One way to solve this would be to explicitly use finite element method. However, you may need to play around with the the boundary conditions for \[Beta][t]
. The code below should give a starting point:
l = 1; P = 380.57402011815145`; \[Sigma] = 0.5; \[Zeta] =
1 - \[Sigma]; B = 100; Ks = 1000; NDSolve[{B \[Theta]''[t] +
P \[Zeta] Sin[\[Theta][t]] +
P \[Sigma] Sin[\[Theta][t] + \[Beta][t]] ==
NeumannValue[0, t == 0],
Ks \[Sigma]^2 Sin[\[Beta][t]] Cos[\[Beta][t]] -
P \[Sigma] Sin[\[Theta][t] + \[Beta][t]] == 0, \[Theta][l] ==
0, \[Beta][0] == 0}, {\[Theta], \[Beta]}, {t, 0, l},
Method -> "PDEDiscretization" -> "FiniteElement",
InitialSeeding -> {\[Theta][t] == 0.1, \[Beta][t] == 0}]