NDSolve: Unable to find initial conditions

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1

@ZSeong Hwang Your posting has no reply window.

In your equations an initial condition for theta’[0] seems to be missing

pre_iypt_2.nb (38.4 KB)

2
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Hello Roland. I put initial condition for theta dot. But still not working.

pre_iypt_2 _OP_1_CP_.nb (38.4 KB)

3

To make your notebook readable and tractable, put a semicolon ; behind each setting line and break the commands into different input cells.

Its is too time consuming, if for checking for syntax errors the notebook has to be edited beforehand.

Roland

4

Parameter fixing

L = 0.001;
d = 0.001;
l = 0.001;
B = 0.1;
M = 0.001;
k = 1;
R = 0.001;
g = 10;
rx0 = L/2;
rz0 = L/2 + d + l/2;
r0 = Sqrt[rx0^2 + rz0^2];
A0 = -rz0 rx0;
Fz0 = B/r0^5 (rx0 - 2 rz0 + 5 A0/r0^2 rz0);

Spherical coordinates by time dependent angles

rx = L/2 Sin[alpha[t]] Cos[psi[t]];
ry = L/2 Sin[alpha[t]] Sin[psi[t]];
rz = L/2 Cos[alpha[t]] + d + l/2;
r = Sqrt[rx^2 + ry^2 + rz^2];

 A spherical coordinate tensor A of rank 2 and a vector F

In[18]:= A = (rx^2 + ry^2) Cos[alpha[t]] - ry rz Sin[alpha[t]] Sin[psi[t]] - 
   rz rx Sin[alpha[t]] Cos[psi[t]];
Fx = B/r^5 (-4 rx Cos[alpha[t]] + rz Sin[alpha[t]] Cos[psi[t]] + 5 A/r^2 rx);
Fy = B/r^5 (-4 ry Cos[alpha[t]] + rz Sin[alpha[t]] Sin[psi[t]] + 5 A/r^2 ry);
Fz = B/r^5 (ry Sin[alpha[t]] Sin[psi[t]] + rx Sin[alpha[t]] Cos[psi[t]] - 
     2 rz Cos[alpha[t]] + 5 A/r^2 rz);

Three equations  of  of second order for alpha , theta of the form

D{rx, ry, rz},t,t]== Fx + fs + Nf

In[48]:= D[rx, t, t] /. {_?NumericQ :> 1} // FullSimplify

Out[48]= Sin[alpha[t]] (Cos[psi[t]] + Sin[psi[t]]) Derivative[1][psi][t] + 
 Derivative[1][alpha][
   t] (Cos[psi[t]] (Cos[alpha[t]] + Sin[alpha[t]]) + 
    Cos[alpha[t]] Sin[psi[t]] Derivative[1][psi][t])

In[46]:= eq1 = M L/
     2 (alpha''[t] Cos[alpha[t]] Cos[psi[t]] - 
      psi''[t] Sin[alpha[t]] Sin[psi[t]] - (alpha'[t]^2 + psi'[t]^2) Sin[
        alpha[t]] Cos[psi[t]] - 
      2 alpha'[t] psi'[t] Cos[alpha[t]] Cos[psi[t]]) == 
   Fx + (fs[t] + k Nf[t] theta'[t]) Sin[psi[t]];


eq2 = M L/
     2 (alpha''[t] Cos[alpha[t]] Sin[psi[t]] + 
      psi''[t] Sin[alpha[t]] Cos[psi[t]] - (alpha'[t]^2 + psi'[t]^2) Sin[
        alpha[t]] Sin[psi[t]] + 
      2 alpha'[t] psi'[t] Cos[alpha[t]] Cos[psi[t]]) == 
   Fy - (fs[t] + k Nf[t] theta'[t]) Cos[psi[t]];

eq3 = M R (alpha''[t] Cos[alpha[t]] - alpha'[t]^2 Sin[alpha[t]]) - 
    M L/2 (alpha''[t] Sin[alpha[t]] + alpha'[t]^2 Cos[alpha[t]]) == 
   Fz + M g + Nf[t];

One algebraic constraint for two unknown functions  Nf and fs, once theta is known


eq4 = theta''[t] - 2 k Nf[t]/(M R) theta'[t] - 2 fs[t]/(M R) == 0;

One  algebraic equation for alpha, once theta and psi are known


eq5 = L/2 Sin[alpha[t]] psi'[t] - R theta'[t] == 0;

Any differential equation  for Nf  and fs  is missing ,  so the given values at one point are void, the system is indefinite with respect to Nf and fs. On the other hand  eq5 is a constraint to the second order system {alpha, psi, theta} . So the system is over-under-determined, an error that NDSolve seems to be unable to detect.

In[37]:= soln = nDSolve[{eq1, eq2, eq3, eq4, eq5, alpha[0] == Pi/2, alpha'[0] == 1, 
   theta[0] == 0, theta'[0] == 1/2, psi[0] == 0, psi'[0] == 1, 
   Nf[0] == -Fz0 - M g, Nf'[0] == 1, fs[0] == 0, fs'[0] == 1}, {alpha[t], 
   theta[t], psi[t], Nf[t], fs[t]}, {t, 0, 10}]

–

Roland Franzius

5

Your initial conditions are not compatible with eq1

(eq1 /. t -> 0) /. {alpha[0] -> \[Pi]/2, psi[0] -> 0, 
  theta[0] -> 0, Derivative[1][alpha][0] -> 1, 
  Derivative[1][psi][0] -> 1, Derivative[1][theta][0] -> 1/2}

False

Your system has no solution.

6

The reason for the message NDSolve::pdord is that you do not specify differential equations for Nf[t] and fs[t]. So we are only dealing with arbitrary predefined functions Nf[t] and fs[t] with given values and derivatives at t = 0.

How is this supposed to be numerically solvable?

I think your problem is not yet sufficiently specified.