Damping and resonance

​​In a nutshell

When a system is forced to oscillate at its natural frequency, the system will resonate and its oscillations will quickly gain amplitude. Damping affects systems in different ways depending on how much damping is being applied. Systems can be lightly damped, heavily damped, critically damped or overdamped.  

Free and forced vibrations

​​Free vibrations

If an object such as a mass on a spring is pulled from its equilibrium position and released it will undergo simple harmonic motion. The mass will oscillate at the systems natural frequency, 

If there were no resistive forces acting on the system then the mass would continue to oscillate forever. This is what is called a free vibration. 

Note: In reality moving things will transfer some energy into frictional forces such as air resistance, but the mass on the spring can still be considered to be a free vibration. 

Forced vibrations

Take the same mass on a spring and attach it to a vibration generator. The mass on a spring is now being driven at the frequency of the vibration generator. This is called the driving frequency.

This is known as a forced vibration as the oscillation has being provided by an external driving force. 

Resonance

When the driving force of an external oscillator is matched to the systems natural frequency, the amplitude of the system increases rapidly.

Note: Natural frequency is the frequency in which an object oscillates without an external driving force.

When this happens the system is said to be resonating. 

The graph for amplitude against driving frequency for a system looks like this: 

1. Amplitude 2. Driving frequency 3. Natural frequency

Example

When a child is pushed on a swing, the external force of the push is timed with the natural frequency of the swing, which causes the amplitude to increase, or the child to go higher. 

Curiosity: Architects who design buildings in earthquake prone areas have to take into account the natural frequency of the building. This is because an earthquake might produce a driving force which is the same or similar to the natural frequency of the building which causes the building to shake apart. 

Damping

Damping happens when energy of the system is lost to the surroundings. In most real life applications this happens through natural damping such as air resistance and friction, however some systems are deliberately damped in order to reduce unwanted oscillations. 

Example

When a car passes over a bump in the road, the suspension absorbs the shock and then dampens the oscillations to return to the cars equilibrium position. It would be uncomfortable (and nauseating) for the suspension to continue oscillating down the road. 

Different types of damping

There are four different types of damping that you need to know about.

Lightly damped systems reduce the amplitude over a relatively large period of time.

Example

Air resistance damping a pendulum.

Heavily damped systems reduce the amplitude in a much quicker period of time compared to the lightly damped system.

Example

The needle in a compass is heavily damped so it doesn't oscillate when it points to magnetic north.

A critically damped system returns to the equilibrium point in the quickest time possible without any oscillations.

Example

A car suspension system is critically damped so the cabin doesn't oscillate when going over a bump in the road.

Overdamped systems take longer to return to the equilibrium than critically damped systems but do not oscillate.

Example

An automatically closing door is overdamped so that it closes slowly but does not oscillate.

Damping and resonance

Damping affects the amount of resonance that occurs in a system. This is important for a lot of systems as oscillations are sometimes unwanted within a system. 

The resonance of the system depends on how much the system is damped by:   

1. Amplitude 2. Driving frequency 3. ​$$f_0$$​​ 4. Resonant frequency Red line Undamped Blue line Lightly damped Green line Heavy damped

In general the heavier the damping, the flatter and less sensitive the system is to resonance.