Chapter 17 - Oscillations and simple harmonic motion

• Define simple harmonic motion:

  Oscillatory motion for which the acceleration of the object is directly proportional to its displacement and acting in the opposite direction.
• What is an isochronous oscillator?

  An oscillator that has the same period regardless of amplitude
• What is damping?


  An oscillation is damped when an external force that acts on the oscillator has the effect of reducing the amplitude of its oscillations.
• Oscillating motion is the repetitive motion of an object around its equilibrium
• What is the phase difference (oscillations)

  The difference in displacement of an oscillating object at different times.
• What is free oscillation?

  The motion of a mechanical system displaced from its equilibrium
• What is forced oscillation?

  An oscillation in which a periodic driver force is applied to an oscillator
• What is resonance?


  The increase in amplitude of a forced oscillation when the driving frequency matches the driving frequency of the oscillating system
• What is the driving frequency?

  The frequency with which the periodic driver force is applied to a system in forced oscillation
• Natural frequency is the frequency of a free oscillation
• Angular frequency is used to describe the motion of an oscillating object and uses the same equations of angular velocity in circular motion
• A graph of acceleration against displacement for any object moving in SHM has a gradient equal to $-\omega^2$. Since the gradient is constant, this implies that the frequency and period of the oscillator are also constant showing that period is independent of amplitude for an object in SHM.
• For the graph x-t, at zero displacement the oscillator is at equilibrium, at t =0 a pendulum is at maximum displacement giving it a cosine shape.

  For the graph v-t, at zero displacement the oscillator has maximum velocity and at zero velocity at maximum displacement.

  For the graph of a-t, this is opposite to the displacement time graph therefore in this case sine. The acceleration can be determined by the gradient of the velocity time graph.

  
• The equations commonly used for displacement are:

  $x=$$A\cos\theta\omega t$ and $x=$$A\sin\theta\omega t$

  • If the object begins oscillating from its amplitude, then the cosine version is used.
  • If the object begins oscillating from equilibrium, then the sine version is used
• The equation for velocity can be simplified to find its maximum giving;

  $v_{\max}=$$\omega A$
• For an object moving in simple harmonic motion the total energy remains constant as long as nothing is lost to frictional forces

  
• In a simple pendulum the potential energy is simply made up of gravitational energy. However, for a simple spring oscillator the potential energy is made up of the sum of its gravitational potential energy and its elastic potential energy - unless the mass is oscillating horizontally.
• Light damping occurs when the damping forces are small and the amplitude of the oscillator gradually decreases over time but the period of the oscillations is almost unchanged.
• Heavy damping is when the damping forces are large and the amplitude of the oscillator decreases significantly and the period of oscillations increases slightly .
• Free oscillation is the motion of a mechanical system displaced from its equilibrium
• Forced oscillation is an oscillation in which a periodic driver force is applied to an oscillator
• The driving frequency is the frequency with which the periodic driver is applied to a system in forced oscillation
• Resonance is the increase in amplitude of a forced oscillation when the driving frequency matches the natural frequency of the oscillating system
• For a forced oscillator with negligible dampening at resonance, the driving frequency equals the natural frequency of the forced oscillator. When an objects resonates, the amplitude of oscillations increases dramatically and can cause objects to fail
• The greatest possible energy transfer from the driver to the forced oscillators occurs at the resonant frequency
• Damping of a forced oscillator reduces the maximum amplitude of the resonance and the degree of damping has an effect on the frequency of the driver when the maximum amplitude occurs. As the amount of damping increases the amplitude of the vibration at any frequency decreases and the maximum amplitude decreases