EQUATION

Damped Natural Frequency

Gives the actual oscillation frequency of an underdamped system: ω_d = √(ω₀² − γ²/4)

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The equation

EQ.DAMPED-NATURAL-FREQUENCY
\omega_d = \sqrt{\omega_0^2 - \gamma^2/4}
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What it solves

Gives the actual oscillation frequency of an underdamped system: ω_d = √(ω₀² − γ²/4). Damping always lowers the oscillation frequency below the natural frequency ω₀.

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When to use it

Writing x(t) for a damped oscillator, computing the period of damped oscillations, or determining whether a system is underdamped (γ² < 4ω₀²) or not.

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When NOT to use it

If γ² ≥ 4ω₀², ω_d is imaginary — the system is critically or overdamped and does not oscillate. Do not use ω_d in the driven-oscillator formula where the driving frequency ω_d (same letter, different meaning) appears — notation varies by textbook.

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Common mistakes

Writing γ²/2 or γ² instead of γ²/4 under the square root. Confusing the damping parameter γ (in x″ + γx′ + ω₀²x = 0) with b/m — they may differ by a factor of 2 depending on the textbook's convention. Not checking whether ω_d is real before using it.

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Topics that use this equation

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Problems using this equation