Resonance: Amplifying Oscillations at the Natural Frequency
What is the Natural Frequency?
Every oscillating system, whether it’s a swing, a guitar string, or a building, has a natural frequency.
Definition
Natural frequency
A natural frequency is a frequency at which the system naturally oscillates when not disturbed by external forces.
Resonance and Amplitude
- When an external periodic force is applied to a system at its natural frequency, the amplitude of oscillation increases significantly.
- This is because the energy from the external force is transferred efficiently to the system.
Example
- Consider a swing being pushed.
- If you push in sync with the swing’s natural frequency, each push adds energy, making the swing go higher.
- If you push at the wrong time, the energy is not transferred efficiently, and the swing’s motion is disrupted.
Graphical Representation of Resonance
- A graph of amplitude versus driving frequency shows a sharp peak at the natural frequency.
- This peak represents the maximum amplitude achieved during resonance.
Note
- In the absence of damping, the amplitude at resonance can theoretically become infinite.
- However, in real-world systems, damping limits the amplitude.
Energy Storage in Resonance: Efficient Energy Transfer
How Does Energy Transfer Work in Resonance?
- During resonance, energy is transferred from the external force to the oscillating system with minimal loss.
- This efficient transfer is why the amplitude increases so dramatically.
Hint
At resonance, the external force is always in phase with the system’s motion, ensuring that energy is added at the optimal point in each cycle.
The Role of Damping
Damping affects how energy is stored and transferred during resonance.
- Light Damping: The system achieves a high amplitude at resonance, and the peak is sharp.
- Heavy Damping: The amplitude is lower, and the peak is broader and shifted to a lower frequency.
Common Mistake
- A common misconception is that damping always reduces the natural frequency.
- In reality, damping affects the amplitude and sharpness of the resonance peak, not the natural frequency itself.
Applications of Resonance
Musical Instruments
Resonance is crucial in musical instruments to amplify sound.
- String Instruments: When a string vibrates at its natural frequency, the body of the instrument resonates, amplifying the sound.
- Wind Instruments: Standing waves form in the air column, resonating at specific frequencies to produce musical notes.
Example
- A guitar string vibrating alone produces a faint sound.
- However, when the guitar body resonates with the string’s vibrations, the sound is amplified and becomes audible.
Building Design
Engineers use resonance to design structures that can withstand external forces like wind or earthquakes.
- Tuned Mass Dampers: These devices are added to skyscrapers to counteract resonance by oscillating out of phase with the building’s motion.
- Earthquake-Resistant Structures: Buildings are designed to avoid resonating with the frequencies of seismic waves.
Destructive Resonance: When Resonance Goes Wrong
Structural Failure
- Resonance can be destructive if not properly managed.
- When a structure resonates with an external force, the resulting large amplitudes can lead to catastrophic failure.
The Tacoma Narrows Bridge
One of the most famous examples of destructive resonance is the collapse of the Tacoma Narrows Bridge in 1940.
Note
- The bridge began oscillating violently due to wind-induced resonance.
- The oscillations grew so large that the bridge eventually collapsed.
- This disaster highlighted the importance of considering resonance in engineering design.
Preventing Destructive Resonance
- Damping Systems: Adding damping reduces the amplitude of oscillations during resonance.
- Avoiding Matching Frequencies: Engineers design structures to ensure their natural frequencies do not match common external forces, such as wind or seismic activity.
Self review
- What happens to the amplitude of a system when it is driven at its natural frequency?
- How does damping affect the resonance peak in a graph of amplitude versus driving frequency?
- Can you think of another real-world example where resonance is either beneficial or harmful?
