Physics (9702) | Damped and forced oscillations, resonance

Study Notes: Damped and Forced Oscillations, and Resonance (9702 Physics)

Welcome to one of the most exciting topics in oscillations! You’ve already mastered Simple Harmonic Motion (SHM), where things swing forever. But in the real world, nothing swings forever—energy is always lost. This chapter investigates these real-world scenarios: how energy loss affects motion (Damping) and what happens when we continuously push an oscillating system (Forced Oscillations), leading to the dramatic phenomenon of Resonance.

Don't worry if this seems tricky at first. We will use lots of analogies to break down these concepts!

1. Damping: The Reality of Oscillations

What is Damping?

In ideal SHM, the total energy of the system remains constant, so the amplitude never changes. However, in reality, all oscillating systems experience damping.

Damping is the process where the energy of an oscillating system is gradually lost to the surroundings, usually converted into thermal energy (heat).

The loss of energy is caused by resistive forces (like air resistance or internal friction/viscosity) acting on the system.

Key takeaway: Because the system loses energy, the amplitude of the oscillation decreases over time. The oscillation is no longer truly simple harmonic, although the period often remains approximately the same if the damping is light.

Visualising Damping: The Displacement-Time Graph

If you plotted the displacement against time for a damped oscillator, the shape is similar to the SHM sine wave, but the peaks get lower and lower. This decrease in amplitude follows an exponential decay pattern.

Mathematically, the amplitude \(x_0\) decays over time \(t\) according to:
$$ x = x_0 e^{-kt} \cos(\omega t) $$
(You do not need to recall this exact formula, but understand the exponential nature of the decay.)

Types of Damping (Syllabus 17.3.2)

We classify damping based on how quickly the system loses energy and stops moving.

(a) Light Damping (Under-damped)

This is where the damping force is small.

• The system oscillates many times before coming to rest.
• The period of oscillation changes very little, but the amplitude gradually decreases exponentially.
• Analogy: A grandfather clock pendulum swinging in the air, slowly losing momentum.

Sketch Description (Displacement-Time Graph): A typical wave shape, but the envelope (the imaginary line connecting the peaks) curves inward exponentially.

(b) Critical Damping

This is the optimum amount of damping.

• The system returns to the equilibrium position in the shortest possible time.
• It stops moving without oscillating.

Analogy: The shock absorbers in your car. When you hit a speed bump, you want the car body to return to level ground immediately, not bounce up and down. Critical damping ensures the fastest, non-oscillatory return.

Sketch Description (Displacement-Time Graph): The displacement starts high (or low) and drops quickly, reaching zero displacement instantly without crossing the x-axis.

(c) Heavy Damping (Over-damped)

This is where the damping force is very large.

• The system returns to the equilibrium position very slowly.
• It stops moving without oscillating, but takes much longer than critical damping.

Analogy: Trying to close a door that has a heavy, well-maintained door-closer mechanism. It moves extremely slowly and never swings back and forth.

Sketch Description (Displacement-Time Graph): Similar to critical damping, but the decay curve is much shallower and takes a longer time to approach zero displacement.

2. Forced Oscillations and Resonance

Natural Frequency (\(f_0\))

Before we discuss forcing, recall the natural frequency (\(f_0\)).

This is the frequency at which a system oscillates if it is displaced once and then left to move freely (i.e., its intrinsic, undamped frequency).

Example: The natural frequency of a simple pendulum depends only on its length and the acceleration due to gravity, \(g\).

Forced Oscillations

If we apply an external periodic force to an oscillating system, we cause forced oscillations.

• The external force is called the driving force.
• The frequency of this external force is the driving frequency (\(f\)).

When an oscillator is forced, it eventually settles down to oscillate at the driving frequency (\(f\)), regardless of its original natural frequency (\(f_0\)).

Resonance (Syllabus 17.3.3)

The most dramatic effect of forced oscillations occurs when the driving frequency matches the natural frequency.

Resonance Definition: Resonance occurs when a system is forced to oscillate at its natural frequency (\(f_0\)), causing the amplitude of oscillation to reach a maximum value.

Analogy: Pushing a child on a swing. The swing has a natural rhythm (\(f_0\)). If you push at exactly that rhythm (\(f = f_0\)), you transfer energy most effectively, and the amplitude (height) of the swing increases dramatically. If you push too fast or too slow, your pushes dampen the motion instead.

During resonance, energy is transferred from the driving force to the oscillator at the maximum possible rate, leading to a large energy build-up and thus maximum amplitude.

The Effect of Damping on Resonance

Damping is crucial because it limits the maximum amplitude reached at resonance.

Consider a graph showing amplitude against driving frequency (\(f\)):

• A system with no damping would theoretically achieve an infinite amplitude at resonance (though this is physically impossible). The peak would be infinitely sharp.
• A system with light damping produces a tall, very sharp resonance curve. The maximum amplitude is large, and it occurs almost exactly at \(f = f_0\).
• A system with heavy damping produces a short, very broad (flat) resonance curve. The maximum amplitude is small, and the peak might even shift slightly to a frequency lower than \(f_0\).

Key Conclusion: The lighter the damping, the sharper and larger the maximum amplitude at resonance.

3. Real-World Implications of Resonance

When Resonance is Desirable (Applications)

We often rely on resonance to amplify weak signals or produce specific effects:

• Musical Instruments: A musician plucks a string (the driving force), and the body of the instrument (like a guitar's wooden body) is designed to have many natural frequencies that match the sound of the string, causing the sound to be amplified.
• Tuning Radio Circuits: To pick up a specific radio station, you adjust the circuit's components so its natural electrical frequency matches the frequency broadcast by the station. This causes the signal from that station to resonate (amplify), while others remain weak.
• Magnetic Resonance Imaging (MRI): This medical tool uses controlled resonance of atomic nuclei within the body to create detailed images.

When Resonance is Undesirable (Dangers)

If resonance causes too much energy to build up in a large structure, it can lead to catastrophic failure.

• Bridges and Buildings: If the natural swaying frequency of a bridge matches the frequency of wind gusts or marching footsteps, the amplitude of oscillation can increase until the structure fails.
• Did you know? The famous collapse of the Tacoma Narrows Bridge in 1940 (USA) was due to the wind creating turbulent vortices that matched the bridge's natural torsional frequency, leading to massive, destructive resonance.
• Machinery: Resonance can cause parts in engines or aircraft to vibrate violently, leading to fatigue and breakage. Engineers use damping materials (like specialized shock absorbers or rubber mounts) to heavily damp critical components, preventing sharp resonance peaks.

Chapter Summary: Key Takeaways

Damping:

• Caused by resistive forces (like friction/air resistance).
• Decreases the amplitude of oscillation exponentially due to energy loss.
• Light damping: Oscillates many times, slow decay.
• Heavy damping: Slow return to equilibrium, no oscillation.
• Critical damping: Fastest non-oscillatory return to equilibrium (ideal for stabilizing systems like shock absorbers).

Resonance:

• Occurs in forced oscillations when the driving frequency (\(f\)) equals the natural frequency (\(f_0\)).
• Leads to a maximum amplitude because energy transfer is maximized.
• Light damping results in a sharp, high resonance peak.
• Heavy damping results in a broad, low resonance peak.