Natural Frequency Calculator
Calculate the undamped natural frequency of common mechanical systems — spring-mass, cantilever beam, simply supported beam, and pendulum. Select a system type below, enter the parameters, and get frequency in Hz, angular frequency in rad/s, and period in seconds.
What is Natural Frequency?
Every physical system that can store and release energy has a natural frequency — the rate at which it oscillates when disturbed and left to itself. Strike a tuning fork and it rings at its natural frequency. Pluck a guitar string and it vibrates at its natural frequency. Tap a bridge deck and it flexes and rebounds at frequencies determined by its stiffness and mass.
The defining characteristic is that no ongoing external energy is needed to maintain the oscillation — it continues (decaying slowly due to damping) purely from the initial disturbance. This distinguishes free vibration at the natural frequency from forced vibration, where an external source continuously drives the motion.
In formal terms, the undamped natural frequency ωn is the angular frequency at which the equation of motion for a conservative (lossless) system has oscillatory solutions. For the simplest possible case — a mass on a spring — this reduces to a single elegant formula. More complex systems have multiple natural frequencies, one for each degree of freedom.
The Spring–Mass Formula
The spring-mass model is the foundation of vibration analysis. Almost any oscillating system — a car suspension, a building floor, a machine on mounts — can be approximated as one or more masses connected to ground by springs, at least for preliminary calculations.
Worked example
A machine component has a stiffness of 5,000 N/m and a mass of 2.5 kg.
How stiffness and mass scale frequency
Because of the square root, the relationship is not linear. Doubling the stiffness increases natural frequency by a factor of √2 ≈ 1.41. Quadrupling the stiffness doubles the frequency. Similarly, quadrupling the mass halves the frequency. This square-root scaling is one of the most practically important things to understand when trying to move a natural frequency away from a problem excitation source.
Beam Natural Frequencies
Beams are distributed-parameter systems — their stiffness and mass are spread continuously along their length rather than lumped at a point. The fundamental natural frequency for common beam configurations is well-established in structural dynamics literature.
Cantilever beam with tip mass
A beam fixed at one end with a concentrated mass at the free tip is a common model for sensors, robotic arms, antenna masts, and overhanging machine components. The effective stiffness at the tip is 3EI/L³, and the beam's distributed mass contributes approximately 23% of its total mass to the effective lumped mass at the tip.
Simply supported beam (fundamental mode)
Pinned at both ends, the fundamental frequency uses the Euler–Bernoulli beam equation. This models floor joists, bridge spans, and simply supported structural members.
Common material properties for reference
| Material | Young's modulus E | Density ρ | Typical use |
|---|---|---|---|
| Structural steel | 200 GPa | 7,850 kg/m³ | Bridges, frames, beams |
| Aluminium (6061) | 68.9 GPa | 2,700 kg/m³ | Aerospace, lightweight structures |
| Timber (pine, along grain) | 9–13 GPa | 500–600 kg/m³ | Floor joists, timber frames |
| Reinforced concrete | 25–35 GPa | 2,400 kg/m³ | Slabs, columns, frames |
| Carbon fibre (CFRP) | 70–200 GPa | 1,550–1,600 kg/m³ | Aerospace, motorsport, high-performance |
Resonance: Why Natural Frequency Matters in Practice
Resonance occurs when a periodic driving force is applied at or near a system's natural frequency. Under these conditions, energy is added to the system each cycle faster than damping can remove it, and oscillation amplitudes grow — sometimes catastrophically.
In an undamped system, resonance produces theoretically infinite amplitude. In reality, all systems have some damping, so amplitude is finite but can still be large enough to cause failure. The amplification factor at resonance (the dynamic amplification factor, or DAF) for a lightly damped system is approximately 1/(2ζ), where ζ is the damping ratio. For ζ = 0.02 (2% damping, typical for welded steel structures), the DAF is 25 — meaning the response is 25 times the static deflection under the same force magnitude.
Design approaches
Frequency separation is the first line of defence: ensure the natural frequency is well away from any expected excitation frequency. A common rule of thumb is a separation factor of at least 1.2 — i.e., the natural frequency should be at least 20% above or below the excitation frequency. For rotating machinery, this means the running speed and its harmonics should not coincide with structural natural frequencies.
Adding damping reduces the peak amplitude at resonance without changing the natural frequency. Viscous dampers, tuned mass dampers (used in skyscrapers and long-span bridges), and viscoelastic materials all increase the effective damping ratio.
Mass and stiffness tuning shifts the natural frequency. Stiffening a structure (adding ribs, increasing section depth) raises it; adding mass lowers it. A tuned mass damper is a special case where a secondary mass-spring system is deliberately tuned to the problematic frequency so that it absorbs energy and reduces the primary structure's motion.
Notable resonance failures and near-misses
The Tacoma Narrows Bridge (1940) collapsed after wind-induced aeroelastic flutter locked onto a torsional mode. The Millennium Bridge in London (2000) had to be closed two days after opening because pedestrian footfall synchronised with its 1 Hz lateral natural frequency, producing uncomfortable swaying — subsequently fixed with 37 fluid viscous dampers. The Broughton Suspension Bridge (1831) collapsed when soldiers marching in step excited its natural frequency, which is why troops are still ordered to break step when crossing bridges. In mechanical engineering, torsional resonance in rotating shafts, and acoustic resonance in pipework and combustion chambers, are recurrent design challenges.
The Pendulum and Other Simple Systems
The simple pendulum — a point mass on a massless string or rod — is the original vibration problem, studied by Galileo and solved analytically for small angles by Huygens in the 17th century. Its natural frequency depends only on length and gravity, not on the mass of the bob.
A 1-metre pendulum has a natural frequency of about 0.498 Hz — nearly exactly half a hertz, which is why it was used as the basis for the original definition of the metre. A 0.248-metre (roughly 25 cm) pendulum oscillates at 1 Hz, making it a convenient timing reference. Pendulum clocks exploit this property directly.
At larger swing angles the formula breaks down — the true period gets longer as amplitude increases, and the full solution requires an elliptic integral. For most engineering purposes the small-angle approximation is adequate below about 15° of arc.
Common questions
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Natural frequency is the rate at which a system oscillates when disturbed and then left to vibrate freely, without any ongoing external force. Every physical structure — a bridge, a guitar string, a machine component, a building — has one or more natural frequencies determined by its stiffness and mass distribution. When an external force drives a structure at or near its natural frequency, the resulting resonance can produce oscillations far larger than the driving force alone would suggest.
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For an undamped system, natural frequency and resonant frequency are the same thing. In a real system with damping, the resonant frequency (the frequency at which maximum amplitude occurs under forced vibration) is slightly lower than the undamped natural frequency. For most engineering structures with light damping the difference is negligible, but in highly damped systems — rubber mounts, for example — the distinction matters.
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Natural frequency can be expressed in two ways. Cyclic frequency (f) is measured in hertz (Hz), where 1 Hz means one complete oscillation per second. Angular frequency (ω, omega) is measured in radians per second (rad/s). They are related by ω = 2πf. The period (T), measured in seconds, is the time for one complete cycle: T = 1/f. Engineering vibration analysis tends to use rad/s internally and Hz for reporting.
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Stiffness and mass pull in opposite directions. Increasing stiffness (a stiffer spring, a shorter or thicker beam) raises natural frequency — the system snaps back faster. Increasing mass lowers natural frequency — there is more inertia to accelerate. The relationship is a square root: doubling the stiffness raises frequency by a factor of √2 (about 41%), not by a factor of 2. To halve the natural frequency you need to quadruple the mass, or quarter the stiffness.
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Resonance is the main concern. If a rotating machine, engine, or environmental excitation source (wind, waves, traffic) drives a structure at its natural frequency, vibration amplitudes can grow until something fails. Classic examples include the Tacoma Narrows Bridge collapse in 1940 (aeroelastic flutter near a natural frequency) and the London Millennium Bridge closure in 2000 (pedestrian footfall synchronised with the bridge lateral natural frequency). Structural engineers calculate natural frequencies early in the design process to ensure they are well separated from likely excitation frequencies.
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Most real structures have multiple natural frequencies, each associated with a characteristic pattern of deformation called a mode shape. The first (lowest) natural frequency is called the fundamental frequency. A simply supported beam, for instance, has modes where it bows into one hump, then two humps, then three, each at a progressively higher frequency. Modal analysis — either by calculation or by experimental testing — maps these frequencies and shapes to inform damping strategies and design changes.
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The lumped spring-mass model (f = (1/2π)√(k/m)) is exact for an ideal massless spring with a point mass. For real beams and structures, it is an approximation — a beam's own distributed mass contributes to inertia, and the stiffness is not perfectly linear. Engineers often use an "effective mass" that accounts for the beam's own distributed mass (for a cantilever with a tip mass, the beam contributes roughly 23% of its own mass to the effective lumped mass). Finite element analysis gives more accurate results for complex geometries.