Library / Structural dynamics and vibration testing / SDY-15
Structural dynamics and vibration testing
Beam frequency from a trial deflection
Rayleigh-quotient estimate of a beam frequency from a trial deflected shape: the square root of the stiffness-weighted curvature integral over the mass-weighted shape integral.
- Ref
- SDY-15
- Rev
- v1
- Field
- Structural dynamics and vibration testing
- Result
- f [Hz]
The Rayleigh quotient gives an upper bound on a natural frequency from any trial shape that meets the support conditions: ω² = ∫EI(y″)²dx / ∫μy²dx. Here U₂ is the integral in the numerator and T₂ the one in the denominator. The better the trial shape resembles the true mode, the closer the estimate. For y = sin(πx/L) on a pinned–pinned beam it is exact.
Rayleigh-quotient estimate of a beam frequency from a trial deflected shape: the square root of the stiffness-weighted curvature integral over the mass-weighted shape integral. A valid trial gives an upper bound.
Calculator
- Angular frequency, √(U₂/T₂)98.7 rad/s
- Period0.06366 s
- Frequency if the mass is increased by 10 %14.98 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.
- A strain-energy term of 9,741 and a kinetic-energy term of 1 give 15.71 Hz; the true fundamental can only be lower.
- A guess that is too stiff overestimates the frequency, so try more than one shape and keep the lowest result.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (Hz) | Local step | Change in result |
|---|---|---|---|---|---|---|
| T2 Integrated μ·(shape)² of the trial shape | 0.1 to 10 kg | 4.967 | 49.67 | 44.71 | +10% | −4.65 % |
| U2 Integrated EI·(curvature)² of the trial shape | 100 to 50,000 N/m | 1.592 | 35.59 | 34 | +10% | +4.88 % |
Sweep of Integrated EI·(curvature)² of the trial shape
| Integrated EI·(curvature)² of the trial shape (N/m) | Frequency estimate (Hz) |
|---|---|
| 100 | 1.592 |
| 6,338 | 12.67 |
| 12,580 | 17.85 |
| 18,810 | 21.83 |
| 25,050 | 25.19 |
| 31,290 | 28.15 |
| 37,530 | 30.83 |
| 43,760 | 33.29 |
| 50,000 | 35.59 |
Sweep one variable
Variables and how each one contributes
Frequency estimate
What the equation gives you. Everything else feeds it.
Integrated EI·(curvature)² of the trial shape
The integrated EI(y″)² of the trial shape, in N/m: the strain-energy term. A more curved trial shape raises the frequency estimate. Raising it by 10 % (to 10,710) moves the result by +4.88 %, from 15.71 to 16.47 Hz.
Integrated μ·(shape)² of the trial shape
The integrated μy² of the trial shape, in kg: the kinetic-energy term. More mass in the trial shape lowers the frequency estimate. Raising it by 10 % (to 1.1) moves the result by −4.65 %, from 15.71 to 14.98 Hz.
Worked example with the values above
- Ratio of energy terms: U₂/T₂ = 9,741 / 1 = 9,740.9 (rad/s)².
- Square root: ω = 98.696 rad/s.
- f = ω/(2π) = 15.708 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