Coupled oscillators

📋 Teacher notes

• •Coupled oscillators are the linear counterpart to the double pendulum. Instead of chaos, we get superposition of normal modes.
• •The system has two normal modes: in-phase (both masses move together) and out-of-phase (they move oppositely). Any motion is a sum of these two modes.
• •When the coupling is weak and the two individual oscillators have nearly the same frequency, you see beats — periodic transfer of energy from one mass to the other.
• •The beat frequency is the difference between the two normal-mode frequencies: f_beat = |ω₂ − ω₁| / (2π).
• •Damping removes energy from the system. The Q-factor tells you how many oscillations occur before the energy drops significantly: Q ≈ 2π × (energy stored) / (energy lost per cycle).
• •In the damped case, the motion eventually settles into the slower normal mode because higher frequencies are damped more strongly.