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I want to solve this no-linear differential equation system:

$1.2 (1 + 20(r(t)^2 + 2^2 + 4 r(t) \sin(\phi(t)))^2) \ddot\phi(t) + \dot \phi(t) + \phi(t) = 50 \sin(4 t)$

$\textbf{If}$ ($|-r(t)\dot\phi^2(t)+9.8\sin\phi(t)|$<$0.7\times 9.8\cos\phi(t)$)$\Rightarrow$ $\ddot r(t)=0$

$\textbf{If}$ ($-r(t)\dot\phi^2(t)=<9.8\sin\phi(t)$)$\Rightarrow$ $\ddot r(t)-r(t)\dot\phi^2+9.8(\sin\phi(t)-0.7\cos\phi(t))=0$

$\textbf{If}$ ($-r(t)\dot\phi^2(t)>9.8\sin\phi(t)$)$\Rightarrow$ $\ddot r(t)-r(t)\dot\phi^2+9.8(\sin\phi(t)+0.7\cos\phi(t))=0$

Besides, When $r(t)>1\Rightarrow\dot r(0)=0$ and $r(t)=1$ or $r(t)<-1\Rightarrow\dot r(0)=0$ and $r(t)=-1$.

So to solve this differential equation system, I use NDSolve method. Nevertheless, i am getting the following error: NDSolve: Encountered non-numerical value for a derivative at t == 0 and i do not know Where it is the error. The code used is below:

 Clear["Global`*"]
sol = NDSolve[{If[
     Abs[-r[t] phi'[t]^2 + 9.8 Sin[phi[t]]] <= 9.8*0.7 Cos[phi[t]], 
     r''[t], If[-r[t] phi'[t]^2 + 9.8 Sin[phi[t]] >= 0, 
      r''[t] - r[t] phi'[t]^2 + 9.8 (Sin[phi[t]] - 0.7*Cos[phi[t]]), 
      r''[t] - r[t] phi'[t]^2 + 
       9.8 (Sin[phi[t]] + 0.7*Cos[phi[t]])]] == 0, 
   1.2 (1 + 20*(r[t]^2 + 2^2 + 4 r[t] Sin[phi[t]])^2) phi''[t] + 
     phi'[t] + phi[t] == 50 Sin[4 t], 
   r[0] == phi[0] == r'[0] == phi'[0] == 0, 
   WhenEvent[r[t] > 1, {r[t] -> 0.99998, r'[t] -> 0}], 
   WhenEvent[r[t] < -1, {r[t] -> -0.99998, r'[t] -> 0}]}, {r, 
   phi}, {t, 0, 100}]

Plot[Evaluate[{r'[t], r[t]} /. sol], {t, 0, 100}, PlotRange -> All, 
 PlotStyle -> {Blue, Black}, PlotLegends -> {"Speed", "Position"}]

If I remove the if condition and i put the following code r''[t] - r[t] phi'[t]^2 + 9.8 (Sin[phi[t]] - 0.7*Cos[phi[t]]) That it is the then condition of the second If ti works...

Thank you for you time

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  • $\begingroup$ The three If that you see in the post it is a piecewise function, that is it depends of what condtion it is true, i will have one equation so the total number of equation it will be $2$. I do not know where you see three equation rather than two, Have i written wrong the code? $\endgroup$ Oct 15, 2021 at 10:13

1 Answer 1

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NDSolve cannot handle the first ode. If you separate r''[t] out of your If-construct it works as expected:

sol = NDSolve[{r''[t] + 
     If[Abs[-r[t] phi'[t]^2 + 9.8 Sin[phi[t]]] <= 9.8*0.7 Cos[phi[t]],
       0, If[-r[t] phi'[t]^2 + 9.8 Sin[phi[t]] >= 0, 
       0 - r[t] phi'[t]^2 + 9.8 (Sin[phi[t]] - 0.7*Cos[phi[t]]), 
       0 - r[t] phi'[t]^2 + 9.8 (Sin[phi[t]] + 0.7*Cos[phi[t]])]] == 
    0, 1.2 (1 + 20*(r[t]^2 + 2^2 + 4 r[t] Sin[phi[t]])^2) phi''[t] + 
     phi'[t] + phi[t] == 50 Sin[4 t], 
   r[0] == phi[0] == r'[0] == phi'[0] == 0, 
   WhenEvent[r[t] > 1, {r[t] -> 0.99998, r'[t] -> 0}], 
   WhenEvent[r[t] < -1, {r[t] -> -0.99998, r'[t] -> 0}]}, {r, 
   phi}, {t, 0, 100}]

Plot[Evaluate[{r'[t], r[t]} /. sol], {t, 0, 100}, PlotRange -> All, 
 PlotStyle -> {Blue, Black}, PlotLegends -> {"Speed", "Position"}]

enter image description here

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  • $\begingroup$ Thanks a lot!! So the thing that was wrong was that in NDSolve, outside of If statatement, must be some function? $\endgroup$ Oct 15, 2021 at 10:23
  • $\begingroup$ NDSolve probably fails when dissolving for the highest derivative of the ode. $\endgroup$ Oct 15, 2021 at 11:32

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