Spring damping

Study the effect of damping on spring oscillations: underdamped, critically damped, and overdamped regimes.

In reality, no oscillation lasts forever: friction always stops the motion eventually. But how does the amplitude decrease? Gently and gradually, or abruptly? The answer depends on the strength of the damping. The FizziQ Web Spring Oscillator simulation lets you explore three fundamentally different behaviors by progressively increasing the damping coefficient: underdamped oscillations that gradually decay, critical damping where the system returns to equilibrium as fast as possible without oscillating, and overdamped motion where the return is sluggish and slow.

Activity overview:

The student uses the FizziQ Web Spring Oscillator simulation with fixed mass and stiffness, then progressively increases damping. They observe and record the position-time curve for each regime and determine the critical damping value.

Level:

High school

FizziQ Web

Author:

Duration (minutes) :

35

What students will do :

- Observe the effect of damping on oscillation amplitude and shape
- Identify the three regimes of a damped oscillator
- Determine the critical damping value experimentally
- Understand the exponential decay of amplitude in the underdamped regime
- Connect the three regimes to practical applications

Scientific concepts:

- Damped harmonic oscillator
- Underdamped regime
- Critical damping
- Overdamped regime
- Viscous damping
- Logarithmic decrement

Sensors:

- FizziQ Web Spring Oscillator simulation

Material needed:

- Computer, tablet, or smartphone with FizziQ Web

Experimental procedure:

Open the Spring Oscillator simulation in FizziQ Web. Set the mass to 1.0 kg, stiffness to 20 N/m, and amplitude to 0.3 m.
Set the damping to 0 N·s/m. Start a recording (REC). Observe: the oscillations continue indefinitely with constant amplitude. This is the undamped reference.
Set the damping to 0.5 N·s/m. Start a recording. Observe: the oscillations continue but the amplitude gradually decreases. This is the underdamped regime.
Increase the damping to 1.0 N·s/m, then 2.0 N·s/m. Start a recording for each. Compare the curves: the amplitude decreases faster with higher damping.
Continue increasing the damping. Find the value at which the mass returns to equilibrium without oscillating (it does not overshoot). This is the critical damping value.
Increase the damping beyond the critical value. Observe: the mass returns to equilibrium even more slowly, without oscillating. This is the overdamped regime.
Superimpose the curves obtained for different damping values. Describe the differences.
For the underdamped regime (damping = 1.0 N·s/m), measure the amplitude of several successive oscillations. Does the amplitude decrease at a constant rate?
Calculate the ratio between two successive amplitudes. Is this ratio constant? If so, the decay is exponential.
Write a conclusion: describe the three regimes and give a concrete example of each (car shock absorber for critical, door closer for overdamped, guitar string for underdamped).

Expected results:

For m = 1 kg and k = 20 N/m, the natural angular frequency is ω₀ = √(k/m) ≈ 4.47 rad/s. The theoretical critical damping is b_c = 2√(km) = 2√(20) ≈ 8.94 N·s/m. For b < b_c: damped oscillations (underdamped) with exponentially decreasing amplitude. For b = b_c: fastest return to equilibrium without oscillation (critical damping). For b > b_c: slow return without oscillation (overdamped). The ratio of consecutive amplitudes is constant, confirming exponential decay.

Scientific questions:

- Why are car shock absorbers tuned to critical damping?
- In the underdamped regime, does the pseudo-period change compared to the undamped case?
- Why is the amplitude decay exponential rather than linear?
- What happens to the energy of the oscillator as the amplitude decreases?
- Could you determine the damping coefficient from the rate of amplitude decay?
- Why does overdamped motion take longer to reach equilibrium than critical damping?

Scientific explanations:

The equation of motion of a damped oscillator is: m × a = -k × x - b × v, where b is the damping coefficient (in N·s/m). The term -bv is the viscous friction force, proportional to velocity and opposing the motion.

The system behavior depends on the ratio between damping b and the critical damping b_c = 2√(km). If b < b_c, the system oscillates with decreasing amplitude: this is the underdamped (pseudo-periodic) regime.

If b = b_c, the system returns to equilibrium as quickly as possible without ever oscillating: this is the critical damping regime. This is the optimal setting for car shock absorbers.

If b > b_c, the system returns to equilibrium without oscillating, but more slowly than at critical damping: this is the overdamped regime. Door closers use this regime to ensure a slow, gentle return.

In the underdamped regime, the amplitude decays as A(t) = A₀ × exp(-b×t / 2m). The ratio between two successive amplitudes is constant, equal to exp(-bT/2m), where T is the pseudo-period.

Extension activities:

Frequently asked questions:

Q: How do I find the exact critical value?
R: Proceed by bisection: if oscillations are still visible, increase the damping; if the mass does not overshoot equilibrium, decrease it. Narrow down until you find the transition.

Q: The pseudo-period seems to change with damping. Is this normal?
R: Yes, the pseudo-period increases slightly with damping. The theoretical value is T' = 2π/√(ω₀² - (b/2m)²), which is longer than the undamped period.

Q: What is the logarithmic decrement?
R: It is the natural logarithm of the ratio between two successive maxima: δ = ln(A_n/A_{n+1}) = bT/2m. It is constant for a given damping and characterizes the damping rate.

Q: Why does the overdamped regime return more slowly even though the damping is stronger?
R: More damping means more resistance to motion. Beyond critical damping, the excessive friction slows the return rather than speeding it up. Critical damping is the optimal compromise.