How to apply Heaviside function over external force?

Dear all,

Suppose we have a harmonic external force in time-domain, and I need to apply this force for only 5 seconds and then get the system response for 100 seconds.

System is defied with all dynamic properties like mass, stuffness and damping ...,

• there is easier way to handle it via Simulink, but we need to script the code in time-domain.

The code we have, applies the external force during the entire time interval which is not our interest. We want to apply the excitation only for a short time interval like 3-5 seconds but not more.

Any ideas?

m = 1; % mass
c1 = 0.5; % damping 
k = 1; % stiffness 

ccr = 2*sqrt(m*k); % critical damping 
zeta1 = c1/ccr; % damping ratio 

w = sqrt(k/m); % w stands for natural frequency 

%% Bode plot and TF generation 
mytf1 = tf([0,01],[1,2*zeta1*w,w^2]) % creating the T.F. 

% time-domain 
v0 = 0.5; %I.C. in velocity in m/s
y0 = 0.05; % I.C. initial displacement in meteres 
t = linspace(0 , 100 , 1000);
wd = w*sqrt(1 - zeta1^2); % damped frequency 
yh = exp(-zeta1.*w.*t).*(y0.*cos(wd.*t)+((v0+zeta1.*w.*y0)/wd).*sin(wd.*t));
% define the Harmonic Excitation
we = 0.25
% frequency of the system - Resonance 
F0 = 0.1; % F0 is the amplitude of the harmonic excitations in meters 
F = F0.*exp(1j.*we.*t); 
yp = (F0 ./ (w^2 - we^2 + 2*zeta1*w*we)) .* exp(1j.*we.*t); 
y = yh + yp;
 
figure(2)
plot(t , y , 'b');
 hold on 
 plot(t , yp , 'r');
 hold on 
 plot(t , yh , 'k');
 hold off 
%  set(gca , 'FontSize' , 12)
 legend({'both transient and steady-state' , 'steady-state' , 'transient'} , 'FontSize' , 12)
grid on 

figure(3)
plot(t , yp)

figure(4)
plot(t , y)

Answer 1

[Walter Roberson]

The laplace of a rectangular pulse can be written like s*(1-exp(-s*h)) where h is the width of the pulse. This can be written s=tf('s') and then the above expression with numeric h value, getting a ss model as a result. Multiply by your tf to get a windowed version of the model.