Mechanics — Linear Systems
Coupled oscillators
Linear coupled oscillators — normal modes & beatsF = −kx − c v
Parameters
First mass
Mass m₁1 kg
Spring k₁1 N/m
Initial x₁0.1 m
Initial v₁0 m/s
Damping c₁0
Second mass
Mass m₂1 kg
Spring k₃1 N/m
Initial x₂0.1 m
Initial v₂0 m/s
Damping c₂0
Coupling k₂0.2 N/m
Spring between the two masses
Trail length300 frames
Calculated
Initial total E0.0100 J
Live kinetic E0.0000 J
Live potential E0.0000 J
Live total E0.0000 J
x₁0.000 m
x₂0.000 m
v₁0.00 m/s
v₂0.00 m/s
Q factor—
Bandwidth— rad/s
Damping curve
Run simulation to see decay curve
Log-energy vs time. A straight line confirms exponential decay (linear damping).
Curriculum
WAECNECOIGCSESATJUPEB
📋 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.
✏️ Exercises