Вычислим моменты инерции

Cz = m*r^2/2
Cxy = m*(h^2/12 + r^2/4)

Гироскопический момент

Hg = Cz*ω

Обобщенные координаты

s = D[β[t], t]
s1 = D[α[t], t]
s2 = D[γ[t], t]

Силы, действующие по осям

fx = 0
fy = (Hg s1 - d s) Cos[β[t]]
fz = 0

fxyz = {{fx}, {fy}, {fz}} // MatrixForm

Кинеческая энергия

Tk = Cxy*s^2/2 + Cxy*s1^2/2 + Cz*(ω + s2)^2/2

Запись уравнений Лагранжа

wx = Integrate [fx,  α[t]]
wy = Integrate [fy, β[t]]
wz = Integrate[fz,  γ[t]]

qxyz = {{D[wx, α[t]]}, {D[wy, β[t]]}, {D[
            wz, γ[t]]}}

ddTkDqdt = {{D[D[Tk, s1], t]}, {D[D[Tk, s], t]}, {D[D[Tk, s2], 
            t]}}

dTkDq = {{D[Tk, α[t]]}, {D[Tk, β[t]]}, {D[
            Tk, γ[t]]}}

df = ddTkDqdt - dTkDq - qxyz

df = ReplaceAll[df, D[s1, t] -> D[q11[t], t]]

Представление в форме Коши

m1 = {Solve[q11[t] - D[α[t], t] == 0, 
            D[α[t], t]],  
          Solve[Flatten[df[[1, 1]]] == 0, D[q11[t], t]]} // 
        MatrixForm

df = 
      ReplaceAll[
        df,  {D[β[t], t] -> q21[t],  β[t] -> q22[t], 
          D[β[t], {t, 2}] -> D[q21[t], t]}]

m2 = {{q21[t] - D[β[t], t]}, {D[q22[t], t] - 
              D[β[t], t]},  {Flatten[df[[2, 1]]]}} // 
        MatrixForm

m2k = {Solve[q21[t] - D[β[t], t] == 0, D[β[t], t]], 
          Solve [D[q22[t], t] - D[β[t], t] == 0,  
            D[q22[t], t]],  
          Solve[Flatten[df[[2, 1]]] == 0, D[q21[t], t]]} // 
        MatrixForm

df = ReplaceAll[df,  {D[s2, t] -> D[q32[t],  t]}]

m3 = {Solve[q31[t] - D[γ[t], t] == 0, 
            D[γ[t], t]],  
          Solve[Flatten[df[[3, 1]]] == 0, D[q32[t], t]]} // 
        MatrixForm

Ввод упрощений

ReplaceAll[m1, D[α[t], t] -> 0]
ReplaceAll[m3, D[γ[t], t] -> 0]
m2 = 
      ReplaceAll[
        m2,  {Cos[q22[t]] -> 1,  D[q21[t], t] -> 0, 
          D[α[t], t] -> Ω}]

Решение

resh0 = 
      Flatten[Solve[Flatten[m2[[1, 3]]] == 0, 
          q21[t]]]
Clear[β]
m2 = ReplaceAll[m2, resh0[[1]]]
resh = 
      Flatten[DSolve[{m2[[1, 1]][[1]] == 
              0, β[0] == 0}, β[t], t]]

Подстановка

resh = 
      ReplaceAll[
        resh, {m -> dr^2  Pi/4*h*ro, 
          r^2 -> dr^2/4}]
resh = 
      ReplaceAll[
        resh, {dr -> 2,  h -> 1, 
          d -> 10000, ω -> 20000 2 Pi/60, 
          ro -> 7.8}]
resh = ReplaceAll[resh, Ω -> 0.2]

\(Построение\ граффика\)

tn = 0
tk = 5

Plot[resh[[1, 2]],  {t,  tn,  tk}]

ReplaceAll[resh, {t -> tk}]