Free Vibration with Damping Modelling

The following code was written based on the theory provided in the following link: http://en.wikipedia.org/wiki/Vibration

clc
clear
C=1e-4;
Cc=10e-4;
K=1;
ksi=C/Cc
m=0.01;
Cc=2*m*(K/m)^0.5;
wn=(K/m)^0.5;
eqn2= 'm*D2x+C*Dx+K*x=0';
init2= 'x(0)=0,Dx(0)=1';
x=dsolve(eqn2,init2,'t')
t=linspace(0,1500,100);
z=eval(vectorize(x));
plot(t,z,'-*')
xlabel('t (sec)')
ylabel('x (m)')
grid on

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Damped and Undamped Natural Frequencies

The following code was written based on the theory provided in the following link: http://en.wikipedia.org/wiki/Vibration

clc
clear
x0=1;
c=2e-5;
k=1e-6;
m=0.01;
cc=2*(k*m)^0.5;
ksi=c/cc;
t0=0;
tt=8;
M=100
dt=(tt-t0)/M;
fn=1;
wn=2*pi*fn;
fi=0;
for i=1:M;
    t(i)=i*dt;
    b=(1-ksi^2)^0.5;
    xx(i)=x0*exp(-ksi*wn*t(i));
    xxx(i)=-x0*exp(-ksi*wn*t(i));
    x(i)=xx(i)*cos(b*wn*t(i)-fi);
end
plot(t,x,'-x')
xlabel('t (sec)')
ylabel('x (m)')
grid on
hold on
plot(t,xx,'r-')
hold on
plot(t,xxx,'r-')

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Forced Vibration with Camping

The following code was written based on the theory provided in the following link: http://en.wikipedia.org/wiki/Vibration

clc
clear
M=40;
fn=2;
F0=1;
K=1e3;

ksi0=0;
ksi00=1;
N=7;
dksi=(ksi00-ksi0)/N;
ksi(1:M,1)=0;
for j=1:N;
ksi(1:M,j+1)=ksi(1:M,j)+dksi;
end

f0=0;
ff0=5;
df=(ff0-f0)/M;
f(1,1:M)=0;
for i=1:M;
f(i+1,1:N)=f(i,1:N)+df;
end

for j=1:N;
for i=1:M;
    r(i,j)=f(i,j)/fn;
    fii(i,j)=atan((2*ksi(i,j)*r(i,j)/(1-r(i,j)^2)));
    fi(i,j)=(180/pi)*fii(i,j);
    bb(i,j)=(1-r(i,j)^2)^2+(2*ksi(i,j)*r(i,j))^2;
    bbb(i,j)=1/bb(i,j)^0.5;
    x0(i,j)=(F0/K)*bbb(i,j);
    xx0(i,j)=(K/F0)*x0(i,j);
end
end

for j=1:N;
plot(r(:,j),xx0(:,j),'-')
hold on
end
xlabel('Frequency Ratio (r)')
ylabel('Amplification Ratio (X k/F0)')
grid on
hold off
figure(2)
for j=1:N;
plot(r(1:M,j),fi(1:M,j),'-')
hold on
end
xlabel('Frequency Ratio (r)')
ylabel('Phase Angle (deg) (fi)')
grid on

The second plot didn't mimic the original one because after Angle 90 deg there should be addition to 180 instead of the negative values.

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