Natural Frequency and Damping Calculator (Single Degree of Freedom)

The natural frequency, damping ratio and damped frequency of a mass on a spring with a damper.

Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.

How this calculator works

A mass m on a spring of stiffness k vibrates at its natural frequency ωn = √(k / m) when disturbed. A damper with coefficient c removes energy. The damping ratio ζ = c / (2√(k m)) tells you how quickly the vibration dies away: below 1 the system still oscillates (underdamped), at 1 it is critically damped, and above 1 it creeps back without oscillating.

Natural frequency matters in machine design because a machine running near a natural frequency can resonate and shake much harder than the force alone would suggest.

Formulas used

  • \omega_n = \sqrt{k / m}
  • f_n = \omega_n / (2\pi)
  • \zeta = \frac{c}{2\sqrt{k m}}
  • \omega_d = \omega_n\sqrt{1 - \zeta^2}

Worked example

With the default numbers in the calculator above:

  1. Natural angular frequency: 63.2 rad/s
  2. Natural frequency and period: 10.1 Hz
  3. Critical damping coefficient: 1265 N s/m
  4. Damping ratio: 0.0791
  5. Damped natural frequency (underdamped): 63 rad/s

Change the numbers in the calculator to see the full working, every step, for your own case.

Common questions

How do I find the spring stiffness of a beam?

Stiffness is load divided by deflection. For a cantilever with an end load, k = 3 E I / L³.

What is resonance?

Large vibration when the forcing frequency matches the natural frequency.

What damping ratio is typical?

Steel structures are often 0.5 to 2 percent. Vehicle suspensions are around 0.2 to 0.4.

Related calculators

Further reading: Meriam and Kraige, Engineering Mechanics: Dynamics; Hibbeler, Engineering Mechanics: Dynamics; Rao, Mechanical Vibrations.

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