Animate these 2D plots?

Questions
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1

Consider the following code:

z1 = {-R*W*Sin[W*t] + l'[t]*Sin[\[Phi][t]] + 
            l[t]*(\[Phi]'[t])*Cos[\[Phi][t]], 
           R*W*Cos[W*t] - l'[t]*Cos[\[Phi][t]] + 
            l[t]*(\[Phi]'[t])*Sin[\[Phi][t]]};
        
        V = m*g*(R*Sin[W*t] - l[t]*Cos[\[Phi][t]]) + 1/2*k*(l[t] - l0)^2;
        T = 1/2*m*z1.z1;
        Lagrange = T - V;
        eqs = D[D[Lagrange, \[Phi]'[t]], t] - D[Lagrange, \[Phi][t]];
        eqs2 = D[D[Lagrange, l'[t]], t] - D[Lagrange, l[t]];
        
        g = 9.7; m = 1; l0 = 1; k = 1000; R = 2; W = Pi/2;
        sol = NDSolveValue[{eqs == 0, eqs2 == 0, l[0] == l0, l'[0] == 0, 
           Derivative[1][\[Phi]][0] == 0, \[Phi][0] == 0}, {l[t], \[Phi][
            t]}, {t, 0, 20}]
        
        {Plot[sol.{1, 0}, {t, 0, 5}, AxesLabel -> {"t", "l"}], 
         Plot[sol.{0, 1}, {t, 0, 5}, AxesLabel -> {"t", "\[Phi]"}]}
2

Try

Plot[Evaluate[#[[2]] & /@ Flatten[sol] /. t -> tt], {tt, 0, 5},  PlotStyle -> {Red, Blue}]
3

I mean like this double-pendulum animation here.How can I make like this for my equation?

https://mathematica.stackexchange.com/questions/48059/animation-of-double-pendulum

4

You could try this, I hope it helps.

z1 = {-R*W*Sin[W*t] + l'[t]*Sin[\[Phi][t]] + 
l[t]*(\[Phi]'[t])*Cos[\[Phi][t]], 
R*W*Cos[W*t] - l'[t]*Cos[\[Phi][t]] + 
l[t]*(\[Phi]'[t])*Sin[\[Phi][t]]};

V = m*g*(R*Sin[W*t] - l[t]*Cos[\[Phi][t]]) + 1/2*k*(l[t] - l0)^2;
T = 1/2*m*z1.z1;
Lagrange = T - V;
eqs = D[D[Lagrange, \[Phi]'[t]], t] - D[Lagrange, \[Phi][t]];
eqs2 = D[D[Lagrange, l'[t]], t] - D[Lagrange, l[t]];

g = 9.7; m = 1; l0 = 1; k = 1000; R = 2; W = Pi/2;
sol = NDSolve[{eqs == 0, eqs2 == 0, l[0] == l0, l'[0] == 0, 
Derivative[1][\[Phi]][0] == 0, \[Phi][0] == 0}, {l[t], \[Phi][
t]}, {t, 0, 20}];

z1a = z1 /. Flatten[sol /. a_[t] -> a];
Animate[
Graphics[{Red, PointSize[.05], Point[z1a /. t -> tt]}, Axes -> True, 
PlotRange -> {{-4, 4}, {-4, 4}}],
{tt, 0, 20}]
5

Or did you mean something like this?

ParametricPlot[z1a /. t -> tt, {tt, 0, 3}, Axes -> True,  PlotRange -> {{-10, 10}, {-10, 10}}]
6

Yes it is works, but there is a disc and the spring pendulum rotating at above the disc. How can I show the disc in this code?

7

Dear Ömer,

I don’t know what you really want to do, what the physical situation is you want to describe and if your equations are correct.

Anyhow, what about this? Is it this what you want to see?

z1 = {-R*W*Sin[W*t] + l'[t]*Sin[\[Phi][t]] + 
    l[t]*(\[Phi]'[t])*Cos[\[Phi][t]], 
   R*W*Cos[W*t] - l'[t]*Cos[\[Phi][t]] + 
    l[t]*(\[Phi]'[t])*Sin[\[Phi][t]]};

V = m*g*(R*Sin[W*t] - l[t]*Cos[\[Phi][t]]) + 1/2*k*(l[t] - l0)^2;
T = 1/2*m*z1.z1;
Lagrange = T - V;
eqs = D[D[Lagrange, \[Phi]'[t]], t] - D[Lagrange, \[Phi][t]];
eqs2 = D[D[Lagrange, l'[t]], t] - D[Lagrange, l[t]];

g = 9.7; m = 1; l0 = 1; k = 1000; R = 2; W = Pi/2;
sol = NDSolve[{eqs == 0, eqs2 == 0, l[0] == l0, l'[0] == 0, 
    Derivative[1][\[Phi]][0] == 0, \[Phi][0] == 0}, {l[t], \[Phi][
     t]}, {t, 0, 20}];

p1 = R {Cos[W t], Sin[W t]};
z1a = z1 /. Flatten[sol /. a_[t] -> a];
Animate[Graphics[{Circle[{0, 0}, R], PointSize[.05], Black, 
   Point[p1 /. t -> tt], Red, Point[z1a /. t -> tt], Blue, 
   Line[{p1, z1a} /. t -> tt]}, Axes -> True, 
  PlotRange -> {{-4, 4}, {-4, 4}}], {tt, 0, 20}]
8

Is it this what you want to see?

I am afraid that no. you wrote that z1 is the derivative of your position-vector. So

xx = \[Integral]z1 \[DifferentialD]t

and

p1 = R {Cos[W t], Sin[W t]};
xxa = xx /. Flatten[sol /. a_[t] -> a];
Animate[Graphics[{Circle[{0, 0}, R], PointSize[.05], Black, 
   Point[p1 /. t -> tt], Red, Point[xxa /. t -> tt], Blue, 
   Line[{p1, xxa} /. t -> tt]}, Axes -> True, 
  PlotRange -> {{-4, 4}, {-4, 4}}], {tt, 0, 20}]

is perhaps more what you want to see.

9

Thanks for reply.It was helpful.