CIE AS & A Level Physics 9702: Topic 17: Oscillations- 17.3 Damped and forced oscillations, resonance Study Notes

Damping

  • 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

Exam Tip
Make sure not to confuse resistive force and restoring force:

  •  Resistive force is what opposes the motion of the oscillator and causes damping
  •  Restoring force is what brings the oscillator back to the equilibrium position

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
  • 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

 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 slamming shut

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

Worked example

A mechanical weighing scale consists of a needle which moves to a position on a numerical scale depending on the weight applied. Sometimes, the needle moves to the equilibrium position after oscillating slightly, making it difficult to read the number on the scale to which it is pointing to. Suggest, with a reason, whether light, critical or heavy damping should be applied to the mechanical weighing scale to read the scale more easily.

  • 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

Resonance

  • In order to sustain oscillations in a simple harmonic system, a periodic force must be applied to replace the energy lost in damping
    •  This periodic force does work on the resistive force decreasing the oscillations
  •  These are known as forced oscillations, and are defined as:
  • Periodic forces which are applied in order to sustain oscillations
  • For example, when a child is on a swing, they will be pushed at one end after each cycle in order to keep swinging and prevent air resistance from damping the oscillations
    • These extra pushes are the forced oscillations, without them, the child will eventually come to a stop
  •  The frequency of forced oscillations is referred to as the driving frequency (f)
  • All oscillating systems have a natural frequency $\left(\mathbf{f}_0\right)$, this is defined as:
                          The frequency of an oscillation when the oscillating system is allowed to oscillate freely
  •  Oscillating systems can exhibit a property known as resonance
  •  When resonance is achieved, a maximum amplitude of oscillations can be observed
  • Resonance is defined as:

                   When the driving frequency applied to an oscillating system is equal to its natural frequency, the amplitude of the resulting oscillations increases significantly

  • For example, when a child is pushed on a swing:
    • The swing plus the child has a fixed natural frequency
    • A small push after each cycle increases the amplitude of the oscillations to swing the child higher
    • If the driving frequency does not quite match the natural frequency, the amplitude will increase but not to the same extent at when resonance is achieved
  • This is because at resonance, energy is transferred from the driver to the oscillating system most efficiently
    • Therefore, at resonance, the system will be transferring the maximum kinetic energy possible
  • A graph of driving frequency $f$ against amplitude $a$ of oscillations is called a resonance curve. It has the following key features:
    • When $f<f_0$, the amplitude of oscillations increases
    •  At the peak where $f=f_0$, the amplitude is at its maximum. This is resonance
    •  When $f>f_0$, the amplitude of oscillations starts to decrease
  •  Damping reduces the amplitude of resonance vibrations
  • The height and shape of the resonance curve will therefore change slightly depending on the degree of damping
    • Note: the natural frequency $f_0$ will remain the same
  • As the degree of damping is increased, the resonance graph is altered in the following ways:
    • The amplitude of resonance vibrations decrease, meaning the peak of the curve lowers
    • The resonance peak broadens
    • The resonance peak moves slightly to the left of the natural frequency when heavily damped
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