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Damping

Most of the oscillations that occur in nature are subject to resistive forces. Such oscillations are known as damped oscillations.

For the case of a mechanical oscillator (e.g. simple pendulum), these forces act in the direction opposite to the velocity of the oscillator. The movement of the oscillator is constrained by the resistive forces.

The total energy of a damped oscillator decreases over time. This means that the amplitude of oscillation decreases over time depending on the strength of the resistive force.

The process of applying a resistive force to a free oscillator is called damping. There are three main types of damping arranged according to increasing damping strength: light, critical and heavy damping.

Light damping occurs when the resistive force is small.

The amplitude of oscillation decreases gradually over time until it reaches zero. The period and frequency of a lightly damped oscillator remain unchanged.

A simple pendulum oscillating in air: the air resistance decreases the amplitude of the oscillating pendulum gradually until the pendulum stops oscillating.

An alternating current passing through a resistor: the amplitude and energy of the alternating current decrease due to the resistance of the resistor.

Graphs of amplitude vs time for lightly damped (A), heavily damped (B) and critically damped (C) systems
Graphs of amplitude vs time for lightly damped (A), heavily damped (B) and critically damped (C) systems

As the damping force increases, the amplitude of oscillation of an object decreases at a greater rate. When the resistive force reaches a critical value, the object returns to zero displacement without being displaced to the other side.

This is known as critical damping. Of all the types of damping, critical damping makes the object reach a state of rest in the shortest time.

A car suspension system: after going over a road bump, the car suspension system quickly damps the oscillations of the springs supporting the tires, returning them to their equilibrium positions.

This reduces the discomfort of the passengers sitting in the car (from excessive oscillation/movement).

Graphs of amplitude vs time for lightly damped (A), heavily damped (B) and critically damped (C) systems
Graphs of amplitude vs time for lightly damped (A), heavily damped (B) and critically damped (C) systems

Heavy damping occurs when the resistive force exceeds that of critical damping.

The object returns to and remains at its equilibrium position without being displaced to the other side (similar to critical damping) but it takes a longer time to do so than in the case of critical damping.

This may be counterintuitive as one would think that a more heavily damped system would return to zero at a faster rate. However, recall that the damping force acts in a direction that is the opposite of the velocity of the object.

This means that the force resists the motion of the object as it moves towards the equilibrium position. More time is therefore required for the object to reach equilibrium.

An example of a heavily damped system is a coiled spring mattress, where the springs slowly return to a new equilibrium position when a person lies on the mattress.

Graphs of amplitude vs time for lightly damped (A), heavily damped (B) and critically damped (C) systems
Graphs of amplitude vs time for lightly damped (A), heavily damped (B) and critically damped (C) systems

All damped oscillating systems lose energy and will eventually stop moving.

To maintain regular oscillation, energy must be supplied to the system in the form of an external periodic driver force.

The oscillation of a pendulum can be maintained by a small impulse after each cycle.

Forced oscillations are oscillations that are maintained by the driver force.

The driver force imposes a new frequency on the system. The system therefore takes on the frequency of the driver force, regardless of the original frequency of oscillation.

A spinning electrical motor is an example of a system undergoing forced oscillations. In this case, the driver force causes the motor to spin with a particular frequency, regardless of air resistance.