Damped Harmonic Oscillator - Interactive Simulation

Interactive simulation of spring-mass-damper system with real-time visualization of motion, displacement curve, and phase trajectory

Displacement (x): 0.00 m
Velocity (v): 0.00 m/s
Acceleration (a): 0.00 m/s²
Time: 0.00 s
Damping Mode: Underdamped

Displacement vs Time

Phase Space (x vs v)

Energy vs Time

Parameters

Preset Damping Modes

Physical Equations

Equation of Motion: mx'' + cx' + kx = 0
Natural Frequency: ω₀ = √(k/m)
Damping Ratio: ζ = c/(2√(km))
Damped Frequency: ωd = ω₀√(1-ζ²)
Current Parameters: ω₀ = 4.47 rad/s, ζ = 0.11

What is a Damped Harmonic Oscillator?

A damped harmonic oscillator consists of a mass attached to a spring and a damper. The spring provides a restoring force proportional to displacement (Hooke's Law: F = -kx), while the damper provides a resistive force proportional to velocity (F = -cv). This system models many real-world phenomena such as car suspensions, building vibrations, and electrical circuits.

Damping Modes

Underdamped (ζ < 1): The system oscillates with gradually decreasing amplitude. The mass crosses the equilibrium position multiple times before settling. This is the most common behavior in everyday mechanical systems.

Critically Damped (ζ = 1): The system returns to equilibrium as quickly as possible without oscillating. This is ideal for applications like car shock absorbers and door closers where you want fast settling without overshoot.

Overdamped (ζ > 1): The system returns to equilibrium slowly without oscillating. The damping force is so strong that it prevents oscillation entirely. This occurs in heavily damped systems like some measuring instruments.

Phase Space Trajectory

The phase space plot shows position (x) vs velocity (v). For a damped oscillator, the trajectory forms an inward spiral as energy is dissipated. Each loop represents one oscillation cycle, with the size decreasing over time. The spiral eventually converges to the origin (x=0, v=0) as the system loses energy.

Energy Dissipation

In a damped oscillator, mechanical energy is continuously converted to heat through the damping force. The total energy E = ½mv² + ½kx² decreases over time, with the rate of energy loss proportional to the damping coefficient. The envelope of displacement follows an exponential decay: x(t) ∝ e^(-ζω₀t).