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Structural dynamics and vibration testing
Uniform beam bending frequencies
Bending natural frequency of a uniform Euler–Bernoulli beam from its flexural rigidity EI, mass per metre μ, length L and a dimensionless root λ set by the end supports.
- Ref
- SDY-11
- Rev
- v1
- Field
- Structural dynamics and vibration testing
- Result
- fn [Hz]
The bending frequencies of a uniform Euler–Bernoulli beam are fₙ = λₙ²√(EI/μ)/(2πL²). The dimensionless root λₙ is set by the supports: nπ for pinned–pinned, 1.8751 (first), 4.6941 (second) and so on for a cantilever, and 4.7300 (first) for clamped–clamped. Rotary inertia and shear deformation are neglected, so the formula overestimates frequencies of thick or short beams.
Bending natural frequency of a uniform Euler–Bernoulli beam from its flexural rigidity EI, mass per metre μ, length L and a dimensionless root λ set by the end supports.
Calculator
- Angular frequency, 2πf35.16 rad/s
- Period of the vibration0.1787 s
- Frequency with the length doubled1.399 Hz
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- With a root of 1.875, a rigidity of 200 N·m², 2 kg/m and 1 m, the frequency is 5.596 Hz.
- Frequency falls with the square of the length: a beam twice as long vibrates at one quarter of the frequency.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (Hz) | Local step | Change in result |
|---|---|---|---|---|---|---|
| λn Dimensionless root for the supports | 1 to 12 | 1.592 | 229.2 | 227.6 | +10% | +21 % |
| L Beam length | 0.2 to 5 m | 0.2238 | 139.9 | 139.7 | +10% | −17.4 % |
| μ Mass per metre | 0.1 to 20 kg/m | 1.77 | 25.03 | 23.26 | +10% | −4.65 % |
| EI Flexural rigidity | 10 to 2,000 N·m² | 1.251 | 17.7 | 16.44 | +10% | +4.88 % |
Sweep of Beam length
| Beam length (m) | Natural frequency (Hz) |
|---|---|
| 0.2 | 139.9 |
| 0.8 | 8.744 |
| 1.4 | 2.855 |
| 2 | 1.399 |
| 2.6 | 0.8278 |
| 3.2 | 0.5465 |
| 3.8 | 0.3875 |
| 4.4 | 0.289 |
| 5 | 0.2238 |
Sweep one variable
Variables and how each one contributes
Natural frequency
What the equation gives you. Everything else feeds it.
Dimensionless root for the supports
The dimensionless root set by the end supports: nπ for pinned–pinned, 1.8751 for the first cantilever mode, 4.7300 for the first clamped–clamped mode. Enters squared, so the frequency rises with the square of the root. Raising it by 10 % (to 2.063) moves the result by +21 %, from 5.596 to 6.771 Hz.
Flexural rigidity
The flexural rigidity, in N·m²: Young’s modulus times the second moment of area. A stiffer section raises the speed or frequency. Raising it by 10 % (to 220) moves the result by +4.88 %, from 5.596 to 5.869 Hz.
Mass per metre
The mass per metre of the beam, in kg/m. A heavier beam lowers the speed or frequency. Raising it by 10 % (to 2.2) moves the result by −4.65 %, from 5.596 to 5.335 Hz.
Beam length
The length, in m. A longer beam is slower to vibrate: the frequency falls with the square of the length. Raising it by 10 % (to 1.1) moves the result by −17.4 %, from 5.596 to 4.625 Hz.
Worked example with the values above
- Root squared: λₙ² = 1.875² = 3.516.
- Stiffness over mass: √(EI/μ) = √(200 / 2) = 10.
- fₙ = 3.516 × 10 / (2π × 1) = 5.5959 Hz.
Assumptions and limits
- A uniform Euler–Bernoulli beam: slender, constant section, small deflections, no rotary inertia or shear deformation.
- For a trial shape, it satisfies the geometric support conditions.
Common mistakes
- Using the formula for thick or short beams, where shear and rotary inertia lower the frequencies.
- Using the wrong dimensionless root for the supports.
- Mixing the mass per metre with the total mass.
Statistics and uncertainty
Frequency scales with the square root of EI/μ and inversely with L², so a 2 % error in length gives a 4 % error in frequency. Section properties and the true support conditions are the dominant uncertainties in practice.
Where it is used
- Benchmarking a finite-element beam model against a closed-form result.
- Estimating bending frequencies and wave speeds for bars, shafts, pipes and cantilevers.
Maths you need
- Second moment of area: the section property that sets bending stiffness
- Mode shape: the pattern of motion at one natural frequency
- Wavenumber: rad/m, 2π over wavelength