Damping is the phenomenon of gradual decrease in oscillation amplitude of a mechanical system due to dissipative forces (friction, air resistance). This is what distinguishes real oscillators from ideal harmonic oscillators. This concept is covered in 12th grade curriculum.
Discover FizziQ
How to measure it in class
With the FizziQ app, the accelerometer allows observation and quantification of damping.
Steps:
- Attach the smartphone to a spring and let it oscillate
- Record the acceleration with FizziQ over several oscillations
- Observe the amplitude decrease over time
- Measure the half-life (time for the amplitude to be halved)
- Compare with a pendulum in air and in water (different damping)
- Calculate the quality factor Q
Scientific activities on this topic
- Damped oscillations of a pendulum: https://www.fizziq.org/en/activities/simple-pendulum/
- Comparison of air/water/oil
- Measurement of quality factor
Learn more
Damping regimes:
Depending on the damping intensity, three regimes are possible:
| Regime | Behavior |
|---|---|
| Underdamped | Damped oscillations (most common) |
| Critical | Fastest return without oscillation |
| Overdamped | Slow return without oscillation |
Equation with damping:
d squared x / dt squared + 2 lambda x dx/dt + omega_0 squared x x = 0
where lambda is the damping coefficient and omega_0 is the natural angular frequency.
Exponential decay:
In the underdamped regime, amplitude decreases exponentially: A(t) = A_0 x e to the power of (-lambda t)
The characteristic time tau = 1/lambda is the relaxation time.
Quality factor:
The quality factor Q characterizes the “quality” of the oscillator: Q = omega_0 / (2 lambda)
The larger Q, the longer the oscillator oscillates.
Formula
Solution in underdamped regime: x(t) = A_0 x e to the power of (-lambda t) x cos(omega t + phi)
with omega = square root of (omega_0 squared - lambda squared) < omega_0
Quality factor: Q = omega_0 / (2 lambda) = pi x n_oscillations
Energy after n oscillations: E_n = E_0 x e to the power of (-2n pi / Q)
where:
- lambda: damping coefficient (s to the -1)
- omega_0: natural angular frequency (rad/s)
- Q: quality factor (dimensionless)
Application examples
- A quality tuning fork has Q approximately equals 1000 (oscillates long after being struck)
- Car shock absorbers are in critical regime to avoid bouncing
- A pendulum in air: Q approximately equals 100, in water: Q approximately equals 5
- Resonance is sharper when Q is larger
FAQ
Q: How to measure the quality factor with FizziQ? A: Count the number of oscillations before the amplitude falls to 37% (1/e) of its initial value. Q approximately equals pi times this number.
Q: Why is damping sometimes desirable? A: It prevents prolonged oscillations. Car shock absorbers prevent the body from continuing to oscillate after a speed bump.
Q: Does frequency change with damping? A: Yes, slightly. The pseudo-period is larger than T_0: T = 2 pi / square root of (omega_0 squared - lambda squared). But for weak damping, the effect is negligible.
Q: Can damping be compensated? A: Yes, by providing energy periodically (clock with escapement, electronic oscillator). This is the principle of maintaining oscillations.
Related concepts
Harmonic Oscillator - Friction - Quality Factor - Transient Regime - Resonance - Mechanical Energy