ONE FORMULA, ONE ANSWER
Forced-vibration steady amplitude calculator
Compute the harmonic steady displacement amplitude of a single-degree-of-freedom mass-spring-damper system. Choose input units and calculate one value within the stated conditions.
Inputs
Calculations run in this browser. Scientific notation such as 1.2e-3 is supported. Do not group thousands.
Result
Steady displacement amplitude X
Enter values and select Calculate.
X = F₀ / √[(k − mω²)² + (cω)²]
Assumptions
Required: a linear spring, viscous damping, and a direct force F₀ cos(ωt) on one degree of freedom. Transients from initial conditions are excluded. With c = 0, the result is the harmonic particular-solution amplitude; free motion does not decay. Undamped resonance with F₀ > 0 has no finite steady amplitude. ω = 0 gives static deflection F₀/k.
Displayed values are floating-point approximations. Design safety factors and unprovided error contributions are not calculated.
Formula references and credits
NaNa Lab created the code, descriptions, icons and preview images. Text, figures and code from the linked references have not been reproduced.
Physical constants: NIST / CODATA 2022. The values of μ₀ and ε₀ are measured recommendations, not exact defining constants. Values and precision notes are in the source record.
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