Forced oscillations, resonance and response
Key idea: H2 Physics lessons on free oscillations, simple harmonic motion, energy interchange, damping, forced response and resonance.
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The core idea
Build the idea
Learn the idea
Big question: Why can a small repeating force produce a large response?
A forced oscillator responds at the driving frequency. Near its natural frequency, energy is transferred efficiently and the steady amplitude becomes large: resonance. More damping lowers and broadens the amplitude peak. Demonstrate this by varying driving frequency, waiting for steady motion and plotting amplitude against frequency while keeping the driving force fixed.
Separate natural and driving frequency
A free oscillator reveals its natural frequency. A forced oscillator receives a continuing periodic force and, after transients fade, oscillates at the driving frequency—not automatically at its natural frequency.
The driver supplies energy on every cycle. The steady amplitude is set by how effectively that energy arrives and how quickly damping removes it. Far from the natural frequency, transfers on different parts of the cycle largely cancel and the response stays small.
Check your understanding: A 2 Hz oscillator is driven steadily at 3 Hz. What is its steady response frequency?
3 Hz, the driving frequency. Its amplitude depends on how close 3 Hz is to its natural frequency and on damping.
Read and explain the resonance curve
Resonance is the large steady response when the driving frequency is close to the natural frequency and energy transfer is especially effective. A lightly damped system has a high, narrow amplitude peak; increasing damping makes the peak lower and broader.
Resonance can be useful in musical instruments, tuning and sensing, or harmful in machinery and structures. Engineers reduce harmful response by adding damping, changing natural frequency or avoiding sustained driving near the peak.
Check your understanding: Why does greater damping broaden the response curve?
It suppresses the sharp build-up near resonance, so the amplitude changes less dramatically across neighbouring driving frequencies.
Key ideas to keep
- The driving frequency sets the steady response frequency.
- Resonance is a large-amplitude response, not an ever-growing amplitude when damping is present.
- Damping changes the height and width of the response curve.
See the reasoning
Worked example
Interpret measured response data
Question: An oscillator gives steady amplitudes 1.2, 2.5, 6.8, 3.0 and 1.5 cm at 4.0, 4.5, 5.0, 5.5 and 6.0 Hz. With extra damping the amplitudes are 1.4, 2.4, 3.2, 2.7 and 1.8 cm. Compare the two responses.
Step 1: Locate each maximum
Why: The sampled maximum estimates the resonance region.
Working: Light damping peaks at 6.8 cm near 5.0 Hz; extra damping peaks at 3.2 cm near 5.0 Hz.
Step 2: Compare peak height
Why: Damping removes energy and limits the steady build-up.
Working: The maximum falls by 3.6 cm, to about 47% of its former value.
Step 3: Compare width
Why: A broader response stays relatively significant farther from the maximum.
Working: With extra damping, amplitudes at 4.0 and 6.0 Hz are closer to the peak fractionally, so the response is broader.
Answer: Resonance is near 5.0 Hz. Extra damping lowers the peak from 6.8 to 3.2 cm and broadens the response.
Check: Both data sets use the same frequency grid, so the comparison does not claim a more precise peak frequency than the measurements support.
Another worked model
Question
Sketch in words lightly and heavily damped amplitude–driving-frequency curves and explain one useful and one harmful resonance example.
Check the worked solution
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.
Use a hint if needed
Practise with support
Try this
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 your answer
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.
Now work without the hint
Practise independently
Your turn
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 your answer
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.
Avoid these traps
Common mistakes
Common mistake
A forced oscillator always moves at its natural frequency.
What is wrong with this reasoning?
Show better thinking
After transients decay it moves at the driving frequency. Its natural frequency determines where the response is largest.
Common mistake
Resonance is always dangerous and occurs at one unchanged frequency.
What is wrong with this reasoning?
Show better thinking
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.
Write for the examiner
Exam guidance
In an explanation, name the energy-transfer condition and describe how damping changes the response graph.
Exam-style practice [8 marks]
Explain how resonance arises in a forced oscillator, sketch the effect of greater damping on amplitude against driving frequency, and apply the ideas to a pedestrian bridge that sways strongly at one walking cadence.
Plan before you answer
- Name natural and driving frequencies.
- Describe energy transfer and the steady response.
- Connect graph changes to engineering actions.
Mark your answer and compare the model
Marking points
Tick each point only if your answer states it clearly.
Model answer
The bridge has natural modes. Repeated footsteps drive it at the walking cadence; when this is close to a natural frequency, force and motion transfer energy effectively each cycle and a large steady amplitude develops. A lightly damped response curve has a high narrow peak, while extra damping lowers and broadens it. Engineers can add dampers, alter stiffness or mass to move the natural frequency, and prevent sustained synchronised forcing near the peak.
Come back in three days
Check what stayed with you
Recall question 1
What sets the steady response frequency?
Check the answer
The driving frequency.
Recall question 2
What condition produces resonance?
Check the answer
Driving frequency close to natural frequency, allowing especially effective energy transfer.
Recall question 3
How does damping change the amplitude response?
Check the answer
It lowers and broadens the resonance peak.
Syllabus and review details
This lesson covers the listed H2 Physics 9478 outcomes. The official topic states no explicit exclusions and defines natural frequency as the frequency of a system in free oscillation. Ideal SHM uses a linear restoring relation and no environmental energy exchange; damping and steady forced response are introduced only when stated.
- GCE A-Level H2 PhysicsTopic 9(j) / Topic 9(k) / Topic 9(l) · 2027Checked against the syllabus · partial topic coverageOfficial 9478 syllabus
Course and syllabus information
- Course
- GCE A-Level H2 Physics
- Edition
- GCE A-Level H2 Physics 2027