# NDSolve::ndcf: Repeated convergence test failure. How to solve?

I am trying to simulate a system of $n$ pendulums with some friction in Mathematica 9. This is the code I am using:

nPendulos = 3;
tiempoFinal = 20;
fps = 30;
g = 9.81;
r = 0.5;
x[0][t_] := 0
y[0][t_] := 0
CondicionesIniciales =
Join[Table[x[n][0] == n, {n, 1, nPendulos}], Table[y[n]'[0] == 0, {n, 1, nPendulos}]];
Restricciones = Table[(x[n][t] - x[n-1][t])^2 + (y[n][t] - y[n - 1][t])^2 == 1, {n, 1, nPendulos}];
EqNewton = Join[Table[x[n]''[t] == λ[n][t]     (x[n][t] - x[n - 1][t]) -
λ[n + 1][t] (x[n + 1][t] - x[n][t]) - r x[n]'[t], {n, 1, nPendulos - 1}],
Table[y[n]''[t] == λ[n][ t] (y[n][t] - y[n - 1][t]) - λ[n + 1][t]
(y[n + 1][t] - y[n][t]) - g - r y[n]'[t], {n, 1, nPendulos - 1}],
{x[nPendulos]''[t] == λ[nPendulos][ t] (x[nPendulos][t] - x[nPendulos - 1][t]) - r x[nPendulos]'[t]},
{y[nPendulos]''[t] == λ[nPendulos][t] (y[nPendulos][t] - y[nPendulos - 1][t]) - g
- r y[nPendulos]'[t]}];
Vars = Flatten@Table[{λ[n], x[n], y[n]}, {n, 1, nPendulos}];

Sol = NDSolve[{EqNewton, Restricciones, CondicionesIniciales}, Vars,
{t, 0, tiempoFinal}, AccuracyGoal -> 2, PrecisionGoal -> 2, MaxStepSize -> 0.01,
Method -> {"IndexReduction" -> {True, "ConstraintMethod" -> {"Projection",
"Invariants" -> Restricciones}}}];


It works fine when the friction is high or nPendulos (number of pendulus) is low. But for example for nPendulos = 4 and r = 0.5 or nPendulos = 3 and r = 0.15, I get things like

NDSolve::ndcf: Repeated convergence test failure at t == 0.93296875; unable to continue.

NDSolve::ndsz: At t == 2.5562500630525498, step size is effectively zero; singularity or stiff system suspected.


I am almost sure that the physics behind the system is right, because the results for example when nPendulos = 2 or 3 are nice (see https://dl.dropboxusercontent.com/u/35192406/3_con_rozamiento.gif with friction or https://dl.dropboxusercontent.com/u/35192406/test2.gif with no friction)

Why is NDSolve failing? How can I make it work?

• Using Subscript[] makes the code very unpleasant to read here. Commented Nov 25, 2013 at 4:53
• @belisarius What can I use instead? Commented Nov 25, 2013 at 5:12
• Perhaps x[n] instead of Subscript[x, n] Commented Nov 25, 2013 at 5:23
• Done! I have modified it Commented Nov 25, 2013 at 5:37
• See the n-pendulum example: reference.wolfram.com/mathematica/tutorial/… Commented Dec 6, 2013 at 2:04

Ok dude, if you want to get convergence on this type of stiffness you need a "more well posed problem", you can't have too much pendulus and little friction at the same time,for avoid confusion this is just a numerical issue. In other words if you want more Pendulos take there biggers! The problem converge on n=4 if r=10.. I personally think this is a numerical limit with 64bit and NDSolve. Engineer advice: model it bigger and scale down smaller.

nPendulos = 4;
tiempoFinal = 20;
fps = 30;
g = 9.81;
r = 10;
x[0][t_] := 0
y[0][t_] := 0
CondicionesIniciales =
Join[Table[x[n][0] == n, {n, 1, nPendulos}],
Table[y[n]'[0] == 0, {n, 1, nPendulos}]];
Restricciones =
Table[(x[n][t] - x[n - 1][t])^2 + (y[n][t] - y[n - 1][t])^2 ==
1, {n, 1, nPendulos}];
EqNewton =
Join[Table[
x[n]''[t] == \[Lambda][n][
t] (x[n][t] - x[n - 1][t]) - \[Lambda][n + 1][
t] (x[n + 1][t] - x[n][t]) - r x[n]'[t], {n, 1,
nPendulos - 1}],
Table[y[n]''[
t] == \[Lambda][n][
t] (y[n][t] - y[n - 1][t]) - \[Lambda][n + 1][
t] (y[n + 1][t] - y[n][t]) - g - r y[n]'[t], {n, 1,
nPendulos - 1}], {x[nPendulos]''[
t] == \[Lambda][nPendulos][
t] (x[nPendulos][t] - x[nPendulos - 1][t]) -
r x[nPendulos]'[t]}, {y[nPendulos]''[
t] == \[Lambda][nPendulos][
t] (y[nPendulos][t] - y[nPendulos - 1][t]) - g -
r y[nPendulos]'[t]}];
Vars = Flatten@Table[{\[Lambda][n], x[n], y[n]}, {n, 1, nPendulos}];
<< DifferentialEquationsNDSolveUtilities
Sol = NDSolve[{EqNewton, Restricciones, CondicionesIniciales},
Vars, {t, 0, tiempoFinal}, AccuracyGoal -> 8, PrecisionGoal -> 10,
MaxStepSize -> 0.001,
Method -> {"IndexReduction" -> {Automatic,
"ConstraintMethod" -> {"Projection",
"Invariants" -> Restricciones}}}, SolveDelayed -> True]
StepDataPlot[Sol]