φ̈ + 2γφ̇ + ω₀²φ = f(t),
We know the analytical solution is:
φ(t) = A e^(-γt) cos(ωᵣt + δ) + A_f cos(ωt + δ_f),
ωᵣ= √(ω₀² - γ²)
where A_f and δ_f don't depend on initial values.
We see that with time, the first term goes to zero.
So we will solve for A_f(ω)
-
f(t) = f₀ cos(ωt)
-
f(t) = f₀ cos(ωt), if cos(ωt) > 0, else 0
-
f(t) = f₀ (1 / π + 1 / 2 cos(ωt) + 2 / (3π) cos(2ωt) - 2 / (15π) cos(4ωt))
