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Mathematics Calculus Equation Solving Graphics and Visualization Wolfram Language Numerical Computation

Use results of NDSolve for other computations?

Daniel Gotsmann

Posted 7 years ago

I have been working on a model for a pendulum(using a Lagrangian) and I am using NDSolve. The solver gives me outputs for alpha an l yet I can´t us these to recalculate x and y. In[2]:= ClearAll["Global`"] Definition of parameters In[3]:= R = 0 Subscript[l, 0] = 0.5 m = 1 M = 2.394 g = 9.81 Subscript[\[Alpha], 0] = 1/2Pi Expression of position(x,y) as a function of [Alpha] In[9]:= x[\[Alpha]_, l_, t_] := -Cos[\[Alpha][t]]R + Sin[\[Alpha][t]](l[t] - \[Alpha][t]R) y[\[Alpha]_, l_, t_] := Sin[\[Alpha][t]]R + Cos[\[Alpha][t]](l[t] - \[Alpha][t]R) Calculation of the components and the norm of the velocity Computation of kinetic energy, potential energy and Lagrangian In[11]:= dx[\[Alpha]_, l_, t_] := \!\( \SubscriptBox[\(\[PartialD]\), \(t\)]\((x[\[Alpha], l, t])\)\) dy[\[Alpha]_, l_, t_] := \!\( \SubscriptBox[\(\[PartialD]\), \(t\)]\((y[\[Alpha], l, t])\)\) v[\[Alpha]_, l_, t_] := Sqrt[dx[\[Alpha], l, t]^2 + dy[\[Alpha], l, t]^2] // FullSimplify; Ekin[\[Alpha]_, l_, t_] := 1/2mv[\[Alpha], l, t]^2 + 1/2M(l'[t])^2 Epot[\[Alpha]_, l_, t_] := mgy[\[Alpha], l, t] + M*g (l[t] - Subscript[l, 0]) L1[\[Alpha]_, l_, t_] := Ekin[\[Alpha], l, t] - Epot[\[Alpha], l, t] Euler-Lagrange equation In[17]:= eq1[\[Alpha]_, l_, t_] = D[D[L1[\[Alpha], l, t], \[Alpha]'[t]], t] - D[L1[\[Alpha], l, t], \[Alpha][t]] // FullSimplify; eq2[\[Alpha]_, l_, t_] = D[D[L1[\[Alpha], l, t], l'[t]], t] - D[L1[\[Alpha], l, t], l[t]] // FullSimplify; eq = {eq1[\[Alpha], l, t] == 0, eq2[\[Alpha], l, t] == 0} Out[19]= {l[t] (-9.81 Sin[\[Alpha][t]] + 2 Derivative[1][l][t] Derivative[1][\[Alpha]][t] + l[t] (\[Alpha]^\[Prime]\[Prime])[t]) == 0, 23.4851 + 9.81 Cos[\[Alpha][t]] - l[t] Derivative[1][\[Alpha]][t]^2 + 3.394 (l^\[Prime]\[Prime])[t] == 0} In[20]:= Solution of equation of motion [AliasDelimiter]with given initial conditions In[22]:= sol = First@ NDSolve[{eq, \[Alpha][0] == Subscript[\[Alpha], 0], \[Alpha]'[0] == 0, l[0] == Subscript[l, 0], l'[0] == 0}, {\[Alpha][t], l[t]}, {t, 0, 20}] Plot In[23]:= Plot[\[Alpha][t] /. sol, {t, 0, 20}] In[24]:= Plot[l[t] /. sol, {t, 0, 20}] In[29]:= ParametricPlot[{x[\[Alpha], t]/.sol, y[\[Alpha], t]/.sol}, {t, 0, 20}, PlotRange -> All]

I have been working on a model for a pendulum(using a Lagrangian) and I am using NDSolve. The solver gives me outputs for alpha an l yet I can´t us these to recalculate x and y.

In[2]:= ClearAll["Global`*"]

Definition of parameters

In[3]:= R = 0
Subscript[l, 0] = 0.5
m = 1
M = 2.394
g = 9.81
Subscript[\[Alpha], 0] = 1/2*Pi

Expression of position(x,y) as a function of [Alpha]

In[9]:= x[\[Alpha]_, l_, t_] := -Cos[\[Alpha][t]]*R + 
  Sin[\[Alpha][t]]*(l[t] - \[Alpha][t]*R)
y[\[Alpha]_, l_, t_] := 
 Sin[\[Alpha][t]]*R + Cos[\[Alpha][t]]*(l[t] - \[Alpha][t]*R)

Calculation of the components and the norm of the velocity Computation of kinetic energy, potential energy and Lagrangian

In[11]:= dx[\[Alpha]_, l_, t_] := \!\(
\*SubscriptBox[\(\[PartialD]\), \(t\)]\((x[\[Alpha], l, t])\)\)
dy[\[Alpha]_, l_, t_] := \!\(
\*SubscriptBox[\(\[PartialD]\), \(t\)]\((y[\[Alpha], l, t])\)\)
v[\[Alpha]_, l_, t_] := 
  Sqrt[dx[\[Alpha], l, t]^2 + dy[\[Alpha], l, t]^2] // FullSimplify;
Ekin[\[Alpha]_, l_, t_] := 1/2*m*v[\[Alpha], l, t]^2 + 1/2*M*(l'[t])^2
Epot[\[Alpha]_, l_, t_] := m*g*y[\[Alpha], l, t] + M*g (l[t] - Subscript[l, 0])
L1[\[Alpha]_, l_, t_] := Ekin[\[Alpha], l, t] - Epot[\[Alpha], l, t]

Euler-Lagrange equation

In[17]:= eq1[\[Alpha]_, l_, t_] = 
  D[D[L1[\[Alpha], l, t], \[Alpha]'[t]], t] - 
    D[L1[\[Alpha], l, t], \[Alpha][t]] // FullSimplify;
eq2[\[Alpha]_, l_, t_] = 
  D[D[L1[\[Alpha], l, t], l'[t]], t] - D[L1[\[Alpha], l, t], l[t]] // 
   FullSimplify;
eq = {eq1[\[Alpha], l, t] == 0, eq2[\[Alpha], l, t] == 0}

Out[19]= {l[t] (-9.81 Sin[\[Alpha][t]] + 
     2 Derivative[1][l][t] Derivative[1][\[Alpha]][t] + 
     l[t] (\[Alpha]^\[Prime]\[Prime])[t]) == 0, 
 23.4851 + 9.81 Cos[\[Alpha][t]] - l[t] Derivative[1][\[Alpha]][t]^2 + 
   3.394 (l^\[Prime]\[Prime])[t] == 0}

In[20]:=

Solution of equation of motion [AliasDelimiter]with given initial conditions

In[22]:= sol = First@
  NDSolve[{eq, \[Alpha][0] == Subscript[\[Alpha], 0], \[Alpha]'[0] == 0, 
    l[0] == Subscript[l, 0], l'[0] == 0}, {\[Alpha][t], l[t]}, {t, 0, 20}]

Plot

In[23]:= Plot[\[Alpha][t] /. sol, {t, 0, 20}]

In[24]:= Plot[l[t] /. sol, {t, 0, 20}]

In[29]:= 
ParametricPlot[{x[\[Alpha], t]/.sol, y[\[Alpha], t]/.sol}, {t, 0, 20}, PlotRange -> All]

POSTED BY: Daniel Gotsmann