I need to obtain second derivative from a 2nd order ode

Hi, I solved following 2 coupled 2nd order odes:

m*x"+c1*x'+k1*x=F1

m*y"+c2*y'+k2*y=F2

I used ode45 and obtained x,x',y and y'.

I need to obtain x" too. Can anyone please tell me how can I do so?

Answer 1

[Friedrich]

So, after rethinking the whole problem it looks that you are doing it 100% right.

You split up the 2nd order ODE into a coupled system of 1st order ODEs. After solving this system you get x, x’, y and y’. Now you want x’’ and y’’. This can be obtained from x’,x,y and y’ since you have a representation of x’’and y’’ through x,x’,y and y’. So this 2nd order ODE gives

m*x"+c1*x'+k1*x=F1

x‘‘ = (F1-ci*x‘-k1*x)/m

Since you know x and x‘ you can get x‘‘.

[Shantanu Chhaparia]

What if the second order derivative is a function of some other second order derivatives. I basically have a 14 dof second order system I want to obtain all the 14 second order derivatives but they are function of other 2nd order dof. So how to go about that?

[Torsten]

Solve your system for the 2nd order derivatives of the variables.

Usually, the system can be written as

M(t,y,y') * y'' = f(t,y,y')

with a matrix M and a vector f both depending only on t,y, and y'

Then

y'' = M\f

Answer 2

[Friedrich]

Hi,

Matlab can solve ODE of first order only. You have to convert your 2nd order ODE to a sytem of coupled 1st order ODE’s. How this is done can be obtained here:

http://www.youtube.com/watch?v=fx3bl4oA_0U

[Arti]

Thanks.

I reduced them to first order as:

dz(1)= z(2);

dz(2)=(F1-c1*z(2)-k1*z(1))/m;

dz(3)= z(4);

dz(4)=(F2-c2*z(4)-k2*z(3))/m;

So, z(:,1)=x, z(:,2)=x' , z(:,3)=y , z(:,4)=y'

Now if I want to obtain x", can MATLAB do it?

Answer 3

[Friedrich]

I would suggest reading this article:

http://www.math.tamu.edu/REU/comp/matode.pdf

Your code should look similar to this:

function example

x_0 = [2,2];
tspan = [0,20];
[t,x] = ode45(@my_func,tspan,x_0);
plot(t,x);

function xbar = my_func( t,x )

F_1 = 10;
c_1 = 2;
k_1 = 1;
m = 5;

xbar = [x(2); (F_1 - c_1*x(2) - k_1*x(1))/m];

end

end

Where x(1) is x’ and x(2) is x’’.

[Arti]

Thanks!!

But c1 shd be multiplied with x' and k1 with x. I need all three of them, x, x' and x".

here is my code:

function dz=znew(t,z)

m=7.5;

c_x=1880;

c_y=2160;

k_x=27300;

k_y=30200;

F_0x=460;

F_0y=350;

w=2;

dz=zeros(size(z));

dz(1)=z(2);

dz(2)=(sin(w*t)*F_0x-c_x*z(2)-k_x*z(1))/m;

dz(3)=z(4);

dz(4)=(sin(w*t)*F_0y-c_y*z(4)-k_y*z(3))/m;

clear all

close all

echo on

z0=[0 0 0 0];

[t,z]=ode45(@znew2,[0 10],z0);

plot(t,z(:,1));

title ('t vs x');

xlabel ('time (sec)');

ylabel ('tool displacement x (m)')

grid on;

echo off