--- title: "Driven Oscillations, Resonance and the Quality Factor" description: "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 q" canonical: https://lightmysky.com/learn/science/driven-oscillations-resonance-and-the-quality-factor-mt_FmpOTEZFy4 source: https://lightmysky.com/learn/science/driven-oscillations-resonance-and-the-quality-factor-mt_FmpOTEZFy4.md retrieved: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # 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. Subject: Science · Area: Forces & Motion · Ages 19 to 20 Page: https://lightmysky.com/learn/science/driven-oscillations-resonance-and-the-quality-factor-mt_FmpOTEZFy4 ## Ready when they can - 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 ## Lesson: Pushing at the favourite frequency 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. **Example.** 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. *(drawing: Q_new is the new quality factor. Keep mass and stiffness fixed. Halving b doubles Q.)* 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. **Recap.** 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.* ## Practice 8 questions on this page, each with its working shown. ## Needs first - [Damped Oscillations and the Three Damping Regimes](https://lightmysky.com/learn/science/damped-oscillations-and-the-three-damping-regimes-mt_JCpYpFVn6s) ## Opens up - [Coupled Oscillators and Normal Modes](https://lightmysky.com/learn/science/coupled-oscillators-and-normal-modes-mt_vnTGOIqZzP) - [Resonance and Power in AC Circuits](https://lightmysky.com/learn/science/resonance-and-power-in-ac-circuits-mt_wviaR0dJud)