Forced Oscillation
The Basics of Oscillatory Motion
Before diving into forced oscillations, let's understand simple oscillations. An oscillation is a repetitive back-and-forth motion around a central point. Imagine a child swinging on a swing set. The swing moves from one highest point to the other and back again. This is an oscillation. Every system that can oscillate has its own preferred rhythm, known as its natural frequency. If you pull a swing back and let it go, it will swing back and forth at this specific frequency all on its own. This is called free or natural oscillation.
However, in the real world, oscillations don't last forever. Forces like air resistance and friction slowly rob the system of energy, causing the oscillations to die out. This effect is called damping. The swing will eventually stop if no one pushes it. This is where the concept of a forced oscillation comes into play.
What is a Forced Oscillation?
A forced oscillation occurs when an external, periodic force—called the driving force—is applied to a system to keep it oscillating. The key difference is that the system is no longer oscillating at its natural frequency; instead, it oscillates at the frequency of the driving force.
Let's return to the swing example. When a child sits on a stationary swing, it is at rest. If a parent gives the child a push at just the right moments, they are applying an external periodic force. The frequency of the swing's motion is now determined by the timing of the parent's pushes, not the swing's natural frequency. The parent is forcing the swing to oscillate.
The Power and Peril of Resonance
The most fascinating and important phenomenon in forced oscillations is resonance. Resonance occurs when the frequency of the driving force is exactly equal to (or very close to) the natural frequency of the system. When this happens, the system absorbs energy from the driving force with maximum efficiency, leading to a dramatic increase in the amplitude of the oscillation.
Think about pushing the swing again. If you push at random times, the swing might not go very high. But if you push in rhythm with the swing's natural motion—just as it reaches the highest point and starts to come back—you add a little bit of energy at the perfect moment each time. Over many pushes, these small amounts of energy add up, and the swing goes higher and higher. You have achieved resonance.
Resonance is not just about playground fun. It has both useful and destructive applications:
- Useful Resonance: Tuning a radio involves adjusting a circuit's natural frequency to resonate with the frequency of a specific radio wave, allowing you to pick up that station clearly. Microwave ovens use resonance to heat food by making water molecules vibrate violently.
- Destructive Resonance: In 1940, the Tacoma Narrows Bridge in the USA collapsed due to resonant vibrations caused by wind. The wind provided a driving force at a frequency that matched the bridge's natural frequency, causing it to twist and shake uncontrollably until it failed. Soldiers breaking step while marching across a bridge is a safety measure to prevent their rhythmic footsteps from causing resonant vibrations.
Comparing Different Types of Oscillations
It's helpful to see how forced oscillation compares to other types of oscillatory motion. The table below summarizes the key differences.
| Feature | Free Oscillation | Damped Oscillation | Forced Oscillation |
|---|---|---|---|
| External Force | Only an initial push | Only an initial push | Continuous periodic force |
| Frequency | Natural frequency of the system | Natural frequency of the system | Frequency of the driving force |
| Amplitude | Constant (in an ideal world) | Gradually decreases to zero | Constant, set by the driver (except at resonance) |
| Energy | Conserved (in an ideal world) | Dissipated as heat/sound | Continuously supplied by the driver |
| Example | An ideal frictionless pendulum | A car's shock absorber after a bump | A child being pushed on a swing |
Forced Oscillation in Action: Real-World Examples
Let's explore some concrete examples to see how forced oscillation shapes our world.
1. The Seismograph: This instrument detects and records ground movements during an earthquake. The main mass inside a seismograph is suspended and has a very low natural frequency. The ground (and the seismograph's frame) shakes during an earthquake, providing the driving force. Because the mass's natural frequency is much lower than the driving frequency of the earthquake tremors, it remains almost stationary due to its inertia. The relative motion between the still mass and the shaking frame is what gets recorded on the paper, creating a seismogram.
2. Playing a Guitar: When you pluck a guitar string, it vibrates at its natural frequency, producing a specific musical note. This is a free oscillation. However, the sound would be very faint. To amplify it, the vibrating string forces the body of the guitar to oscillate at the same frequency. The large surface area of the guitar body then forces the surrounding air molecules to oscillate, creating a much louder sound wave that we can hear. The guitar body is being forced to oscillate by the string.
3. The LC Circuit in a Radio: Inside a radio tuner is a circuit with an inductor (L) and a capacitor (C) that has a natural frequency determined by $f = \frac{1}{2\pi\sqrt{LC}}$. When you turn the tuning knob, you are changing the capacitance (C), which changes the circuit's natural frequency. You are searching for the frequency that resonates with the frequency of the radio wave from your desired station. When they match, resonance occurs, and the signal is amplified, allowing you to hear that station clearly.
Common Mistakes and Important Questions
Q: Is forced oscillation the same as resonance?
No, this is a common mix-up. Forced oscillation is the general phenomenon of a system being driven by an external force. Resonance is a special case of forced oscillation that happens only when the driving frequency matches the natural frequency, leading to a maximum amplitude. All resonant oscillations are forced, but not all forced oscillations are resonant.
Q: Can forced oscillation happen without damping?
In theory, yes, but in the real world, damping is almost always present. Damping is actually crucial in the context of resonance. Without any damping, the amplitude at resonance would theoretically become infinitely large, which is impossible. Damping limits the maximum amplitude at resonance and makes the system safer and more controllable.
Q: Why does a system oscillate at the driver's frequency and not its own?
Think of the driving force as a stubborn metronome that sets the beat. The system is being "told" when to move and how fast to move by this external force. After a short initial period (called the transient state), the system settles into a steady rhythm dictated by the driver. Its natural tendency is overridden by the continuous input of energy from the outside.
Conclusion
Footnote
This article uses several key scientific terms. Their definitions are provided below for clarity.
[1] Amplitude: The maximum extent of a vibration or oscillation, measured from the position of equilibrium. For a swing, it is the maximum height it reaches on either side.
[2] Damping: The effect of friction or other resistive forces that cause the amplitude of an oscillation to decrease over time.
[3] Frequency: The number of oscillations occurring per unit of time. It is often measured in Hertz (Hz), where 1 Hz equals one oscillation per second.
[4] Natural Frequency: The specific frequency at which a system will oscillate freely after being disturbed, determined by the system's physical properties (like mass and stiffness).
[5] Periodic Force: A force that repeats itself at regular intervals of time.