Here is a mass-spring-damper system with an impulse response, h
and an arbitrary forcing function, f
(cos(t)
in this case). I am trying to use Matlab's FFT function in order to perform convolution in the frequency domain. I am expecting for the output (ifft(conv)
) to be the solution to the mass-spring-damper system with the specified forcing, however my plot looks completely wrong! So, i must be implementing something wrong. Please help me find my errors in my code below! Thanks
clear
%system parameters
m=4;
k=256;
c=1;
wn=sqrt(k/m);
z=c/2/sqrt(m*k);
wd=wn*sqrt(1-z^2);
w=sqrt(4*k*m-c^2)/(2*m);
x0=0; %initial conditions
v0=0;
%%%%%%%%%%%%%%%%%%%%%%%%%
t=0:.01:2*pi ;%time vector
f=[cos(t),zeros(1,length(t)-1)]; %force f
F=fft(f);
h=[1/m/wd*exp(-z*wn*t).*sin(wd*t),zeros(1,length(t)-1)]; %impulse response
H=fft(h);
conv=H.*F; %convolution is multiplication in freq domain
plot(0:.01:4*pi,ifft(conv))
To see what is expected run this code. Enter in cos(t); 4; 1; 256 for the inputs. You can see that it reaches a steady state at an amplitude much different than the plot generated from the above FFT code.
%%%FOR UNDERDAMPED SYSTEMS
func=input('enter function of t---> ','s');
m=input('mass ');
c=input('c ');
k=input('k ');
z=c/2/sqrt(k*m);
wn=sqrt(k/m);
wd=wn*sqrt(1-z^2);
dt=.001;
tMax=20;
x0=0;
v0=0;
t0=0;
t=0:dt:tMax;
X(:,1)=[x0;v0;t0];
functionForce=inline(func);
tic
for i=1:length(t)-1
a=([0, 1, 0; -wn^2, -2*z*wn, 0; 0,0,0]*[X(1,i);X(2,i);X(3,i)]+[0;functionForce(X(3,i));0]);
Xtemp=X(:,i)+[0;0;dt/2] + a*dt/2;
b=([0, 1, 0; -wn^2, -2*z*wn, 0; 0,0,0]*[Xtemp(1);Xtemp(2);Xtemp(3)]+[0;functionForce(X(3,i));0]);
Xtemp=Xtemp+[0;0;dt/2] + b*dt/2;
c=([0, 1, 0; -wn^2, -2*z*wn, 0; 0,0,0]*[Xtemp(1);Xtemp(2);Xtemp(3)]+[0;functionForce(X(3,i));0]);
Xtemp=Xtemp + [0;0;dt] +c*dt;
d=([0, 1, 0; -wn^2, -2*z*wn, 0; 0,0,0]*[Xtemp(1);Xtemp(2);Xtemp(3)]+[0;functionForce(X(3,i));0]);
X(:,i+1)=X(:,i)+(a+2*b+2*c+d)*dt/6+[0;0;dt];
end
toc
figure(1)
axis([0 tMax min(X(1,:)) max(X(1,:))])
plot(t,X(1,:))