Use Matlab to plot x(t) for damped system

[Northbysouth]

Homework Statement

Plot x(t) for a damped system of natural frequency w_n= 2 rad/s and initial conditions x_0= 1 mm and v_0 = o mm/s, for the following values of the damping ratio: z= 0.01, 0.2, 0.6, 0.1, 0.4 and 0.8

Homework Equations

The Attempt at a Solution

I began by defining the values of my constants. I realize that z is not a constant per se but I'm not particularly knowledgeable about matlab

w_n=2; %rad/s
x_0=1; %mm
v_0=0; %mm/s
z= 0.01; %this value varies, thereby changing the response of the plot

Next I defined the time interval as well as some intermediate variables and the solution to a damped system.

t=[0:0.1:2*3.41];
w_d=w_n*sqrt(1-z^2);
x=x_0*exp(z*w_n*t_i)*sin(w_d*t_i);

With the above code I'm told:

Error using *
Inner matrix dimensions must agree.

I spoke to my professor about this and he explained that my exp and sin portions of x were 1x63 and 1x63 matrices, which doesn't work out. I've got the following lines of code which he gave me, but I'm still confused.

x=zeros(63,1);
for i=1:63
x(i)=x_0*exp(z*w_n*t(i))*sin(w_d*t(i));
end
plot(t,x)

Trying to use this code as it is just repeats:

Error using *
Inner matrix dimensions must agree.

Any help would be greatly appreciated

 

[Simon Bridge]

Error using *
Inner matrix dimensions must agree.

Matlab does everything as vectors - this allows for some shortcuts but also means you have to be mindful of the operations.

The "*" is a matrix inner-product operation.
I don't think that's what you want.

I suspect you need the ith term in the result to be the ith term in one vector multiplied by the ith term in another vector.
For that you use the ".*" (dot-star) operation instead.

i.e.
t=1:0.125:6;
v=sin(t);
u=6-t;

t,u, and v are 1x49 vectors.

If I now do y=u*v I get an error because the dimensions are wrong for the dot product.
If I do y=u*v' (making the dimensions right) I get 50.176 ... because that's the dot-product.
But that's not what I want: I want the amplitude of the sine function to decrease with t.

This means that I need $y=(y_1,y_2,\cdots,y_{49}): y_i=u_iv_i$

To get that I have to do: y=u.*v.

 

[Northbysouth]

Thank you so much. I'll admit I still don't fully understand. My final code was

w_n=2; %rad/s
x_0=1; %mm
v_0=0; %mm/s
z=0.8; %this value varies, thereby changing the response of the plot

t=[0:0.1:20];
w_d=w_n*sqrt(1-z^2);

x=x_0*exp(-z*w_n*t).*sin(w_d*t);

plot(t,x)

Which by changing z for each value produced the graphs I was looking for.

 

[Simon Bridge]

Well done.

It can take a while to wrap your head around how you do things in these math-script programs, but it's usually worth the effort. All the program is doing is storing your numbers in matrices - so that's what you have to get used to.

eg. You could have made z into a vector, then the output would be a matrix where each row is for a different z value. I've had to do this where I had about 50-100 such plots to do but you don't have anything like that kind of number so you may as well just do them one at a time like what you did ;)

Cheers.

 

[rezeew]

.

I would recommend checking the dimensions of your variables and making sure they are compatible for the operations you are trying to perform. In this case, it seems like your t variable is a 1x63 matrix while your x variable is a 63x1 matrix, so they cannot be multiplied together.

I would also suggest using the "hold on" function in Matlab to plot multiple curves on the same graph, so you can easily compare the responses for different values of z.

Here is a possible solution for your problem:

%Define constants
w_n=2; %rad/s
x_0=1; %mm
v_0=0; %mm/s

%Define time interval
t=[0:0.1:2*3.41];

%Initialize x matrix with zeros
x=zeros(63,6);

%Loop through different values of z
for i=1:6
%Calculate damped natural frequency
w_d=w_n*sqrt(1-z(i)^2);
%Calculate x(t) for the current value of z
x(:,i)=x_0*exp(z(i)*w_n*t).*sin(w_d*t);
end

%Plot x(t) for different values of z
plot(t,x)
legend('z=0.01','z=0.2','z=0.6','z=0.1','z=0.4','z=0.8')
xlabel('Time (s)')
ylabel('Displacement (mm)')
title('Response of a Damped System for Different Values of z')
grid on