Help with solving a system of coupled equations of motions in time domain?

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Jake 2023-3-28

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https://ww2.mathworks.cn/matlabcentral/answers/1936824-help-with-solving-a-system-of-coupled-equations-of-motions-in-time-domain

编辑： Torsten 2023-3-29

Hi,

I've posted a similar post here, and I learned much through that post and comments. Now, I'm building on it and the system of equations has become a bit more complex again. I will explain it here again. Following were the set of equations I had, as I mentioned in the earlier post (linked).

But now, suppose I have another time-varying function added to each equation above. For instance, each DoF noted above includes another function f_1(t) (in fact, my system includes several of these, but I'm using one here for explanation). For example, as below.

So, I understand that these additional "functions" affects the system of equations at each time step. For example, f_1(t) here is the excitation loads due to air, as the object moves. i.e., one has to model it separately, and add a function/script to calculate that at the respective time step. Also, f_1(t) depends on the state variables which I am trying to solve.

For example, following what I had posted earlier (above linked), the equation system now changed to,

function sdot = learnF(A, B, s, data) % Equations to solve
sdot = [s(2) + f_1(inputs); % Added arbitrarily only to explain!!
        data.m*s(4)*s(12) + Ctn*A(2,2,1)*s(4)*s(12) + A(2,4,1)*s(8)*s(12) + A(2,6,1)*s(12)^2;
        s(4);
        -data.m*s(2)*s(12) - A(1,1,1)*s(2)*s(12) - A(2,4,1)*s(8)
        s(6);
        -data.m*data.g;
        s(8);
        data.I(4,4)*s(10)*s(12);
        s(10);
        -data.I(4,4)*s(8)*s(12);
        s(12);
        (data.I(4,4)-data.I(6,6))*s(8)*s(10) + A(1,1,1)*s(2)*s(4) ...
        - A(2,2,1)*s(2)*s(4) - A(2,4,1)*s(2)*s(8) - A(2,6,1)*s(2)*s(8) - A(6,4,1)*s(8)];
end

I added f_1(inputs) just arbitrarily (without deriving things) and I do know the placing could be wrong. The point is, now I have time-varying function within the function, which I have to solve using an ode solver. So my questions are;

Did I understand the approach correctly here? If yes, how can I model this f_1(t) to use inputs from sdot? (I can feel that I am wrong here).
I can, instead, use a for loop (time loop) and solve the equations step by step, but then I cannot use, for example, ode45 because I cannot solve ODE for one time step using that. Is there a work around for this?
IF I can still use ode45 and solve this, can I get outputs of f_1(t) after the simulation?

My sincere apologies for the long and possibly confusing post, but I'm kind of helpless here. I'd appreicate any feedback!

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Answer 1

Torsten 2023-3-28

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https://ww2.mathworks.cn/matlabcentral/answers/1936824-help-with-solving-a-system-of-coupled-equations-of-motions-in-time-domain#answer_1203059

移动：Torsten 2023-3-28

在 MATLAB Online 中打开

How did you come up with the sdot vector for the equations so far ?

If you use a mass matrix, you simply have to add the additional force terms to the sdot vector.

A force term f1(t) in equation 1, e.g., has to be reflected as

sdot(2) = data.m*s(4)*s(12) + Ctn*A(2,2,1)*s(4)*s(12) + A(2,4,1)*s(8)*s(12) + A(2,6,1)*s(12)^2 + f1(t)

A force term f2(t) in equation 2 has to be reflected as

sdot(4) = -data.m*s(2)*s(12) - A(1,1,1)*s(2)*s(12) - A(2,4,1)*s(8) + f2(t)

and so on.

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Jake 2023-3-29

编辑：Jake 2023-3-29

在 MATLAB Online 中打开

Hi @Torsten, Thank you for the response!

The sdot vector is derived following what I have done in the previous post, and with the help on Matlab example, "Solve Equations of Motion for Baton Thrown into Air". I use a mass matrix and so far, my mass matrix is still the same (although, I understand that these have to be changed accordingly), as I have noted in the earlier post. i.e., My mass mtarix is,

function M = learnMass(A, B, data) % Mass matrix function
m = data.m;
I = data.I;
M = zeros(12,12);
M(1,1)  = 1; M(3,3) = 1; M(5,5) = 1; M(7,7) = 1; M(9,9) = 1; M(11,11) = 1;
M(2,2)  = m + A(1,1,1);
M(2,6)  = A(3,1,1);
M(2,10) = A(5,1,1);
M(4,4)  = m + A(2,2,1);
M(4,8)  = A(4,2,1);
M(4,12) = A(6,2,1);
M(6,2)  = A(1,3,1);
M(6,6)  = m + A(3,3,1);
M(6,10) = I(5,5) + A(5,3,1);
M(8,8)  = I(4,4) + A(4,4,1);
M(10,2) = A(1,5,1);
M(10,6) = A(3,5,1);
M(10,10)= A(5,5,1);
M(12,4) = A(2,6,1);
M(12,8) = A(4,6,1);
M(12,12)= I(6,6) + A(6,6,1);
end

I agree that the f1(t) and f2(t) terms have to be reflected in the respective sdot lines as you mentioned. I just added the f1(t) in sdot(1) just to explain, but I agree it was a poor explanation.

But my question is how to model the f1, f2 parts. What I meant by f1(t) is that f1 is a function that depends on time - which I also wanted to implicitly mean that it depends on the velocity and/or displacement (which are the state variables that I get as sdot). I'm not sure if I make much sense here, so I'll try to explain further.

Suppose the EOM I'm solving are a motion of a car, and the module f1 represents an autopilot system. Then, at a particular time-step (t = t), autopilot system needs the inputs, which are the solutions of the EOM (velocity, displacement etc.) at the previous time-step (t = t-1). Further, f1 is modelled by another equation, which I believe I have to write a separate script. Am I missing something?

Jake 2023-3-29

Thanks. I don't quite follow the first alternative, but the second one should be fine to implement.

Torsten 2023-3-29

编辑：Torsten 2023-3-29

在 MATLAB Online 中打开

For the first (quick-and-dirty) alternative:

if t < t0
    f1 = ...;
else if t >= t0 && t <= t1
    f1 = ...;
else if ...
    ...
else
    ...
end

But the second way is recommended.

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Help with solving a system of coupled equations of motions in time domain?

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Help with solving a system of coupled equations of motions in time domain?

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