Forced vibrations and resonance - Physics (9630) - Oxford AQA A-Level

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Welcome to the exciting world of Forced Vibrations and Resonance! This chapter explains how external forces can make things shake, rattle, and even break if the timing is just right. Understanding these concepts is crucial—they explain why tuning a radio works and why engineers must worry about bridges collapsing!

1. Review: Free Vibrations and Natural Frequency

Before forcing a system, we must know how it behaves naturally.

A system undergoing free vibration (or natural oscillation) oscillates only under the influence of the restoring forces (like tension or gravity) with no external driving force applied after the initial displacement.

The system vibrates at its own unique rate, known as the natural frequency ($f_0$).

Example: If you pluck a guitar string and let it vibrate freely, the sound it makes corresponds to its natural frequency ($f_0$).

Quick Review of Damping

In reality, free oscillations always decrease in amplitude over time due to energy being dissipated (lost) to the surroundings (usually as heat or sound). This process is called damping. Air resistance and friction are common sources of damping.

Key Takeaway: Every physical object has a natural frequency ($f_0$) at which it 'prefers' to oscillate if left alone.

2. Forced Vibrations and Driving Frequency

What happens when we don't let the system oscillate freely? We introduce a persistent, periodic external force.

What is a Forced Vibration?

A forced vibration occurs when an external periodic force is continuously applied to an oscillating system. This external force is often called the driving force.

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

When forced, the system is made to vibrate at the driving frequency ($f$) of the external force, not its own natural frequency ($f_0$).

Analogy: Imagine a swing. The person on the swing has a natural frequency ($f_0$). If you push the swing (the driving force) regularly, the swing oscillates at the rate you are pushing ($f$), regardless of its natural rate.

Key Term: The steady amplitude achieved by the forced oscillation depends heavily on the relationship between the driving frequency ($f$) and the natural frequency ($f_0$).

3. The Phenomenon of Resonance

Resonance is arguably the most important concept in this chapter—it's what happens when the driving force is perfectly timed.

Definition of Resonance

Resonance occurs when the driving frequency ($f$) equals the natural frequency ($f_0$) of the oscillating system.

$$f = f_0$$

When this match happens, energy transfer from the driver to the oscillating system is at its maximum efficiency. This causes the amplitude of the oscillations to build up dramatically.

If there were absolutely no damping (which is impossible in reality), the amplitude would theoretically increase forever, leading to catastrophic failure.

Why does the Amplitude Get so Large?

When the driving frequency matches the natural frequency, the external force is always applied in the same direction as the system is moving. This means that:
1. Maximum energy is added to the system during each cycle.
2. The work done by the driving force continuously increases the system's total energy and, therefore, its amplitude.

Memory Trick: Resonance happens when Rates are equal ($f = f_0$), leading to Really big amplitudes.

Did you know? The Tacoma Narrows Bridge collapse in 1940 is a famous (though often simplified) example. While wind forces caused a complex oscillation, the general principle of large-amplitude build-up due to external energy matching a natural frequency illustrates the danger of mechanical resonance.

4. The Crucial Role of Damping

In a real-world scenario, damping dictates how dangerous or useful resonance can be.

Damping and Resonance Graphs (Qualitative Treatment)

We often use a graph plotting amplitude against driving frequency ($f$) to show resonance. The peak of this curve occurs at $f = f_0$.

The amount of damping significantly affects the shape and height of this resonance curve (the sharpness of resonance):

1. Low Damping:

The resonance peak is very tall (high maximum amplitude).
The peak is very narrow and sharp.
Resonance only occurs over a very small range of frequencies near $f_0$.

2. High Damping (Heavy Damping):

The resonance peak is short (low maximum amplitude).
The peak is broad and flat.
The maximum amplitude is shifted slightly to a frequency less than $f_0$.

In heavily damped systems, resonance is barely noticeable. The system absorbs energy poorly, regardless of whether the frequencies match.

Accessibility Check: Don't worry if sketching this graph seems tricky at first. The key idea is this:
Damping fights the build-up of amplitude. Less damping = bigger fight = higher peak.

Key Takeaway: Damping reduces the maximum amplitude at resonance and makes the resonance curve broader (less sharp).

5. Real-World Applications and Examples

Resonance can be helpful (good) or destructive (bad).

A. Beneficial Resonance (We want this!)

1. Tuning a Radio Receiver

An electronic circuit (like the one inside your radio) acts as an oscillating system. It has a natural electrical frequency ($f_0$).

The radio wave signal coming from the broadcaster acts as the driving force with frequency ($f$).

When you turn the dial, you change the components (like capacitance), thereby changing the circuit's natural frequency ($f_0$) until it matches the driving frequency ($f$) of the desired station.

At resonance, the circuit oscillates with maximum current (maximum amplitude), allowing the radio to pick up that specific station clearly.

2. Musical Instruments (Stationary Waves)

Musical instruments rely on resonance to amplify sound.

In a wind instrument (like a flute), blowing into the mouthpiece creates a range of driving frequencies. The column of air inside the tube has several natural frequencies where stationary waves can form.
When the driving frequency matches one of the air column's natural frequencies, resonance occurs, creating a high-amplitude stationary wave (a loud note). This is why a small vibration from your lips can create a loud sound.

B. Destructive Resonance (We want to avoid this!)

1. Mechanical Structures (Buildings and Bridges)

Every bridge, building, and even plane wing has a natural frequency. External vibrations (wind, traffic, earthquakes, or marching soldiers) act as the driving force.

Engineers must ensure the building’s natural frequency ($f_0$) is very different from any common driving frequency ($f$) they might encounter.
If resonance occurs, the massive build-up in amplitude can stress the material beyond its breaking point, leading to structural failure.
This is why large mechanical systems often incorporate heavy damping elements (like shock absorbers) to keep amplitudes low, even if accidental resonance occurs.

2. Machinery and Engines

Engines and rotors vibrate as they run. If the rotational speed (driving frequency) hits the natural frequency of the engine mount or the chassis, severe resonant vibrations can occur, damaging the equipment.

Quick Review Box: Key Concepts in Forced Oscillations

Natural Frequency ($f_0$): The frequency at which a system oscillates freely (if undamped).

Driving Frequency ($f$): The frequency of the external force continually pushing the system.

Forced Vibration: An oscillation driven by an external periodic force.

Resonance: Occurs when $f = f_0$, resulting in maximum amplitude due to maximum energy transfer.

Damping Effect: Reduces the maximum amplitude at resonance and makes the resonance curve broader (less sharp).