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Structural Resonance Calculator

This structural resonance calculator computes the natural frequencies (modes 1–5) of an idealised beam or shaft (Euler–Bernoulli), a tensioned string, or a rectangular plate. Pick a boundary condition, choose a material from the built-in database, watch the animated mode shape, and check whether an excitation frequency falls in a resonance danger zone.

ℹ These are exact textbook formulas for an ideal, prismatic, uniform member with idealised supports — not a measurement and not a structural-safety verdict. Real members differ because of joints, welds, damping, added mass, non-uniform sections, anisotropy and temperature; the published material properties below are typical values that vary by grade and source (carbon-fibre especially — verify your own E, ρ and geometry). For a real assessment, measure the structure and consult a qualified engineer. Everything runs in your browser; nothing is uploaded.

Optional — the forcing frequency to check against the modes.

Mode shape

The curve is the relative deflected shape (not to scale). For a plate it shows a cross-section profile. Supports: ▮ = clamped, ▲ = pinned; free–free has no supports.

Mode table

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How It Works

Every elastic structure has natural frequencies — the frequencies at which it “wants” to vibrate. Drive it near one and the response is amplified, sometimes dramatically; that amplification is resonance, and it is behind everything from a ringing tuning fork to a vibrating machine guard to the famous Tacoma Narrows bridge failure. This structural resonance calculator computes those frequencies for three classic idealised systems using their standard textbook equations: a beam vibration model (Euler–Bernoulli theory), a tensioned string, and a thin rectangular plate. If you also need to investigate the spring-mass equivalent of a mounted component, the spring-mass resonance calculator covers single-degree-of-freedom systems with damping and transmissibility.

For a slender beam or shaft the transverse natural frequencies come from Euler–Bernoulli theory: fn = (βnL)² / (2π) · √(E·I / (ρ·A·L⁴)), where E is Young’s modulus, ρ density, A the cross-sectional area, I the second moment of area, L the length, and βnL are the dimensionless eigenvalue coefficients set by the boundary condition. The published βnL roots are: cantilever frequency modes at 1.875, 4.694, 7.855, 10.996, 14.137; simply-supported nπ; and fixed–fixed and free–free (which share the same non-trivial roots) 4.730, 7.853, 10.996, 14.137, 17.279. A stiffer or lighter beam rings higher; a longer beam rings much lower (frequency scales with 1/L²). The ratio of bending stiffness (EI) to mass per unit length (ρA) is the key specific stiffness that controls where the modes fall.

A tensioned string is simpler: fn = (n / 2L)·√(T/μ), where T is tension and μ the linear mass density — an exact integer harmonic series of string frequency modes, which is why strings sound musical and why modal analysis of string instruments maps directly to this formula. A simply-supported rectangular plate uses Kirchhoff thin-plate theory: f(m,n) = (π/2)·√(D / ρt)·((m/a)² + (n/b)²), with flexural rigidity D = E·t³ / (12(1−ν²)); the first five distinct plate natural frequency modes are shown. The resonance danger zone check compares an excitation frequency you enter against each computed modal frequency and flags any that fall within your chosen percentage band — a quick way to spot whether a motor, pump or fan speed will excite the structure. Remember to check integer harmonics of the forcing frequency, not just its fundamental — gearboxes and pumps produce rich harmonic content that can excite higher vibration modes. For measurement-based verification, the mechanical resonance frequency tool lets you detect resonance from a live audio signal.

Material reference data

The built-in materials use typical published values. Treat them as starting points: real moduli and densities vary with grade, alloy, moisture, temperature and direction. Carbon-fibre is shown as a single midpoint but spans a wide range in practice depending on fibre orientation and layup fraction. Always confirm against your own material’s datasheet or test data. If you need to calculate the uncertainty introduced by those variations, the frequency error and uncertainty calculator can help propagate input tolerances through to an output frequency range.

MaterialE (GPa)ρ (kg/m³)ν (plate)
Structural steel20078500.30
Aluminium (6061)6927000.33
Wood (softwood, along grain)~11~600~0.30
Concrete~30~24000.20
Carbon-fibre composite~70–150~1600~0.30

Sources: typical engineering reference values (e.g. The Engineering ToolBox, MatWeb and standard mechanics-of-materials texts such as Gere & Goodno). Wood is highly anisotropic (the figure is softwood along the grain); concrete E depends on mix and strength; carbon-fibre depends entirely on fibre, resin and layup. Use a Custom material to enter your own E and ρ.

Frequently Asked Questions

How accurate are these natural frequencies?
They are exact for the idealised model — a perfectly prismatic, uniform member with idealised, rigid supports and the material properties you enter. Real structures depart from that: bolted or welded joints are not perfectly rigid or perfectly free, sections taper, mass is added, materials are anisotropic, and damping shifts and broadens the peaks. Expect the real frequencies to land in the right ballpark but not on the exact number. Use the result to compare designs and screen for resonance, then verify by measurement.
Which boundary condition should I choose?
Match how the ends are restrained. A diving board or a bracket fixed at one end and free at the other is a cantilever. A beam resting on two supports that can rotate freely is simply supported (pinned). A beam rigidly built into walls at both ends is fixed–fixed. A free-floating object with no supports (like a part hanging by a soft sling for testing) is free–free. Real joints are usually somewhere between pinned and fixed, so computing both brackets the likely answer.
What does the resonance danger-zone check tell me?
It compares an excitation frequency you enter — say a motor at 3000 rpm = 50 Hz — against each computed natural frequency and flags any that fall within your chosen percentage band. If a forcing frequency sits near a natural frequency, vibration is amplified (theoretically without limit when there is no damping), which causes noise, fatigue cracks and failures. A common rule of thumb is to keep operating frequencies at least 20–25% away from any natural frequency. Remember to check the harmonics of the excitation too, not just its fundamental.
Where do the material values come from, and can I use my own?
They are typical published engineering reference values (standard mechanics-of-materials texts and material databases). They are not a proprietary part-number database and they are not guaranteed for your specific material — grade, alloy, moisture, temperature and direction all matter, and carbon-fibre in particular spans roughly 70–150 GPa depending on layup. Choose the Custom material option to enter your own measured E, density and (for plates) Poisson’s ratio.
Why does the calculator warn about “stocky” or “thick” members?
Euler–Bernoulli beam theory and Kirchhoff plate theory both ignore shear deformation and rotary inertia, which is fine for slender beams and thin plates but increasingly wrong for short, deep beams or thick plates. In those cases the higher modes are over-predicted, and Timoshenko (beam) or Mindlin (plate) theory is needed. The tool flags the case so you know to treat the result as indicative rather than precise.
Can I use this for structural safety decisions?
No. This is an educational estimator for idealised members, not an engineering analysis of your actual structure, and it says nothing about stress, fatigue life or failure. For anything where safety matters, model the real geometry (for example with finite-element analysis), measure the built structure, and have a qualified, licensed engineer review the design.
How does beam length affect natural frequency, and how do I convert RPM to Hz?
For a beam, natural frequency scales with the inverse square of length — double the length and the first natural frequency drops to one quarter. That steep dependence is why even small errors in span matter. For the excitation check, convert a machine speed in revolutions per minute (RPM) to Hz by dividing by 60: a motor at 3000 RPM produces a forcing frequency of 50 Hz. Also check harmonics (2×, 3×, etc.) because those can excite higher modes.
What is the difference between Euler–Bernoulli and Timoshenko beam theory?
Euler–Bernoulli theory, used here, assumes plane cross-sections remain plane and ignores shear deformation and rotary inertia. It is accurate when the beam is slender (length-to-depth ratio greater than roughly 10). Timoshenko beam theory adds shear flexibility and rotary inertia terms, which lower the predicted natural frequencies for higher modes and short, deep beams. If the tool warns that your member is “stocky,” the first mode is still reasonably accurate but treat higher modes with caution and consider a finite-element or Timoshenko model.
How does structural damping affect these resonance frequencies?
The undamped natural frequencies computed here are almost identical to the damped natural frequencies for lightly damped structures (damping ratios below about 5%), so the frequencies themselves are accurate. Damping does not change the resonant frequency much, but it dramatically limits the peak amplitude at resonance. A higher modal damping ratio (more energy dissipated per cycle) means a lower, broader resonance peak and a lower dynamic amplification factor — which is why welded joints, viscoelastic materials and tuned mass dampers are used to control structural vibration.
Can I use the plate calculator for a floor or wall panel?
Yes, as a rough estimate, but with important caveats. Real floor and wall panels have boundary conditions closer to partially clamped rather than the simply-supported (pinned) edges assumed here, which raises the actual natural frequencies. They also carry distributed loads, have composite layups, and are connected to framing that adds stiffness and mass. The result gives a useful order-of-magnitude check — if your first mode falls in the footfall excitation range (1–8 Hz for floors), that is a signal worth investigating — but a detailed finite-element analysis or specialist floor vibration assessment is required for design decisions.

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