: (a:h:b)'; u(1, :) : u0;for i : 1:n
k1 - h*feval{f, xti , u(i,:))';k2 : h*feval(f, x{i}+h/2' u{i,:l+kL/2,\';k3 : h*feva1 (f , x{i)+b/2, u{i, : )+k2/2}';k4 : h*fevai (f, x{i) +h, u(i, : } +k3) ';u{i+1,:) (i,:) + k!/6 + k2/3 + k3/3 + k4/6;
end
Inoat Command>> f : inline (' [ (u (2) ] ;01 ' , 't', 'u') ;
>> plot{t. u{:,1))>> xlabel ('Length. (n)')>> y1abe1 {'Ternperature ( \circC) '}>> title{'Temperature Profile of the Rod at j-nsuiation case')>> xlimi [0, 10] )
>> ylin(140, 2001)
MATLAB OuWt
?ilil
tfifi
16*
Temp*rature Frofle rrf the Fod a! in:ulatinn c*se
01:l{56783iilLength, {mi
; \ t ,ln< rrLJ
r$
d'{l}
5 r*nF--
Gr-l
sn
r{U
22:SoluttOn
GOvernir'rA 26yltgt7ion o* bho bempe.rabure dis vribubiort
Por & fod Whieh is ex+osed vo room Sempera.DurvQt
Nhete, M:
To satve a selond orderbo redUc-e- AhO problem boQ4ua-aiofl ,
P0L
KA
ODE ntLrnoriallg,& €a€Pem of Aloo
"o#- Po,(T-r,) -o
+ #-#ff-G):o=+ g-: M(r-zo)
ox-
Ooundan\ Cond''vion
eU )c:OrT:4O" C
ab L: lO,T:ZodC
ue neeAf irs V ordet
Lev, 'T: Zt
*dr _ d=f _dx drd'T _ dz,
<z ) zt(O)-T (O): 40
M (=,-2o)-:
dr-z dx
Evaluabjnq M
For e2a(nle.ss 9beel, K: lV N/n-K
Lea, /l-s consrd or , fo - 25 N/n'-KP- r'D: 4'(o'o2)A: ry/4 (o.o2)tzs*@.o?) _ 2g4. t17 z, (o.ot)'M_
i
-t\lhr€; tf; O-
problembo gL(ass@mparO
@ndi?ion
1fru,s, zr(o):
(Here, L is )n
L To 40I 50.32 6t.63 +4
4 8B.t
5 lO4.z
G 122.7
1 144.9
a 169.5
9 199.2
IO zO4.l
2C To 40
I 48,2
2 57.2
3 67.2
4 78.7
5 gl.g(o 1o7.3
7 125.2
a 146,2
I t +o.9
to 200. I
" .T(to)- 20OJ"e
23:boudary value Probten, Do solvo bhe
nibh ahe \ni Aiat value Urchnique ue ne.ed
bho soc-ond e'ondi vio''t aD inivial efuDe and
bhe bourtdary Value. ulvh Vho giren bounda'U
b6l\xsar inresPolaVion
Ag n, Leb,zt(O)= IOOO z"(O):789
C - -T(to): 234 .I"C
drt _| -7Ogd^ ' fu=O
Iem)
Lert' ='(0):600r- To 40I 46.3
z 53.4
c 6r.54 vo.B
5 Bl.6
6 94.ga lo9.l3 t26.5
I t47. I
to t+t.4
-"- T- (to1 : IVI'4'c
24:Problem # 9 {b} {Date: 24109/11}
MATUIB Functionfunction lx, ul : RK4*sys (f. tspan, uC, n)a : tspan{l); b : tspan(2}; h : ib-a)ln; (a:h:b)'; u(1. :) : u0;for i : 1:n
k1 - h*feva]'(f, x(i), u{i,:))';k2 : h*feva] {f , xtL')+h/2f uii-,: l+k1/2}' ;k3 : h*ferral (f, x{i)+n/2t u{i, : )+k2/2)' ;k4 : h*feval-(f, x(i)+h, u(i,:)+k3)';u(i+1,:) : u {i,:) + kll6 + k2/3 + k3/3 + k4/6;
eld
Input Cammsnd>> f : inline(' [(u(2));0]','t','u');
>> plot(t*100, ui:,1) )
>> xla-bel ('LenEth, {cm) '}>> y1abel {'Temperature ( \circC) ';>> titie('Temperature Profile of the Rod exposed to Room Temper:ature')>> xlim( [0, 10] )
>> ylim( [40, 200] )
MATIAB Output
?m
r fttl
fi1?3,{56789'iilL*nglh icm)
.f- r*:
E
i 1:oqr
F; tnnF
*it
6fi
Trgrnpsvalure Frc'trl* c'{the R*d *eptsed tn Fc'r,r'n Tenipera!ure
'I
t:
4*
25:Proble,m #to (oate,..25/og/ t t)
Fr'nd bho froo-ond displaceme-nD of ?hu varia-blecross- Eq.Vion shorb CqnVilane,r rnaAe Of e@
L: 42!/, e-: l Q", b- 4" t D: l//fi= 2OOOO tb-in
comPa're t/o*r nu-mericd rosul> uibh ec<-aettO-n al6Vi CO-l One. For nz,Lrner?crat S olubi on S rrb - di ridet'he beo-m a.bteo-sb into 3 seAme,nvs.
Solut ion
Ctoverrun$ e4y-nViort,
=t#: M@
Inibia(
a-V 2L:
Colrdl ?iort,
o,{b-oLb"-o
For a catAitever be,am wibh var'obla c/oss-sec.D)on,momonD o+ inerAia is e_ {z*nc:tsr on of 2{,
nidvh ab any babi orL,,x-: + @- b)+b
.' . Momenv of rnerbia aD x. r - i, ,{ffta*b) "4u
26:Tu*, ut+b- pr@
+ E [+rl7(a-u)+*']#:M4 3ox6l$-", L+?(ro- 4) * ^fl'# - 2eo o o
f+ o, Sbeel , E - 3ox (oc tat /in" (zote ei]
* {+ ez-l-r4}"#:oor=+ &- o'ol
d*t- r t r.\)'A- l+ @z-4*olu
T:o sdve hhe- saconc] order ODt=
Vo redLLce ahe e4luabion in bo &order e4pa7i on
numei,u\6, Ne needsbsZern of vvo firsz
Lev, b: =td?-- d;zt -'=dx- - dx- - 'z1-2 ; 7t (o) : Alo) : O
o.ol
{+@2-4-+!
5o/ vin$ using Mabt"ab , wi ah r1o of se4rnonD - 3
) z" (o): A)'@):o
TF.u€, bho Free encl
displa'ee rnenD : O-O22"
27:AnatTbical Solubian
Qi ve.n, Gover n'( nA EqUaDi on,
d'a _ o.o I
dx-" t+c- z-Q*43u
+ffi- o-o, t4 -+(x-o*-u
I-n begra7lng,
#'- o- ot (*) L4- ? V- o43-'*c4 "
Agat n, I-n be$rattinV,
b: o.o t (A @ 14 - + (*-+"lj- t
-t- c1r+ q
l-t-sing condibiort, ln@ &@
Ct: - O'1xtO a
a4: - O"O245
b:o_ ot^${4- Z@o*-t+ +- ' s.endx--o' c.245
Nou), At), X-: L:42, b: O- O 2205
-Thrt;, Vhe f re.oend disp (acemonfs : O'O22O5"'
2B:
Deflection of the Variable Cross Section Short Cantilever Beam
-0.025
-0.02
{.015
-0.01
-0.005.g
co-a(){t)
oo 0.005
0.01
0.015
0.02
0.025
Problem # 10
29:Problem # tl (ootp: z6/og/tt)
Qiven, EI:25xlO4 N-m2
Find bho deflenvion of bhe above qabtlOvqfollowing buo consid eraDions;
(o) g^al dof tecaion theorg(to) t-arge daf tec?ion bheor|
Use a, Sbe+ srze of L/g A,rd shou bhe Dwo solerbio n in a-
gra-ph as ndt a,s in a. Table.
Solubion
Governlng Uyuottion for €rra(( de+tecAion dl@rA
#:#M@Intbi6ll condi viort
ab x:o,{ b- O
L b'-o
ODE n?Lrner't@ll?4, NQ
Uo a- €bs Dern of a/0To solvQ. vho eecond orderbo reduce ahe qpaVion inordq q/uaAtcrt
LeD' U: =,
#:*+-zz ; z,(o)*b(o) :o
#:#:# \ zz(o) :td'@):o
bo-seA en bhe.
needli rs?
=20kN
30:Govern'ny e4uabion lor large. dar tecvion *te,or6,
' # : # M v() D * (*t)"fs /' :,' Y "f;:'q" "
6 ,a:o
T6 sotvea- second order OOZ numericaltg, uo nrr;S
bo redue *to probLem Uo a- gbsvem oF auo firsa
order eqpra?ion,
Leb A: =rd& _ 1t, : -n i <t(o):U(o) -odx dx z-
# : # : #[, * =;fu/', z.l4 :Ul@) =o
Expr*ion For Moment:
vlI
r(xru)
.'. M V4:- zOooo/- (N-4
31:Problcm # 1,1 il)ate: 26/{}9/11}
IIIATLAB Functionfunct.ion [x, u] : RK4_sys{f, tspan, u0, n)a: tspan(1); b: tspan(2); h: (b-a)/n; x: (a:h:b)'; utl. :) : u0;for -i : 1:n
k1 : h*feva}{f, x(i), u{-i,:)),;k2 : h*feva1 {f. x (i} +h12, u {i, : } +k712]i , ;k3 : ir*ferral{f, xlz)+n/2, u{i_,:}+k2/2)ttk4 : h*feval(f, x(i)+h, u(i,:)+k3)';u(i+l,:) : u {i,:) + k1/6 + k2/3 + k3/3 + k4l6;
end
Irp* Command>> f : inline{' [{u12)} ; |2AE3*L/2584} ]','t','u,} ;
>> f : inline (' [ {u 12) } ; { {2083*t/2584} * {1+u 12) ^2} ^ l3/2} ) ]','t','u' ) ;
>> p1-ot{t, u{:,1}, t, vi:,1))>> xlabel i'tength, {m} ')>> ylabel { 'Dellect j-on, {m) ' }
>> tit]e {'Deflection of Bearn {comparision of rwo Defl-ection Theory) ' )
>> axis ('ij')>> grid on)> legend ( 'LarEe' , 'Snal_l! )
MATMB Au@ut
DeflectionSmall deflection Theorv Large Deflection Theorv
0 U 00.4000 0.0009 0.00090. 8000 0.0068 0.00681 - 2000 0 - 0230 0 - 02311 - 6000 0-0546 0.05472.0000 0. 1067 0 - 1073
0.02
32:
Defletkx of Beam (Comparision of Two Deflection Theory)Probbm # 11
0.04
E
i€ c.oe()oGoo
0.08
0,1
4.12o.2 0.60.4 0.8 1 1.2
Length, (m)1.81.61,4
33:Problsm # 12 (OaYe". 26/oOt t)
ion bo Phe *ollow)^g I"V P
-oc <.2C< oC
C<b<w
Find ahe__, aaah|biql soluo
#rs#-o l#+2#:c J
Ini>i&l Condi ,ion$,
tL,@,c): f Q)ttz(x,o): gUA
$olwbion
Tha g*n pOE,s an be wr'taben in fute form
J€#l ro art*l :{:}l#1. L' ojl#
+ {u4" LAltr'J:{o} O
LeV, tLS aSSUrne bh' solZtAion as
l"l == [ P] {oJ
{:,7,31Lpl 1-s a- ffiobrtr- UhOeO ulUtmrtS oto Aho e,t1erl
vocrtors of maVrix- A
34:{ur3:Lpl{vfi{"4:LpJ{"4
N oal,
Frorn Ofpl 1"3+ [A]Lpl{"}: to]
=+ {v"3+ [pT' [e]Lpll"l: to]+ {"4o [of {"1 :{o3
tDl is a- diago na[ mayrir uhOee e,lene]4 Ps &re
aha etgenvatue ol tAl
Eigi^ value of tAl: Nt:4, fu;:-4
Eige-n voctors of tAl .
For fut:4, {ilfur, fuz:-4, f?]
-rhurc, tol :rz :^l , [pJ:[1 ']N'u
{il* r: o^l{&"1 :{ :}
7hus, bhe equa7ions
6lrr +_ .L El2tAD &rot92 _/J OA,&P ' &X-
35:
For bhis b6ee of un@&pled paraial diffen e-nhal?4pabions, solubion is knoun a,s , Travelting Nave.So/ u-bion', uhidt is as fol(ords:
V, (re,v): Q@-4D)
vr@,D): Y@f4b)
ttlhe;e, p and Y are buo arbiVrarU tuncblons, uhie.h
will be deaernined from ahe appliu.abn oF I.C-s
Qenora( Solubions of Dhe given OEQs
ut(hb): 2 6(K*4u)- 2V (ry q
tr^@v)= Ae4q+Y@"<o)
beorne-
-Q IF:o
J
{LI:tpt{31 = [? -il{?,823}AJou,
Inirtal UndiaionC
rnibiaL cond ivion@
tb@a)=
ut @ p) : * tr t*- 4b) -2gl*- 4')l- tLtZ@*4b)-f (s-- oo)J
ILza@*4b)*F(*-t4ril
36:
I z s(91
Y@-Ft: +[zs(e- r @l
* t,t"- 4D)-zg(r- u)J*
u,@a):g@: y') (o)* VV)..
ft il{rg}:{;g:l+t4 -zl
fl@:%:+Lr@-2tn",lJl7
-:l
37:Proble,rrt # I 3 (OaPe'. 8/ to/ t t).
Find bheE"noral s oluvion o€ bha PDEs
#+ff*X*-oY;*4H*€*L-o
9of avion
Thogr^ q/L@Dio1s @n be uri7ben in bho form'
{'#}-[;;]&] :{:}
=) I"fi+LAf t,&l:{.1 C
LeD, bho gluDion fu' aesl'l;rned a-s
{"3: fpl t"lLpl is a. maDr\rc uhoee ulumns aro Uhe -8*VeoDore of 1.76bftx- tAl
{w3: [P] t"'J
Iq: tpl 1"4
3B:
tAl t Pl 1"1: {oJ
ef 'Ia] t11 1"4: to]ol {v.3: I o]
+
L
tPJL"8
=+ loo3*
=+ lvr3n
@
1.
f rofi,
I olttlorJ
) to,,-Lov
Noao, FronO
Alort),
aro bhecnal mavrizc ushoee e-lemeintzs
f LA]
ol LAJ , fu,:3, Lz--lof LA7
{;}
tr]?l 'f
:f', -;lJ
I Dl is a dia"g,
e.igen vafues o
Ei6en vatues
Eigen vechof,s
For, fu: e
For fu:-l
rol:[3
[z:] {#} :{:}
39:
Aloal, Ahe qluaAions ahocome,
ovt -- zol9,aD ' u &*L: o I
'#-#:o J
For bhis %pn o+ uncoupled Varai6rl differenui'al
e4luafr)ong, sOlUVion is knonn as "Travell'1ny blave
Sol uaion>, uhioh is as folloajs '
v,@d: @@-eQv'@,D): ( (x+ b)
Qe-ne-raL .S olubians of bhe give-n OE@s
ur @, b) - 0 (o- ou) - V (f*u)
uz@tD): 2 Q @-e0+ 2V(ffE
uhue, O ard Y ore Dao arbiVrarb funcdoytE, uhi*',can be deverm\ned frotn ?he applioaa\ort o{ I.cs
Nou:'
{I:J : f eJlY;,}: [; -:J{i,ff\
40:Problom #14 (Oavo: 9/ to/ t)Find bho eolrr?ion5 of ahe follotnting non-linqr aeb
ol ea,unnons.
%'-{: 4 Take, x-o:bo: 2.828x3+ t: 16
Sol ubion
Q\ven, EqnVions.
t-6':4 tx-'*bt : 16 J
thus, F, tnU) - 7'' -A" - 4fr-@,e: tu2+6-16
Norl , zhe ilacobian maDrix-,
l--: +l fzx -"b1rilr: lL; /: L,* ,;J
L oL ob)
o-r t ri|o
of i aera
Aloau , Dha soLMion is
{ t, }n*' :
dven, bb
t"rlK - [ dn]
ushete, K- ho 'bion- |
41:lberabion N.o I (K:o\,
, r3'r-- t:::2 i2z2]
ro'Jlf,:"? 33?31
fo.oaa o.oEzlJ-4 1L-o.oes o.oaal f o. oos J
Lr'J: [:;: -i3::]{ ?,} : J t2 '45\L ozJ - Ltz"4o J
f o.otg o.ozsl f -o.ora?L-o. ro, o. I o, J t o.24e J
[o,] - fo' e24 -<'agell: Lu.s24 +"*eaJ
{z} :173J}
f ru? _ I o.rs jLu,J- Lz.<v6J-
+{eJ
rabion Alo. A (N:r)
foo+s o.ozsI Io.oooal- t- o.to2 o.tot J1-o.o oo J
] t::l :1""J
3.1622.4\9
[n"1 J o.t62 1tu,J
: \' iqgJ
=) {T,}: {:12;
ffi} : {;z:..\
f z ozal: 1 2 s2uJ-
f ola l-I z4v6J=+ {;,}Iberauion No z (K:t\
Ibe
( 2c:tu:
Sotu-btons a,ro
42:Problem # 15 (oava: 9/tot tt)
Finql bhosolxhions of vho fotlowin$ nonlineqr se?
eqt-taDions
r'+28:zz lZr-'-x16+et6: ll J
ra.ke,ti:::'u
AherqK:no. of i ber - |
Aloaj , bhe Aacobian mabrix-,
[g eal f ,* 4t6 I[d]=l';,:B"l:1, . L) IL-J
L;; frl ?*-u u-*l
1he Sotu7ions are g\vert Ug.
t^r1s' {*,}K- tn iJ-' t ti J^
o.z9t Io.tool1.75\9,5 Jt0.7641t. t2o J
lT,]: t','] tdTL t-:Z: -
{aJ:t','} -r:i;: :":lJJ t-'+ {T}: I iz],\ ra'f = t:;?;
Soluuion@ons,
x"+2t:2'2*r_^%+o\: /l
Aloru, P, @3) : t+28'*-22fr@,C: Lx:-*16t36-tl
rberabion No. I (t'<:o).
43:
In\ : -f r.a++ 1 l-o.o24 0.2261 J- 3.994J
\v,J: lr.orJ - L o.tor - o oeJ Lu nurJ
+ fT"1 : {'inon'I - [a2J- L::tr':::")tsii: lE ':"]
f- o.org o.2t+ 1 f o.zoglL o-oss -o o uJ to.o' J
Laul:L: ,lJq?: f o-<1\qJ:to..J
-h# -Z:7,',1,1 t;]
{s:J : lsl
Iberabion No. 2 (N: t\
Lberaaion No. e (rc=z)
{&}:t:Hl-=) {tr\:t z3
{a}: {z}+ {frJ
: lA3
( x.:2lu:s
So(nbtons Are
44:Pnoblom # 16 (Dat o'. to/ to/ tt)
Load De\le?ion Be_haviou<r of Spr-g1
Fr:5g8xatGC6O*?
ry: 657Kt * 919 rfFs: 6 g /-s* I% rZ
LoaAPt:- (2C wtit'
Pz- 3gB wiv
Find bho epring derPleLbion s .
I **"
Solr,r,Dion
To give-n proble,v4 is a- nantine.a-r algebtro,'te q/uaoiortbbpo prab@n, ulherg, Ltr ond (bore bha lln*rdisp{ocefnenv of bho bato l-be-a.n
To soluo vho probtem for epring daf { s.cbiort, firsD uenee$ ao forrn bhO Frcndamerrfu,l goVerrtlng q/D4 in VerlrtSo+ {ar d_/rd as -
45:For I-beo.m I
aa F: Pt+Fsr- Fst : O
+ - l2o + 657r-t+9 pre-598x1
- 6o6o%i - O
+ 59 lut+6a6ou?- LntT (u;u)
{-srs(ts-qu} *- t20. . . (i)
For I- bean 2
Pt
Fr-
Fsz i I
I1i-Fs>ii
ItL_r----li ---l" (r,)
NoN, s
F, (uo un): -t20 - 597ut - 6OG Ouf + 65V (u3-tt) *g t9(4-u)fr(u,, ue) - 99 B* 697 (us- u)- gr9 (rra- u)u_69 r.tg - /96 a3
fia ilacobiaru marfir
f --t _f z+s7(u;us)L tSeouf- t255 2V57@,*uga65+ 1L a J -
I_ zro + (ur - us)I asr * 27ot (urw)L oB tfr*7 t4
g > F: Pt- Fsg - Fs": Q
+ gg3 - 657rz-t9 I Qx3, - 6e xe_tg6 4:o
=+ 65+(ua-,r,) + e f g (LB- rr.)'
+69 u.s -t-l g 6 uZ : 3gB
>-fr.,
46:Aloao, ahe eo(ubion is 6ruen, b?4,
{L}^.: {,?}"-nhere, K:no' of
[a"r'{:"[i veravion- |
t/-sin6 trlaUlob,
{u3- { o. o a491Lo.4szt J
tnu8) tur: Ltt: O'OO4. unib
Lz : ?.Lg-LLr: O-42V2 tmiO
tuO:W: 0.<321 UniD
47:Probfien # t6' {kls:I0/l$ll l}
MATLAB Functionfur,c:icir x : Neertonsys{F, J, x0, tol, kmax}
xold = x0; iter = 1;
whrle {iter <= }cnaxiy : -feva1(J, xold) \ feval(F, xold);xnew=xold+y';dif = nom(xnew - xold),'disp( [iter xnew dif] );
if dif <: tolx : xnebr; disp l ':lar.: i : €::.:..-- :,:s --a:r-.'er:{ea:' ) , re:l::-,'
xol-d = xnew;garC
iter=iter+1,'
erli
disp (' ];en:.r n:e:l:,: : dli r,.-, ::::-.-: : :: :-i' )
x = xnew;
InWt Command>> F = inline('[-120-598*x(1)-6060rxi1)^'3+657*(x{2}-x(1))^3+919*(x(2)-x(1))^3;
398-657* (x (2) -x (1) ) -919* {x (2} -x {1) } ^3-69*x (2) -I96* x(2) ^31' ) ;>> J = i-n1ine('L-ZlSl' (x(1)-x(2))^2-18180*x(1)^2-1255, 2'157* (x(1)-x(2\\^2+65'7;
27 51 * {x ( 1 ) -x {2) | ^2+ 651, -27 5'7 * {x ( 1} -x (2) ) ^2-588*x (2) ^2-7 26}' ) ;>> x0 = [0, 0];>> tol = 0.00001,'>> maxit : 50;>> x = ilewtonsys(F, J, x0, tol, maxit)
MATLAB OutputIteration x (1) x(2) dx1.0000 0.3637 0.8773 0 -9497
2.0000 0. 1856 0. 6132 0.3185
3.0000 0.0618 0.48s0 0 _ 1783
4.0000 0.0101 0.4371 0.0705
5.0000 0-0050 0 -4322 0-0071
6.0000 0.0049 0 .4321 0 .0001
7.0000 0.0049 0.4321 0.0000
Newton method has converged
x=
0-0049 0-432r
Part B
Solution of Bound"ry Value
Problems
by Finite Difference Method
48:Pnoblom # 17 (Dat,oz l6/ lO/tt)
A"*A+t-O (a@-otaal='e
Solvo bhoM e'bhod.
Oou*nda;u Value Problern bg Finive Differenec,
Sol ytbi on
Disc.rOV'izAVion of "ha
Domatrt
t1llil
ioMe,sh Lengbh, k:
GsvernjnT Eq/uztViort
_fadx|+ bt l: o
b-o, 4-O :l
o
frrsing cen nal difreYenee
o+ ordar O(te)J
Ua*r - 2At* Ar,* dw* * rt-o
U;*, + (P3- z) Ai + Ai*, - - h'
bi_t - Ai -t bi*t- -l tr ,.
e.x4rrassioYr- for A"
+ V*,-?ktt+Bla +bc+ t:oh'
++:+
49:Sbenctl
l=i ni bO Di ffesert@, Mebh o d
di+re.renaal eq/uabi ofr ,
EVencil for bhe BNern'mA
For in?qnal nodes
L:2,
i:3,
t:4,
A:A,%e+-lA"-Uy*U<: - t
Ae- lI4+Ug:- I
i:I, Ur:oL: 5, w:o
JfittAt aho rQue.4 €eb of e4/uaDiorr e
-b"*Uo --lU"-Us+ $a: -l
As- Sa: - I
50:AJor,,r> rna.brir,
tlsin6 Gaacss Elinitnablory ui7h ro^) pivonnB,
S2: -2%g:-OA4:-2
lhuq 9olubron
1:fi}
Vho globa( coelfictenD
LAI{Y}:{*
f-r tollu,
lr rllt:
zLi V;
O o
I_?
2 -33 -24 o
51":Problem {. 17 {Date: 16/1011 1}
MATIAB .m Filefunction X : GaussPlvot {A, B}
[n, n1 ] : size(A) ;lm1, pl : size {B) ;
i f - -: n1LLL
'errcr ('": :r' :': : 1,'),endif
^ *: aluL! ,
error (
gn,:1
c: lABl ,
{^- : - 1.-_1
I pivot, k I : flrax{absiCii:n, i)}};
if k > 1iamnl - Cli .i'Lur.!t/f vti, -lt
C(i' :) : Cti+k-l, '-);C(itk-1, :) = LemPl;
onri
rn(i+1:n, i) : -C{i+1:n. i) /C(i, i);Cii+1:n, :) : C(i+1:n, :) + m{i+1:n, i)*Cii. rl,
enC
X (n, 1:p) : C (nr n+1:n+p) lC ln, n) ;
fcr!:n-1:-1:1X(i, 1rp) (C(i, n+1:n+p) - C{i, i+1:n)*X(i-+1:n, 1:p) ) /C{L, i);
end
Inpat Qommand MATLAB Aatput>>A: i_1 i 0 C:
1-1 1
0 1-11 -1 1 0 -1A: 1 -1 1 -1
-11C01-1-Ll-1 1n 1 _1
>>r:t-1 y:-1-11 -2
r: -3-1 -2_1
-1
52:?roblon # lB (oate : t6/ tot t t)
A simplg g&pporbd beam column is s ub6ect'ed bo &
t'rartgvet'se and a.,x-ib-l toa.dinl b6 Pa'ramebe,rs P&9'ThodeflecAion of bhe Sbru-ct/Ure is governod bY the
fottowing DEg
tt#-- pU+* q/---+q%^
Given, Axto-l Load, P: /OkN
InDensiby at bransv ers o lo aAi n B ,g: 4o o o N/n
Len7bh, L= 5 m
EI:26xlo4 N -m2
Find bhe de{lecbiort of bl'te beqrn col&rnn b6 FDM"
q,
P+7
L_l
9olubion
t__. " " -=T---I
K1 V; l(:, '/.t Vi K;
L:5nt , ;
Lonyah, 0': +9- : l nMesh
53:Govarnirq Eauaaion
,t#+pa-+q%L**q/t= o
+ Er ry+ pbc- * qLxr * *gnf: o
+ Er (Arr,-2Ai*Ac-)*pW - *qtei +zeri :O
* uc-, - (z - ?rl 6i r bi*,- + k ri + *q,ti/ru :o
+Ai-,-#ar*ai+t: #-ffiSbeneil
7t
For inbernal nodes
i: 2,
i: e,
i: 4,
(: 5,
5t2ar-EtSrUz:6o
a"-*ue+b'a:kae-*ro'as:#aq-*ae*aa:A
For bouno)artA nodes
i:|, ur : o
i:6, aa- Q
thtt^S, Ahe- redUCed se-b
5t- % \rrlds5t
Itbz 26
of e4lua7ions,2:dTI
-- 693
lt03 zG Uq *Ae :6551 1.
Aq- 26 6a - 65
bal Cooff'ciena
-Pt I o26
r-*r5t
Equa2iorL,
br
Ae
Aq
AE
54:
65_a_653
G52
65
o- o. og55
- o. tb69- o.p6g- o. 0855
o
LetUq5t
Gto \'lg1a ri x-
o
o
I
5t26Oo
26
I
IJ^;s'tng Gaass Etiminabion
[?;.] Eii:::L1 wl: 1- o,s*sfI urj L" o*os)
ui7h rot Piv oblng'
' {u}:
55:
E
io(J(1}G(l,o
0.2
0.15
0.1
0.05
-0,05
-0"1
-0.15
4.2
Deflectbn of the BeamProblem #18
33.54Lenght, x (m)
56:Problem #19 (DaLez I7/tA/ II)
eq-+ rx,#-+G O --sin (qn')dx4
I \-^ dx, I
Find bhe, e4uivalenD F D e4Pabion
Sol ubion
Given, qtuaaion,
#*t#+&At:sin (o*)
Aloru #ln: f la,r- 2Li-f Qi-,f + o(tt)
Qn-,
I'(f, tz
/'','t.,h
i-r
+ 6i*r-+ P, Qi*,@r-'
I
Vt vi' Qitt
I >{,_-
/"T
{}2
d40 I-w I *,: # [6n'- 4@i''r- 6 Q i - a do-'" I t4'qh1
fkentral differenc> exPrr2ssicn ol order z?l
Thut, Zho givan equaflon ,
t^ f O*- 4 @c:,t-r 6Qt - 4Lc-t+ Qi-,]- # fu,*,-z@i* fri-')
+ e46i:sin qo4)
) Qi*r+ (d0."--4) @io,+ (6 -2d6 +-fro") 9t + (aa-+)97''
+Qrt: t"Asin @6)
+ PrQt' n fu 01-t+ bc^-t: ha sin(a'4)
ih(-2 ( (+t
57:Problem # 20 (Oa+,ez t7 / to/ t t)
Qiven : L:6 rn, g:4kN/m, EL- 26xto4 N -rnz
Find
I. Def lscLicn cwrve of tho Poppt2,3- Ccrrtb"tlevor boa-rn
bg trDM
2. Disbri bubion Cf qxia-l Sbrezs3 o"long bhe be-a'm
qxis congide,ring bhe beam X-secl,,ion sq&are
3. DiSAriUuDiort of shu-r sbress o"long Lhe boa.m
o-rcis fottowing bhz ccndiDilon "f @4. How a.re bhe disbri bubions of axta,l a;'rd shm"r
sbrosS ctlon,4 bettrn depth a1> mid_ span secbion
SoluDion
Dis craAizabion of aho donnain
'.tAo
,,.C
I
I
::1\
't.a
il:--
)t(,)IU
tI
I
"rrlt i
.>
/^/./fy4, [1 L/,
+*-----
6'O6
ll
/,/l̂A I I
;:6m
-* I rtrMosh LonBvh, h'=
GoverninT futuabion
_{T: q@)
d4 ET
+ ba*r- Abrir+ 6Aa- a1t-,+ Ai-t:
=+ Ai+r- 4Aiu,n 6bt- Ui-,+bi-z:
SVancil
65
58:
#e,@l
For inyornal nodas
i:2,):9,
t: 4,
i: 5,
i: 6,
Ao- 4y; 6bn-44uoW<: O
a; 4a"+ 6!o- 4Aq*bs: O
A,-4bu+6Ur.- 4Au*M:#Ae-4Uo*6Uo-4Aa+Ur: #Vq-a4s* 6b- A+*Ua:*
5e:For boundartd nodes
L: l, / r: O i:7, Ar: O
A'lr:t--o ET b" Ii:*- o
abtrbrt l:e ;W/=oZA" I i=r h' I i=7"
4 Ao:at + Aa*2U+*lFo+ Aa:Ae
-l4b<c, 7he recJtced eeb of e4/ua,Viong,
7ryr-4bu+Uq : O
- 4br+6bu-r-,A<u4E : O
Ur- 4r4e+6bq-44u*Ua : *6Ue- 4r6a-r6Ae-aAa: #
b4- 4Ae* aba = *b
1hus, frhe globat eooff icie.nD ^6pitx-
qtuoFion,
ool
TsI
G5I
66
: -4 ? 3l [X] I-: : :-'ol1,;,1
:]D e( -4 t ) La, J L
v-4
I
oo
60:t,tsirB Gaass Elimi nabian,
lV"laulaqlruLba
{,a}:
o.olo2o.0282o. o4leo-0o68o. ot73
o
= 1 0.o4r3
o.olo2o -0282
o.o368o.o t7 e
Do+te,caion aA different node
Ur: on
w: o.olo2m
Ue: O. O282rn
lJq: O'o 4bm
Ae: o'o 368m
Ib: o'ot7gm
U+: o rn
61:
-0.05
-0,04
4.03
-0.02
= {.01E
co.F()o(uo
tleflection of a Popped Cantilever BeamProblem # 20
3Length, L {m)
0.01
0.0-2
0.03
0.04
0.05
62:Axtal J7ress alonl ahe, ax,ts ol Dho beam
MarryTyyx ftry-wal rgDrass is given bg,
McM _M
- zbi-Ai4foi
o: r r/c s ulhere, M- benStrtgTnorneltv
9 - sernionmodulus
A/or,t; , M: EI #Mi:=tIWJ
EIr: _ s LAi*,
=+ oi : 26 x lO4 x 6 (Af,- ZUt *Ar*,)
L= I , q - 21xto4^ 6 (bo-- 2bt+ A) : Ol ,e2 kPa
L: 2, Or: )6x16 ^6 (b -2 tdzr|,d : l2.lV kPa
i:3, oe: 26x(6* A (U2-2WtA1): - 7.64 kPa
i:4, cq: z6xt&^6 (bg- 2UqnAu): - 2v.46 kPq
t=5t %- 26xt6^a (bq- zba+Aa) 2g.4 uPa
i: 6, % - 2Gnt6 .x. e (bE- 2ba*b) : O .43 kpcr
i=7, c+: 26x(d\nG (Ua - zyt*Ae) : 53.gA kpa.
ffor sq/uare , _. . \ o --)- Isecvton D- G I
63:
AxblStess {max} along the axis of the Beam
3Length, L (m)
Problem # 20
ffi:thaar Sare.ss alonT bhe, arcis ol bho Baam
Shaar fore- is given bty,
v - ='r du4
'- r dx-g
For, nodo 2- 6, W /n: EI
For, nodo i:t, Vf/n: EI t{i-t47Ai*r-r75Ui+z-Ai+sg ou?
f.For nat"d diftereneo Jo r rnula ot oto"il
For , node i=7, Vi/A: EI 3b;o-t4 tdi-6+24 tdi-t te\r:,+;tdi2h3
fAacuword diffar ence formuta of otdl
For Qluaro saCbiort, A:lxl : I m"
i: I , v, /A: 2 6xt o4 IA ,- t47 U"+"o g; Uq] /S -- l 6 -6 kPq
(,':2, Vz/A - 26xtd [A"* 2Ua2Ae*A"] /t : - o 'exPo"
(: g, ve/a:2;xto4 t-Ar+2U;2b,;A1/t: - o"o t<Pa-
i:4, V,,/A- 2axtd FUz+2U;2Ue*Mf /z: - /. I kPa"
L: 5, Ve/A - )GNd FAu*2Ar,- zAu*br] /z : z. G kPa
i: G, valA : 26xtf f-U4+2fu- 2b+r Us] /z : 6 .4 RPa
i: v, vz/tl: 26xtoa se6e- l4bq* AAn- Bua+ e'arf /z: I O.2 kPa
65:
$o-v,PuiooL
g>LctoEa
Shear Stress (avg) along the axis of the BeamProblem # 20
3Length, L (m)
66:Flercural 9brees a)ong ahe bam dopvh
Flex,ural s?rase is given bt4
o-: + s1\lrot M: bending rnomenD
ab nnid span, i-4, M:EILffi+ fvl: 26 xt 6 (Au-zbo*bdF - 4 576 Po-
- 4576xOaa 6:0., o:
av u: o-l , cr:
-ot/r"
-4lvaxo. I _ _ 5.5 kPa( /rz
-4 o76xO.zaa A: O.2, v:
eb b:O.3, c:
ab b:o,A, ci-:
ab b: g'5, o-:
(O.gaWa(/tz
-45vOx0.3---T/t" :
-457 6x0.4 _t /rz
-45v6x0,5 _t/e
- l6.6kPa
- 2l .96Apa-
- 27.4 6L<pa.
67:
FlexuralStress along the Beam DepthProbbm # 20
E
.go-(uo
-0.5
-4.4
4.3
-a-2
-0.1
0.1
0.2
0.3
o.4
0.5
Flexural Stress, o (kPa)
68:Shoar S7rose alon1 bhe beam depDh
5he,ar s?a"ass f sr aocvion ,* Aiven br4'
D - # whore, V- ehmr foreo
9: efea moTnenb ofinerbio'
For r^cAangutar soc-Diorl ,
(-/ 2r \ 4 u/
--\ - t?t 3 _ 1 _!__ _u"\ [aV rrid span _T--:-T(4 -b)2' 12 \a L)'/ \/*-lgaP^l
) a: -7BvS(+-U")
ab, b:O, [: -/gGg'5Pa'ab b:O.t, T--tBgO-ZzPa
aa b: O-2 ) U: - t654 " Ob Pa-
ab A:o'9, T- -12 60'49 Pa
ab A- o.4, z : - VA9 -oz Pc^
eb A: O.5 v- O Pa-
69:
Horizontal Shear Sbess along the Beam Depth
E
-co-(l)o
-0.
{-4
4.3
-a_2
-0.1
0
0.1
4.2
0.3
0_4
0.
Problem # 20
70:P robl en # 2l (D qt,e: 23/ t0/ I t)
A canyi lever sub4ecDod tto q" linoarlg d;spnibwrcdIoaAin6
qir"n, L= 6 rn
EL = zdxlo4 N-m'q,lq):J%@-RNlm
Find bha do{locbion ourvo of bha benm-
q,(x)
DiscreVizaVion ol bhe doma\rt
71:
-d-b: q@d4 Er-
biu;4Ai*,oObf -_Abi-,f bi-r: # q@
a6-x-\- Ui*r 4bi*,+ 4i - br-[*=-ffi
Sbencil
Ai*,
-4 -7
For jnbernal nodes
Vt-,
L= 2,
i: 3,
i: 4,
L: 9,
i-- 6,
Ao- 4 At 6gr- 4A,* A,a:,# +V,- 4bru^r66-4Aq+Us: k #
J- 2,
Q"- 4ryun GU',- 4bu*U: & *"ve- a uq-t 6 us- 4 aa*a+ : * +ffi teo
Vq- 4t4g+6 Aa-4Ar+U&: #
72:
u'li:,:o-^, Ui*,-Act I-hl4 Ao:U"
L:7, =r U" li:7: O
)Et-
) Aa-2A+*be- o
+ as-2a+-u
Er Uo'l*r: o
+ET
=> Ae +2A6 2Uz* U.g : Q
=+ Ug : 4\o-4ba- As
:QL=l
.:Q[:7
73:In bh'e Dtdpe of problern, bho add',Et6p6l numbarof nodes rq7lro an cddi7rcnAl Q4/uADion ac solreall Phe nodal poinP. -Tha lp6pkuard differonc-aformula &^ be apalia6 bo reduce Dhq ftLLmbe.r of'ttmd?)inar\ nadal po'tnb bub Ute-./efoc-ess mak@\s
bhe mglobal coolliaenb mabrirc umplica-b"! 7hab,s
^hb ue applA ahe Zoverning eqfuqD|orL Ao Ahe
boundar^U noc)al po'rnb 3o have bhe addi vi onal e4/uabion'
Apply,ing Govern'tne eqna lor i:Z
ue_ aba+ 6a+_ 4ue*us: o
+ - 4ba*2Ar -.C)
thus, 7he re.duced sob of q/uazions,
=r4r- 4U.* Aq
-4ar+ e,a; aga+%e
e2I
65e
260I
t56I
zaO
V^- aAs* 6br-4b"*A"
Us- aAq+ 6 Ue-4lbnbt :AA-aUs+ OU6 2A+:
- 4Aa+2A7 : O
74:*(hus,
bhe global coellicienb m&r\x,
ooO
I
-22
{u}:
o. otg3o.o354o. o zao
- o. ot60
- o.o 862
- o.lv25
7
-4l
o
a
o
O
oI
-45
-4
Aq
AE
Ua
A+
I
5ZI
653
260(
tooI
260O
-4 I
a-4-46l-4olOO
oI
-46
-4o
r o"ots3
I o 0354
) o.o2Go
J- o ot'oI
l- o-o 862lL- o -tt z5
U'Us
Aq
bs
Va
Az
At
l,Ls)ng Gaotss Elininabion niUh rou) p)vo"ing
75:
*1
-s.8
{.6
-0.4
4.2
hfiection of BeamProblem #21
3Length, L {m}
E
ioo{DFoo 4.2
0,4
0.6
0.8
1
76:Problem #22 (Do,be: 2"/ tOt t t)
4 eanVf @ver subjecbedI oa.dinB p ar o16 and i De
e)asbic supporD.
k=tRNlm
dis?ri bubedrosdng on ar1
q= 4kN
-->K
Do uatif or ^lUfree ond is
Solubion
Dis creb't zabion of Fhe do rna'tn
qUe '.) ^ab Ug----i
/,, /{ jL
'i/.t 'xt
r--+ , /It I ,t. L1
!
Mash Lengvh, k:*: trn
!fu- q/dB- Et
=) A*;Aai*,+ 6Ur
=) Ui-r- 4Un366t
- Ai-,+Air:-4- 4Ui*,+Vi*r: *u
'tj
t4
q/
77:
i: 2,
i:3,i- 4,
t: 5,
i:6,
SbenciI
For in?erna( nodas
U,-AA,+ 6gz- 4Au*A',: Q
Ur 4A"*6Ue-4U<*As: O
6r- 4 6s+6brt - aAs +rg : #Us-4t64+6bo-4batUz: #Ua- 4A<-t 6Uu- 4 U+*b s: #
i:l , ar: ou'li:,:o
-\-
=+
=O(,:lAt*r-bd-t
zh"
ao: U"
78:
i:7, EI A,"li:+: - kAr
+ El Ai*'-2t\i',+2ui-aUi-" t
zhb tli:r: - lou U+
+-Uo+ 2Ua-zAsrg s: - #t-=+ Us-2Au- #*2Ae-Ug:O
EI A"li:T: - kU+nG
+ ET Htu-2hc+Aa1 - _ - 6xtos 6,t "' I t,:7
=+ Aa-2V++UA:- #+ aa-#u+*as: o
For bhis bgpe of problem ne need ano?her eq*bo salre all vhe nodal poi nb,, App(UitA goverrtingeqVL Do node i: V,
As-ary^t 6U+-AAa+Ug : #
-lht'LE, hte reduced €eD ol e4uaVions,
zAr_4 AutArt-4 A"+ @Ae-4Ao*Ve
U,- 4Au+-6aa-aUu+Uo
Ue-4U',+6 As-4M*U=Ua_ aAs +6A; a bz*Ve
Ae- 4A^* 6U*-4ba*bg
As-zAa-# +2As-Us
Ua- ffiU+*ae
thlts, hhe. globa( coefficie.nD maDrirc q/,@don,
79:
/-E(OJ(
oo
I
65I
6e(
(oO
l
EToo
7
-4l
oo
o
oc
oCooo
I
*l
brUe
bq
As
Ao
b+As
bg
-4t6-4
-46t-4olooooOC
ooto
-4 I
6-4-46t-4| -2
OOooocIO
-4t6-4
ln /rgo
25+ t-too I
O
cI
65I
65I
65o
oo
BO:
U'Us
W1
Us
Aa
Uv
Ua
{u3-
U^sing Gau^ss Etiminabion uizh ral PivoVin?t
o.l t54o.eggro-791o
l. 22Bz
t - 6640
z.06Va
z.47sg
2-7e rc
oo.t to4o.egg7o.793l. 2 282l- c64O2.O6V8
81:
E
i.q()6)F(uo
Defiection of BeamPrablem#22
3Length, L (m)
82:Probten #23 (Dabe,'- 24/ to/tt).
A glondor colurnn sub6ecbed bo
/oading ^ \,
rllan ogtal comPre,esve
-x
Find bhe critiu,l buckling loacl
9olu.Dion
Ueqobtzo-b'ton ol bho doma,rn
t\e*h Le,nBvh,
,t#: Mkg-- Pt6
*ffi"(#)a:o
'r) ,.*
X1
^-+
f Momenaeocai on
ab anaMx-:-PUJ
+#*tu^A:o*W*73gi:o
[. A,:rl v-l
o
83:For inbor nal noclaS
Ur+ }itt-4/"*AeA'+ 01A'4Us*bq
{=2,
i:9,i- 4,
:o:o:obe* 0:K_4a{as
U r_,* (fu"1r<z1gn+Un* ; OL: TI,
For boundarLnodes
i:t, Sr :o
i: t'tt-t t An*,: O
Far bhis A6Ve of problern minirnl-Lm numbe.r oJge\rnenD req/utrea, v1 : 2 ,
For , T1: 2
(nit'"-4Ar- o+ Ti/'i: z
+ 7i ('/z)" - 2
=+ fr-+PAffiET L2
EEIl-v.
--'l lzL-=>
84:For n: ?
I c"l
1,iulhere.rAl=7fr; /A
Itxr-z)aafue:oLu"u (t3h'- ue: o
lGtr-M1.leue:o:3{' l-tb;t4,+Ftre: ?)UFo
f ,tr _7i -,/h, I I a"l+ L _,/h, ,/n,_r]1rrl:
=+ lrol- r ufl{ul:tol1\f okJ, f or E igen valuee of t Al
lr/^'-',v., -/A, l:O
| - '/^' "/h'-7) I
=) (z/a" - At)" - ,/Ao - Q
+ 4/a4 * 7J - a/A"fu^ " -',/ho - o
=) 71 - </n"V3 +o/ao : C)
. 2tl41 _/\r h,z,?: #:+ and >'i
-\ o--gEr )Pz--
tt- E
421\-/: --:-;- -
tzh'L21 ET
t2-I-
85:For n: fr4
('-"h"
U,+ (71/'"-2)Ue*@4: c
Ae+ (frt'"- Uq: O
t/a'aa
-L)A"*r[e :Q
l("ta'-71)A,-'+ I -th(U,*(
L
:. c)
s- tla'go: o
Ptre-Ma/ro
vr-/c)6t/zrYu*
+ [fAl- ^3Lr]l LZI:{q
1n'-7?
-t/h.o
-t/h^
2/tr -7?
- t/h-
O
-tfu,,
zB-l. -ry6zl
" rrC
AJ oal, tor E-igan value-g ot [n7
2/h'-fu'" -t/h. O
- t/h. "k-'N^ " -t/h"
-t/0"' 1rc-fr
{r"1 : {;}Ir^J
nhere,[Affr4
2/r-
:oo
B6:
+ (2/tr -71) Lutn - </rc71 + 737- r/n n'/hv3 : o
+ 6/a6*a/rt +2/^"71 -e/n4fi +q/rcLo -79-TaG
+t/hoVj:O
+ Z9 - a/f 71 + ro/b-N-'- q/re - O
,9ol ving for Vf
M--6tt-4/t:+ Pr: - Wa-4xt6 EI
2fut" - 3zLU
=) Pr: EZ EI
(fT +z)/h" :({Z+z)xl6 r.t
-UIz-2)xtbLN
g.o7 26EL-=-L3
54.62v4 p7
L2_
^2_M:
t:
^a_,I /vo
4 Pu: t:
87:
TVaneform bhe t-apla,co eqryVim lnbo polarcoordinabo st6ebom.
Solubion
Laptoco Equabion
K(*-#)_Fs@,C:o-> o"tl- , o"u . .a_ _ ^-t eE*ar" *;i- -u
rr Ou OU On OU A8^l
ol/'l' -ao-: o r ox- * ng o x
=+ w: #(cos6).Si FrP)pu ou or au a6eu or oa s0 ob
+ &u:#(s)ns)*# (+9)
NoN, *: n_y *+ yr _agauz - Ar ax- - t, olL
#u o a o1l sirr 6 cls1 au sin dcosdCO.S-8 - =-- -'
.:- f =-;- --- --': orz cos-6' - tr&6 * od rt
o'tx sin 0 cos) E u s\nt6 . &u& uin'6b- Ofigforr, ot^ sin9as]' O0 r.
BB:
., O'u. _OurrL_Or _*OL{.^. aOaWL on a6 06 ob
o" u ^,. -2_., _._ Ota gi n6cos 0: Or" sln-A * ;;06
_ 4+ sin6casd o ,'u s)n6coE6OO tr. dno6 i, ou eo,f6 , O't t, us'6 a u sirr6cosd
Or a^ o0" f" A0 ro
. a't) o'zu _,_ fl _ orz;- I ou I o2u. , g-J- -erc-r o1z* t- E- * T # r ;' G- f=: 0
Govorn'tn4 equa-Vion lor sbend6 sDabe heabconducVion q/uabion irt- polar coardin afu -bsbQtn,
e"! {_ I eu -o._! g+ * "q,q
_: OOtr' il^ Of f' A8' f-
tror onlA radral dis?ri bubion,
#*+#*###-o
89:Problorn #zs (oqte; 29/lo/tt
Transf ortn bha bhreo dr'rrranei ona.l Lap lqc.oeq/nbion inbo %tfndrica.t coordina7e arrd lor*te condibion of ra-dial di.s?ribubion onlt6.
Laplace Eq' pion \ nL bh reo di ne,n€,\ on
xffi*#-*y)+ s (ob,=): o
+ o"^*_n.In"uu- * A :e' e49 eA" o=z f<
Q^l) 2n #u: *. ers a - araO
SoluYisn
NoN, o"ttEE
putl
sin6 s,os O ou,_' EO
sirt'6 &'t*r &6o
sia 6 cosdr'sin'6r'gra 6
.ou-J--
-
't6
o'tl:o'u- sin-o**boA. Or2
, &'& sin 6c.os6- oraa
oou sinO eosd
l^^
o0
ottl
r2
ei n 0 cos9f Fron previo?r,s problen]
o7L-t-
-
or
sindc.os0 oA sin0co,s6r
. ou eos'}'or r
oO
o"lLo6"
tr'ust6
,r2_
&L , si nO us7 frron pravious Probternf
gz2 Ozz
90:afu g"u. . o'u, o'u- I Du I E"u- d"zr. -dtr* tO"*G-: er"u r or *V @-E
&ove,rninB qlVL for eaead6 etube hent:
oou I Eu I o"A . a-uor' ' tr^ Er f t &O" &zz
[=or Dnlg radial d k*rib&bion,
#**#*#+:o
?V'o'z) - Qr<.
91:
Dove,l6p bho qrnpbDe FiniDe. Differe,nce rnodali"Bol bhe eveady-sDabo radial heaD cond ua,Dian in o-hottow ct4lindorf ephoro (a<r-<b), in uhich conve&bion:Qre albwed Vo anbie.nbe from bo1h ,tnner andouber wall of Zhe c6tind er/sphqo.
SoluDion
Diserottrzabion of yhe domain
=) (,-h)T;i -2Tru(, *h)T*,
+ &tTr-t- 2TZ + RzTi*t: - V
I J. ,:-, t:t la /i, i; ,./-i --er i)t,:, r
Grrer!'rnA E4uapion
d"T ld-I-*8(l) -(a o- J l, c:t4lin6e'I7* r dr ' K -\) ' IL2, sphere
+ Tt*,-{(+Ta-r * +. T*t_-p-t __ ?i _ohz r; 2k K -\/
-ft
M e*hh:
3EK
Lengbh,b-a"
.4__
gbonc"tl
boun&r^ Condi7ions
a.t r:&, K# : A*.G-T*)
=+K ry:ha(n-r"o)=) r. * 2hh ---- tz'_rt 7f t;_T;+r:FT4
+T-t-t+MT-Tt*r: MTq. @
) -at, + Mrr- +T*t : MT& . . .
92:
ThuE, bho se-v ol qpabionsz
i:1, To -r MTi - -T2
i:2, tZtT - 2Tat R-zTe
i:3, b tTz-ZTg+R'"7+
93:
: MTq-
B.zk":-K
--gsftK:
tutTn-- 2Tn+ U--ln*t :-ryI- -Tn -t-MIn *ttTn..z: MT*
L:fl,
(:t1*l t
fu, l'he,d,fbl fuR&;aionel 6ndiri orL,
'MTrcl_Bt I
-#rl-["rlL^|-AftI-6e1KI-Mrrc )
94:
Coneide,r ebenA6-efu,be radial hoqt @ndrrcDion ino- hollow cg,linder (zc.mSr<?c n) Energ]6 is gen erabedaV o- rabe ef g : OxlOG N/mu, uhile bhe lnside surfcrce.rs nd nbatned ab o- qnsDqnD @mpera"vure. Tq:loo'C,and bhe ou-ber ula]l dissip azes heav u6 qnvqlionuibh a heav bronsfer qettic)enD h": oooN/mroc invoAn arnbienb qtt z,ero pemperabure. b|diuidinfl the,ea6linder invo qlnt 5 @Btnonve find ahemdialbOmperaturo &ebribubion b6 FDI,A. Take_ bhoc}linder rnqberial , K- OO N/noC,
Solubion
Given,l,(: 5O n/m". - O.5 N/"rfc
A: 5OO l^J/m'oc: O.O Ohtf"r.7
8 - Sxlon N/*u: i N/cn3
DiwroVizabion of bhe. donain
\ ,!*h LengDh,
\- h:*:(cm
e5:Qovernin1 Eq/uab"ron
d?r-- r dr a :odr- t- F* K
_\ Ti*:?T,i"rit *f Tt*t--Ti-t *€_: Ohz r; zgn- R
+ (r- *)7,_, - zr-+ (l * h)Ti*,- - r o
Saeneil
t-*c.
(-l
For i n?ernaf n od es
i:2,
(:3,
i:4,L: 5,
___L
- 2Tzt tru :-lo
*t,-zTsi*ro :-to
g l{ --to Te- 2T<+#fu :-lo
#Ta- 2To* #-Ta:-{oFcr boundar\ noda>
i:t, 7i -too, c\6,-K# lr:n: lL(rt -Te) l*n
h
6 tr
96:For addi vi ona( condiVi on,
i:G, ftTo-2Ta*#Tz:-/oG to bal coettiae.nD rnavrix- qpazbn,
I
56
o
o
oo c c o o-2* o o o o
\
-r2
-r3
T1
-f-3
T6
-r+
loo
- lo
-lo
-to
-lo
-(0o
gB-2illz
o
C
Itl
I
t:
t(
It
Eti
cccoooooo
=Ia
az-Oot
coob c o
-2tO
#ot5
-2 l1
-o.2 -/t4
I
U"s)ng Galss on uiah roN pi vobin$,
q-r2
Tg-Ta \-z-T.-
t.o
-T+
bi/14,
OOq l,(
)z-
:o..
\v-t5.7
6.?
o.6r"8l,b7.4
t.7
mtrl
'lct4t6:l7(I6=I C.E
e6
'oo4{ {. (
o1"vo,67.55.
t5at.eG
lOo41. (
62"70,67.55.
I
f
t(
(
a?i6
tO/1,6
tr:l
7.4 I
;lI
zl4l7)
{r3:
97:
Radial Temperature Distribution of Hotknr Cylinder
6. rrru
s-FoLe 14OE(uo-E{DF 130
Problem #27
9B:Problem #zg (Dava.. Oo/Ioltt)-./fuapnD Oho probte;rn 27 for hollon ophere.
Solubton
Disrye7\zabian of bhe domain
ft 12_re14rg
a-oMosh Lengbh, A_: t: tcm
!=*z dr *-A :Adr' r dr /< \J
To.,-27.t +4b.r 2 -Tt*t -Tr-t
h' zh"
-) Qri-, -zU.*Tr*r+t(rr*,_lr)
+ (,- ,)T-r - 2).+(r* +)Tz*t: -to
4
16
+fll.<
--,o
:o
99:9he,ne,il
(-r
For inbernal nodas
-zC
i:2, trr-27"**.u.q5(: e, ft T"- zTo+4- (4
: -lO
- -lO
--lO--ro
L:4,
i: 5,
tror boundat"tA node-s,
d':lt Zl:lOOt-i:6, -x ff1,:n: h(Tr
+ - gTc*r--Ti-t 1.2h Il:6
Te-G: Q-276
Te- o.2T6c -Tv :
A 6_5 Te-274r €re
1ntro-2Teu a Ta
-rdli:a: lt(ri -E) l;:n
o
+:=>
Global coe)f ici ena 146*ri x" EqTuo.fion t
o-2
e4
c
-2-0.2
olol
T1-2
To
Ta
-Tn2
T6
-lq
100:Addi Vional
&b, i: 6,
' t001-tD I
,; L
_-to (
-/o I
ol
o87-l
oo
ccC
oI6
oC
C64
-267I
c4e
-2L6
oo
O
ooc
U-S'tng Galrss E(irninabion niph rou: pivoping,
-rt
-r2
-rs
-T4
kT6
T7
100t6b l
172.1r7 5.5t69.5t56.G
loot 5g.l172.1
t76.et69.5t56.6t98.2
lr3:
I2a,
O
cond i hi on,
?ru-2Ta*+77:- to
oC5q-256
o
o
101:
Radial Temperature Distribution of Hollow Sphere
: 150(J
;EpMaTELooE(l,F 130
Problem #28
102:
Dovolop bhe @mploae Finibe Diffe.l^er1ce madel ir.,Bo+ bhe ovendry eDaDe radio-l haav @ndacaion in o-solid rod/spherq, (r: a),in uhich anvec-bion.e arealloned bo ambie.nDs Frorn Vhe oube.r utall oP Vharod /uph ere.
Solubi on
DiscroDizabi on o+ uhe dema\n
Me,sh Lengbh,,Q-h": -n-
d,r 4_ p dr __ BU) _ /^dr2 r dr ' K -\-/
*Tt+rZT4^-rV-r - P T"t-Te-t0t 2' 2h
p__{ t, 4 rocl' L z, ephere
*3i :oh<v
> (,- +)rr-, - 2T* (r * !?)Tt*,: - t#
-/\,tt \.
\
rA',/\n--- \
) renTt-t - 2I[ -+ [22-Tr*( : - Y o
103:Jfuenal
r-)F<-,t :_____> (2.. I/
i+t
=> -Tr-t+ MTi -r-f-i+r: MTa Otror bhe, inbernal poirtb t:l
+q*+!t* fl - odr' r dr K
N oN., r daT - / dE/d, \l- or dr -\ r /lr:oo
AppluirA l= uoPlratP rulqrdu/t.rr #ff)l o'arrI r /: --T; Io:o: AV lr=o
- 3h'K
-Thus bha seb of qpaDiorl;,(:lt To-2T f Tz
i:2, l?'tT -27"*R'4i4, fotTt-276-+R';f4
[: fi,
i: fltl,
4dd i bional eond i bi o n .,
&, , i- n+I, tZfl-n - 2Tn+t rand , Ar) [,: t , -To :72
-/hurs, hhe dobal ooef* ic,ie.nb
,*fu-At
l_tull_tul
:) ,K I
)*^"1L a"sI
L^;l
2 0O o-2 R2 O o&t
.-2 R.2
o o oocoo o o-
O
-2R'1
o
oO
o
-2R2OR'1 -2 R2
-l lvl, I
OOoooo
\"T2
-Ts
j
T/1
Tnrr
Tn*z
aoo
R't
oO
t.3:
104:
tZ I^ -,- 2Tn+ R'z-tn+t : -Y- -rn -rMh*ttG*z:tv\k
R4Tnt2:-W
rnabrix- qTtabion,
=) tAl {1
105:Probtem # OO (Dabe^. 3t/ to/ t)A lO cm diano2er sDee) Ssrrf rod of bhe,rma(condtrcbivivt6 40 N/m"C is heooed e_lenbricallt4 bt4
bha PaesaBe oP e)ecvrie ourrrena uhidt Beneraaesene'rg6 niahin ilre bax of a raDe ffi 4xo6 N/n3Hatt is dissipabed fran bha s?,tr f are. of frie barbA *nvecrtion wiVh a coeiliaenD of 4OO tA/mTocinDo o-n a-tnbien? of ZO"C. p.1div?ding pho radiz.r.sinDo e-q/Lta( Pive. e%monDs.
(t) Develop bha ample7e, fini ?a -diffe re.ne.SoluDi an Poqm f o r a^adiat pe.rn p. d isrr i bubion .
(z) rtnd vhe vor?abion ofdire,c-Uion and shou ia
bemp along bhe.'ad,a{on a- B raph .
Sclubion
Qiven,{
DiscreVi<aPion of bha donain
l-{: 4O Nlm"C : O.4ln)/on'C
[a: 4 0O N/.'oc: O"T4nfc; "C
B - 4xloG N/n3: 4 N/c^'
Me'sh Lengbh'4 r'2 h:=- -l cn
I
- bcm
-r'--
106:
d'r *{-gl-r fl :()dr' r dr K \
_\ T*,- ZTl +T;_1_.7+ (t-+)T-,-
SDencil
z\.* (,*h)t*,:
l
4,,
tr -
rr/t( | L-l a_l.<zk- -o
lo
:- lo
:- lo
:-lO
€G --lo
(-1
For ) niernal nodes
4:2,
i: e,
i- 4,
(: b,
For boutndar?\ nodes
i:6, - K**f,, 6: t"(rt-r*)li:n
+-K 2h tf,:
*n -2Tz**ru3ET Iz - 2Ts-tT',
*tu- 2T,t+* r"=tro-2k
+ * To - o.2Ta -T+ : ll-4
107:For add i aional cond ibion ,
[:6, *r"-2Ta*#n:-roFor i:t ) 2#*+-o
-Q*r-2Tc'+T;.
+ T"-2Tt +Tz: - 5
+-2\ t2T2:- e
*+- -o/"t
7ht(8,
-2L2
o
C
co
bhe gtobal
2 O O
-2*oe .64 -z-I
o 8-2=cct
oooo o c
COOo coooo+ o o6
-29 og . l(ro -z lo
t -o'2 -t
coeffici env mabrirc enYuabiart,
TT2
-T3
T4
Tg
-ro
-77
-5-ro-to-10
-to
-to4o
ttsinV GarLGs E tim)ndaiot,292.e290z&2.52vo2e2.5260
t'J:
108:
300
290
280
270
I zooFE? 250E(l)qcb z+oF
230
220
210
20a
Radiat Temperature DistributionProblem # 30
22_53Radial Distance, r icm)
109:Probtem # gl (Dabe:gtltottt)
/R'e'paap bho probtem go {.or o- sbeel ephere
Solubion
Discrebiza\ort of bhe domatn
rt 12- ro r.t ro
Me,eh Lengvh, tL:?: tcm
-d=T 2 dr
"F*T;;*f--:o-ry\-277 +Ta:1=+ *+rr*:-rt-, + ff _Otrl 2h K -\=+ C,- *)T*t- 2Ti r(r* t)Titt:-{o
Sbe-ncil
-la
-_a
/(
110:For )nbernal node,,=
(:2, -2Tza ZTe :-lO2: z: *rr- 2Tet?t, :-to
i-/L 2-^-)' L\, ,) e LO --T4* gTg :-lO
A.6[= O, tfo -2Ts+i-fn : -lo
For baundartA noo)es
i: 6, - K# l r:.: h(71 -rx,) l r: n
+-o:Tlo^- ^(tn-2o)
:+ To - O-2Te -Tz : 4
For add iaional @nd;7i on '
i:6, t r-2Ta* *rr:- lo
Far, i:l) 2#**: o
4c Ti.t*T,<-2Tt * fl :O---zz- Crt
. K
+ T; - 2T +Tn:-5
=+ - 2-fi + 2Tz :- 5
mavrizc7h bho global coejf icien?
2o-22
| _.)21
2CoCCOOco
Using Gaass Eildna2ion,
T-a-fe
Trl
%T6
T7
tBg.2186.67tgl.67179.93tGt.67
1Ug,
-2oC
o
ooO
T-G
-Ts
-fa
G
h-T7
ooc"-lo o o ot* o o ol-2+ocl*-2+ o
Io +-2?lO l -o.2-,_]
tjs 2 Its 6.6T I
t,t GT /Lt73.es (
t6r 6v /
4G.67 I
2o. ss )
{r3:
t46_67
112:
()o
FoL3E(DoE(l}F
Radial Temperature Distribu$onProblem # 31
22.53Radial Disbnce, r {cm)