Resonance Condition
States that maximum steady-state amplitude occurs when the driving frequency ω_d is close to the natural frequency ω₀
The equation
What it solves
States that maximum steady-state amplitude occurs when the driving frequency ω_d is close to the natural frequency ω₀. At resonance the denominator of the amplitude formula is minimized and limited only by damping (γ).
When to use it
Identifying the driving frequency that produces maximum response in any linear oscillator. The exact resonance peak for amplitude is at ω_d = √(ω₀² − γ²/2), which approaches ω₀ for high-Q systems.
When NOT to use it
The simple ω_d = ω₀ rule is exact for velocity amplitude but only approximate for displacement amplitude. In highly damped systems (Q < ½), there is no resonance peak — the system is overdamped.
Common mistakes
Assuming amplitude goes to infinity at resonance — damping always limits the peak: A_res = F₀/(mγω₀). Confusing resonance frequency (amplitude peak) with natural frequency ω₀ — they differ by a damping correction. Applying the resonance condition to the transient response rather than the steady state.