How to solve a system of coupled, non-homogenous differential equations of motions?

Hi,

First, I must say that I'm still learning the mathematics behind the governing equations, and I know I have a long way to go. But I cannot even begin to understand how one can go about solving the system of equations of motion. I added the following to express the type of equation system that I'm trying to solve. Again, it is not the complete system, and I want to learn how to approach it.

Suppose (a_1 - a_7) and b_1, b_2 are known. The (0) here means that it is the value at "zero frequency" . Further, the coefficients are known for a series of frequencies. So, I think the system of equations are to be solved at each frequency using the known coefficients. I have not included the complete system, and the last equation is also not complete. I was simply explaining the structure. But I can add the complete system here as well.

I looked through a lot of questions that were asked here and followed some tutorials too. But I still cannot understand how to go about this. I would really appreciate any help or advice (or any links to read/learn more). My apologies if the question is irrelevant here, but I’m really helpless at this point.

Answer 1

[Torsten]

Do you know how to transform a system of 2nd order equations to a system of first order equations ?

So this is the first step:

Set

x = x1, xdot = x1p, y = y1, ydot = y1p, phi = phi1, phidot = phi1p, psi = psi1, psidot = psi1p,...

and write your system as

dx1/dt - x1p = 0
dy1/dt - y1p = 0
dphi1/dt - phi1p = 0
dpsi1/dt - psi1p = 0
a1(0)*dx1p/dt + a2(0)*y1p*psi1p + a3(0)*phi1p*psi1p + a4(0)*psi1p^2 - F1 = 0
...

(same for equations 2 and 3).

Now you have a 1st-order system of differential equations of the form

F(z,zdot,t) = 0

with vectors z and zdot as

z = [x1 x1p y1 y1p phi1 phi1p psi1 psi1p ...]
zdot = [dx1/dt dx1p/dt dy1/dt dy1p/dt dphi1/dt dphi1p/dt dpsi1/dt dpsi1p/dt ...]

Use ODE15I to solve it.

[Jake]

Thank you so much for the detailed response, Torsten! I will try this and learn as I go.

[Jake]

Hi again @Torsten,

I understood the approach here, but I wanted to know how to use/include the F terms here (i.e. F1, F2,..). These are unknowns. I'm sure this is a silly confusion, but I'd appreciate your thoughts.

[Torsten]

So the functions on the right-hand side (x,y,phi,psi,...) are known ? This is necessary to calculate the F-terms.

But I doubt it.

[Jake]

Only known terms are the coefficients and the initial conditions. i.e. Initial conditions of the six velocities (3 linear, and 3 rotational) are known.

(Say, for instance, xdot0=1 and all other velocities ydot0, zdot0, phidot0, thetadot0, psidot0 = 0). In my head, I thought this would be an iterative process but I have no idea anymore :(

[Torsten]

I have no idea anymore :(

Me too. Force terms (F1, F2,...) are usually given and not unknowns.