define o(t) = 80 + S(t), where S(t) is a sinusoidal function = -11 Cos[[Pi t)/12]+3.25691 Sin[Pi t/12]

-- well your parenthesis are not balanced, so it isn't clear what the exact function is supposed to be, but it doesn't matter. S(t) is sinusoidal with some min/max range.

>> settle in at about 50

Assume you mean as t -> oo

At t = oo, y = 50

At t = oo, dy/dt = ?

Plug these two values into your DE at t = oo, and express in terms of S(t) and d(t). Then solve for d(t) = f( S(t) ). This should remove the oscillation.

-- well your parenthesis are not balanced, so it isn't clear what the exact function is supposed to be, but it doesn't matter. S(t) is sinusoidal with some min/max range.

>> settle in at about 50

Assume you mean as t -> oo

At t = oo, y = 50

At t = oo, dy/dt = ?

Plug these two values into your DE at t = oo, and express in terms of S(t) and d(t). Then solve for d(t) = f( S(t) ). This should remove the oscillation.