13. Damped Harmonic Oscillations — Logarithmic Decrement and Quality Factor¶

Module

Oscillations, Damping & Acoustic Methods (PG) · Module 4

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Aim¶

To study the decay of amplitude in a damped harmonic oscillator, and to determine the logarithmic decrement and quality factor characterising the damping.

Theoretical Background¶

Every real oscillator loses energy over time to friction, air resistance, or other dissipative mechanisms, so its oscillations gradually die away rather than continuing forever at constant amplitude, as an ideal (undamped) simple-harmonic oscillator would. For the common case of 'light' damping — where the dissipative force is proportional to velocity and not too large — the amplitude decays exponentially with time while the system still oscillates at a frequency only very slightly lower than its undamped natural frequency.

The logarithmic decrement, δ, defined as the natural logarithm of the ratio of any two successive amplitude peaks, provides a simple, dimensionless measure of how quickly the oscillation dies away: a larger δ means faster decay (heavier damping) per cycle. Because the amplitude ratio between successive peaks is the same throughout the entire decay for exponential damping, δ can equally well be measured from peaks that are several cycles apart, then divided by the number of intervening cycles, to reduce the effect of any single reading error.

A closely related and widely used figure of merit is the quality factor, Q, which for light damping is well approximated by Q ≈ π/δ; physically, Q is (up to a factor of 2π) the number of radians of oscillation the system completes before its energy decays to 1/e of its initial value, so a high-Q oscillator (a good pendulum clock, a struck tuning fork, or a superconducting microwave cavity) rings for a very long time, while a low-Q oscillator (a car's suspension, deliberately) dies away within a cycle or two.

This same mathematics of exponential amplitude decay, logarithmic decrement, and quality factor recurs throughout physics and engineering — from mechanical vibration damping and electrical RLC circuits to the linewidth of atomic spectral lines and the ringdown of laser cavities — making this simple mechanical experiment a gateway to a very general and important concept.

Governing Formulas¶

Exponential amplitude decay:

\[ A_{n} = A_{0}\, e^{-\gamma t}, \qquad \gamma = \dfrac{b}{2m} \]

Logarithmic decrement (from successive peak amplitudes):

\[ \delta = \ln\!\left(\dfrac{A_{n}}{A_{n+1}}\right) = \gamma T \]

Quality factor (light damping):

\[ Q \approx \dfrac{\pi}{\delta} \]

Labelled apparatus layout — original schematic prepared for this guide (not a photograph). A mass–spring (or torsional) oscillator fitted with an adjustable dashpot or eddy-current damper, together with a means of recording the amplitude of successive oscillations (e.g. an attached pointer and scale, or a motion sensor).

Simulator Controls¶

Control	Symbol	Range	Unit
Damping coefficient	b	0.05 – 1.00	kg/s
Oscillating mass	m	50 – 300	g
Spring/restoring constant	k_spring	5 – 40	N/m

Readouts:

Quantity	Symbol	Unit
Logarithmic decrement	δ	dimensionless
Quality factor	Q	dimensionless

Procedure¶

Launch the Damped Oscillations panel; set the mass m and spring constant k_spring to their default values.
Starting from the smallest damping coefficient b, note the logarithmic decrement δ and quality factor Q shown on the readout.
Increase b in equal steps, holding m and k_spring fixed, and tabulate δ and Q at each step.
Plot δ against b and verify the expected roughly linear increase; plot Q against b and verify the expected decrease.
From the slope of the δ vs b graph, estimate the period T of the (lightly damped) oscillation, and compare with T = 2π√(m/k_spring).

Observation Table¶

Table: Logarithmic decrement and quality factor with increasing damping

S. No.	b (kg/s)	m (g)	k_spring (N/m)	δ (log. decrement)	Q (quality factor)
1
2
3
4
5
6

Graph¶

Plot Logarithmic decrement, δ (y-axis) against Damping coefficient, b (x-axis).

Expected trend: an increasing, roughly linear trend (δ = bT/2m, with T only weakly dependent on b for light damping).

δ vs b, expected to increase roughly linearly for light-to-moderate damping

15×15 grid — choose your own scale per square; plot from your observation table

Calculation¶

Compute the slope of the δ vs b graph: slope ≈ T/2m.
Rearrange to estimate the period T ≈ 2m × slope, and compare with the expected T = 2π√(m/k_spring).
For each trial, verify Q ≈ π/δ against the readout value of Q.

Result¶

Table: Damping characteristics of the oscillator

b (kg/s)	δ	Q	T estimated (s)	T expected (s)
1
2
3

Maximum Permissible Error¶

Since δ is a logarithm of an amplitude ratio, its absolute error is approximately the sum of the fractional errors in the two amplitude readings — reading amplitudes several cycles apart and dividing by the cycle count reduces this error.

Precautions¶

Read amplitude peaks on the same side of the equilibrium position each time (e.g. only the rightward extremes), to avoid any small asymmetry in the setup biasing the result.
Where possible, measure the amplitude ratio between peaks separated by several cycles and divide by the number of intervening cycles, rather than relying on a single successive pair.
Keep the initial amplitude small enough that damping can be treated as linear (proportional to velocity) throughout the run.

Self-Check Quiz¶

1. The logarithmic decrement δ is defined as:

A) The ratio of the period to the amplitude
B) The natural logarithm of the ratio of two successive peak amplitudes
C) The total number of oscillations before stopping
D) The undamped natural frequency

Answer: B — δ = ln(A_n/A_{n+1}) by definition, and for exponential decay this ratio (and hence δ) is the same for every successive pair of peaks.

2. A high-quality-factor (high-Q) oscillator is one that:

A) Stops oscillating almost immediately
B) Continues oscillating for many cycles before its energy decays significantly
C) Has no restoring force
D) Oscillates at zero frequency

Answer: B — High Q corresponds to light damping — the system 'rings' for a long time, completing many radians of oscillation before its energy falls to 1/e of its initial value.

3. If the damping coefficient b is increased (all else fixed), the logarithmic decrement δ:

A) Decreases
B) Increases
C) Stays exactly constant
D) Becomes negative

Answer: B — δ = γT = (b/2m)T, so increasing b (which increases the decay rate γ) increases δ.

Viva-Voce Questions¶

Why is the ratio of successive amplitudes constant throughout an exponentially damped oscillation, rather than the difference?
Derive, in outline, the approximate relation Q ≈ π/δ for light damping.
Why might you prefer to measure amplitudes several cycles apart rather than just two successive peaks?
How would critically damped or overdamped motion differ qualitatively from the lightly damped case studied here?
Give an example of a real system where a high Q is desirable, and one where a low Q is deliberately engineered.
How does the period of a lightly damped oscillator compare with the period of the same system if it were undamped?