Library / Structural dynamics and vibration testing / SDY-13
Structural dynamics and vibration testing
Flexural travelling-wave phase speed
Speed at which the crests of a bending wave travel along a long uniform beam at angular frequency ω.
- Ref
- SDY-13
- Rev
- v1
- Field
- Structural dynamics and vibration testing
- Result
- cp [m/s]
The phase speed of a bending wave on a long uniform beam is cₚ = (EI/μ)1/4√ω: it increases with the square root of frequency, so beams are dispersive and high frequencies outrun low ones. The group speed, at which energy travels, is twice the phase speed at the same frequency.
Speed at which the crests of a bending wave travel along a long uniform beam at angular frequency ω. It rises with the square root of frequency, so beams are dispersive.
Calculator
- Wavenumber, ω/cₚ2 rad/m
- Wavelength, 2π cₚ/ω3.142 m
- Group speed, 2cₚ40 m/s
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.
- At 40 rad/s the crests move at 20 m/s and the energy at 40 m/s, with a wavelength of 3.14 m.
- Quadrupling the frequency only doubles the speed, so impacts spread into a train of waves instead of a single pulse.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m/s) | Local step | Change in result |
|---|---|---|---|---|---|---|
| ω Angular frequency | 1 to 2,000 rad/s | 3.162 | 141.4 | 138.3 | +10% | +4.88 % |
| μ Mass per metre | 0.1 to 20 kg/m | 11.25 | 42.29 | 31.05 | +10% | −2.35 % |
| EI Flexural rigidity | 10 to 2,000 N·m² | 9.457 | 35.57 | 26.11 | +10% | +2.41 % |
Sweep of Angular frequency
| Angular frequency (rad/s) | Phase speed (m/s) |
|---|---|
| 1 | 3.162 |
| 2.586 | 5.085 |
| 6.687 | 8.178 |
| 17.29 | 13.15 |
| 44.72 | 21.15 |
| 115.6 | 34.01 |
| 299.1 | 54.69 |
| 773.4 | 87.94 |
| 2,000 | 141.4 |
Sweep one variable
Variables and how each one contributes
Phase speed
What the equation gives you. Everything else feeds it.
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 +2.41 %, from 20 to 20.48 m/s.
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 −2.35 %, from 20 to 19.53 m/s.
Angular frequency
The angular frequency of the wave, in rad/s. A higher frequency travels faster on a beam. Raising it by 10 % (to 44) moves the result by +4.88 %, from 20 to 20.98 m/s.
Worked example with the values above
- Fourth root of the stiffness-to-mass ratio: (EI/μ)^(1/4) = (100)^(1/4) = 3.1623.
- Square root of the frequency: √40 = 6.3246.
- cₚ = 20 m/s.
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