Oscillations: SHM, damping and resonance
Key idea: Connect measured periodic motion to the defining SHM model, its phase and energy relationships, then distinguish damping from driven resonance.
Before you start: Circular Motion objective chainEnergy & Fields objective chain
By the end, you can
- Describe and investigate free oscillations, and use amplitude, period, frequency, angular frequency, phase and phase difference.
- Recognise SHM from a = −ω²x and use its displacement, velocity, acceleration and graphical relationships.
- Describe kinetic–potential energy interchange and compare light, critical and heavy damping in practical systems.
- Interpret forced-response curves, damping effects and useful or harmful resonance.
Starting-point self-check
1. Check your starting point
Attempt all four groups without notes and mark the first definition, phase sign, energy location or response-curve feature you could not justify. Use the recorded topic diagnostic above when you want scoring and a personalised repair plan.
Forced oscillations, resonance and response 9(j)–(l)
Question 1
A lightly damped oscillator has amplitudes 1.0, 2.2, 4.8, 2.5 and 1.3 cm at driving frequencies 1.6, 1.8, 2.0, 2.2 and 2.4 Hz. Identify resonance and predict the effect of greater damping.
Check the model response
The largest measured response is 4.8 cm at about 2.0 Hz, so resonance is near 2.0 Hz where driving and natural frequencies are close. Greater damping lowers and broadens the peak and may shift its maximum slightly lower.
repair
2. Repair the eight common breaks
Use only the repair matching an error, then redraw the equilibrium line, phase cycle, energy bars or response axes before retrying.
Forced oscillations, resonance and response 9(j)–(l)
Check this idea
Misconception: A forced oscillator always moves at its natural frequency.
Repair: After transients decay it moves at the driving frequency. Its natural frequency determines where the response is largest.
Check this idea
Misconception: Resonance is always dangerous and occurs at one unchanged frequency.
Repair: Resonance can be useful, for example in tuning. Damping lowers and broadens the response and can shift the maximum, while engineering can avoid or control harmful large amplitudes.
worked example
3. Follow four worked models
Follow how each solution fixes the timing reference, equilibrium direction, energy boundary and driving frequency before calculating.
Forced oscillations, resonance and response 9(j)–(l)
Model 1
Sketch in words lightly and heavily damped amplitude–driving-frequency curves and explain one useful and one harmful resonance example.
Check the model response
The lightly damped curve has a high narrow maximum close to the natural frequency; greater damping produces a lower, broader maximum. Resonance is useful when tuning a receiver or producing a large musical response, but harmful when periodic forcing drives excessive bridge, building or machinery vibration; add damping or change the system frequency.
guided practice
4. Guided practice
Use each hint only to select a time-to-phase conversion, SHM equation, energy location or response-curve comparison.
Forced oscillations, resonance and response 9(j)–(l)
Question 1
A machine component has natural frequency 12 Hz and is driven at 7 Hz. Describe its steady frequency and what happens as the drive rises towards 12 Hz, first with light and then greater damping.
Hint: Separate the frequency of the steady motion from the frequency at which its amplitude is largest.
Check the model response
Its steady forced oscillation is at the driving frequency, initially 7 Hz. Amplitude rises towards a maximum near 12 Hz. Light damping gives a large sharp peak; greater damping gives a smaller broader peak.
independent practice
5. Independent practice
Solve without repair notes and state the equilibrium reference, phase convention and energy or driving assumptions used.
Forced oscillations, resonance and response 9(j)–(l)
Question 1
Explain why adding damping can protect a structure from periodic forcing and why the same design choice may be undesirable in a resonant sensor.
Check the model response
Damping dissipates energy, lowering and broadening the resonance peak so forcing near the natural frequency causes less structural amplitude. A resonant sensor may rely on a high, sharp response for sensitivity and frequency discrimination, so excessive damping reduces its useful signal.
Practice exit check
6. Practice assessment
Use this as extra closed-book practice, then complete the separate recorded assessment in your plan.
Forced oscillations, resonance and response 9(j)–(l)
Question 1
Describe an amplitude–driving-frequency graph for a forced oscillator, including resonance, natural and driving frequency, the effect of greater damping, and one useful or avoided application.
Check the model response
The steady oscillator moves at the driving frequency. Its response rises to a maximum when that frequency is close to the natural frequency, defined by free oscillation. Greater damping lowers and broadens the peak and can shift it slightly lower. Resonance may be used in tuning or avoided in structures and machinery by adding damping or changing natural or drive frequencies.
Re-test practice
7. Delayed re-test practice
Return after at least three days and solve these fresh contexts without reopening earlier responses. The recorded plan enforces the delay and uses a separate re-test family for selected-response skill-group evidence.
Forced oscillations, resonance and response 9(j)–(l)
Question 1
A driven system's narrow peak is at 5.0 Hz. Predict the steady-motion frequency when driven at 4.2 Hz, then explain how and why its curve changes after a damper is fitted.
Check the model response
After transients, it oscillates at 4.2 Hz, the driving frequency. The damper removes mechanical energy, so the amplitude peak becomes lower and broader and may shift slightly below its former position, reducing sensitivity to forcing near resonance.