Resonance

Resonance can occur when there is a periodic external driving force causing a system to oscillate. When the frequency of the driving force gets closer to the natural frequency of the system, it begins to vibrate with a much greater amplitude, which continues to increase up to a maximum when the two frequencies are equal.

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There are several factors that lead to resonance, as we shall explore below.

Natural frequency

The natural frequency of an object is the frequency at which it oscillates during a free vibration, which is when there are no external forces to affect its motion and thus no energy transfer between the object and its surroundings.

Let’s consider the mass-spring system as an example. If you were to hold the mass away from its equilibrium position so that the spring is stretched and then release the mass, the system would oscillate at its natural frequency. If no energy is transferred to the surroundings, the mass continues to oscillate at this frequency forever. In reality, however, the mass is eventually slowed down by frictional forces, such as air resistance.

Figure 1. A mass-spring system oscillating at its natural frequency. Source: Physics LibreTexts (CC BY-SA 4.0).

Forced vibrations

A periodic external driving force acting on a system causes a forced vibration. The frequency of the force is called the driving frequency. If this is equal to the natural frequency of the system, it causes extremely large oscillations, and at that point, the system is resonating.

An example of a system that can be caused to resonate by a driving force is a swing in a playground. If someone pushes it at time intervals that coincide correctly with its back-and-forth motion – i.e., if the frequencies of the pushes and the swing are the same – the system resonates, and the swings become much higher.

A child on a swing is being pushed by their parent every 2 seconds so that the swings finish very high – the system is resonating.

What is the frequency of the driving force (the parent pushing)? And what is the time interval t between the parent exerting maximum force and the child reaching the top of the swing?

The frequency is related to the time period of a periodic motion by:

$$ƒ = \frac{1}{T}; ƒ = 0.5 \space Hz$$

The parent should push when the swing is moving through the equilibrium position. This is a quarter of the time period of the swing’s motion.

$$t = \frac{T}{4}; t = 0.5 \space s$$

Phase difference

It is clear from the previous example that if the person pushes the swing at the wrong time, for instance, when the swing is travelling towards them, the system will not resonate. To increase the height of the oscillations, the most force should be applied when the swing is travelling away from the person pushing it and passing through the equilibrium position, which is when its speed is greatest. The time taken for the swing to move between the highest point and the equilibrium position is equal to a quarter of the overall time period, and so this needs to be the difference in time between when the swing is at maximum displacement and when the force is applied.

At higher and lower values of the driving frequency, compared to the natural frequency of the object being acted on, energy is transferred to the system much less efficiently, and the amplitude is much lower.

• At low driving frequencies, the force oscillates much more slowly than the object, and there is a phase difference of 0 between them.
• At resonance, the phase difference is $$\frac{\pi}{2}$$, which causes the greatest transfer of energy, as the force is always acting in the same direction as the motion, and the force is largest when the object is passing through equilibrium with maximum velocity. This is also called velocity resonance – the graphs for the driving force and the velocity of the oscillator have the same shape.
• When the driving frequency continues to increase past the point of resonance, the phase difference increases to π, and the force gets completely out of phase with the oscillator. At this point, the oscillator is unable to keep up with the driving force.

Figure 2. The phase difference between the driver and the oscillator plotted against the ratio of their frequencies. Source: Physics LibreTexts (CC BY-NC-SA 4.0).

Damping

Real oscillating systems do not vibrate forever, as they lose energy to their surroundings. This is usually due to damping forces, which are frictional forces, such as air resistance or the friction between moving parts of a system. They act to decrease the amplitude of oscillations and thus minimise the effect of resonance.

The effects of different types of damping on amplitude

Light damping: the object takes a long time to stop. An example of this is air resistance acting on a swinging pendulum to slowly decrease the amplitude of the oscillations.

Figure 3. A lightly damped system. Source: Wikibooks IB Physics/Oscillations and Waves (CC BY-SA 3.0).

Heavy damping: a large force is applied against the motion of the vibrating object. In this case, it takes much less time to stop the oscillations. The amplitude decreases greatly during each period. The graph for displacement against time would have a similar shape to the graph for light damping but would decrease to zero much more quickly. An example of heavy damping would be using an object with a very large surface area in place of the bob on a simple pendulum. The air resistance would be much greater and would decrease the amplitude more quickly.

Critical damping: this is the exact amount of resistive force needed to stop a system oscillating in the shortest possible time.

Overdamping: overdamped systems have a greater resistive force acting on them than critically damped ones but take longer to return to their equilibrium. Overdamping is used for very large, heavy doors to get them to close slowly instead of slamming shut.

Figure 4. Displacement against time in a critically damped system (A) and an overdamped system (B). Source: Lumen Physics Damped Harmonic Motion (CC BY 4.0).

The effects of damping on resonance

The increase in amplitude caused by resonance is affected by damping in the oscillating system. Lightly damped systems have a very sharp peak in amplitude at resonance: the system is very sensitive to when the resonant frequency is reached. As the damping forces are increased, the peak of the resonance curves gets flatter, and the peak starts to occur slightly before resonance – the resonant frequency decreases.

Figure 5. Resonance curves for systems with varying amounts of damping. The amplitude of the oscillating systems is on the y-axis, B is a constant that represents the degree of damping, and w0 is the resonant frequency of the object. Source: Daniel A. Russell, Acoustics and Vibration Animations (CC BY-NC-ND 4.0).

Damping effects can be very useful in some cases, as resonance can cause problems in large structures, such as bridges. People and objects moving over the bridge can cause the bridge to vibrate slightly, and if the frequency happens to match the natural frequency of the bridge, it can oscillate violently and even break apart. This can be prevented by designing the bridge so that there is more friction between the parts that would move to reduce the amplitude of the oscillations.

Applications of resonance

Although resonance can be a source of danger, it also plays an important role in many useful applications, including, for instance, MRI scans and musical instruments.

MRI scans

MRI scans are based on the effect of nuclear magnetic resonance (NMR). Hydrogen nuclei have their own natural frequency, and if a very strong magnetic field is applied to one, it can absorb energy from electromagnetic radiation (in the radio frequency range) that is equal to this frequency. The controlled de-excitation of the hydrogen nuclei enables them to be located based on the radiation they emit, which allows for the body tissue to be mapped.

Musical instruments

Musical instruments also rely on resonance. For example, when a guitar string is plucked, a wave is produced, and the superpositions of that waveform a stationary wave on the string (the frequency of the stationary wave determines the pitch). The strings themselves would only make a quiet sound as they move through the air easily, thus causing little vibration of air molecules. The vibrations of the strings are transferred to the whole guitar body by the bridge at the end of the strings. The body is designed to have the same resonant frequencies as the frequencies of the stationary waves on the strings. This means that the guitar resonates when it is plucked, causing the surrounding air molecules to vibrate enough to produce a louder sound.

Resonance - Key takeaways

• Resonance describes the oscillations of very large amplitude that happen when a system is driven by a periodic force at the natural frequency of the system.
• The phase difference between the driver and the oscillator at resonance is $$\frac{\pi}{2}$$.
• Damping decreases the maximum amplitude resonance and causes the resonance frequency to decrease.

Flashcards inResonance 12

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What is resonant frequency?

Resonant frequency is the driving frequency at which a system resonates.

What is resonance?

Resonance describes the oscillations of very large amplitude that happen when a system is driven by a periodic force at the natural frequency of the system.

Why is resonance a problem?

Resonance can be a problem in large structures when caused by oscillations, as it can cause the structures to break apart.

What is an example of resonance?

A child being pushed on the swing at the right moments.

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