Calculator Inputs
Choose a method, enter known values, and calculate damping ratio, natural frequency, decay rate, periods, and response behavior.
Response Plot
The graph shows a normalized free-vibration response based on the calculated damping ratio and natural frequency.
Example Data Table
| Method | Known Inputs | Estimated Damping Ratio | Estimated Natural Frequency | Notes |
|---|---|---|---|---|
| Mass + c + k | m = 10 kg, c = 8 N·s/m, k = 500 N/m | 0.0566 | 7.0711 rad/s | Light underdamping with visible oscillation. |
| Overshoot + Settling | Mp = 20%, Ts = 2.5 s, 2% band | 0.4559 | 3.5097 rad/s | Useful in control-system design work. |
| Log Decrement + Td | δ = 0.25, Td = 0.75 s | 0.0398 | 8.3877 rad/s | Derived from measured decay data. |
| Decay Rate + ωd | σ = 1.2 1/s, ωd = 7.5 rad/s | 0.1580 | 7.5954 rad/s | Good for experimental modal estimates. |
Formula Used
1) From mass, damping coefficient, and stiffness:
ωn = √(k / m)
cc = 2√(km)
ζ = c / cc
ωd = ωn√(1 − ζ²) for underdamped systems.
2) From percent overshoot and settling time:
ζ = −ln(Mp/100) / √(π² + ln²(Mp/100))
Ts ≈ 4 / (ζωn) for a 2% band
Ts ≈ 3 / (ζωn) for a 5% band
3) From logarithmic decrement and damped period:
δ = ln(x1 / x2)
ζ = δ / √(4π² + δ²)
ωn = ωd / √(1 − ζ²), where ωd = 2π / Td
4) From decay rate and damped frequency:
ωn = √(σ² + ωd²)
ζ = σ / ωn
How to Use This Calculator
- Choose the calculation method that matches the data you already know.
- Enter the required values only for that method.
- Adjust initial displacement, initial velocity, precision, and plot settings if needed.
- Click Calculate Now to show the result above the form.
- Review damping ratio, natural frequency, damped frequency, time constants, and settling times.
- Use the CSV or PDF buttons to save the result summary.
- Inspect the plot to understand decay speed and oscillation behavior.
Frequently Asked Questions
1) What does damping ratio describe?
Damping ratio shows how strongly a system resists oscillation. Low values mean longer oscillations. A value of one means critical damping, and values above one indicate overdamped motion.
2) What is natural frequency?
Natural frequency is the rate at which a system wants to vibrate when disturbed. It depends on system mass and stiffness, or it can be inferred from response measurements.
3) When should I use the overshoot method?
Use percent overshoot and settling time when analyzing second-order control or vibration responses. This approach is especially helpful when you have transient performance targets instead of physical system parameters.
4) Is logarithmic decrement valid for all systems?
No. Logarithmic decrement is commonly used for underdamped systems where peaks can be measured across successive cycles. It is not suitable for overdamped motion without oscillatory peaks.
5) Why is damped frequency lower than natural frequency?
Damping removes energy from the system, which reduces the oscillation rate. That is why the damped frequency is lower than the undamped natural frequency in underdamped cases.
6) What does the response plot represent?
The plot shows a normalized free-vibration response based on your calculated parameters and chosen initial conditions. It helps visualize decay, oscillation, and how quickly motion settles.
7) Can I use this for design checks?
Yes, it is useful for quick design checks, education, and preliminary vibration studies. For safety-critical engineering work, verify assumptions and use validated domain-specific models.
8) What units should I use?
Use consistent units. For example, mass in kilograms, stiffness in newtons per meter, damping in newton-seconds per meter, time in seconds, and frequency in radians per second.