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Damped Harmonic Oscillator Reference cheat sheet - grade 11-12

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Physics Grade 11-12

Damped Harmonic Oscillator Reference Cheat Sheet

A printable reference covering damping force, damped frequency, decay constant, quality factor, and underdamped motion for grades 11-12.

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A damped harmonic oscillator is a system that vibrates while losing mechanical energy to a resistive force such as friction or air resistance. This reference helps students connect the motion equation, exponential decay, frequency change, and energy loss in one organized place. It is useful for solving spring, pendulum, circuit analogy, and wave damping problems in upper high school physics.

The core model uses a restoring force proportional to displacement and a damping force proportional to velocity. For light damping, the object still oscillates, but its amplitude decreases like A(t)=A0e−γtA(t) = A_0 e^{-\gamma t}. The most important ideas are the damping ratio, damped angular frequency, exponential energy decay, and whether the system is underdamped, critically damped, or overdamped.

Key Facts

  • The damped oscillator equation for linear damping is md2xdt2+bdxdt+kx=0m\frac{d^2x}{dt^2} + b\frac{dx}{dt} + kx = 0.
  • The natural angular frequency without damping is ω0=km\omega_0 = \sqrt{\frac{k}{m}}.
  • The damping constant used in many solutions is γ=b2m\gamma = \frac{b}{2m}.
  • For underdamped motion, the displacement is x(t)=A0e−γtcos⁡(ωdt+ϕ)x(t) = A_0 e^{-\gamma t}\cos(\omega_d t + \phi).
  • The damped angular frequency is ωd=ω02−γ2\omega_d = \sqrt{\omega_0^2 - \gamma^2} for an underdamped oscillator.
  • The motion is underdamped when γ<ω0\gamma < \omega_0, critically damped when γ=ω0\gamma = \omega_0, and overdamped when γ>ω0\gamma > \omega_0.
  • The amplitude envelope decreases according to A(t)=A0e−γtA(t) = A_0 e^{-\gamma t}.
  • The mechanical energy decreases approximately as E(t)=E0e−2γtE(t) = E_0 e^{-2\gamma t} because energy is proportional to amplitude squared.

Vocabulary

Damping
Damping is the loss of mechanical energy from an oscillator due to resistive forces such as friction, drag, or internal resistance.
Damping force
A damping force is often modeled as Fd=−bvF_d = -bv, where bb is the damping coefficient and the negative sign means the force opposes velocity.
Natural angular frequency
The natural angular frequency ω0=km\omega_0 = \sqrt{\frac{k}{m}} is the angular frequency the system would have with no damping.
Damped angular frequency
The damped angular frequency ωd=ω02−γ2\omega_d = \sqrt{\omega_0^2 - \gamma^2} is the oscillation rate of an underdamped system.
Critical damping
Critical damping occurs when γ=ω0\gamma = \omega_0, giving the fastest return to equilibrium without oscillation.
Quality factor
The quality factor Q=ω02γQ = \frac{\omega_0}{2\gamma} measures how weakly damped an oscillator is, with larger QQ meaning slower energy loss.

Common Mistakes to Avoid

  • Using ω0\omega_0 instead of ωd\omega_d for the actual oscillation frequency is wrong because damping lowers the frequency in underdamped motion.
  • Forgetting that amplitude and energy decay at different rates is wrong because A(t)=A0e−γtA(t) = A_0 e^{-\gamma t} but E(t)=E0e−2γtE(t) = E_0 e^{-2\gamma t}.
  • Treating every damped system as oscillatory is wrong because overdamped and critically damped systems return to equilibrium without repeated oscillations.
  • Dropping the negative sign in Fd=−bvF_d = -bv is wrong because the damping force must oppose the direction of motion.
  • Confusing the damping coefficient bb with the decay constant γ\gamma is wrong because they are related by γ=b2m\gamma = \frac{b}{2m}, not equal in general.

Practice Questions

  1. 1 A mass spring system has m=2.0 kgm = 2.0\,\text{kg} and k=50 N/mk = 50\,\text{N/m}. Find the natural angular frequency ω0\omega_0.
  2. 2 For a damped oscillator with m=1.5 kgm = 1.5\,\text{kg} and b=0.60 kg/sb = 0.60\,\text{kg/s}, calculate γ=b2m\gamma = \frac{b}{2m}.
  3. 3 An oscillator has ω0=10 rad/s\omega_0 = 10\,\text{rad/s} and γ=3 s−1\gamma = 3\,\text{s}^{-1}. Find ωd\omega_d and classify the motion.
  4. 4 Explain why a critically damped car suspension is usually preferred over an underdamped or overdamped suspension.

Understanding Damped Harmonic Oscillator Reference

Damping is not just a smaller amplitude drawn on a graph. It changes the transfer of energy during every cycle. A resistive force points opposite to the direction of motion, so it removes energy whether the mass moves left or right.

The force is greatest when speed is greatest. Near the turning points, the speed is nearly zero, so the damping force is weak there.

This explains why energy loss is strongly connected to the motion through the central equilibrium position. In a real spring experiment, rubbing at the support, internal stretching of the spring, and air drag can all contribute.

The exponential envelope has an important consequence. Equal time intervals do not remove equal amounts of amplitude. During the first few seconds, the amplitude falls by a large amount.

Later, it falls more slowly in absolute distance, though it keeps losing the same fraction over each equal interval. The time required for the amplitude to fall to about thirty seven percent of its starting value is called the decay time. It equals one divided by the damping constant.

For energy, the comparable time is half as long because energy depends on the square of amplitude. Students often confuse these two decay rates.

Damping causes a small shift in the period of an underdamped oscillator. The oscillator takes slightly longer to complete a cycle than it would with no resistance. With light damping, this difference can be too small to notice on a classroom graph.

With stronger damping, it becomes clear. The motion still crosses equilibrium repeatedly only while the damping remains below the critical level. At the critical level, the object returns to equilibrium in the shortest possible time without crossing it.

Greater damping makes the return slower because the resistive force prevents motion too strongly. Door closers, vehicle suspension systems, and measuring instruments are designed by balancing these effects.

The quality factor gives a useful way to describe how long an oscillator keeps its vibration. A high quality factor means little energy is lost per cycle, so the resonance peak is narrow and the oscillations persist. A low quality factor means energy is lost quickly, producing a broader and less selective response.

In simple light damping, quality factor is approximately the natural angular frequency divided by two times the damping constant. Keep track of which frequency a problem gives. Angular frequency is measured in radians per second, while ordinary frequency is measured in cycles per second.

When reading graphs, measure successive peak heights to test exponential decay, then measure the time between peaks to find the damped period. Real data will rarely fit perfectly because damping may not be exactly proportional to velocity.