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Structural dynamics and vibration testing
Circular membrane fundamental frequency
Fundamental frequency of a circular membrane with a fixed edge, from the first zero of the Bessel function J₀ (about 2.4048), the tension T, the mass per unit area σ and the radius R.
- Ref
- SDY-18
- Rev
- v1
- Field
- Structural dynamics and vibration testing
- Result
- f01 [Hz]
The fundamental of a fixed circular membrane is f₀₁ = j₀₁√(T/σ)/(2πR), where j₀₁ ≈ 2.4048 is the first zero of the Bessel function J₀. The mode is axisymmetric: the whole membrane moves in the same sense and only the edge is still. Higher modes need other zeros of Bessel functions.
Fundamental frequency of a circular membrane with a fixed edge, from the first zero of the Bessel function J₀ (about 2.4048), the tension T, the mass per unit area σ and the radius R.
Calculator
- Wave speed, √(T/σ)20 m/s
- Angular frequency, 2πf240.5 rad/s
- Frequency of a square membrane of the same area39.89 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 drum of radius 0.2 m with 100 N/m tension and 0.25 kg/m² vibrates at 38.27 Hz.
- Frequency is inversely proportional to the radius: a drum twice as big is an octave lower.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (Hz) | Local step | Change in result |
|---|---|---|---|---|---|---|
| R Radius | 0.05 to 2 m | 3.827 | 153.1 | 149.3 | +10% | −9.09 % |
| T Edge tension per unit length | 10 to 1,000 N/m | 12.1 | 121 | 108.9 | +10% | +4.88 % |
| σ Mass per unit area | 0.05 to 2 kg/m² | 13.53 | 85.58 | 72.05 | +10% | −4.65 % |
Sweep of Radius
| Radius (m) | Fundamental frequency (Hz) |
|---|---|
| 0.05 | 153.1 |
| 0.2938 | 26.06 |
| 0.5375 | 14.24 |
| 0.7813 | 9.798 |
| 1.025 | 7.468 |
| 1.269 | 6.033 |
| 1.512 | 5.061 |
| 1.756 | 4.359 |
| 2 | 3.827 |
Sweep one variable
Variables and how each one contributes
Fundamental frequency
What the equation gives you. Everything else feeds it.
Edge tension per unit length
The tension, in N for a string, or the edge force per unit length, in N/m, for a membrane. More tension raises the frequency with its square root. Raising it by 10 % (to 110) moves the result by +4.88 %, from 38.27 to 40.14 Hz.
Mass per unit area
The mass per unit area of the membrane, in kg/m². A heavier membrane lowers the frequency. Raising it by 10 % (to 0.275) moves the result by −4.65 %, from 38.27 to 36.49 Hz.
Radius
The radius, in m. A larger drum lowers the frequency in inverse proportion. Raising it by 10 % (to 0.22) moves the result by −9.09 %, from 38.27 to 34.79 Hz.
Worked example with the values above
- Wave speed: √(T/σ) = √(100 / 0.25) = 20 m/s.
- Bessel zero over circumference factor: 2.4048 / (2π × 0.2) = 1.9137 1/m.
- f₀₁ = 38.274 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