How to use this calculator
Enter the mass, spring stiffness and damping coefficient (0 for no damping), then the starting displacement and velocity. The graph shows how the mass moves after release.
Formulas
Equation of motionmx″ + cx′ + kx = 0
Natural frequencyωn = √(k/m) fn = ωn2π
Dampingζ = c2√(km) ωd = ωn√(1 − ζ²)
Worked example
2 kg on an 800 N/m spring with c = 8 N·s/m (the default)
- ωn = √(800/2) = 20 rad/s, or 3.18 Hz.
- cc = 2√(800 × 2) = 80 N·s/m, so ζ = 8/80 = 0.1: underdamped.
- ωd = 20√(1 − 0.01) = 19.9 rad/s.
Assumptions and limits
- Linear spring and viscous damping.
- Free vibration only (no applied force).
Frequently asked questions
What does ζ tell me?
Below 1 the system oscillates; at 1 (critical damping) it returns fastest without overshoot; above 1 it creeps back slowly. Car suspensions run around 0.2–0.4.
How is this a differential equations problem?
m x″ + c x′ + k x = 0 is a second-order linear ODE with constant coefficients. Its characteristic roots are −ζωn ± ωn√(ζ² − 1).