1. What Forced Oscillations Mean
In many real situations, an oscillating system does not move on its own. Instead, it is pushed or driven by an external force. This type of motion is called a forced oscillation. Here, the system follows the rhythm of the force acting on it, not its own natural rhythm.
I like to think of forced oscillations as someone ‘guiding’ the system from outside — giving it repeated pushes to keep it moving continuously.
2. Definition of Forced Oscillations
Definition: Forced oscillations are oscillations in which an external periodic force continuously drives the system, causing it to oscillate at the frequency of the external force.
Without this driving force, the system may not oscillate at all, or its motion may quickly fade due to damping.
3. Why Forced Oscillations Occur
Real systems often lose energy because of damping. To keep them oscillating steadily, energy must be supplied from outside.
A few common reasons why forced oscillations happen:
- To counter damping (like keeping a swing moving by giving small pushes).
- To maintain continuous vibrations in machines and instruments.
- To drive systems at frequencies that suit a purpose, like tuning radio circuits.
4. Mathematical Form of Forced Oscillations
A forced oscillator can be expressed using an equation of the type:
\( F(t) = F_0 \sin(\omega_d t) \)
where:
- \( F_0 \) is the amplitude of the driving force
- \( \omega_d \) is the driving angular frequency (frequency of the external push)
The system responds to this force by oscillating with the same frequency \( \omega_d \), even if this frequency is different from its natural frequency.
4.1. Driven Frequency vs Natural Frequency
In forced oscillations, the system does not oscillate at its natural frequency. Instead, it follows the frequency of the external force.
Only the amplitude depends on how close \( \omega_d \) is to the natural frequency. This becomes very important in resonance.
5. Steady-State Motion in Forced Oscillations
After some initial irregular motion, the system settles into a steady pattern in which it oscillates exactly with the driving frequency. This phase is called the steady-state response.
In this state:
- The system oscillates with constant amplitude.
- The frequency equals the driving frequency.
- Energy gained from the driving force balances the energy lost due to damping.
6. Amplitude of Forced Oscillations
The amplitude of a forced oscillation depends on:
- the driving frequency,
- the natural frequency of the system,
- and the amount of damping present.
When the driving frequency is close to the natural frequency, the amplitude becomes very large. This special case is known as resonance, which is studied separately.
7. Examples of Forced Oscillations
Forced oscillations appear in many familiar and practical systems:
- A swing being pushed at regular intervals.
- A loudspeaker cone vibrating due to electrical signals.
- Washing machines vibrating because the motor drives the drum.
- Buildings shaking during an earthquake due to ground vibrations.
- Vibrations in bridges caused by wind or traffic.
In all these cases, an external, repeating force drives the system into oscillation.
8. Importance of Forced Oscillations
Understanding forced oscillations is essential because almost all real oscillating systems are influenced by external forces. This concept plays a major role in:
- designing stable buildings and bridges,
- tuning instruments and electronic circuits,
- controlling vibrations in machines,
- studying sound production and wave behaviour.
The idea might look simple, but it forms the foundation for understanding resonance and many wave-related phenomena.