14. Forced Oscillations and Resonance — Barton's Pendulums¶

Module

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

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Aim¶

To study the steady-state response of a damped oscillator to a periodic driving force, and to determine the resonant frequency and quality factor from the resonance curve.

Theoretical Background¶

When a damped oscillator is driven by an external periodic force of frequency ω, rather than left to oscillate freely, it eventually settles into a steady state in which it oscillates at exactly the driving frequency ω (not at its own natural frequency ω₀), with an amplitude and phase lag relative to the driving force that both depend sensitively on how close ω is to ω₀ and on the amount of damping present.

Barton's pendulums is the classic demonstration of this: a heavy 'driver' pendulum is set swinging, and its motion is coupled (via a stretched string) to a row of lighter pendulums of various lengths hung from the same string. Each lighter pendulum experiences the same driving frequency (that of the driver), but the one whose own natural frequency ω₀ most closely matches the driving frequency swings with by far the largest amplitude — visually and dramatically demonstrating resonance.

The steady-state amplitude, as a function of driving frequency, traces out a characteristic peaked curve (the resonance curve): it is small for driving frequencies far from ω₀, rises to a maximum near ω = ω₀, and the sharpness of this peak — the ratio of the resonant frequency to the width of the peak at half its maximum height — is precisely the quality factor Q introduced in the damped-oscillation experiment, now measured in an entirely different (frequency-domain) way. Lightly damped systems (high Q) show a very sharp, tall resonance peak; heavily damped systems (low Q) show a broad, shallow one.

Resonance is far from a mere laboratory curiosity: it governs the tuning of radio receivers, the design (and avoidance of unwanted resonance in) bridges and buildings under wind or seismic loading, magnetic resonance imaging, and the vibration behaviour of virtually every mechanical structure subjected to periodic loads.

Governing Formulas¶

Steady-state amplitude of a driven, damped oscillator:

\[ A(\omega) = \dfrac{F_{0}/m}{\sqrt{\left(\omega_{0}^{2}-\omega^{2}\right)^{2} + \left(2\gamma\omega\right)^{2}}} \]

Phase lag between response and driving force:

\[ \phi = \arctan\!\left(\dfrac{2\gamma\omega}{\omega_{0}^{2}-\omega^{2}}\right) \]

Quality factor from the resonance curve (sharpness of the peak):

\[ Q = \dfrac{f_{0}}{\Delta f} \]

where \(\Delta f\) is the full width of the resonance curve at half its maximum power (amplitude²).

Labelled apparatus layout — original schematic prepared for this guide (not a photograph). A heavy 'driver' pendulum coupled via a taut string to a row of lighter pendulums of different lengths (Barton's pendulums), together with a variable-frequency drive and an amplitude/phase read-out for the pendulum of interest.

Simulator Controls¶

Control	Symbol	Range	Unit
Driving frequency	f_drive	0.2 – 3.0	Hz
Damping (of driven pendulum)	γ	0.05 – 1.00	s⁻¹
Natural frequency of driven pendulum	f0	0.5 – 2.0	Hz

Readouts:

Quantity	Symbol	Unit
Steady-state amplitude	A	cm
Phase lag	φ	degrees

Procedure¶

Launch the Forced Oscillations & Resonance panel; set the natural frequency f0 and damping γ of the driven pendulum to their default values.
Sweep the driving frequency f_drive across a range spanning below and above f0, and record the steady-state amplitude A and phase lag φ at each setting.
Plot A against f_drive; the curve should peak sharply near f_drive = f0.
Determine the full width Δf of the resonance peak at half its maximum power (amplitude²), and calculate Q = f0/Δf.
Repeat the frequency sweep for a larger damping γ and compare the resulting (now broader, shorter) resonance curve and its Q.

Observation Table¶

Table: Steady-state amplitude and phase lag as the driving frequency is swept through resonance

S. No.	f_drive (Hz)	Amplitude A (cm)	Phase lag φ (°)
1
2
3
4
5
6
7
8

Graph¶

Plot Amplitude, A (y-axis) against Driving frequency, f_drive (x-axis).

Expected trend: a peaked curve, maximum near f_drive = f0, narrower and taller for lower damping.

A vs f_drive — the resonance curve; locate the peak and its half-power width

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

Calculation¶

From the A vs f_drive graph, read off the peak frequency (an estimate of f0) and the peak amplitude A_max.
Find the two frequencies on either side of the peak where A² = A_max²/2 (half-power points); their difference is Δf.
Compute Q = f0/Δf, and compare with the Q obtained independently from the damped-oscillation (time-domain) experiment for a similar damping setting.

Result¶

Table: Resonance characteristics of the driven pendulum

γ (s⁻¹)	f0 (Hz), from peak	Δf (Hz)	Q = f0/Δf
1
2

Maximum Permissible Error¶

The half-power width Δf is usually the less precisely determined quantity, since it depends on locating two points on a curve that may be sparsely sampled near the half-power level.

Precautions¶

Allow the driven pendulum to settle into its steady-state response at each driving frequency before recording the amplitude — the initial transient response is a mixture of the driving frequency and the system's own natural frequency and must be allowed to die away.
Sample the driving frequency more finely near the expected resonance, where the amplitude changes rapidly, than far from it.
Keep the driving amplitude F0 constant throughout the entire frequency sweep, so the resonance curve reflects only the frequency dependence, not a changing drive strength.

Self-Check Quiz¶

1. At resonance (driving frequency equal to the natural frequency), the phase lag between the response and the driving force is approximately:

A) 0°
B) 45°
C) 90°
D) 180°

Answer: C — From the phase formula, when ω = ω₀ the denominator (ω0²−ω²) vanishes, giving φ = arctan(∞) = 90°.

2. A sharply peaked, tall resonance curve corresponds to:

A) Heavy damping (low Q)
B) Light damping (high Q)
C) Zero driving force
D) No damping information can be inferred from the curve's shape

Answer: B — Light damping allows the amplitude to build up to a very large value right at resonance while falling off quickly away from it, giving a narrow, tall peak and hence a high Q = f0/Δf.

3. The steady-state response of a driven oscillator settles at a frequency equal to:

A) The oscillator's own natural frequency, regardless of the driving frequency
B) The driving frequency, not the oscillator's natural frequency
C) The average of the two frequencies
D) Zero

Answer: B — In the steady state, the driven oscillator is forced to oscillate at exactly the frequency of the periodic driving force, even though its amplitude and phase depend on how this compares to its own natural frequency.

Viva-Voce Questions¶

Why does the phase lag between the driven oscillator and the driving force pass through 90° exactly at resonance?
How is the quality factor obtained from a resonance curve related to the quality factor obtained from a free (damped) decay measurement?
In Barton's pendulums, why does only the pendulum whose length matches the driver's length swing with a large amplitude?
Why must you wait for the transient response to die away before reading a 'steady-state' amplitude at each driving frequency?
Give a real engineering example where avoiding resonance is a critical design consideration.
How would the resonance curve change if the damping were reduced to nearly zero?