Vibration Natural Frequency Calculator
The natural frequency of a spring-mass system is the frequency at which it will oscillate if disturbed and left free to vibrate. This is the fundamental parameter in vibration analysis and is used to check for resonance with rotating machinery, acoustic sources, or seismic excitation. Enter the system stiffness, mass, and optional damping coefficient to compute the undamped and damped natural frequencies, critical damping, and damping ratio.
Natural frequency formulas (SDOF system)
omegan = sqrt(k / m) (rad/s)
fn = omegan / (2 × pi) (Hz)
cc = 2 × sqrt(k × m) (N.s/m)
zeta = c / cc
omegad = omegan × sqrt(1 - zeta2) (damped)
Where: k = stiffness (N/m), m = mass (kg), c = damping coefficient (N.s/m), cc = critical damping coefficient, zeta = damping ratio, omega_d = damped natural frequency.
Resonance avoidance guidance
- ISO 10816-3 and API 670 require machinery vibration to remain below specified alarm and shutdown levels.
- Operating speed should be at least 20% above or below the natural frequency of any structural support or rotating component.
- If resonance is unavoidable, add damping (e.g. tuned mass dampers, viscoelastic materials) or change the stiffness or mass to shift the natural frequency.
- For rotating machinery, check the Campbell diagram: natural frequencies versus operating speed lines to identify resonance crossings during start-up and shut-down.
Vibration natural frequency calculator: frequently asked questions
What is the natural frequency formula for a spring-mass system?
The undamped natural frequency is omega_n = sqrt(k / m) in rad/s, where k is the spring stiffness (N/m) and m is the mass (kg). The frequency in Hz is fn = omega_n / (2 * pi). For example, a mass of 10 kg on a spring of stiffness 40,000 N/m has omega_n = sqrt(40000/10) = 63.25 rad/s and fn = 10.07 Hz.
What is critical damping?
Critical damping is the minimum damping that prevents oscillation. The critical damping coefficient is cc = 2 * sqrt(k * m) = 2 * m * omega_n. If the actual damping coefficient c = cc, the system returns to equilibrium without oscillating. If c is less than cc, the system oscillates (underdamped). If c is greater than cc, the system does not oscillate (overdamped).
What is the damping ratio?
The damping ratio zeta = c / cc = c / (2 * sqrt(k * m)). Zeta = 0 is undamped (pure oscillation); zeta = 1 is critically damped; zeta greater than 1 is overdamped. For structures and machinery, typical damping ratios range from 0.01 (metals with little joint friction) to 0.15 (reinforced concrete). The damped natural frequency is omega_d = omega_n * sqrt(1 - zeta^2).
What are equivalent stiffness values for common systems?
A cantilever beam with end load: k = 3EI/L^3. A simply supported beam with central load: k = 48EI/L^3. A coil spring: k = Gd^4/(8D^3N), where G is shear modulus, d is wire diameter, D is coil diameter, and N is active coils. A column in axial compression: k = AE/L.
What is resonance and why is it dangerous?
Resonance occurs when an excitation frequency matches the natural frequency of the system. At resonance, the amplitude of vibration grows without bound in an undamped system, or reaches a maximum proportional to 1/(2*zeta) for a damped system. Resonance in structures can cause catastrophic failure (e.g. Tacoma Narrows Bridge, turbine blade resonance). Machines should operate at least 20% away from natural frequencies.
Official sources
- ISO 10816-3: Mechanical vibration, evaluation of machine vibration by measurements on non-rotating parts: ISO 10816-3.
- NIST Digital Library of Mathematical Functions: dlmf.nist.gov.
Reviewed by the CalculatorHub team, edited by James Graham, 14 June 2026. See our methodology.