Natural Frequency Calculator
Select your system type, enter the physical properties, and instantly get the natural frequency in Hz, angular frequency in rad/s, oscillation period, equivalent RPM, and static deflection. Supports spring-mass systems, cantilever beams with concentrated or distributed mass, simply-supported beams, and fixed-fixed beams. Switch between metric and imperial, and use the step-by-step panel to follow every calculation.
Formula
Worked example
A spring-mass system with k = 1000 N/m and m = 10 kg: omega_n = sqrt(1000/10) = 10 rad/s, fn = 10/(2*pi) = 1.5915 Hz, T = 1/1.5915 = 0.6283 s, static deflection = 10*9.807/1000 = 98.07 mm.
What is natural frequency?
Natural frequency (fn) is the rate at which a physical system oscillates when disturbed and then left free - without any ongoing forcing. Every structure or mechanical component has at least one natural frequency, determined by its stiffness and mass distribution. When an external excitation source - a motor, footstep, wind gust, or earthquake - drives the system at or near its natural frequency, resonance occurs and vibration amplitudes grow dramatically, which can cause fatigue damage or structural failure. Understanding and calculating natural frequency is therefore a core task in mechanical and structural engineering.
Formulas for each system type
The spring-mass system is the foundation: omega_n = sqrt(k/m), where k is the spring stiffness in N/m and m is the oscillating mass in kg. For a cantilever beam with a concentrated tip mass, the equivalent stiffness is k_eff = 3EI/L^3, so omega_n = sqrt(3EI / (m*L^3)). For a uniform cantilever vibrating under its own weight, the first-mode formula uses the eigenvalue beta_1*L = 1.8751: omega_n = (1.8751)^2 / L^2 * sqrt(EI/mu), where mu is the mass per unit length. A simply-supported beam with a midspan mass uses k_eff = 48EI/L^3; with distributed mass the first mode is omega_n = pi^2/L^2 * sqrt(EI/mu). A fixed-fixed beam uses eigenvalue beta_1*L = 4.7300: omega_n = (4.73)^2/L^2 * sqrt(EI/mu). In all cases fn = omega_n / (2*pi) and the period T = 1/fn.
Static deflection and the gravity cross-check
A convenient shortcut links static deflection under gravity (delta_static) to natural frequency: fn = (1/2pi)*sqrt(g/delta_static). For a spring-mass system, delta_static = mg/k. For beams, it is the midspan or tip deflection under the distributed weight. This relationship lets you verify a calculated frequency by measuring deflection with a simple ruler, which is useful when checking analytical models against physical prototypes. The cross-check is exact for simple systems and approximate for beams because the Rayleigh quotient approach behind it slightly overestimates stiffness.
Resonance risk and separation margins
A resonance occurs when excitation frequency matches or approaches natural frequency. The ISO 10137 serviceability standard for floor vibrations and many machinery installation codes recommend a separation factor of at least 1.2 between natural frequency and any expected forcing frequency - meaning the structure's natural frequency should be at least 20% away from the nearest excitation harmonic. For rotating machinery, the critical speeds (1x, 2x, and sometimes 3x running speed in Hz) should be identified and compared with all structural natural frequencies in the drive-train and foundation. Where separation cannot be achieved, damping treatments or dynamic vibration absorbers are used to control peak amplitudes.
Typical natural frequencies in engineering
| System | Typical fn (Hz) | Notes |
|---|---|---|
| Suspension bridge | 0.05-0.5 | Highly flexible; wind and pedestrian excitation risk |
| High-rise building (>20 storeys) | 0.1-0.5 | Wind-sensitive; tuned mass dampers often used |
| Footbridge | 1-4 | Pedestrian footfall excitation at 1.5-2.5 Hz is critical |
| Car body structure | 20-40 | Stiffness-critical; ride comfort targets |
| Car suspension (wheel hop) | 10-15 | Unsprung mass resonance |
| Car suspension (body bounce) | 1-2 | Primary ride frequency for passenger comfort |
| Rotating machinery foundation | 15-60 | Avoid 1x and 2x running speed |
| Printed circuit board | 50-300 | Shock and vibration qualification per MIL-STD-810 |
| Aerospace panel (aircraft skin) | 200-800 | Acoustic fatigue range |
| Steel beam floor (office) | 4-8 | Serviceability limit per ISO 10137 |
Indicative first-mode natural frequencies for common structures and components. Actual values depend on geometry, material, mass, and boundary conditions.
Frequently asked questions
What is the difference between natural frequency and resonant frequency?
For an undamped system the two are identical: the system resonates exactly at its natural frequency. When damping is present, the resonant frequency (where response amplitude peaks) shifts slightly below the undamped natural frequency by a factor of sqrt(1 - 2*zeta^2), where zeta is the damping ratio. For most lightly damped engineering structures (zeta < 0.1) the difference is less than 1%, so the terms are often used interchangeably.
What units is natural frequency measured in?
Natural frequency is most commonly expressed in hertz (Hz, cycles per second). Angular natural frequency omega_n is expressed in radians per second (rad/s), where omega_n = 2*pi*fn. Rotational machinery engineers also use revolutions per minute (RPM = fn * 60). All three representations carry the same information and can be converted freely.
How does adding mass affect natural frequency?
Increasing mass lowers natural frequency because omega_n = sqrt(k/m). Doubling the mass reduces the frequency by a factor of sqrt(2), or about 29%. This is why mass-loading a structure - for example, adding concrete fill to a steel deck - is one of the practical methods used to push a floor's natural frequency below the footfall excitation range.
How does increasing stiffness affect natural frequency?
Increasing stiffness raises natural frequency: omega_n = sqrt(k/m), so doubling k increases fn by sqrt(2), about 41%. Adding bracing, shortening spans, or using a stiffer material (higher Young's modulus) all raise the natural frequency. In practice, stiffness and mass changes are used together to achieve a target frequency range.
What is the Young's modulus for common beam materials?
Structural steel is typically 200-210 GPa, stainless steel about 193 GPa, aluminium alloys 69-72 GPa, titanium alloys 105-120 GPa, glass-fibre reinforced polymer 20-45 GPa, carbon-fibre composites 70-200 GPa (depending on fibre orientation), concrete 25-35 GPa, and softwood timber 8-15 GPa. Always use the datasheet value for the specific alloy or grade.
What is area moment of inertia and how do I find it?
Area moment of inertia (also called second moment of area), I, measures how a cross-section resists bending about a given axis. For a solid rectangle of width b and height h bending about the horizontal axis, I = b*h^3/12. For a solid circular section of diameter d, I = pi*d^4/64. For an I-beam or hollow section use the parallel-axis theorem or section tables from the steel manufacturer (e.g., AISC, SCI). The units are m^4 in SI or in^4 in imperial.