an engineer's guide to matlab,edward,2nd,[solution]by hamed .pdf

7/15/2019 An engineer's guide to MATLAB,edward,2nd,[solution]by Hamed .pdf

http://slidepdf.com/reader/full/an-engineers-guide-to-matlabedward2ndsolutionby-hamed-pdf 1/48



Success isn’t a matter of being the best & winning the

race. Success is a matter of handling the worst & still

finishing the race.

As you know I am a student of this faculty and it’s my first course in

MATLAB.I have no permission to authorship a book due to my low educational

degree. With respect to these points, please do not consider this present aseducational textbook. I take no responsibility against wrong solution (if any) of

problems.

Hamed Parsa

Junior student of

E.E.E. departmentMarch 2010

Email: [email protected]



Chapter 1 Chapter 1 Chapter 1 Chapter 1



>> r=2.5;

>> I=(pi/8-8/(9*pi))*r^4

>> c=5;

>> k=(4*c-1)/(4*c-4)+0.615/c

>> B=0.6;>> K=3/(1-B)^3*(0.5-2*B+B*(1.5-log(B)))

Homework Problems not solved.

>> A=1.7;B=1.2;

>> D=1.265*((A*B)^3/(A+B))^(1/5)

>> n=6;M=1/sin(pi/n);h=(1+M^2)/(4*M);

>> alpha=acos(sqrt(h^2+2)-h);

>> answer=M*(1-M^2)*sin(alpha)/(1+M^2-2*M*cos(alpha))^2

>> L=3000;d=45;V=1600;

>> p=0.03*L/d^1.24*(V/1000)^1.84



>> v=0.3;E=3e+7;d1=1.5;d2=2.75;F=100;z =0.01

>> a=(3*F/8*2*((1-v^2)/E)/(1/d1+1/d2))^(1/3);

>> pmax=3*F/(2*pi*a^2);>> Qz=-pmax/(1+z^2/a^2)>> Qx=-pmax*((1-z/a*atan(a/z))*(1-v)-0.5*(1+z^2/a^2)^-1)

>> v=0.3;E=3e+7;d1=1.5;d2=2.75;F=100;L=2;z=0.001;

>> b=sqrt(2*F/(pi*L)*2*((1-v^2)/E)/(1/d1+1/d2));h=(1+z^2/b^2);

>> pmax=2*F/(pi*b*L);

>> Qz=-pmax/sqrt(h);

>> Qy=-pmax*((2-1/h)*sqrt(h)-2*z/b);

>> Qx=-2*v*pmax*(sqrt(h)-z/b);

>> Tyz=0.5*(Qy-Qz);

>> e=0.8;a=(1-e^2);

>> NL=pi*e*sqrt(pi^2*(a)+16*e^2)/a^2

>> h=1.25; d0=0.25; d1=0.625; E=3e+7; d2=d1+h*tan(pi/6);

>> y=log((d2-d0)*(d1+d0)/((d2+d0)*(d1-d0)));

>> k=pi*E*d0*tan(pi/6)/y

>> alpha=1.2e-5;E=3e+7;v=0.3;Ta=500;Tb=300;a=0.25;b=0.5;r=0.375;

>> Tc=Ta-Tb; c=log(b/a); d=log(b/r); k=a^2/(b^2-a^2);

>> T=Tb+Tc*d/c; h=alpha*E*Tc/(2*(1-v)*c);

>> Qt=h*(1-k*(b^2/r^2+1)*c-d);

>> Qr=h*(k*(b^2/r^2-1)*c-d);



>> p=0.3;k=1.4; % pe/po=p

>> Y=sqrt(k/(k-1)*(p^(2/k)-p^((k+1)/k)))

>> x=0.45;y=sqrt(16*x^2+1);

>> K=1.2/x*(y+1/(4*x)*log(y+4*x))^(-2/3)

>> n1=0; n2=1;

>>Pi1=(sqrt(8)/9801*gamma(4*n1+1)*(1103+26390*n1)/(gamma(n1+1)^4*396^(4*n1)))^(-1);

>>Pi2=(1/Pi1+(sqrt(8)/9801*gamma(4*n2+1)*(1103+26390*n2)/(gamma

(n2+1)^4*396^(4*n2))))^-1;

>> % These two statement cannot be broken up as shown. They have beenpresented in two lines because of page width restrictions.

>> pi-Pi1 %show the approximation of pi with a different less than 10^-7

ans =

-7.64235e-008>> pi-Pi2 %show the approximatoin of pi with a different less than 10^-15

ans =

-4.44089e-016

>> k=1.4; r=10; rc=3;

>> eta=1-1/r^(k-1)*(rc^k-1)/(k*(rc-1))

>> k=1.4 ; M=2 ; %A/A*=X

>> X=1/M*(2/(k+1)*(1+(k-1)/2*M^2))̂ ((k+1)/(2*(k-1)))



>> a=-1:2:13;b=1:2:15;

>>a+b; %part a) >>a-b; %part b)

>>a'*b ; det(a'*b); %part c)

>>a*b'; %part d)

>> x=[17 -3 -47 5 29 -37 51 -7 19]; a=sort(x);>> b=sort(a(1:4),'descend');c=sort(a(5:9),'descend'); y=[b c];>> %we can use fliplr function too.

>> b=fliplr(a(1:4));c=fliplr(a(5:9)); y=[b c];

>> y=[0 -0.2 0.4 -0.6 0.8 -1.0 -1.2 -1.4 1.6];

>> z=sin(y);h=sort(z)h =

-0.985 -0.932 -0.841 -0.564 -0.198 0 0.389 0.717 0.999>> a = min(h(1:5)); b = max(h(1:5)); %part a) >> c=sqrt(h(6:9)); %part b)

>> x=logspace( log10(6) , log10(106) , 8);

>> y=x(5); %part a) >> h=[x(1) x(3) x(5) x(7)]; %part b)



>> z=magic(5);

>> z(: , 2)=z(: , 2)/sqrt(3); % part a) I >> z(5 , :)=z(3 , :) + z(5 , :); % part a) II >> z(: , 1)=z(: , 1) .* z(: , 4); % part a)III

>> q=z-diag(diag(z))+diag([2 2 2 2 2]); % part a) IV

>> diag(q*q'); % part b)

>> c=q.^2; % part c)

>> max(max(c)); min(min(c)); % part d)

>> w=magic(2);

>> a=repmat(w, 2, 2); % part a)

>> b=repmat(w, 3, 1); % part b) >> c=repmat([w w'], 3 ,1); % part c)

>> da=[w w ;w w];db=[w ;w ;w];dc=[w w';w w';w w']; % part d)

>> x=magic(3);

>> new=[x(2,:) ; x(3,:) ; x(1,:)]; % part a

>> new=[x(:,3) , x(:,1) , x(:,2)]; % part b

>> a=1;b=1.5;e=0.3;phi=0:360;

>> s=a*cosd(phi)+sqrt(b^2-(a*sind(phi)-e).^2);>> plot(phi,s)



>> t=1/sqrt(19);Pt=4.3589;n=linspace(2,25,300); %to/T=t

>> Po=1+2*cumsum(sin(n*pi*t).^2./(n*pi*t).^2);

>> plot(n,Po)



>> x=1:0.5:5;

>> n=1:100;>> [xx,nn]=meshgrid(x,n);

>> sn=prod(1-xx.^2./(nn.^2-2.8));

>> sj=a./(sin(pi*a)*sqrt(a^2+x.^2)).*sin(pi*sqrt(a^2+x.^2));>> en=100*(sn-sj)./sj;

>> x=[72 82 97 113 117 126 127 127139154 159 199 207];

>> B=3.644;

>> Delta=(1/length(x))*sum(x.^B)

Delta =3.0946e+028



>> phi=linspace(0,90,10);theta=linspace(0,360,24);b=2;

>> [theta2,phi2]=meshgrid(theta,phi);>> x=b*sind(phi2).*cosd(theta2);

>> y=b*sind(phi2).*sind(theta2);>> z=b*cosd(phi2);

>> mesh(x,y,z);

>>% The result after using the rotate icon in the figure window.

>> x=linspace(0.1,1,5); y=pi*x*sqrt(2); n=0:25;>> t=2*pi^4./y.^3.*(sinh(y)+sin(y))./(cosh(y)-cos(y));

>> [xx,nn]=meshgrid(x,n);

>> S= sum(1./(nn.^4+xx.^4));>> compare=[t' S' (t-S)'];



>> x=[1 2 3 4 5 6]; k=0:25;n=2;

>> [xx,kk]=meshgrid(x,k);>> p=sum((-1).^kk.*(xx/2).^(2.*kk+n)./(gamma(kk+1).*gamma(kk+n+1)));

>> besselj(2,x);>> r=besselj(2,x);

>> compare=[p' r']

>> k=1:13;n=7;

>> s=sum(cos(k*pi/n))

>> w=0.5*[-1 -1; 1 -1;-1 1; 1 1]; q=0.5*[1 -1 -1 1; 1 1 -1 -1; 1 -1 1 -1; 1 1 1 1];>> I1=w'*w; I2=q'*q;

>>e=pi/6;a1=1;a2=2; a3=3;>>A1=[cos(e) -sin(e) 0 a1*cos(e);sin(e) cos(e) 0 a1*sin(e);0 0 1 0;0 0 0 1];>>A2=[cos(e) -sin(e) 0 a2*cos(e);sin(e) cos(e) 0 a2*sin(e);0 0 1 0;0 0 0 1];

>>A3=[cos(e) -sin(e) 0 a3*cos(e);sin(e) cos(e) 0 a3*sin(e);0 0 1 0;0 0 0 1];>>T3=A1*A2*A3;

>>qx=T3(1,4);qy=T3(2,4);

>>%For finding orientation with respect to Fig2.8 orientation of X3 is%arctan(uy/ux) and orientation of Y3 is arctan(vy/vx)

>>X3=atand(t(2,1)/t(1,1))

X3 =90.0

>>Y3=atand(t(2,2)/t(1,2))Y3 =

0.00>>%They are parallel to Y0 and X0 axes respectively.

>> X= [17 31 5; 6 5 4; 19 28 9; 12 11 10];



>> H=X*inv(X'*X)*X';>> diag(H)

Part a)>> n=1:2:399; t=linspace(-0.5,0.5,200);

>> u= sin(2*pi*n'*t);

>> fa=4/pi*cumsum(1./n*u);>> plot(t,fa)

part b)

>> n= 1:200; t=linspace(-1,1,200);

>> u= sin(2*pi*n'*t);

>> fb=1/2+1/pi*cumsum(1./n*u);>> plot(t,fb)

part c)

>> n= 1:200; t=linspace(-1,1,200);>> u= sin(2*pi*n'*t);

>> fb=1/2-1/pi*cumsum(1./n*u);

>> plot(t,fb)

part d)

>> n= 1:200; t=linspace(-1,1,200);

>> u= cos((2*n-1)'*pi*t);>> fd=pi/2-4/pi*cumsum(1./(2*n-1).^2*u);

>> plot(t,fd)

part e)>> n= 1:200; t=linspace(-1,1,200);

>> u= cos(2*n'*pi*t);>> fe=2/pi+4/pi*cumsum(1./(1-4*n.^2)*u);

>> plot(t,fe)



part f)>> n=2:2:106;t=linspace(-2,2,200);

>> ff=1/pi+1/2* sin(pi*t)-2/pi*cumsum(1./(n.^2-1)*cos(pi*n'*t));>> plot(t,ff)

part g)>> n=1:250;t=linspace(0,4*pi,350);

>> u1=1./(1+n.̂ 2)*cos(n'*t);>> u2=n./(1+n.̂ 2)*sin(n'*t);

>> fg=(exp(2*pi)-1)/pi*(1/2+cumsum(u1-u2));

>> plot(t,fg)

part h)

>> a=0.25;n=1:2:105;t=linspace(2,-2,200);>> fh=4/a^2*cumsum(sin(n*pi*a)./(pi*n).^2*sin(n'*pi*t));

>> plot(t,fh)

>> n=1:25;a=sqrt(3);theta=(10:10:80)*pi/180;

>> S1n=1./(n.^2+a^2)*cos(n'*e);

>> S1inf=pi*cosh(a*(pi-e))/(2*a*sinh(pi*a))-1/(2*a^2);>> S2n=n./(n.^2+a^2)*sin(n'*e);>> S2inf=pi*sinh(a*(pi-e))/(2*sinh(pi*a));

>> e1=100*((S1n-S1inf)./S1inf)

>> e2=100*((S2n-S2inf)./S2inf)

>> eta=0:1/14:1;E=0:1/14:1;n=1:2:length(E);a=2;

>> u1=1./(n.*sinh(n*pi*2)); u2=sinh(n'*eta*pi*2); u3=sin(pi*n'*E);>> [eta1,u11]=meshgrid(eta,u1);

>> T=(u11.*u2)'*u3;>> mesh(E,eta,T)



>> n=1:50;a=0.25;t=0:0.05:2;eta=0:0.05:1;>> u1=sin(n*pi*a)./n.^2;u2=sin(n'*pi*eta);u3=cos(n'*pi*t);

>> [eta1,u11]=meshgrid(eta,u1);

>> c=2/(a*pi*(1-a));

>> u=c*(u11.*u2)'*u3;

>> mesh(t,eta,u)>>% The result after using the rotate icon in the figure window.



Homework problems not solved.



>> v=0.4;E1=3e+5;E2=3.5e+4;Uo=0.01;>> a=0.192;b=0.25;c=0.312;t=1+v;h=1-v;

>> S=[1 a^2 0 01 b^2 -1 -b^2

-t h*b^2 t*E1/E2 -h*b^2*E1/E2

0 0 -t h*c^2]S =

1.0000 0.0369 0 0

1.0000 0.0625 -1.0000 -0.0625-1.4000 0.0375 12.0000 -0.3214

0 0 -1.4000 0.0584

>> y=[0 0 0 -Uo*E2*c];>> w=inv(S)*y';x=w';

>> A1=x(1);B1=x(2);A2=x(3);B2=x(4);

>> Qee1=-A1/b^2+B1;>> Qee2=-A2/b^2+B2;



>>a=(1+sqrt(5))/2;b=(1-sqrt(5))/2;c=1/sqrt(5);

>>n=0:15;

>>Fn=c*(a.^n-b.^n);>>disp([repmat('F',16,1) num2str(n') repmat(' = ',16,1) num2str(Fn') ]);>>fprintf(1,'F %2.0f = %3.0f\n ',[n ;Fn]);

% To go next line without execution use shift+Enter

>>ft=input('Enter the value of length in feet :');

disp([num2str(ft) ' ft = ' num2str(ft*0.3048) ' m'])

Homework Problem not solved.

>>D=input('Enter a positive integer < 4.5x10^15 : '); % shift+Enterdisp(['The binary representation of ' num2str(D) ' is ' dec2bin(D)] )

z=input('Enter the complex number:'); % shift+Enter

disp(['The magnitude and phase of ' num2str(z) ' is']); % shift+Enter disp(['Magnitude = ' num2str(abs(z)) ' phase angle = '

num2str(angle(z)*180/pi) ' degrees']);

>>%The last ‘disp’ statement cannot be broken up as shown. It has been

presented in two lines because of page width restrictions.



function y=myexp1(h,a,b)for n=1:length(h);if h(n)>a && h(n)<b;

h(n)=1;

else

h(n)=0;

end

end

disp([h])

end

function myexp2(x)

k=0;

for m=2:length(x)

xn=1/m*sum(x(1:m));

Sn2=1/(m-1)*(sum(x(1:m).^2)-m*(xn).^2);

k=k+1;y(k)=Sn2;

end

disp(['Sn2 = [' num2str(y) ' ]'])

end

function myexp3(a,b)

k=1; c=0.5*(b+a/b); %where c=b(n+1) while abs(b-c)>1e-6

k=k+1;

b=0.5*(b+a/b);

c=0.5*(b+a/b);

end

disp(['n-iteration=' num2str(k)])



end

part a)*

function y=myexp4(x0)

x0=0:200;

r=x0.^2+0.25;x=0:5:200;

for t=1:(length(x)-1);

y(t)= r(5*t);

end

y=[r(1) y];

plot(x,y,'ks')

end

part b)

function y=myexp5(x0)

x0=0:200;

r=x0.^2+0.25;

x=0:5:200;y=[];k=0;

while (length(x)-1)~=length(y)k=k+1;

y(k)= r(5*k);

end

y=[r(1) y];

plot(x,y,'ks')

end

function chisquare(x,e)

k=0;e=[e 5];x=[x 0];for n=1:length(e)

if e(n)<5

e(n:n+1)=cumsum(e(n:n+1));



x(n:n+1)=cumsum(x(n:n+1));

else

k=k+1;

p(k)=n;

end ;end

h=x(p);u=e(p); %e modified= u and x modified=h if (u(end)-5)<5

u(end)=u(end)-5+u(length(u)-1);

u(length(u)-1)=[];

h(end)= h(length(h)-1)+h(end);

h(length(h)-1)=[];

end

y=sum((h-u).^2./u);

disp(['e modified = [' num2str(u) '], x modified = [' num2str(h) '], X2='

num2str(y) ])end

>>%Thist ‘disp’ statement cannot be broken up as shown. It has beenpresented in two lines because of page width restrictions.

function myexp6(p,s)a=length(p) ;b=length(s);

if a>=b

t=s+p(a-b+1:a);

h=[p(1:(a-b)) t]

elseif a<b

t=p+s(b-a+1:b);

h=[s(1:b-a) t]

end end




Chapter Chapter Chapter Chapter 5 55 5



>>h=[6 3 2 1.5 1.2 1.1 1.07 1.05 1.03 1.01]; %h=D/d

>>a=[0.33 0.31 0.29 0.26 0.22 0.24 0.21 0.2 0.18 0.17];>>c=[0.88 0.89 0.91 0.94 0.97 0.95 0.98 0.98 0.98 0.92];

>>%Fifth order polynomial

>>aa=polyfit(h,a,5);cc=polyfit(h,c,5);

>>%Obtain Orginal Value Of Kt

>>kt=c.*(h/2-0.5).^(-a);

>>%Compare

>>k1=polyval(cc,h).*(h/2-0.5).^(-polyval(aa,h));

>>k1-k

ans =

Columns 1 through 6-1.0971e-008 -1.8084e-005 8.9483e-4 -0.011004 0.087181 -0.09516

Columns 7 through 10

-0.059276 -0.060901 0.016685 0.16796

>>%Fit data with a spline

>> n=linspace(1.01,6,100);

>> aa=spline(h,a,n);

>> cc=spline(h,c,n);

>> k=0;

>> for h=[6 3 2 1.5 1.2 1.1 1.07 1.05 1.03 1.01]k=k+1;

a2(k)=interp1(n,aa,h);

c2(k)=interp1(n,cc,h);

end

>>k2=c2.*(h/2-0.5).^(-a2)

>>%Compare

>>k2-k

>> -kt+kt2

ans =

0 0.0005 0.0009 -0.0062 0.0101 -0.0271 0.0321 -0.0637 -0.0408 0>>%2

ndmethod (spline) is better.



>> Qx=100;Qy=-60;Qz=80;Txy=-40;Tyz=50;Tzx=70;

>> C0=Qx*Qy*Qz+2*Txy*Tyz*Tzx-Qx*Tyz^2-Qy*Tzx^2-Qz*Txy^2;

>> C1=Txy^2+Tyz^2+Tzx^2-Qx*Qy-Qy*Qz-Qz*Qx;>> C2=Qx+Qy+Qz;

>> disp([C0 C1 C2])

-844000 11800 120

>> F=inline('Q.^3-120*Q.^2-11800*Q-(-844000)','Q');

>> x=linspace(-100,200,500);

>> plot(x,F(x));

>> Q1=fzero(F,170);Q2=fzero(F,50); Q3=fzero(F,-100);

>> T12=(Q1-Q2)/2;T23=(Q2-Q3)/2;T13=(Q1-Q3)/2;

>> F=inline('tan(x)-x','x');

>> x=linspace(0,15,5000);

>> plot(x,F(x),'k',[0 15],[0 0],'r')

>> axis([0 15 -5 5])

>> r1=fzero(F,[-1 1]);r2=fzero(F,[4.2 4.6]);r3=fzero(F,[7.6 7.8]);

>> r4=fzero(F,[10.85 10.95]);r5=fzero(F,[14 14.1]);

>> disp([r1 r2 r3 r4 r5]);

>> %Part a

>> H=inline('2*cot(x)-10*x+0.1./x','x');

>> x=linspace(0,15,5000);

>> plot(x,H(x),'k',[0 15],[0 0],'r');

>> axis([0 15 -4 4])

>> r1=fzero(H,[0.4 0.6]);r2=fzero(H,[3.2 3.3]);r3=fzero(H,[6.3 6.4]);>> r4=fzero(H,[9.44 9.46]);r5=fzero(H,[12.57 12.6]);

>> disp([r1 r2 r3 r4 r5]);

>> %Part b

>> H=inline('2*cot(x)-x+1./x','x');

>> x=linspace(0,15,5000);



>> plot(x,H(x),'k',[0 15],[0 0],'r');

>> axis([0 15 -4 4])

>> r1=fzero(H,[1 2]);r2=fzero(H,[3.5 4]);r3=fzero(H,[6.5 7]);

>> r4=fzero(H,[9.5 10]);r5=fzero(H,[12.6 13]);

>> disp([r1 r2 r3 r4 r5]);

>> G=inline('besselj(0,x).*bessely(0,2*x)-

besselj(0,2*x).*bessely(0,x)','x');

>> x=linspace(0,20,5000);

>> plot(x,G(x),'k',[0 20],[0 0],'r');

>> r1=fzero(G,[2 4]);r2=fzero(G,[6,8]);r3=fzero(G,[8 10]);

>> r4=fzero(G,[12 14]);r5=fzero(G,[14 16]);

>> disp([r1 r2 r3 r4 r5]);

>> %Mo/mo= m

>> % Case 1

>> H=inline('0*x.*(cos(x).*sinh(x)-

sin(x).*cosh(x))+cos(x).*cosh(x)+1','x');

>> x=linspace(0,20,5000);

>> plot(x,H(x),'k',[0 20],[0 0],'r');;>> axis([0 20 -20 20])

>> r1=fzero(H,[1 4]);r2=fzero(H,[4,6]);r3=fzero(H,[6 8]);

>> r4=fzero(H,[10 12]);r5=fzero(H,[13 16]);

>> disp([r1 r2 r3 r4 r5])

>> % Case 2 >> H=inline('0.2*x.*(cos(x).*sinh(x)sin(x).*cosh(x))+cos(x).*cosh(x)+1','x');

>> plot(x,H(x),'k',[0 20],[0 0],'r');

>> axis([0 20 -20 20])


>> r4=fzero(H,[10 12]);r5=fzero(H,[12 14]);

>> disp([r1 r2 r3 r4 r5])

>> % Case 3

>> H=inline('x.*(cos(x).*sinh(x)-sin(x).*cosh(x))+cos(x).*cosh(x)+1','x');

>> plot(x,H(x),'k',[0 20],[0 0],'r');

>> axis([0 20 -20 20])




>> r4=fzero(H,[10 12]);r5=fzero(H,[12 14]);

>> disp([r1 r2 r3 r4 r5])

>>

>>G=inline('tanh(x)-tan(x)','x');

>> x=linspace(0,15,5000);>> plot(x,G(x),'k',[0 15],[0 0],'r');

>> axis([0 15 -1 1])

>> r1=fzero(G,[-0.5 0.5]);r2=fzero(G,[3.5 4.5]);r3=fzero(G,[6.5 7.5]);

>> r4=fzero(G,[9 10.5]);r5=fzero(G,[13 14]);

>> disp([r1 r2 r3 r4 r5])

>> % Case 1

>> G=inline('besselj(0,x).*besseli(1,x)+besseli(0,x).*besselj(1,x)','x');

>> x=linspace(0,15,5000);

>> plot(x,G(x),'k',[0 15],[0 0],'r');

>> axis([0 15 -22 22])

>> r1=fzero(G,[0 1]);r2=fzero(G,[2 4]);r3=fzero(G,[5 7]);>> disp([r1 r2 r3])

>> % Case 2

>> G=inline('besselj(1,x).*besseli(2,x)+besseli(1,x).*besselj(2,x)','x');

>> plot(x,G(x),'k',[0 15],[0 0],'r');

>> axis([0 15 -22 22])

>> r1=fzero(G,[0 1]);r2=fzero(G,[4 6]);r3=fzero(G,[7 10]);

>> disp([r1 r2 r3])

>> % Case 3

>> G=inline('besselj(2,x).*besseli(3,x)+besseli(2,x).*besselj(3,x)','x'); >> plot(x,G(x),'k',[0 15],[0 0],'r');

>> axis([0 15 -22 22])

>> r1=fzero(G,[0 1]);r2=fzero(G,[5 7]);r3=fzero(G,[7 10]);

>> disp([r1 r2 r3])



>> % Case 1

>> G=inline('tan(x)-x+4*x.^3/(2*pi^2)','x');


>> axis([0 15 -10 20])

>> r1=fzero(G,[0 1]);r2=fzero(G,[2 4]);r3=fzero(G,[4.75 5]);

>> r4=fzero(G,[7.86 7.87]);r5=fzero(G,[10.998 11]);

>> disp([r1 r2 r3 r4 r5])

>> % Case 2


>> plot(x,G(x),'k',[0 15],[0 0],'r');

>> axis([0 15 -10 20])

>> r1=fzero(G,[0 1]);r2=fzero(G,[2 4]);r3=fzero(G,[4.75 5]);>> r4=fzero(G,[7.86 7.9]);r5=fzero(G,[11 11.1]);

>> disp([r1 r2 r3 r4 r5])

>> % Case 3


>> plot(x,G(x),'k',[0 15],[0 0],'r');

>> axis([0 15 -10 20])

>> r1=fzero(G,[0 1]);r2=fzero(G,[2 4]);r3=fzero(G,[5 5.5]);

>> r4=fzero(G,[7.86 7.96]);r5=fzero(G,[11 11.1]);

>> disp([r1 r2 r3 r4 r5])

>> %case 1

>> G=inline('(1+0.4*x).^2.*(x.^2-x.^3)-0.2','x');


>> axis([0 1 -1 1])

>> r1=fzero(G,[0.4 0.7]);r2=fzero(G,[0.7 1]);>> disp([r1 r2]);

>> %using "roots" function

>> n=0.4; m=0.2; %co=n c1=m

>> y=roots([-n^2 (n^2-2*n) (2*n-1) 1 0 -m]);

>> disp(y(y>0))

>> %case 2



>> G=inline('(1+7*x).^2.*(x.^2-x.^3)-4','x');

>> plot(x,G(x),'k',[0 1],[0 0],'r');

>> axis([0 1 -1 1])

>> r1=fzero(G,[0.4 0.7]);r2=fzero(G,[0.7 1]);

>> disp([r1 r2]);

>> %using "roots" function >> n=7; m=4; %co=n c1=m

>> y=roots([-n^2 (n^2-2*n) (2*n-1) 1 0 -m]);

>> disp(y(y>0))

>> G=inline('2*5^2*sind(b).^2.*tand(b-35).^2-tand(b).*tand(b-35)-

tand(b).^2','b');>>%This ‘inline’ statement cannot be broken up as shown. It has been

presented in two lines because of page width restrictions.

>> x=linspace(0,90,5000);

>> plot(x,G(x),'k',[0 90],[0 0],'r');

>> axis([0 90 -0.5 0.5])

>> r1=fzero(G,[0 10]);r2=fzero(G,[10 40]);

>> r3=fzero(G,[40 60]);r4=fzero(G,[60 85]);

>> disp([r1 r2 r3 r4])

>> H=inline('(sum(x.^B.*log(x))./(sum(x.^B)-1/14*sum(log(x)))).^-1-B','x','B');

>> B=linspace(-1,1,length(x));

>> x=[72 82 97 103 113 117 126 127 127 139 154 159 199 207];

>> plot(B,H(x,B),'k',[-1 1],[0 0],'r');

>> syms B

>> y=H(x,B);

>> G=inline(vectorize(y),'B');

>> r=fzero(G,[0.1 0.2]);

>> H=inline('x.*log(sqrt(x.^2-1)+x)-sqrt(x.^2-1)-0.5*x','x');



>> x=linspace(1.01,4,500);

>> plot(x,H(x),'k',[0 4],[0 0],'r');

>> r=fzero(H,[2 3])


This problem is simple but has much long number as entering data.

For first part, just make an inline function with variable r and tau.

In the second part we have the values of p and tau. Multiply both side of

equation by r and now you have just one variable and you can use fsolve to

find the value of r then put this value and tau in equation of part one you

will find z(r,tau).

For part c do same procedure that done for part two, but this time put tau

as variable.

>> %case 1

>> G=inline('(2*log10(2.51./(1e5*sqrt(y))+0.27/200)).^-2-y','y');>> y=linspace(0,1,200);

>> plot(y,G(y),'k',[0 1],[0 0],'r')

>> r=fzero(G,[0.01,1]);

>> r=fzero(G,[0.01,1])

>> %case 2

>> G=inline('(2*log10((1e5*sqrt(y))/2.51)).^-2-y','y');

>> plot(y,G(y),'k',[0 1],[0 0],'r')

>> r=fzero(G,[0.01,1])




>> %As you see this equation has unexpected parenthesis or bracket.

>> %I can't find the correct equation, but solve the problem. >> %the procedure is correct, but the answer is different. It’s a simple problem.

>> G=inline( '(1-(1-cos(x))).*2.*0.6.^1.5.*cos(x)' ,'x');>> I=1/(2*pi)*quadl(G,-a,a)

>> H=inline('c1./(y.^5.*(exp(c2./(y.*T1))-1))','y','c1','c2','T1');

>> c1=3.742e8;c2=1.439e4;sig=5.667e-8;T1=300;

>> Int=quadl(H,1e-6,150,[],[],c1,c2,T1)

>> sig1=Int/T1^4;

>> %error

>> (sig-sig1)/(sig1) *100;

>>%you can do this procedure for other values of T.

>> H=inline('cos(x-y).*exp(-x.*y./(pi^2))','x','y');

>> I=dblquad(H,0,pi/2,pi/4,pi);



Ode45 differential equation solver (numerically)

function xp=myexp3(t,x)

L=150;m=70;k=10;co=0.00324;g=9.8;

xp=zeros(2,1);



xp(1)=x(2);

xp(2)=-k/m*(x(1)-L).*(x(1)>=L)-co*sign(x(2)).*(x(2).^2)+g;

end

>> [t,x]=ode45('myexp3',[0 15],[0 0]);

>> %the first column of x is distance and the second is velocity;

>> tnew=linspace(0,15,5000);>> distance=spline(t,x(:,1),tnew);

>> velocity=spline(t,x(:,2),tnew);

>> L=150;m=70;k=10;co=0.00324;g=9.8;

>> acceleration=-k/m*(distance-L).*(distance>L)-

co*sign(velocity).*(velocity.^2)+g;

>>%The last statement cannot be broken up as shown. It has been

>> %presented in two lines because of page width restrictions.

>>%the sign of answers depends on some other principles.>> %part a)

>> %using interpolation to find a value that lie between two others,

>> interp1(tnew,distance,11.47);>> %part b)

>> interp1(tnew,distance,5.988); interp1(tnew,velocity,5.988);

>> %part c) >> interp1(tnew,acceleration,11.18)


M=0;B=10;a=0.1;

xp=zeros(2,1);

xp(1)=x(2);

xp(2)=-a*x(2)+sin(x(1))-B*sin(x(1)).*(1-1./sqrt(5-4*cos(x(1))));

end

>> [t,x]=ode45('myexp5',[0 50],[pi/4 0]);

>> r=linspace(0,50,1000);

>> y=spline(t,x(:,1),r);

>> plot(r,y) %for plot theta as function of t

>> plot(x(:,1),x(:,2)) % for plot theta versus it’s derivative




p=0.375;q=7.4e-4;

xp=zeros(2,1);xp(1)=x(2);

xp(2)=-sign(x(2)).*p*(x(2).^2)-q*x(1);

end

>> [t,x]=ode45('myexp4',[0 150],[10 0]);

>> interp1(x(:,1),t,0);

>> [t,x]=ode45('myexp4',[0 300],[20 0]);

>> interp1(x((60:80),1),t(60:80),0);

>>%It seems that the answers of problem in book aren’t precise because

using another method to solve problem. You can check the correct answerafter plot it and zoom in. You can plot the curves with this script.

>> ode45('myexp4',[0 300],[20 0]);

function xp = halfsin(t,x,z)

if t<=5

h=sin(pi*t/5);

elseif t>5h=0;

end

xp=ones(2,1);

xp(1)=x(2);

xp(2)=-2*z*x(2)-x(1)+h;

end >> k=0;

>> for z=0.05:0.05:0.95

[t,x]=ode45(@halfsin,[0 35],[1 0],[],z);k=k+1;

f(k)=sum((x(:,1)-1).^2);

end

>> [x,y]=min(f);

>> z=0.05:0.05:0.95;

>> z(y);



>>%part a)

>> k=0;

>> for B=[0.02,0.05 , 0.08 0.11 0.15 0.18 0.23 0.3]syms x

H=B./sin(x)+1./cos(x);

H=inline(vectorize(H),'x');

t=fminbnd(H,0,pi/4);

k=k+1;

y(k)=t;

end>> disp('Corresponding min value of k with respect to B');disp([num2str(y')]);

>> %part b)

>> H=inline('0.16./sin(x)+1./cos(x)-1.5','x');>> plot(x,H(x),'k',[0 pi/4],[0 0],'r');

>> axis([0 pi/4 -1 1])

>> r1=fzero(H,[0.2 0.5]);r2=fzero(H,[0.5 0.7]);

>> disp([r1 r2]);

>> H=inline('sqrt(1.4/0.4)*sqrt(p.^(2/1.4)-p.^(2.4/1.4))','p');

>> p=linspace(0,1,1000);>> plot(p,H(p),'k',[0 1],[0 0],'r')

>> [H,index]=max(H(p));

>> maxvalue=p(index)maxvalue =

0.5285

>> k=1.4;

>> verify=(2/(k+1))^(k/(k-1))

verify =

0.5283

Note:

We use fsolve to solve nonlinear system of equations.

We have to define equations like a matrix in editor window as function.



For “fsolve” we have to indicate our variables as elements of a vector for

example if we have 3 equations and variables ‘x’, ’y’, ’z’, we can’t use all

of them in our equations. We select a variable like ‘h’ and use h(1) , h(2) ,

h(3) as our variables that ‘h’ refers to the vector after execution of fsolve

function .

%For part (a ) first define function in editor window as below:function w=myexp(e)

w=[e(1)*(1-cos(e(2)))-3;e(1)*(e(2)-sin(e(2)))-1];

end

>> option=optimset('display' , 'off');

>> z=fsolve(@myexp,[1 1],option);

>> %part b)

>> H=inline('3*(x-sin(x))-(1-cos(x))','x');

>> theta=fzero(H,1);>> k=3/(1-cos(theta))

function u=myexp2(x)

sig=5.667e-8;T1=373;T2=293;

u=[T1^4-x(1)^4-x(2)/sig; x(1)^4-x(3)^4-x(2)/sig;x(3)^4-T2^4-x(2)/sig];

end

>> option=optimset('display' , 'off');

>> z=fsolve(@myexp2,[10 10 10],option);

>> %part b)

>> A=[1 0 1/sig;1 -1 -1/sig;0 1 -1/sig];

>> B=[T1^4 0 T2^4];

>> x=inv(A)*B';

>> %where x(1)=Ta^4 x(2)=Tb^4

>> Ta=(x(1))^(1/4);Tb=(x(2))^(1/4);Q=x(3);

>> %part a)

>> syms x

>> limit((1-sin(2*x))^(1/x),x,0);



>> %part b)

>> limit(log(x^n)/(1-x^2),x,1)

>> syms a b w >> x=a+b*cos(w);

>> f=(1-exp(-x))/(1+x^3);

>> g=inline(vectorize(f),'a','b','w');

>> a=1.2;b=-0.45;w=pi/3;

>> g(a,b,w)

>> k=0;

>> for b=linspace(0,4*pi,10)

syms x

y=int((2*x+5)./(x.^2+4*x+5),x,0,b);

k=k+1;

h(k)=y;

end

>> disp(h)



Chapter Chapter Chapter Chapter 6 66 6



a. Cycloid

>> phi=linspace(-pi,3*pi,200);ra=[0.5 1 1.5];

>> for n=1:3

x=ra(n)*phi-sin(phi);y=ra(n)-cos(phi);subplot(2,2,n);

plot(x,y);

axis equal;

end

b. lemniscate

>> phi=linspace(-pi/4,pi/4,200);

>> x=cos(phi).*sqrt(2*cos(2*phi));

>> y=sin(phi).*sqrt(2*cos(2*phi));>> plot(x,y)

>> axis equal

c. Spiral

i

>> phi=linspace(0,6*pi,200);

>> x=phi.*cos(phi);

>> y=phi.*sin(phi);

>> plot(x,y)>> axis equal

ii

>> k=0.1

>> x=exp(k*phi).*cos(phi);

>> y=exp(k*phi).*sin(phi);

>> plot(x,y)

>> axis equal

d. cardioid


>> x=2*cos(phi)-cos(2*phi);

>> y=2*sin(phi)-sin(2*phi);

>> plot(x,y)

>> axis equal



e. Astroid


>> x=4*cos(phi).^3;

>> y=4*sin(phi).^3;

>> plot(x,y)

>> axis equal

f. Epicycloid

case 1


>> Rr=3;ar=[0.5 1 2];

>> for n=1:3

x=(Rr+1)*cos(phi)-ar(n)*cos(phi*(Rr+1));

y=(Rr+1)*sin(phi)-ar(n)*sin(phi*(Rr+1));subplot(2,2,n);

plot(x,y)

axis equal end

case 2

>> Rr=2.5;ar=2;phi=linspace(0,4*pi,200);

>> x=(Rr+1)*cos(phi)-ar*cos(phi*(Rr+1));

>> y=(Rr+1)*sin(phi)-ar*sin(phi*(Rr+1));

>> plot(x,y)

>> axis equal

g. Hypocycloid

>> Rr=3;ar=[0.5 1 2];phi=linspace(0,2*pi,200);

>> for n=1:3

x=(Rr-1)*cos(phi)+ar(n)*cos(phi*(Rr-1));

y=(Rr-1)*sin(phi)-ar(n)*sin(phi*(Rr-1));subplot(2,2,n);

plot(x,y)

axis equal

end



6.2

>> x=linspace(-15,15,50);

>> y=log(abs(sin(x)./x));>> plot(x,y);

>> x=linspace(-12,7,5000);

>> y=0.001*x.^5+0.01*x.^4+0.2*x.^3+x.^2+4*x-5;

>> n=find(y>0);

>> t=x(n);h=y(n);

>> plot(t,h)

>> dt=linspace(0,5,400);

>> ht=1:0.25:3;

>> [hht,ddt]=meshgrid(ht,dt);

>> c=0.5*ddt.^3-1.5*hht.*ddt.^2+(1+hht.^2).*ddt;

>> plot(dt,c)

>> axis([0 5 0 8])


% we can solve this problem without using for loop, but it maybe have toolong script.

function nonoverlapping(d,w,L)

b=atan((w/2)/(d+L));

a=atan(w/(2*d));

r1=sqrt(d^2+(w/2)^2);

r2=sqrt((d+L)^2+(w/2)^2);



t=2*pi/floor(pi/a);

for n=0:(floor(pi/a)-1)

p1=cos(n*t)*d ;p2= r1*cos(n*t+a);p3= sin(n*t)*d ;p4= r1*sin(n*t+a);x=[0 p1 p2 p2+L*cos(n*t) r2*cos(-b+(n*t)) r2*cos(-b+n*t)-L*cos(n*t) p1];

y=[0 p3 p4 p4+L*sin(n*t) r2*sin(-b+(n*t)) r2*sin(-b+n*t)-L*sin(n*t) p3];

plot(x,y,'k')hold on

end

axis equal

end

%case 1 >>v=0.3; % z/a=u

>>F=inline('-((1-u.*atan(1./u))*(1-v)-0.5*(1+(u).^2).^-1)','u','v');

>>plot(u,F(u,v));

>>plot(u,F(u,v));

>>hold on

>>xlabel('u');

>>ylabel('\it\sigma_{\rmx}\rm/ \itp_{\rmmax}');

>>%case 2

>>G=inline('-(1+u.^2)','u');>>plot(u,G(u));

>>hold on >>xlabel('u');

>>ylabel('\it\sigma_{\rmz}\rm/ \itp_{\rmmax}');

>>%case 3

>>H=inline('0.5*(-((1-u.*atan(1./u))*(1-v)-0.5*(1+(u).^2).^-1)- -

(1+u.^2))','u','v');

>>plot(u,H(u,v));

>>hold on

>>xlabel('u');>>ylabel('\it\tau_{\rmxy}\rm/ \itp_{\rmmax}\rm= \it\tau_{\rmyz}\rm/

\itp_{\rmmax}');

>>%These last statements cannot be broken up as shown. They have been

>> %presented in two lines because of page width restrictions.



%case 1

>>v=0.3; % z/b=u >>u=linspace(0,10,1000);>>F=inline('-2*v*(sqrt(1+u.^2)-u)','u','v');

>>plot(u,F(u,v));

>>hold on

>>xlabel('u');

>>ylabel('\it\sigma_{\rmx}\rm/ \itp_{\rmmax}');

>>%case 2

>>G=inline('-((2-1./(1+u.^2)).*sqrt(1+u.^2)-2*u)','u');

>>plot(u,G(u));

>>xlabel('u');>>ylabel('\it\sigma_{\rmy}\rm/ \itp_{\rmmax}');

>>%case 3

>>H=inline('-1./sqrt(1+u.^2)','u');

>>plot(u,H(u));

>>xlabel('u');

>>ylabel('\it\sigma_{\rmz}\rm/ \itp_{\rmmax}');

>>%case 4 >>K=inline('0.5*(-((2-1./(1+u.^2)).*sqrt(1+u.^2)-2*u)- -1./sqrt(1+u.^2))','u')

>>plot(u,K(u));>>xlabel('u');

>>ylabel('\it\tau_{\rmyz}\rm/ \itp_{\rmmax}');

>> A=[9.1209 9.1067 8.9939 8.9133 8.5194 8.3666];

>> B=[3.5605 3.5385 3.4777 3.4292 3.2621 3.1884];

>> SAE=10:10:60;

>> h=linspace(10,60,1000);>> Aa=spline(SAE,A,h);

>>Ba=spline(SAE,B,h);

>>T=linspace(0,500,1000);

>>To=255.2+5/9*T;

>> c=10.^(Aa-Ba.*log10(To));

>> u=10.^(c-1);



>> %case 1

>> subplot(1,2,1);

>> plot(To,u)

>> %case 2

>> subplot(1,2,2);

>> plot(log10(To),log10(log10(10*u)))

>>B=[0.02 0.05 0.08 0.11 0.15 0.18 0.23 0.3];

>>y=linspace(1,40,500);

>> [yy,BB]=meshgrid(y,B);

>>K=BB./sind(yy)+1./cosd(yy);

>>%recall from exercise 5.32 for finding min vaules >>k=0;

>>for B=[0.02,0.05 , 0.08 0.11 0.15 0.18 0.23 0.3]

syms x

H=B./sin(x)+1./cos(x);

H=inline(vectorize(H),'x');

t=fminbnd(H,0,pi/4);

k=k+1;

h(k)=t;

end >>h=h*180/pi;

>>u=linspace(h(1),h(end),500);

>>for n=1:length(h)

w(n)=interp1(y,K(n,:),h(n));

end

>>p=spline(h,w,u);

>>plot(u,p,'k--')

>>hold on >>plot(y,K,'k')

>>axis([0 40 1 2])

>>xlabel('\rm\lambda');

>>ylabel('\rm\kappa');

>>legend('Minimum','location','southwest')



>> rs=1.5*sin(pi/5)/(1-sin(pi/5));

>> t=linspace(0,2*pi,3000);

>> plot(1.5*cos(t),1.5*sin(t),'k',0,0,'k+')>> axis equal >> hold on

>> for n=0:4

a=cosd(72*n)*(1.5+rs);

b=sind(72*n)*(1.5+rs);

x=a+rs*cos(t);y=b+rs*sin(t);

plot(x,y,'k',a,b,'k+')

end

>>%we can make a function to do this procedure for desire value of ‘rb’and ‘n’ as below.

function ciraroundcir(rb, n)

rs=rb*sin(pi/n)/(1-sin(pi/n));

t=linspace(0,2*pi,3000);

plot(rb*cos(t),rb*sin(t),'k',0,0,'k+')

axis equal

hold on

for m=0:(n-1)a=cos(2*pi/n*m)*(rb+rs);

b=sin(2*pi/n*m)*(rb+rs);

x=a+rs*cos(t);y=b+rs*sin(t);

plot(x,y,'k',a,b,'k+')

end

end

>> t=linspace(0,2*pi,100);

>> plot([3+cos(t)],[1+sin(t)])>> axis equal

>> hold on

>> fill([3+cos(t)],[1+sin(t)],'g')



>> plot([0 5],[-5 5])

>> h=linspace(0,5,100);

>> i=fliplr(h);

>> fill([h,i],[2*h-5 , (zeros(1,length(i))-5)],'r')

>> axis([-2 5 -5 4])

>> r=0.5:0.5:3;>> x=1+cos(t)'*r;

>> y=1+sin(t)'*r;

>> plot(x,y,'b',1,1,'b+')

>> plot([-2 5],[0 0],'k ',[0 0],[-5 4],'k')

>> xlabel('\itx_{\rm1}')

>> ylabel('\itx_{\rm2}')

>> title('Blue circles:function minimized. Green area: feasible region')

>> x=linspace(0,pi,200);

>> plot(x,[sin(x) ;abs(sin(2*x))/2],'k')

>> hold on

>> fill([x,fliplr(x)],[sin(x),abs(sin(2*x))/2],'c')

>> axis([0 pi 0 1])

>> plot([0 pi],[0.5 0.5],'k--')

an engineer's guide to matlab,edward,2nd,[solution]by hamed .pdf

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