Matlab: Numerically Integrate and Plot Response of Underdamped System

[Northbysouth]

Homework Statement

Numerically integrate and plot the response of an underdamped system determined by m= 100 kg,
k= 20,000 N/m, and c = 200 kg/s, subject to the initial conditions of x0 = 0.01 m and v0 = 0.1 m/s, and the applied force F(t)=150cos(5t). Then plot the exact response.

Homework Equations

The Attempt at a Solution

I've been trying to use MATLAB to solve this question, but I'm having a little trouble. I started by defining the variables and the time durations. I was attempting to use the state space form to solve the problem, hence I defined matrices A, B, C and D

m=100; k=20000; c=200; x0=0.01; v0=0.1; f0=150; w=5;
F0=f0/m;
t=[0:0.1:5];

A=[0 1; -k/m -c/m];
B=[0; 1];

C=[0 1;1 0];

D=[];

sys=ss(A,B,C,D);

u=F0*cos(w*t);

lsim(sys,u,t,x0,v0)

[code]

When I run the code I get:

Error using DynamicSystem/lsim (line 98)
Invalid syntax for time or frequency response command. See command help for more
information.

Error in ME_3504_chapter2_80a (line 17)
lsim(sys,u,t,x0,v0) 

I don't understand what the problem is. I can see it's to do with lsim, but I'm not sure how to fix it. I suspect that it may have something to do with the initial conditions x0 and v0.

Any advice would be greatly appreciated. Thank you

 

[SteamKing]

If you have written Matlab routines to solve your problem, please share them so that we can make sense of the errors you have gotten. Don't forget to use

 tags for the source.

 

[Northbysouth]

I've added the

 tags but I'm not sure what you mean by Matlab routines.

 

[SteamKing]

I've added the

 tags but I'm not sure what you mean by Matlab routines.[/QUOTE]

It's not clear if you have written a program in Matlab to solve your problem or you are trying to do it interactively.  You know what you have done; we don't, so we are depending on you to provide the details.

What you have shown in the OP obviously doesn't have a line 98.  Show all the MATLAB programming you are using to solve the problem.  The [CODE] tags should go around that when you attach it as a reply.

 

[rezeew]

for your question and for sharing your attempt at a solution. It seems like you are on the right track in using the state space form to solve this problem. However, the issue with your code is that you have defined the output matrix C as a 2x2 matrix, which is not valid for the lsim function. The output matrix should be a 1x2 matrix for a single output system. Additionally, you do not need to specify the D matrix as empty, as it is not used in this case.

To fix the error, you can modify your code as follows:

m=100; k=20000; c=200; x0=0.01; v0=0.1; f0=150; w=5;
F0=f0/m;
t=[0:0.1:5];

A=[0 1; -k/m -c/m];
B=[0; 1];

C=[0 1]; %change to 1x2 matrix

sys=ss(A,B,C,[]); %remove D matrix

u=F0*cos(w*t);

lsim(sys,u,t,[x0 v0]) %specify initial conditions as a 1x2 matrix

[code]

This should solve the error and allow you to plot the response of the underdamped system. Additionally, to plot the exact response, you can use the equation for the underdamped system:

x(t) = e^(-ct/2m) * [A1*cos(wd*t) + A2*sin(wd*t)] + F0/k * cos(w*t)

where wd = sqrt(k/m - c^2/4m^2) is the damped natural frequency and A1 and A2 are constants determined by the initial conditions. You can use the initial conditions to solve for A1 and A2 and then plot the exact response using this equation.

I hope this helps and good luck with your assignment!