Simple Harmonic Motion

• The condition for SHM is that the acceleration of the object must be proportional to the negative displacement.
• Energy in an SHM system:

  • As the object moves towards its midpoint, the restoring force does work on the spring and so $E_p$ is transferred to $E_k$
  • At the midpoint, $E_p =$$0$ and $E_k$ is at a maximum.
  • At both sides of maximum amplitude, $E_k =$$0$ and $E_p$ is at a maximum.
• A free vibration means that no energy is transferred to or from the surroundings, so an object continues to oscillate at the same amplitude.
• A system can be forced to vibrate via periodic external forces.

  The frequency of this force is known as the driving frequency.

  When the driving frequency is equal to the resonant frequency of the system, the system gains more and more energy from the driving force, and amplitude rapidly increases.

  This is called resonance.
• Damping occurs when energy is lost to the surroundings, due to forces such as air resistance. Systems are deliberately damped to decrease the effect of resonance.
• Critical damping: Reduces the the amplitude in the shortest possible time

  Overdamping: The system takes longer to return to equilibrium than a critically damped system

  Light damping: The system will oscillate about equilibrium with decreasing amplitude.
• Lightly damped systems have a sharp resonance peak. The more damped a system is, the flatter the resonance peak and the lower the amplitude for the resonant frequency.