Library / Structural dynamics and vibration testing / SDY-19
Structural dynamics and vibration testing
Longitudinal frequencies of a fixed–free rod
Axial (compression-wave) natural frequencies of a uniform slender rod fixed at one end and free at the other, from Young’s modulus E, density ρ, length L and the mode number n.
- Ref
- SDY-19
- Rev
- v1
- Field
- Structural dynamics and vibration testing
- Result
- fn [Hz]
A uniform slender rod with one end fixed and one end free vibrates axially at fₙ = (2n − 1)√(E/ρ)/(4L). The frequencies are the odd multiples of the fundamental. The wave speed in the rod is c = √(E/ρ), about 5000 m/s in steel. The result needs long wavelengths compared with the section size. E is entered in GPa.
Axial (compression-wave) natural frequencies of a uniform slender rod fixed at one end and free at the other, from Young’s modulus E, density ρ, length L and the mode number n.
Calculator
- Wave speed, √(E/ρ)5,000 m/s
- Fundamental (n = 1)625 Hz
- Wavelength of this mode, 4L/(2n − 1)2.667 m
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.
- Mode 2 of a 2 m rod vibrates axially at 1,875 Hz; the wave speed is 5,000 m/s.
- Axial frequencies are far above bending frequencies for the same rod, which is why bending usually dominates the low-frequency response.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (Hz) | Local step | Change in result |
|---|---|---|---|---|---|---|
| n Mode number | 1 to 8 | 625 | 9,375 | 8,750 | +1 | +66.7 % |
| L Rod length | 0.5 to 5 m | 750 | 7,500 | 6,750 | +10% | −9.09 % |
| ρ Density | 1,000 to 20,000 kg/m³ | 1,186 | 5,303 | 4,117 | +10% | −4.65 % |
| E Young’s modulus | 1 to 400 GPa | 132.6 | 2,652 | 2,519 | +10% | +4.88 % |
Sweep of Rod length
| Rod length (m) | Axial natural frequency (Hz) |
|---|---|
| 0.5 | 7,500 |
| 1.063 | 3,529 |
| 1.625 | 2,308 |
| 2.188 | 1,714 |
| 2.75 | 1,364 |
| 3.313 | 1,132 |
| 3.875 | 967.7 |
| 4.438 | 845.1 |
| 5 | 750 |
Sweep one variable
Variables and how each one contributes
Axial natural frequency
What the equation gives you. Everything else feeds it.
Mode number
The mode or harmonic number (a whole number from 1). Raises the frequency in direct proportion (for the string) or in steps. Raising it by 1 (to 3) moves the result by +66.7 %, from 1,875 to 3,125 Hz.
Young’s modulus
Young’s modulus, in GPa (steel about 200). A stiffer material raises the axial frequencies and the impedance. Raising it by 10 % (to 220) moves the result by +4.88 %, from 1,875 to 1,967 Hz.
Density
The density, in kg/m³ (steel about 7850). A denser material lowers the frequencies. Raising it by 10 % (to 8,800) moves the result by −4.65 %, from 1,875 to 1,788 Hz.
Rod length
The span or length, in m. A longer span lowers the frequency. Raising it by 10 % (to 2.2) moves the result by −9.09 %, from 1,875 to 1,705 Hz.
Worked example with the values above
- Wave speed: √(E/ρ) = √(200 × 10⁹ / 8,000) = 5,000 m/s.
- Odd multiple over four lengths: (2n − 1)/(4L) = 3 / 8 = 0.375 1/m.
- fₙ = 1,875 Hz.
Assumptions and limits
- Small motion, a uniform string, membrane or rod, and perfectly fixed or free ends as stated.
- Negligible bending stiffness for strings and membranes, and long axial wavelengths compared with the rod section.
Common mistakes
- Using tension in N for a membrane, which needs N/m, or the area mass in place of the mass per metre.
- Using the string result for a stiff or thick wire, where bending stiffness raises the high harmonics.
- Entering Young’s modulus in Pa when GPa is expected.
Statistics and uncertainty
Frequency follows the square root of tension or modulus over mass, and inversely with length, so length and mass are usually the dominant errors. Real supports are never perfectly fixed, which lowers measured frequencies.
Where it is used
- Tuning and design of strings, cables, drum skins, diaphragms and axial rods.
- Checking axial wave speeds and impedances at joints in impact and pile tests.
Maths you need
- Wave speed: c = √(tension/mass per length), or √(E/ρ)
- Standing waves: modes that fit the boundaries
- Young’s modulus: stress over strain