Driven Oscillations, Resonance and the Quality Factor
After transients fade, a damped linear oscillator follows the driving frequency. Light damping gives a displacement-amplitude peak near the undamped natural frequency and a long free ring-down. High quality factor means weak damping; sufficiently strong damping removes the positive-frequency displacement peak.
What a learner can do afterwards
- Sketches amplitude against driving frequency for several damping strengths
- Relates the quality factor to the sharpness of the peak and to how long free oscillations last
- Names a case where resonance is wanted and one where it has to be suppressed, with the physical reason
1 · Read
A spring oscillator of mass m and stiffness k has undamped natural frequency omega_n = sqrt(k/m). A sinusoidal force of fixed amplitude F0 drives it. After damped transients fade, steady motion follows the driving frequency. Its displacement amplitude tends to F0/k at very low frequency and to zero at very high frequency.
Light velocity-dependent damping gives a tall, narrow displacement-amplitude peak near omega_n. The large driven response near this peak is called resonance. Increasing damping lowers and broadens the peak; sufficiently strong damping removes its positive-frequency peak. Resonance can occur with positive damping.
For velocity-dependent damping coefficient b, the quality factor is Q = m omega_n/b. High Q means weak damping, a narrow peak and slow decay of free oscillations. Less energy lost per cycle explains both the tall driven peak and the long ring-down. At fixed m and k, Q is inversely proportional to b. Low Q need not produce a displacement peak.
Musical instruments use resonance to strengthen selected notes. Radio tuners use a narrow response to select a station. Bridges may need resonant vibration suppressed because large motion can cause stress or discomfort. This pairs a wanted response in an instrument with an unwanted response in a bridge.
For light positive velocity-dependent damping, the displacement-amplitude peak lies slightly below the undamped natural frequency omega_n. At fixed force amplitude, driving at this peak gives the largest steady displacement. Driving exactly at omega_n is a close approximation, not the exact maximum.
In a linear bridge model, increasing damping reduces steady displacement at fixed nonzero driving frequency and force amplitude. Increasing the force or decreasing damping increases that displacement. This model comparison is not a repair instruction. If a real bridge sways under marching feet, stop coordinated marching and follow safety instructions. Engineers must assess possible changes.
Steady motion follows the drive. Light damping gives a near-natural-frequency displacement peak and a long ring-down.
Original lesson and assessment text authored for this dataset.
Original readings and flashcard answers written for this dataset.
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