please help me solve this system of ODE

I am trying to optimize the passive suspension design. in order to do so, i need to get my objective function ready. but for that i will need to solve the following system of ODE

 Ms*D2Zs = -Ks*(Zs-Zus)- Cs*(DZs-DZus);

AND

 Mus*D2Zus = Ks*(Zs-Zus)+Cs*(DZs-DZus))-Kt*(Zus-Zr)-Ct*(DZus-DZr);

note: by "D2" i mean second differential with respect to time and "D" means first differential with respect to time

where Ms,Mus,Ks,Cs,Kt,Ct are all knowns

Zr is a function of time as,

 Zr = H*(sin((2*pi*V*t)/L));

where, V,L,H are also known

i need to obtain an objective function,

 F = W1*F1 + W2*F2;

where W1,W2 are weighing constants and

F1 is RMS of D2Zs

and F2 is RMS of Zus-Zr

i tried several methods on the internet but none gives me numerical solution so that i can get RMS and then obtain the objective function

Please help me

[Torsten]

What is your problem in using ODE45 to solve your differential equations ? How is RMS of D2ZS and RMS of Zus-Zr computed ?

Best wishes

Torsten.

[Ced]

Hi

Steps:

1. Rewrite your system of ODES as a first order system, so that you have dx = f(t,x) as a row vector

2. Write a matlab function of the form in 1

3. call one of the matlab solvers. I usually start with ode45, but your suspension might result in a stiff ode, so you might want to try ode15s if you run into any problems.

The documentation is quite detailed:

How to solve an ode with matlab

4. Compute RMS and evaluate objective function

[pranav mankikar]

I am sorry, it may seem very basic question... I am a newbie.. I am not sure how do i convert it to 1st order as i have the variable of second ode in my first ode... Its all getting very confusing.. trust me i have tried several times myself, went and read several online guides on the net.. it would be very kind of u if u elaborate with respect to my particular system of ODEs Thanking you in anticipation

[Ced]

Converting your system into one of first order just means that you only have first derivatives, not second ones.

I'll make a smaller example than the one above, but it should be easy to apply it to the original question.

Let's say you have:

 Ms*D2Zs = -Ks*(Zs-Zus)
 Mus*D2Zus = Cs*(DZs-DZus)

Then, let's rename the variables: x1 = Zs; x2 = DZs; x3 = Zus; x4 = DZus;

Now you have (which are already only first derivatives!):

Ms*Dx2 = -Ks*(x1-x3);
Mus*Dx4 = Cs*(x2-x4);

The other two equations are trivial (these are necessary because you want to get x1 and x3 as well):

Dx1 = x2
Dx3 = x4

So... all together, you have the following first order system:

Dx1 = x2
Dx2 = -Ks*(x1-x3)/Ms;
Dx3 = x4
Dx4 = Cs*(x2-x4)/Mus;