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NAME:_StevenSmith______________
LABDAYandTIME:_SummberB_10:10AMMF_
Instructor:____ProfessorZhu________
Problem1:
a.)T=2pi/omega0so2pi/1.4=1.49
apha=arctan(c*omega)/(omega0^2omega^2)soarctan(1*1.4)/(2^21.4^2)=.601radians
functionLAB06ex1
clc
omega0=2;c=1;omega=1.4;
param=[omega0,c,omega];
t0=0;y0=0;v0=0;Y0=[y0;v0];tf=50;
options=odeset('AbsTol',1e10,'relTol',1e10);
[t,Y]=ode45(@f,[t0,tf],Y0,options,param);
y=Y(:,1);v=Y(:,2);
figure(1)
plot(t,y,'b');ylabel('y');gridon;
t1=25;i=find(t>t1);
C=(max(Y(i,1))min(Y(i,1)))/2;
disp(['computedamplitudeofforcedoscillation='num2str(C)]);
Ctheory=1/sqrt((omega0^2omega^2)^2+(c*omega)^2);
disp(['theoreticalamplitude='num2str(Ctheory)]);
%
functiondYdt=f(t,Y,param)
y=Y(1);v=Y(2);
omega0=param(1);c=param(2);omega=param(3);
dYdt=[v;cos(omega*t)omega0^2*yc*v];
b.)Itisanexponentiallydecreasingoscillationwhereitdiesoutaroundt=10
functionLAB06ex1
clc
omega0=2;c=1;omega=1.4;
param=[omega0,c,omega];
t0=0;y0=0;v0=0;Y0=[y0;v0];tf=50;
options=odeset('AbsTol',1e10,'relTol',1e10);
[t,Y]=ode45(@f,[t0,tf],Y0,options,param);
y=Y(:,1);v=Y(:,2);
figure(1)
plot(t,y,'b');ylabel('y');gridon;
t1=25;i=find(t>t1);
C=(max(Y(i,1))min(Y(i,1)))/2;
disp(['computedamplitudeofforcedoscillation='num2str(C)]);
Ctheory=1/sqrt((omega0^2omega^2)^2+(c*omega)^2);
disp(['theoreticalamplitude='num2str(Ctheory)]);
alpha=atan(c*omega/(omega0^2omega^2));
yc=yCtheory*cos(omega*talpha);
NAME:_StevenSmith______________
LABDAYandTIME:_SummberB_10:10AMMF_
Instructor:____ProfessorZhu________
figure(2)
plot(t,yc,'b');gridon;title('complementarysolution');
%
functiondYdt=f(t,Y,param)
y=Y(1);v=Y(2);
omega0=param(1);c=param(2);omega=param(3);
dYdt=[v;cos(omega*t)omega0^2*yc*v];
Problem2:
a.)
functionLAB06ex2
omega0=2;c=1;
OMEGA=1:0.02:3;
C=zeros(size(OMEGA));
Ctheory=zeros(size(OMEGA));
t0=0;y0=0;v0=0;Y0=[y0;v0];tf=50;t1=25;
fork=1:length(OMEGA)
omega=OMEGA(k);
param=[omega0,c,omega];
[t,Y]=ode45(@f,[t0,tf],Y0,[],param);
i=find(t>t1);
C(k)=(max(Y(i,1))min(Y(i,1)))/2;
Ctheory(k)=1/sqrt((omega0^2omega^2)^2+(c*omega)^2);
end
figure(2)
plot(OMEGA,C,OMEGA,Ctheory,'ro');gridon;
xlabel('\omega');ylabel('C');
%
functiondYdt=f(t,Y,param)
y=Y(1);v=Y(2);
omega0=param(1);c=param(2);omega=param(3);
dYdt=[v;cos(omega*t)omega0^2*yc*v];
b.)Themaximumvalueforamplitudeisapproximately1.8.0.517isthecorrespondingCvalue.
c.)C=1√(ω0^2−ω^2)^22+c2ω2,whenwetakethederivativewefindthatomega=1.87
d.)Itcanbeseenthattheamplitudeoftheforcedoscillationhasincreasedcomparedto
problem1.
functionLAB06ex1
clc
omega0=2;c=1;omega=1.87;
param=[omega0,c,omega];
t0=0;y0=0;v0=0;Y0=[y0;v0];tf=50;
options=odeset('AbsTol',1e10,'relTol',1e10);
NAME:_StevenSmith______________
LABDAYandTIME:_SummberB_10:10AMMF_
Instructor:____ProfessorZhu________
[t,Y]=ode45(@f,[t0,tf],Y0,options,param);
y=Y(:,1);v=Y(:,2);
figure(1)
plot(t,y,'b');ylabel('y');gridon;
t1=25;i=find(t>t1);
C=(max(Y(i,1))min(Y(i,1)))/2;
disp(['computedamplitudeofforcedoscillation='
num2str(C)]);
Ctheory=1/sqrt((omega0^2omega^2)^2+(c*omega)^2);
disp(['theoreticalamplitude='num2str(Ctheory)]);
alpha=atan(c*omega/(omega0^2omega^2));
yc=yCtheory*cos(omega*talpha);
figure(2)
plot(t,yc,'b');gridon;title('complementarysolution');
%
functiondYdt=f(t,Y,param)
y=Y(1);v=Y(2);
omega0=param(1);c=param(2);omega=param(3);
dYdt=[v;cos(omega*t)omega0^2*yc*v];
e.)bychangingtheinitialconditionsitcanbeseen
thatthecomplementarysolutionischangedbutinthe
longtermtheamplitudeisnotaffected.
functionLAB06ex1
clc
omega0=2;c=1;omega=1.87;
param=[omega0,c,omega];
t0=0;y0=1;v0=1;Y0=[y0;v0];tf=50;
options=odeset('AbsTol',1e10,'relTol',1e10);
[t,Y]=ode45(@f,[t0,tf],Y0,options,param);
y=Y(:,1);v=Y(:,2);
figure(1)
plot(t,y,'b');ylabel('y');gridon;
t1=25;i=find(t>t1);
C=(max(Y(i,1))min(Y(i,1)))/2;
disp(['computedamplitudeofforcedoscillation='num2str(C)]);
Ctheory=1/sqrt((omega0^2omega^2)^2+(c*omega)^2);
disp(['theoreticalamplitude='num2str(Ctheory)]);
alpha=atan(c*omega/(omega0^2omega^2));
yc=yCtheory*cos(omega*talpha);
figure(2)
plot(t,yc,'b');gridon;title('complementarysolution');
%
NAME:_StevenSmith______________
LABDAYandTIME:_SummberB_10:10AMMF_
Instructor:____ProfessorZhu________
functiondYdt=f(t,Y,param)
y=Y(1);v=Y(2);
omega0=param(1);c=param(2);omega=param(3);
dYdt=[v;cos(omega*t)omega0^2*yc*v];
Problem3.
a.)Omega0isthevalueyieldingthemaximumamplitude
functionLAB06ex2
omega0=2;c=0;
OMEGA=1:0.02:3;
C=zeros(size(OMEGA));
Ctheory=zeros(size(OMEGA));
t0=0;y0=0;v0=0;Y0=[y0;v0];tf=50;t1=25;
fork=1:length(OMEGA)
omega=OMEGA(k);
param=[omega0,c,omega];
[t,Y]=ode45(@f,[t0,tf],Y0,[],param);
i=find(t>t1);
C(k)=(max(Y(i,1))min(Y(i,1)))/2;
Ctheory(k)=1/sqrt((omega0^2omega^2)^2+(c*omega)^2);
end
figure(2)
plot(OMEGA,C,OMEGA,Ctheory,'ro');gridon;
xlabel('\omega');ylabel('C');
%
functiondYdt=f(t,Y,param)
y=Y(1);v=Y(2);
omega0=param(1);c=param(2);omega=param(3);
dYdt=[v;cos(omega*t)omega0^2*yc*v];
b.)Whenomegaissetto2,theamplitudeapproachesinfinity.
NAME:_StevenSmith______________
LABDAYandTIME:_SummberB_10:10AMMF_
Instructor:____ProfessorZhu________
Problem4.
functionLAB06ex1
clc
omega0=2;c=0;omega=1.8;
param=[omega0,c,omega];
t0=0;y0=0;v0=0;Y0=[y0;v0];tf=100;
options=odeset('AbsTol',1e10,'relTol',1e10);
[t,Y]=ode45(@f,[t0,tf],Y0,options,param);
y=Y(:,1);v=Y(:,2);
figure(1)
plot(t,y,'b');ylabel('y');gridon;
t1=25;i=find(t>t1);
C=(max(Y(i,1))min(Y(i,1)))/2;
disp(['computedamplitudeofforcedoscillation='num2str(C)]);
Ctheory=1/sqrt((omega0^2omega^2)^2+(c*omega)^2);
disp(['theoreticalamplitude='num2str(Ctheory)]);
alpha=atan(c*omega/(omega0^2omega^2));
yc=yCtheory*cos(omega*talpha);
figure(2)
plot(t,yc,'b');gridon;title('complementarysolution');
%
functiondYdt=f(t,Y,param)
y=Y(1);v=Y(2);
omega0=param(1);c=param(2);omega=param(3);
dYdt=[v;cos(omega*t)omega0^2*yc*v];
a.)functionLAB06ex1
clc
omega0=2;c=0;omega=1.8;
param=[omega0,c,omega];
NAME:_StevenSmith______________
LABDAYandTIME:_SummberB_10:10AMMF_
Instructor:____ProfessorZhu________
t0=0;y0=0;v0=0;Y0=[y0;v0];tf=100;
options=odeset('AbsTol',1e10,'relTol',1e10);
[t,Y]=ode45(@f,[t0,tf],Y0,options,param);
y=Y(:,1);v=Y(:,2);
figure(1)
plot(t,y,'b');ylabel('y');gridon;
C=1/abs(omega0^2omega^2);
A=2*C*sin((omega0omega)*t/2);
holdon
plot(t,A,'r',t,A,'g');
t1=25;i=find(t>t1);
C=(max(Y(i,1))min(Y(i,1)))/2;
disp(['computedamplitudeofforcedoscillation='
num2str(C)]);
Ctheory=1/sqrt((omega0^2omega^2)^2+(c*omega)^2);
disp(['theoreticalamplitude='num2str(Ctheory)]);
alpha=atan(c*omega/(omega0^2omega^2));
yc=yCtheory*cos(omega*talpha);
figure(2)
plot(t,yc,'b');gridon;title('complementarysolution');
%
functiondYdt=f(t,Y,param)
y=Y(1);v=Y(2);
omega0=param(1);c=param(2);omega=param(3);
dYdt=[v;cos(omega*t)omega0^2*yc*v];
b.)T=4pi/(omega0+omega)4pi/3.8=3.307,byzoominginwecanseethisvalueisaproximatly
3.5.
c.)thelengthofthebeatsare4pi/(omega0omega)4pi/.2=20pi.anditishalfthatso10pi
Byzoominginwecanseethisvalueiscorrectat31.4
d.)wecanseethatbydecreasingomegawealsodecreasethemaximumamplitude
NAME:_StevenSmith______________
LABDAYandTIME:_SummberB_10:10AMMF_
Instructor:____ProfessorZhu________
e.)omega=.5itseemstobeoscillatingbackandforthtooquicklytoproducethebeats.
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