hw-3-solution

# hw-3-solution - \$313!? assent ECH s“ Homework 3, problem...

This preview shows pages 1–14. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: \$313!? assent ECH s“ Homework 3, problem 1 Solve for the approximate deﬂection of a simply—supported prestressecl cable with triangular static distributed load using the ﬁnite element Galerkin method. Use a mesh of four equal—length ﬁnite elements, and number the nodes and the elements left to right. (a) Compare the midpoint deﬂection computed analyt— ically and approximately. (b) Compute the transverse force analytically and compare with the ﬁnite element result. Homework 3, problem 2 Find the natural frequencies and normal mode shapes for the roller—pin sup~ ported cable with uniform mass density using a mesh of three ﬁnite elements of equal length. Number the nodes and the elements left to right. Integrate ele— ment Wise quantities exactly. (a) Plot the normal mode Shapes. (b) Calculate the relative error of the natural frequencies. P.” epﬂ®ﬁ®ﬂe a? l;_L__.m HOmework 3, problem 3 Consider the free vibration of a pin—pin supported cable. Construct the “Solid- Works Simulation” ﬁnite element model to compute the seventh natural fre- quency of free Vibration using the cable-uniaxial rod analogy. Use both draft— and high—quality tetrahedra. Find the uniform mesh size necessary for attaining accuracy better than 1% in angular frequency. Compare the lnumerical results to the analytical formulas for the prestressed cable free vibration. Produce a brief report. ‘ yrs» mm; (EEC “224mm Pu’ﬂ-ﬂ) ﬂax Mia: Mam-T“ WM Law ,[K]{w_} + {L} :50} Wyammmm ‘ (I: v v f v (73% , 7‘ 117%!)33 {ﬂit/V2) ﬁg? 35 [,1 [l A UL M MA at ojﬂmﬂ z. W Ia *- iéﬂ/‘A~©) ﬁ ~ij [I L L/ A L i V I [kw ‘3 M Wm)? 70 pyjmww V , D ‘1 . i, I l I UL. r [Mzﬂm NMNMXW WWW zépgwfydx ‘D L L. » a t: :1 Mail" 9"»; V WWW 0 X2; 3 3 2/???) Y Kata) i j/ q £44 + gm {:3 / _‘ 0/“ 273%! w’: ~3§)&ZFG 7,0 L l m. 2 ,wcr i)£%ﬁ¥+€a fl. 3 jammy My} 2 X31; 3%? 31% Li, w/rﬁé): {L2 :2 "~' ﬂak 3W Lira? '29}? v g 5;» x3 H. (@Mﬂﬂié” 5am paws ,4?" JV 3*" 5/1 ,mea gm,- M Tmmg Wg’ iii} 3 L2 ﬂﬂk’géwvmﬁﬁ” 3&5??an («/3 '3 law 3‘" 5.3/2" #32 P I2 P V W; Mm wmmaﬂemmmwmmrmWrmwmmwmg « memummw /7¢22”AWWMME ﬁﬁiWﬁﬁW lg" mag?" 7" 1/ 3 V21] _ WmeWWWMWumstwwmwmmu Wtwmmzmmwmwwwmmmmmm _ -' .mwxg“ W65” WMij Wm WEXHﬁr “Mm-f i5- Mérﬁwa m. W} L, a}; ,1 / / {bf/"5] 69W”; #7ng +— yszWZU wow/Or @510 : 71/1ij Wgaww mu [5)? my; ' 8 0/9" VJ 19“) 3 Wg’waea» ﬁﬁijbg 2P/*4E)(;;;z)%"+%)(ﬁz)\$7 ‘ W Pu’wzgié WMWMME 631“ [VH1 646 Force [qL] 0.1 0.2 0.3 0.4 0.5 x [L] 0.6 0.7 0.8 0.9 7 x. ﬁx" '4‘ 1;: Eg‘.‘ ' Nam: \$X/L ~ 3 Nmtx’) =1 “lax/HE fig/w: @ WW,Wmewiwwimw ‘ ‘ é ;( {ﬁg in mm v Key, v D] chmw awe? WWW 5/ 19" if} 3 I. j V § “22 3 f'l'iﬁméaém‘i‘ ' L via 5’13"”; M fifya Wk m wwﬁﬁr‘L; 3,“) D “r; 0 MM “W i L m w 2: [a w J «:2 W I 1 3% “N23 5 \4 *" (/31 M L, % :13 ﬂ“? V. m m [\AmrmﬁéK W) (was; vwf‘osmca @3081 Q \$47.2, Q 10133 D‘QQQO “G “153157” 0 370\$ mama 02%! a <99?“ D: o 0%} C) o 0 055000 Q O O \eﬂsg >4: 00"“)?- _ 31:03 1 \$33: Lé‘ﬁé wzm 2 02915888 3;, I t m; u; L war—«SOSWWSTF 1, wzztlﬂﬂﬂ ‘fL IL 43%? 41 E ()3. 1:; {33% ")x 5‘: 30m AL wwmﬂHC/WE Wm” M @390: ﬁxgﬁu '1‘ f: ‘ “§f‘§;}w¢af:&mémk @E—Z’Qu £34 ii. WW5 QM Wat) '2 E exvﬂﬁt) S FATE/1k Rd 3‘. : [g P WAVE, affmm CAL clear r_w close 2;: L=l;q:l;F1=11*q*L/48;F2=5*q*L/48; xx1=[O:L/200:L/2]; f=@(X,L)(—q*X“2/L+q*L/4); XX2=[L/2:L/200:L]; xx=Lxxl xx2(2:end)];n=length(xx); f0: i=1:length(xxl) yy(i)=f(xxl(i),L>; yy(n—i+l)=-yy(i); grid ,. :;hold plot<xx,yy,“~ﬂ',?1"cwigvi*,2); plot([0 L/4],[Fl Fl], #k“',‘,irﬁﬁif";’,2); plot([3*L/4 L],-[F1 Fl],vw§:‘,f 'T& Lﬁ:35,2); plot([L/4 L/21,[F2 F2],*~,og,§l;mfriqu3,2); plot([L/2 3*L/4],-[F2 F2],§~.:2,‘TLVﬁ1 mﬁh¥,2); ‘,. “0 var“, A, i]; 171 ,»,y ;>; xlabel(vf ’»;’);ylabel(“a/“v syms 5 i ; u “ Nl=—3*X/L+l;N2L=3*X/L;N2R=—3*X/L+2;N3L=3*X/L-l;N3R=‘3*X/L+3;N4=3*X/L‘2; kll=int(diff(Nl,X)*P*diff(Nl,X),X,O,L/3); mll:int(Nl*u*Nl,X,O,4/3); m12=int(Nl*u*N2L,X,O,L/3); m22=int(N2L*u*N2L,X,O,L/3)+int(N2R*u*N2R,X,L/3,2*L/3); m23=in:(N2R*u*N3L,X,;/3,2*L/3); m33=in1(N3L*u*N3L,X,L/3,2*L/3)+int(N3R*u*N3R,X,2*L/3,L); K=3*P/;*[l —l O;-l 2 *1;O —l 2]; M=[m11 ml2 O;ml2 m22 m23;0 m23 m33]; L- ' Mn:[2 1 O;l 4 1;O 1 4]; , Kn=[l -l O;—l 2 —1;O -l 2]; [Vn Dn]:eig(Kn,Mn); x=[0;1/3;2/3;1]; modeﬁl:[Vn(:,l);O];modeﬁl=mode_1/mode_l(1,1); mode;2:[Vn(:,2);O];mode:2=mode:2/mode:2(1,1); mode_3=[Vn(:,3);O];modeﬂ3:mode_3/mode_3(1,1); xx:[0:l/lOOO:1]; far i=l:length(xx) Modeﬂl(i)=~sin(l*pi*xx(i)/2/l—pi/2); ModeﬂZ(i)=—sin(3*pi*xx(i)/2/l—pi/2); Mode_3(i)=—sin(5*pi*xx(i)/2/l—pi/2); figure;hold “4; p..o:(x,mode_l,‘ﬁwﬂ,E p;0d(x,mode_2,“i Lg," plot(x,mode~3,‘~*“‘,‘ V” legend(’3 L» iﬁ,‘::,w ’“ ‘ * , b ’), plo:(xx,Mode‘l,‘ww ,*,i_ax _, plot(xx,Mode)2,V' 3, ;; av*':,“ plot(xx,Mode!3,bﬁiiW,S 377<W‘” Xlabel(\$a [Q]');ylabel(7‘ '5); title('\$5ri“ 31*-: “3?? 9';N ...
View Full Document

## hw-3-solution - \$313!? assent ECH s“ Homework 3, problem...

This preview shows document pages 1 - 14. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online