Second-Order & Oscillators

The second-order linear ODE is one of the most important equations in all of physics and engineering. The general form for a damped, unforced oscillator is:

Here m is the mass, b is the damping coefficient, and k is the spring constant. This equation models everything from car suspensions to electrical circuits (RLC), to the vibration of buildings during earthquakes.

Watch a spring-mass system oscillate in real time. The left side shows the physical system, while the right side plots the displacement x(t) as it evolves. Adjust the mass, spring constant, and damping to see how the behavior changes between underdamped (oscillating), critically damped, and overdamped regimes.

Key insight: The character of the solution depends entirely on the discriminant b² - 4mk. When negative, the system oscillates (underdamped). When zero, it returns to equilibrium as fast as possible without oscillating (critically damped). When positive, it decays exponentially without oscillation (overdamped).

To solve a second-order linear ODE with constant coefficients, we substitute the trial solution x = e^rt and obtain the characteristic equation:

The roots of this quadratic determine the behavior of the solution:

• Two complex roots (b² < 4mk) — underdamped oscillation with exponential decay envelope
• Repeated real root (b² = 4mk) — critically damped, fastest non-oscillatory return to equilibrium
• Two distinct real roots (b² > 4mk) — overdamped, slow exponential decay without oscillation

Compare all three damping regimes side by side. The dashed lines show the exponential envelope that bounds the underdamped oscillation. Use the slider to place your own curve at any damping ratio and see how it compares to the reference cases.

Key insight: Critical damping (ζ = 1) is the boundary between oscillation and pure decay. It represents the fastest return to equilibrium without overshooting -- which is why it is used in door closers, car shock absorbers, and analog meter movements.

When we apply an external periodic force F₀cos(ωt) to the oscillator, the equation becomes:

The steady-state response has an amplitude that depends on the driving frequency ω. When ω approaches the natural frequency ω₀ and damping is small, the amplitude grows dramatically -- this is resonance. Resonance explains why soldiers break step on bridges, how radio tuners work, and why opera singers can shatter glass.

The top plot shows the frequency response curve: the steady-state amplitude as a function of driving frequency. The bottom plot shows the actual displacement over time. Sweep the driving frequency ω through the natural frequency to see the dramatic amplitude growth at resonance.

Key insight: The resonance peak becomes sharper and taller as damping decreases. With zero damping, the amplitude at resonance would be infinite -- in practice, nonlinear effects or material failure intervene first. The quality factor Q = 1/(2ζ) measures how sharp the resonance peak is.