What is the natural frequency of a second-order system?

The undamped natural frequency ωn (rad/s) is the frequency at which the system oscillates when there is no damping. It depends only on the mass and spring constant: ωn = √(k/m).

What does the damping ratio tell you?

The damping ratio ζ indicates how quickly oscillations decay. ζ 1 means the system returns to equilibrium without oscillating (overdamped).

How is the damped natural frequency different from the natural frequency?

The damped natural frequency ωd is the actual oscillation frequency when damping is present. It equals ωn√(1−ζ²), which is always lower than ωn. For an overdamped or critically damped system ωd is zero — there is no oscillation.

What units should I use?

Use SI units: mass in kg, spring constant in N/m, and damping coefficient in N·s/m. The results for ωn and ωd are in rad/s; divide by 2π to get Hz.

Can I use this for electrical RLC circuits?

Yes. Map inductance L → mass, 1/capacitance (1/C) → spring constant, and resistance R → damping coefficient. The formulas are mathematically equivalent.

Natural Frequency and Damping Ratio Calculator

Enter mass, spring constant, and damping coefficient to find the natural frequency, damped frequency, damping ratio, and response type of a second-order system.

System Parameters

Mass (m)

Spring Constant (k)

N/m

Damping Coefficient (c)

N·s/m

Results

Enter parameters above and click Calculate.

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Summary

This calculator determines the key dynamic properties of a second-order mass-spring-damper system. Enter the physical parameters and instantly get the undamped natural frequency, damping ratio, damped natural frequency, and a classification of the system response (underdamped, critically damped, or overdamped). All calculations follow standard control systems theory.

How it works

Enter the system mass (m) in kilograms.
Enter the spring constant (k) in newtons per meter.
Enter the damping coefficient (c) in newton-seconds per meter.
The calculator computes ωn = √(k/m), ζ = c / (2√(mk)), and ωd = ωn√(1−ζ²).
The response type is classified based on ζ: underdamped (ζ < 1), critically damped (ζ = 1), or overdamped (ζ > 1).
Click Calculate (or press Enter) to generate the results.

Use cases

Analyze mechanical vibration in automotive suspension systems.
Design control loops for robotics and servo actuators.
Check whether a structural system will oscillate or decay after disturbance.
Validate hand calculations for mechanical engineering coursework.
Quickly compare damping scenarios by adjusting the damping coefficient.
Determine the damped oscillation frequency for a known physical system.

Frequently Asked Questions

Last updated: 2026-06-11 · Reviewed by Nham Vu