Find numerical solution to this system of DE

Question

I am trying to solve this system

$$\left( \begin{array}{ccccc} 2 k & -k & 0 & 0 & 0 \\ -k & 2 k & -k & 0 & 0 \\ 0 & -k & 2 k & -k & 0 \\ 0 & 0 & -k & 2 k & -k \\ 0 & 0 & 0 & -k & 2 k \\ \end{array} \right) \left( \begin{array}{c} y_1(t) \\ y_2(t) \\ y_3(t) \\ y_4(t) \\ y_5(t) \\ \end{array} \right)+\left( \begin{array}{ccccc} m & 0 & 0 & 0 & 0 \\ 0 & m & 0 & 0 & 0 \\ 0 & 0 & m & 0 & 0 \\ 0 & 0 & 0 & m & 0 \\ 0 & 0 & 0 & 0 & m \\ \end{array} \right) \left( \begin{array}{c} y_1''(t) \\ y_2''(t) \\ y_3''(t) \\ y_4''(t) \\ y_5''(t) \\ \end{array} \right)=\left( \begin{array}{c} F(t) \\ 0 \\ F(t) \\ 0 \\ F(t) \\ \end{array} \right)$$ but the following code doesn't seem to work..

n = 6;

M = m IdentityMatrix[n - 1];

K = Table[0, {j, 1, n - 1}, {i, 1, n - 1}];

For[j = 1, j <= n - 1, j++, 
 For[i = 1, i <= n - 1, 
  i++, {If[j == i, K[[j, i]] = 2 k, Nothing], 
   If[i == j + 1, K[[j, i]] = -k, Nothing], 
   If[i == j - 1, K[[j, i]] = -k, Nothing]}]]

yy[t] = Table[Subscript[y, i][t], {i, 1, n - 1}];

FF[t] = Table[
   If[(i/n) == 1/6 || i == n/2 || (i/n) == 5/6, F[t], 0], {i, 1, 
    n - 1}];

F[t_] = Piecewise[{{Subscript[F, 0] t/(T/2), 
     0 <= t <= T/2}, {-Subscript[F, 0], 
     T/2 <= t <= T}}];

m = 1;
k = 1;
Subscript[F, 0] = 1;
T = 1;

NDSolve[{M.D[yy[t], {t, 2}] + K.yy[t] == F[t], yy[t] == 0 /. t -> 0, 
  D[yy[t], t] == 0 /. t -> 0}, yy[t], {t, 10 T, 20 T}]

Answer 1

Mathematica is checking to make sure that you have enough equations and unknowns, but the way that you've written them out, it only thinks that you have three equations. This is because you've set them up as {5-component vector} == 0, rather than {5-component vector} == {5-component vector}.

There are a couple of ways to fix this. One is to use Thread:

soln = NDSolve[{Thread[M.D[yy[t], {t, 2}] + K.yy[t] == FF[t]], Thread[yy[t] == 0] /. t -> 0, Thread[D[yy[t], t] == 0] /. t -> 0}, yy[t], {t, 10 T, 20 T}]

(N.B.: I have replaced F[t] with FF[t] here, which I suspect is what you really want given your original equations at the top.) This evaluates with no errors, giving the following result:

Plot[Evaluate[{Subscript[y, 1][t], Subscript[y, 2][t],  Subscript[y, 3][t], Subscript[y, 4][t], Subscript[y, 5][t]} /. First[soln]], {t, 10, 20}]

(Note that $y_2 = y_4$ and $y_1 = y_5$ given your initial conditions and equations, which is why only three graphs are visible in the plot below.)

Alternately, you can write out the equations with vectors on both sides:

soln = NDSolve[{M.D[yy[t], {t, 2}] + K.yy[t] == FF[t], 
   yy[t] == {0, 0, 0, 0, 0} /. t -> 0, 
   D[yy[t], t] == {0, 0, 0, 0, 0} /. t -> 0}, yy[t], {t, 10 T, 20 T}]

This yields the same result.