5.26 Damped & Undamped Oscillating Systems

• In practice, all oscillators eventually stop oscillating
  • Their amplitudes decrease rapidly, or gradually

• This happens due to resistive forces, such friction or air resistance, which act in the opposite direction to the motion of an oscillator
• Resistive forces acting on an oscillating simple harmonic system cause damping
  • These are known as damped oscillations

• Damping is defined as:

The reduction in energy and amplitude of oscillations due to resistive forces on the oscillating system

• Damping continues until the oscillator comes to rest at the equilibrium position
• A key feature of simple harmonic motion is that the frequency of damped oscillations does not change as the amplitude decreases
  • For example, a child on a swing can oscillate back and forth once every second, but this time remains the same regardless of the amplitude

Damping on a mass on a spring is caused by a resistive force acting in the opposite direction to the motion. This continues until the amplitude of the oscillations reaches zero

Types of Damping

• There are three degrees of damping depending on how quickly the amplitude of the oscillations decrease:
  • Light damping
  • Critical damping
  • Heavy damping

Light Damping

• When oscillations are lightly damped, the amplitude does not decrease linearly
  • It decays exponentially with time

• When a lightly damped oscillator is displaced from the equilibrium, it will oscillate with gradually decreasing amplitude
  • For example, a swinging pendulum decreasing in amplitude until it comes to a stop

A graph for a lightly damped system consists of oscillations decreasing exponentially

• Key features of a displacement-time graph for a lightly damped system:
  • There are many oscillations represented by a sine or cosine curve with gradually decreasing amplitude over time
  • This is shown by the height of the curve decreasing in both the positive and negative displacement values
  • The amplitude decreases exponentially
  • The frequency of the oscillations remain constant, this means the time period of oscillations must stay the same and each peak and trough is equally spaced

Critical Damping

• When a critically damped oscillator is displaced from the equilibrium, it will return to rest at its equilibrium position in the shortest possible time without oscillating
  • For example, car suspension systems prevent the car from oscillating after travelling over a bump in the road

The graph for a critically damped system shows no oscillations and the displacement returns to zero in the quickest possible time

• Key features of a displacement-time graph for a critically damped system:
  • This system does not oscillate, meaning the displacement falls to 0 straight away
  • The graph has a fast decreasing gradient when the oscillator is first displaced until it reaches the x axis
  • When the oscillator reaches the equilibrium position (x = 0), the graph is a horizontal line at x = 0 for the remaining time

Heavy Damping

• When a heavily damped oscillator is displaced from the equilibrium, it will take a long time to return to its equilibrium position without oscillating
• The system returns to equilibrium more slowly than the critical damping case
  • For example, door dampers to prevent them from slamming shut

A heavy damping curve has no oscillations and the displacement returns to zero after a long period of time

• Key features of a displacement-time graph for a heavily damped system:
  • There are no oscillations. This means the displacement does not pass 0
  • The graph has a slow decreasing gradient from when the oscillator is first displaced until it reaches the x axis
  • The oscillator reaches the equilibrium position (x = 0) after a long period of time, after which the graph remains a horizontal line for the remaining time

ANSWER:

• Ideally, the needle should not oscillate before settling
  • This means the scale should have either critical or heavy damping

• Since the scale is read straight away after a weight is applied, ideally the needle should settle as quickly as possible
• Heavy damping would mean the needle will take some time to settle on the scale
• Therefore, critical damping should be applied to the weighing scale so the needle can settle as quickly as possible to read from the scale