Question

Consider the system 0.28 +0.2x + 2x = f(t), where f(t) is a unit step function, and the initial conditions are zero. (a) Use MATLAB to plot the step response of the system. Attach the MATLAB plot and the script used to create the plot. Using only the printed MATLAB plot, extract by hand (i.e. use a ruler) the (b) maximum percent overshoot, (c) rise time, and (d) peak time, and then using the extracted values, estimate the (e) damping ratio . (f) natural frequency Wy, and (g) damped natural frequency

Answer 1

MATLAB CODE:

%This program calculates the step response of a single degree of freedom system

a=input('Enter coefficient of ddx : ');
b=input('Enter coefficient of dx : ');
c=input('Enter coefficient of x : ');
wn=sqrt(c/a);
zeta=b/(2*a*wn);
tf=10;
npts=1000;
t=linspace(0,tf,npts);
to=input('Enter time instant for step input : ');;
k=(wn)^2;
Fm=input('Enter a force magnitude : ');
Fint=Fm*ones(1,npts);
qtest=t>to;
fmult=[qtest];
Fo=Fint.*fmult; % The force matrix.
wd=wn*sqrt(1-zeta^2);
phi=atan2(zeta,sqrt(1-zeta^2));
A=Fo(1,:)/k;
B=Fo(1,:)/(k*sqrt(1-zeta^2));
x=A-B.*exp(-zeta*wn*t).*cos(wd*t-phi);
plot(t,x)
title(['Response for wn=',num2str(wn),', Fmax=',num2str(Fm(1)),', and time=', num2str(to)]);
ylabel('Response x')
xlabel('Time, seconds')
grid

INPUT FOR TEST RUN:

Command Window >> Untitled3 Enter coefficient of ddx : 0.2 Enter coefficient of dx : 0.2 Enter coefficient of x : 2 Enter time instant for step input : 1 Enter a force magnitude : 5

OUTPUT OF ABOVE TEST RUN:

- X Insert File Edit View DOS Figure 1 Tools Desktop Window Help A E DE O Response for wn=3.1623, Fmax=5, and time=1 Response x 0 1 2 3 7 8 9 10 4 5 6 Time, seconds