Phy216

• Coupled oscillators


  Oscillators connected in such a way that energy can be transferred between them
• Coupled oscillations


  • Occur when two or more oscillating systems are connected in such a manner as to allow motion energy to be exchanged between them
• Coupled pendulum motion


  1. Pendulum A displaced and released

  2. Amplitude of A decreases, amplitude of B increases

  3. Amplitudes become equal

  4. Motion of B transferred back to A

  5. Energy shuttles back and forth between A and B
• Lower normal mode


  • Pendulums A and B drawn in same direction by equal amount and released
  • Oscillate at same frequency and constant amplitude
  • Distance between them equals relaxed length of coupling spring, spring exerts no force
  • Pendulums oscillate in phase
• Equations of motion for lower normal mode:
• Second normal mode


  • Pendulums A and B drawn aside by equal amount but in opposite directions and released
  • Coupling spring stretched and compressed, exerts forces
  • Motion of A and B mirror images of each other
  • Oscillations have same frequency and amplitude but are 180 degrees out of phase
• Equation of motion for second normal mode:
• Coupling spring increases the restoring force and frequency over uncoupled oscillation
• Superposition of normal modes

  Restoring force on A = -kxA - k(xA - xB)

  Restoring force on B = -kxB - k(xB - xA)
• Second normal mode of oscillation of coupled system


  • Same frequency and amplitude but are 180° out of phase with each other
• 𝑥𝐴 = −𝑥𝐵
• Superposition of normal modes


  1. Restoring force on A

  2. Restoring force on B

  3. Combine equations for A and B

  4. Separate equations into lower and higher modes
• Normal coordinates


  Changes in X occur independently of Y and vice versa
• Initial conditions: 𝑥𝐴 = 𝐴0, 𝑑𝑥𝐴/��𝑡 = 0; 𝑥𝐵 = 0, 𝑑𝑥𝐵/𝑑𝑡 = 0