Calculator Inputs
Use SI units for best consistency: mass in kg, stiffness in N/m, frequency in rad/s, displacement in m, velocity in m/s.
Example Data Table
| Case | Method | Known Inputs | Computed c | Interpretation |
|---|---|---|---|---|
| Machine mount | Mass + ζ + ωₙ | m = 10 kg, ζ = 0.12, ωₙ = 15 rad/s | 36 N·s/m | Lightly underdamped system with decaying oscillation. |
| Suspension test | Mass + ζ + k | m = 8 kg, ζ = 0.25, k = 1800 N/m | 60 N·s/m | Faster decay with reduced overshoot. |
| Peak decay study | Logarithmic decrement | m = 5 kg, k = 1200 N/m, x₁ = 0.08 m, xₙ = 0.052 m, n = 3 | 6.67 N·s/m | Estimated directly from measured amplitude loss. |
| Envelope method | Amplitude over time | m = 4 kg, k = 900 N/m, x₁ = 0.06 m, x₂ = 0.03 m, Δt = 1.8 s | 3.08 N·s/m | Uses exponential decay between two times. |
Formulas Used
Base vibration model
The standard viscously damped single-degree-of-freedom model is:
m x'' + c x' + k x = 0
Direct damping coefficient from ratio and natural frequency
c = 2mζωₙ
Natural frequency from stiffness and mass
ωₙ = √(k/m)
Critical damping coefficient
c_c = 2√(km)
Logarithmic decrement method
δ = (1/n) ln(x₁/xₙ)
ζ = δ / √(4π² + δ²)
c = 2ζ√(km)
Amplitude decay over time method
x(t) ∝ e^(-(c/2m)t)
c = 2m ln(x₁/x₂) / Δt
Damped frequency for underdamped motion
ω_d = ωₙ √(1 - ζ²)
How to Use This Calculator
- Choose the method that matches your available measurements.
- Enter mass first, then add the required damping, stiffness, or decay data.
- Optionally adjust initial displacement, velocity, plot duration, and plot density.
- Press the calculate button to display results above the form.
- Review the solved damping coefficient, related vibration parameters, and response type.
- Inspect the Plotly chart to see how the displacement evolves with time.
- Use the CSV or PDF buttons to export the current calculation summary.
FAQs
1. What does the damping coefficient represent?
It measures how strongly a system resists motion through velocity-dependent energy loss. Larger values mean faster decay of motion and smaller oscillation amplitudes.
2. What unit is used for the damping coefficient?
In SI units, the damping coefficient is usually expressed as N·s/m. This is equivalent to kg/s for linear viscous damping models.
3. When should I use the logarithmic decrement method?
Use it when you have peak amplitude measurements from an underdamped oscillation. It is especially useful in lab tests where decay data is easier to record than force data.
4. What is critical damping?
Critical damping is the exact amount of damping that returns a system to equilibrium as fast as possible without oscillation. It separates underdamped and overdamped behavior.
5. Why is damped frequency blank sometimes?
Damped frequency exists only for underdamped systems with ζ less than 1. Critical and overdamped systems do not oscillate, so no oscillation frequency is reported.
6. Can I use this for suspension and vibration isolation problems?
Yes. The calculator fits many single-degree-of-freedom engineering cases, including machine mounts, vehicle suspension approximations, structural vibration checks, and lab oscillators.
7. Why must the earlier amplitude be larger than the later one?
For positive damping, the motion envelope decays with time. If a later amplitude is larger, the data suggests growth, noise, wrong peak selection, or active energy input.
8. Does this calculator handle nonlinear damping?
No. It assumes linear viscous damping in the classical second-order model. Coulomb, quadratic, and strongly nonlinear damping require different governing equations and fitting methods.