9478 / 2027
Oscillations overview

Lesson 5 of 6

Compare damped responses

  • I can compare light, critical and heavy damping and explain practical uses including car suspension.Syllabus

    Syllabus 9478, 9(i). Compare light, critical and heavy damping and explain practical uses including car suspension.

Make a guess. You are not marked.

A car's suspension is critically damped. The car goes over a bump. How does the car body move afterwards?

  1. It creeps back very slowly over many seconds without bouncing.
  2. It bounces several times with shrinking height before settling.
  3. It returns to its rest height in the shortest time without bouncing.
Show the answer

It returns to its rest height in the shortest time without bouncing.

Critical damping is the amount that returns the system to equilibrium as quickly as possible with no oscillation. Passengers feel one smooth return, not a series of bounces.

Damping transfers mechanical energy from the oscillator to other forms, often internal energy in the apparatus and surroundings. The response depends on the degree of damping.

The ideal free spring-cart model has constant amplitude and no energy loss. With damping, a resisting force acts as well as the restoring force. The ideal acceleration relationship a = -ω2x alone no longer describes the whole resultant.

In an undriven oscillating response, decreasing displacement peaks show decreasing mechanical energy. Energy is transferred, not destroyed. Passing through zero displacement is not evidence of zero mechanical energy: the body can still be moving.

Compare the same release conditions

The three model responses below use the same mass and stiffness. Each starts at +0.0500 m displacement with zero velocity, then receives no continuing drive. This time origin is release from an extreme, unlike the earlier centre-crossing sine model.

Compare the same release from rest

Each model uses the same mass, spring stiffness and initial state: x = +0.0500 m, v = 0. Every trace starts with a horizontal tangent. All panels share both axis scales; the damping is the only changed model parameter.

Light damping: oscillatory return

Light damping: oscillatory returnThe lightly damped response starts from positive 0.0500 metres with zero slope, crosses equilibrium repeatedly and has progressively smaller peaks. At one second it is about negative 0.02188 metres. The horizontal time axis lies at zero displacement. It uses the same zero-to-five-second and plus-or-minus 0.0500-metre scales as the other two responses.012345-0.0500+0.050Displacement x / mElapsed time t / s

Crossing equilibrium does not mean the motion has stopped. Kinetic energy can still be present there.

Critical damping: quickest nonoscillatory return

Critical damping: quickest nonoscillatory returnThe critically damped response starts from positive 0.0500 metres with zero slope and approaches zero from above without crossing it for this release-from-rest state. At one second it is about positive 0.004579 metres. It approaches equilibrium faster than the overdamped response under these same conditions. It is not assigned an exact finite stopping time. Both axes match the other damping panels.012345-0.0500+0.050Displacement x / mElapsed time t / s

For this common release from rest, the boundary between oscillatory and overdamped responses gives the fastest return without oscillating.

Heavy damping: slower nonoscillatory return

Heavy damping: slower nonoscillatory returnThe overdamped response starts from positive 0.0500 metres with zero slope and approaches zero from above more slowly than the critical response. At one second it is about positive 0.01844 metres. It has no crossings for this release-from-rest state and no invented finite stopping time. Both axes match the other damping panels.012345-0.0500+0.050Displacement x / mElapsed time t / s

More damping does not always produce a faster return. Here it slows the approach while still removing mechanical energy.

These are predicted responses, not measured records. A curve becoming visually close to zero does not specify an exact stopping time. Critical and heavy damping approach zero from above for these release-from-rest conditions.

All three plots have the same time and displacement scales, initial displacement and horizontal initial tangent. Their different subsequent responses show the effect of damping under the same release-at-rest conditions.
  • Light or underdamping: the body crosses equilibrium repeatedly, with decreasing oscillation amplitude.
  • Critical damping: the boundary case gives the most rapid return towards equilibrium without oscillating for this release-from-rest comparison.
  • Heavy or overdamping: the return is nonoscillatory but slower under the same initial conditions. More damping does not always mean a quicker return.

The nonoscillating traces approach equilibrium; the model does not assign an exact finite stopping time merely because the curve becomes close to zero. The comparison is tied to the stated initial displacement and zero initial velocity, not a claim that critical or overdamped systems can never cross equilibrium under any possible initial motion.

Check your understanding

Three systems with the same mass and stiffness are released from the same positive displacement at rest. Which damping case returns towards equilibrium most rapidly without oscillating in this comparison?

  1. Light or underdamping.
  2. The heaviest overdamping.
  3. Critical damping.
Show the answer

Critical damping.

Critical damping is the boundary case giving the fastest nonoscillatory return for these common release-at-rest conditions. The more heavily damped system returns more slowly.

Apply the comparison to a suspension

A car's springs allow movement after a disturbance such as a road bump. Dampers transfer mechanical energy to internal energy and limit continued bouncing.

Too little damping allows repeated motion. Excessive damping can make the return slow. A near-critical model illustrates the aim of rapid settling without continued oscillation.

A measuring-instrument pointer provides another application: damping can give a rapid, readable approach to the final position instead of repeated overshoot. The desired response involves both avoiding oscillation and reaching a useful reading promptly.

Practise an exam question

Write your answer first, then tick the mark points it makes.

Exam-style question

Data-based question: A footbridge that sways [20]

Some footbridges sway sideways when crowded. When we walk, each foot pushes slightly sideways, first one way and then the other, so the sideways push repeats once every two steps. At about 2 steps per second this is about 1 Hz. Engineers model the sideways motion of one footbridge as a mass-spring oscillator with an effective mass of 1.2 x 105 kg and a natural frequency of 1.0 Hz. They push the bridge sideways once and record the amplitude after each complete cycle, first without and then with dampers fitted.

Table 1. Amplitude after each cycle of free oscillation
cycle012345
no dampers / mm20.018.016.214.613.111.8
with dampers / mm20.010.05.02.51.30.6

A crowd crossing the bridge puts about 60 J of energy into the sideways oscillation in each cycle. Walkers feel unsafe if the maximum sideways acceleration is more than 0.20 m s-2.

  1. (a) Define simple harmonic motion. [2]
  2. (b) Calculate the effective stiffness k of the bridge. [2]
  3. (c) For the 20.0 mm oscillation, calculate the maximum speed and the maximum acceleration of the bridge. [2]
  4. (d) Calculate the energy of the oscillation when the amplitude is 20.0 mm. [2]
  5. (e) Use Table 1 to show that, without dampers, the amplitude falls by the same fraction each cycle, and calculate the fraction of the energy lost per cycle. [3]
  6. (f) Explain why a crowd walking at about 2 steps per second can make the bridge sway strongly. [3]
  7. (g) The amplitude stops growing when the energy lost per cycle equals the energy put in per cycle. Use this idea to evaluate whether the dampers make the bridge comfortable for this crowd. [5]
  8. (h) State one assumption in the model used in (g). [1]

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