Calculator · Dynamics & Control
Mass–Spring–Damper Calculator
The full single-degree-of-freedom system m·ẍ + c·ẋ + k·x = F(t): frequencies, damping, regime, poles and the equivalent transfer function from force to displacement.
Physical parameters
c = 0 is valid: an undamped system.
Results
| Natural frequency ωₙ = √(k/m) | 1.4142 rad/s |
|---|---|
| Natural frequency fₙ = ωₙ/2π | 0.22508 Hz |
| Natural period Tₙ | 4.4429 s |
| Critical damping c_c = 2√(km) | 2.8284 N·s/m |
| Damping ratio ζ = c/c_c | 0 |
| Damping regime | undamped |
| Damped natural frequency ω_d | 1.4142 rad/s |
| Damped period T_d | Not defined for this regime |
| Logarithmic decrement δ | Not defined — successive peaks exist only in the underdamped regime |
| Quality factor Q = 1/(2ζ) | Unbounded — the undamped system dissipates no energy |
| Static compliance 1/k | 0.5 m/N |
| Poles of X(s)/F(s) | 0 + 1.4142j, 0 − 1.4142j |
| Equivalent transfer function X/F | 1 / (1s² + 0s + 2) [m/N] |
| Static displacement under a 1 N step | 0.5 m |
Diagnostics
- · Info Simple poles on the imaginary axis give a bounded, non-decaying mode: the system is marginally stable, not asymptotically stable.
Technical reference
m·ẍ + c·ẋ + k·x = F(t) (governing equation of motion)
ωₙ = √(k/m), ζ = c / (2√(km)) = c/c_c
ω_d = ωₙ√(1 − ζ²) (underdamped only)
- m
- lumped mass[kg]
- c
- viscous damping coefficient[N·s/m]
- k
- spring stiffness[N/m]
- ωₙ
- undamped natural frequency[rad/s]
- ζ
- damping ratio[–]
- ω_d
- damped natural frequency[rad/s]
This page gives the fuller SDOF characterization — damped period, logarithmic decrement, quality factor, compliance, poles and X(s)/F(s) transfer-function information — built on the same ωₙ and ζ as the Natural Frequency Calculator, which focuses on ωₙ and fₙ while showing damping ratio, regime and ω_d as context. Use that page for the focused frequency calculation; use this one for the additional decay, compliance, pole and transfer-function quantities.
Assumptions
- · Linear, time-invariant, single degree of freedom: m·ẍ + c·ẋ + k·x = F(t).
- · Viscous damping only; m > 0, c ≥ 0, k > 0.
- · ω_d, the damped period and the logarithmic decrement are reported as not defined outside the underdamped regime — never as a fabricated finite value.
- · The transfer function X(s)/F(s) assumes zero initial conditions.
Continue
Free, harmonic and base-excitation response of the same physical system.