1st you should write Cos and Sin instead of cos and sin and try again.

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int = Integrate[1/(x^3 - 1), x];
Map[Framed, int, Infinity]

Rotating Disc With Spring Pendulum;

We need to find equations of motion for a disc with spring pendulum. The disc is massless , but it rotates with "w" constant angular speed.Spring pendulum is attached to the disc and spring pendulum move with the disc.Spring pendulum has a "m" mass ,and spring pendulum is free to springing. The discs move is dependent to time , so we have rheonomic constraint.The discs radius is "R" and , we describe its motion with Theta = wt ,but Theta is not degree of freedom. In this case , we have two degrees of freedom for this system , and these are Phi (Spring pendulums rotation angle.) , l (Springs extension quantity.)

z1=Derivative of my position vector.

And indeed:

NDsolve at the beginning has to find out the value of l''[0] and phi''[0].

But with eqs as defined above have a look at

eqsneu = eqs /. t -> 0 /. {L[0] -> l0, L'[0] -> Lp, L''[0] -> Lpp, \[Phi]'[0] -> fp, \[Phi]''[0] -> fpp}

and

In[23]:= Collect[eqsneu[[2]] /. Solve[eqsneu[[1]], Lpp][[1, 1]], fpp] // FullSimplify

Out[23]= (g l0 m)/(Cos[\[Phi][0]] + Sin[\[Phi][0]]) == 0

showing that it is impossible to find a vlaue for phi''[0].

Are you sure that your T and V is correct?

Can we find it with Runge-Kutta method? I reduced them in to first order differential equations like that;

sol1 = EulerEquations[Lagrange, \[Phi][t], t]
sol2 = EulerEquations[Lagrange, l[t], t]
sol5 = \[Phi]''[t] = x'[t];
sol6 = l''[t] = y'[t];
sol3 = Solve[sol1, \[Phi]''[t]]
sol4 = Solve[sol2, l''[t]] 

Now I have two first order equations that are ;

{Derivative[1][x][t] -> (
  g Sin[\[Phi][t]] + Cos[2 \[Phi][t]] Derivative[1][y][t] + 
   2 Derivative[1][l][t] Derivative[1][\[Phi]][t] - 
   2 Sin[2 \[Phi][t]] Derivative[1][l][t] Derivative[1][\[Phi]][t] - 
   Cos[2 \[Phi][t]] l[t] Derivative[1][\[Phi]][t]^2)/(
  l[t] (-1 + Sin[2 \[Phi][t]]))

Derivative[1][y][t] -> (
 k l0 + g m Cos[\[Phi][t]] - k l[t] - 
  m Cos[2 \[Phi][t]] l[t] Derivative[1][x][t] - 
  2 m Cos[2 \[Phi][t]] Derivative[1][l][t] Derivative[1][\[Phi]][t] + 
  m l[t] Derivative[1][\[Phi]][t]^2 + 
  m l[t] Sin[2 \[Phi][t]] Derivative[1][\[Phi]][t]^2)/(
 m (1 + Sin[2 \[Phi][t]]))

Now I obtained my graphs.Thank you so much. :)

Thanks for reply.I found this code from hereThanks [@Hans Dolhaine ][1] for animate , but I can see just pendulum without spring and disc.How can add the disc and spring in this 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 = 15; 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, 30}];

z1a = z1 /. Flatten[sol /. a_[t] -> a];
Animate[Graphics[{Red, PointSize[.05], Point[z1a /. t -> tt]}, 
  Axes -> True, PlotRange -> {{-10, 10}, {-10, 10}}], {tt, 0, 30}]