Fractional change in damped frequency

For a damped oscillator x'' + 2γx' + ω0²x = 0, the natural (undamped) angular frequency is ω0 and the damped frequency is

ω′ = √(ω0² − γ²) ,    γ = b/2m

Quality factor: Q = ω0/2γ.

Derivation

1. Write ω′ = ω0 √(1 − γ²/ω0²).
2. For light damping γ << ω0, use the binomial expansion (1 − ε)^(1/2) ≈ 1 − ε/2:

ω′ ≈ ω0 (1 − γ²/2ω0²)

3. Change in frequency:

ω0 − ω′ = γ²/(2ω0)
(ω0 − ω′)/ω0 = γ²/(2ω0²)

4. From Q = ω0/2γ: γ/ω0 = 1/2Q, so γ²/ω0² = 1/4Q².
5. Substitute:

(ω0 − ω′)/ω0 = (1/2)(1/4Q²) = 1/(8Q²)

Hence the fractional change in frequency due to damping is 1/8Q². For Q = 10 it is only 0.125%, so a high-Q oscillator oscillates at nearly its natural frequency.