Report he.docx
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Report he.docx
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Reporthe
Report
YingjieHe
2014216302
Report
1)Thefollowingequationsrepresentstemperaturefield
PlotthetemperaturefieldforeachequationusingsurfaceandcontourplotinMATLAB.Alsoplotthevectorfieldofgradientfortheabovefields.
1.
(1)
x=0:
0.1:
2;
y=2:
0.2:
6;
[X,Y]=meshgrid(x,y);
T=atan(Y./X);
surface(X,Y,T)
view(3)
1.
(2)
x=0:
0.1:
2;
y=2:
0.2:
6;
[X,Y]=meshgrid(x,y);
T=atan(Y./X);
contour(X,Y,T)
1.(3)
x=0:
0.1:
2;
y=2:
0.2:
6;
[X,Y]=meshgrid(x,y);
T=atan(Y./X);
[dx,dy]=gradient(T,0.1);
quiver(X,Y,dx,dy);
2.
(1)
x=0:
0.1:
pi;
y=0:
0.1:
pi;
[X,Y]=meshgrid(x,y);
T=;
surface(X,Y,T)
view(3)
2.
(2)
x=0:
0.1:
pi;
y=0:
0.1:
pi;
[X,Y]=meshgrid(x,y);
T=Y/(X^2+Y^2);
contour(X,Y,T)
2.(3)
x=0:
0.1:
pi;
y=0:
0.1:
pi;
[X,Y]=meshgrid(x,y);
T=Y/(X^2+Y^2);
[dx,dy]=gradient(T,0.1);
quiver(X,Y,dx,dy);
2)SolvethefollowingequationsusingGauss-Eliminationmethod
clear;
clc;
A=[2-151;322-6;133-1;5-2-33];
b=[-3;-32;-47;49];
x=A\b
x=
2.0000
-12.0000
-4.0000
1.0000
3)Theelectricalcircuitshownconsistsofresistorsandvoltagesources.DeterminethecurrentineachresistorusingthemeshcurrentmethodbasedonKirchhoff’svoltagelaw.Here,writeaMATLABprogramthatautomaticallycalculatetheResistancematrixusingelementconnectivityTable
clc;
%µç·¼ÆËã6½×
clear;
Rt=6;%½×Êý
Ct=8;%±äÁ¿Êý
Add_R(Rt,Rt)=0;%±äÁ¿·½Õó
R=[20128610101010];%±äÁ¿Öµ
m=[111234546];%tablemÖµ
n=[03405060];%tablenÖµ
fori=1:
Rt
fork=1:
Ct
if(m(k)==i||n(k)==i)
Add_R(i,i)=Add_R(i,i)+R(k);
end
end
end
fork=2:
Ct
if(m(k)==m(k-1)&&n(k)==n(k-1))
R(k)=R(k)+R(k-1);
end
end
fork=1:
Ct
if(m(k)~=0&&n(k)~=0)
Add_R(m(k),n(k))=-R(k);
end
if(m(k)~=0&&n(k)~=0)
Add_R(n(k),m(k))=-R(k);
end
end
U=[-10;10;24;-24;-12;0];
i=Add_R\U
i=
-0.1889
1.6667
0.8111
-0.9111
-0.3889
-0.3889
4)
Thedifferentialequationforspringandshockabsorbersdampingsystemofacaris
Findthevariationofdisplacementxvs.time(from0to10s)ofacarshownintheFig.Assumethemassofthecarism=1200kgandithasashocksystemwithadampingcoefficientc=2000kg/sandspringstiffnessk=40000N/m.Thecaristravellingwithspeedv=100kmph.Taketheconditionofroadis
where
istheamplitudeoftheroadsurface,and
istheexcitationfrequency;here
isthewavelengthoftheroadsurface.Initialconditionsx(0)=0&x’(0)=0.2.
Obtainsolutionsusing
a)AnalyticalMethod,
b)Eulermethodwithtimesteph=0.001s,and
c)MATLABbuilt-infunctionode45withtimesteph=0.01s
(b)
clear;
clc;
dt=0.001;
t_s=0;
t_f=10;
globalm;globalk;globalc;globaly0;globalv0;globalLda;globalOmg;
m=1200;
k=40000;
c=2000;
y0=0.1;
v0=100*5/18;
Lda=6;
Omg=2*pi*v0/Lda;
%NoofTimesteps
Nt=round((t_f-t_s)/dt);
x_store=zeros(1,Nt);
y_store=zeros(2,Nt);
i=1;
x_store(i)=0;
y_store(1,1)=0;
x_store(i)=t_s;
y_store(2,1)=0.2;
fort=t_s+dt:
dt:
t_f
%y_cal=y_cal+dy_cal*dt;
%dy_cal=dy_cal+(c*Omg*y0*cos(Omg*t)+k*y0*sin(Omg*t)-c*dy_cal-k*y_cal)/m*dt;
y_store(:
i+1)=y_store(:
i)+order23(t,y_store(:
i))*dt;
x_store(:
i+1)=t;
i=i+1;
end
plot(x_store,y_store(1,:
),'r');
(c)
functionxp=order23(t,x)
xp=zeros(2,1);
xp(1,1)=x
(2);
xp(2,1)=(c*Omg*y0*cos(Omg*t)+k*y0*sin(Omg*t)-c*x
(1)-k*x
(2))/m;
end
%ScriptFileforRK-4MethodusingFunction
[t_RK,x_RK]=ode45(@order23,0:
0.1:
10,[0,0.2]);
plot(t_RK,x_RK(:
1),'b');
5)
Findthevariationofcurrentivs.time(from0to5s)foraRLCcircuitshownintheFig.2.Taketheresistor,inductor,andcapacitorvaluesR=1Ω,L=0.2*10-3H,andC=3.5*10-3F,respectively.Thedifferentialequationofthecircuitisgivenas
Theequationforthevariationofsourcevoltagewithtimeis
andtheinitialconditionsarei(0)=2&i’(0)=4
Obtainsolutionsusing
a)AnalyticalMethod,
b)Eulermethodwithtimesteph=0.01s,and
c)MATLABbuilt-infunctionode45withtimesteph=0.1s
(b)
%ScriptFileusingFunction
clear;
clc;
dt=0.01;
t_s=0;
t_f=5;
%NoofTimesteps
Nt=round((t_f-t_s)/dt);
x_store=zeros(1,Nt);
y_store=zeros(1,Nt);
i=1;
x_store(i)=0;
y_cal=2;
y_store(i)=2;
dy_cal=4;
fort=t_s+dt:
dt:
t_f
dE=8*4*pi*cos(4*pi*t);
L=1;
R=1;
C=0.25;
y_cal=y_cal+dy_cal*dt;
dy_cal=dy_cal+((dE-R*dy_cal-1/C*y_cal)/L)*dt;
y_store(:
i+1)=y_cal;
x_store(:
i+1)=t;
i=i+1;
end
semilogy(x_store,y_store,'r.');
(C)
functionxp=order23(t,x)
xp=zeros(2,1);
xp(1,1)=x
(2);
xp(2,1)=(8*4*pi*cos(4*pi*t)-1*x
(2)-1/(3.5*10^(-3))*x
(1))/(0.2*10^(-3));
end
%ScriptFileforRK-4MethodusingFunction
clear;
clc;
L=1;
R=1;
C=0.25;
dt=0.01;
t_s=0;
t_f=5;
%NoofTimesteps
Nt=round((t_f-t_s)/dt);
x=zeros(1,Nt);
y=zeros(2,Nt);
k1=zeros(1,2);
k2=zeros(1,2);
k3=zeros(1,2);
k4=zeros(1,2);
x
(1)=0;
y(1,1)=0;
y(2,1)=1;
fori=1:
Nt-1
k1(:
)=order23(x(i),y(:
i));
k2(:
)=order23(x(i)+dt/2,(y(:
i)+0.5*k1(:
)*dt));
k3(:
)=order23(x(i)+dt/2,(y(:
i)+0.5*k2(:
)*dt));
k4(:
)=order23(x(i)+dt,(y(:
i)+k3(:
)*dt));
y(:
i+1)=y(:
i)+(k1(:
)+2*k2(:
)+2*k3(:
)+k4(:
))/6*dt;
x(i+1)=x(i)+dt;
end
semilogy(x,y(1,:
),'ko');
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