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Physics

Damped Oscillator Calculator

Solve damped spring-mass response and classify damping regime.

Last validated: 2026-02-14

Damped Oscillator Calculator analyzes spring-mass motion when damping removes energy from the system. Mass stores kinetic energy, the spring constant controls restoring force, and the damping coefficient represents friction, drag, or deliberate shock absorption. The undamped natural frequency describes how fast the system would oscillate without losses, while the damping ratio classifies behavior: underdamped systems oscillate while decaying, critically damped systems return quickly without overshoot, and overdamped systems return slowly without oscillation. Initial displacement and velocity set the starting state. This model is useful for vibration study, suspension intuition, and control-system checks, but real systems may include nonlinear springs, changing damping, external forcing, and structural limits.

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Input Pattern

Enter values in the left panel, keep units explicit, run the calculation, then copy or share the result. Invalid fields are highlighted immediately.

Damped Oscillator Inputs

Model: m x'' + c x' + k x = 0

Result

Natural frequency wn: 5.16397779 rad/s

Damping ratio zeta: 0.16137431

Regime: Underdamped

x(t): -0.0125867501 m

Damped period: 1.23289279 s

Approx settling time (5%%): 3.600000 s

How to use this tool

  1. Enter Mass m (kg), Spring constant k (N/m), Damping c (N*s/m), x(0) (m), v(0) (m/s) for the damped oscillator calculator, keeping units, dates, or text format consistent with the form labels.
  2. Confirm all units and known variables before running the calculation so the formula is applied consistently.
  3. Click "Run the tool" and review Damped Oscillator Inputs, Result for the primary output.
  4. Verify units and assumptions, especially before using the result for design, lab, or safety-sensitive work.

Worked Example

Auto-generated from the tool's current default or entered inputs.

Example Inputs

  • Mass kg: 1.5
  • Spring constant: 40.0
  • Damping coeff: 2.5
  • Displacement 0: 0.08
  • Velocity 0: 0.0
  • Time s: 2.0
  • Omega n: 5.163977794943222
  • Zeta: 0.1613743060919757

Expected Outputs

  • Mass kg: 1.5
  • Spring constant: 40
  • Damping coeff: 2.5
  • Displacement 0: 0.08

Interpretation

Confidence and limitations

Formula References

Assumptions

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