Library / Structural dynamics and vibration testing / SDY-17
Structural dynamics and vibration testing
Rectangular membrane frequencies
Natural frequencies of a rectangular membrane with all edges fixed, from the edge tension T, the mass per unit area σ, the side lengths a and b and the mode indices i and j.
- Ref
- SDY-17
- Rev
- v1
- Field
- Structural dynamics and vibration testing
- Result
- fij [Hz]
A rectangular membrane fixed on all edges vibrates at fij = ½√(T/σ)√[(i/a)² + (j/b)²], where i and j count the half-waves along each side. Unlike a string, the frequencies are not whole multiples of the fundamental. Bending stiffness is neglected and the tension is uniform and equal in all directions.
Natural frequencies of a rectangular membrane with all edges fixed, from the edge tension T, the mass per unit area σ, the side lengths a and b and the mode indices i and j.
Calculator
- Wave speed, √(T/σ)20 m/s
- Fundamental (i = j = 1)11.18 Hz
- Ratio to the fundamental1.844
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 (1, 2) of a 2 m by 1 m membrane vibrates at 20.62 Hz.
- The modes are inharmonic, which is why a drum gives an unpitched sound compared with a string.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (Hz) | Local step | Change in result |
|---|---|---|---|---|---|---|
| T Edge tension per unit length | 10 to 1,000 N/m | 6.519 | 65.19 | 58.67 | +10% | +4.88 % |
| j Mode index along side b | 1 to 6 | 11.18 | 60.21 | 49.03 | +1 | +47.5 % |
| σ Mass per unit area | 0.05 to 2 kg/m² | 7.289 | 46.1 | 38.81 | +10% | −4.65 % |
| b Side length b | 0.5 to 5 m | 6.403 | 40.31 | 33.91 | +10% | −8.53 % |
| i Mode index along side a | 1 to 6 | 20.62 | 36.06 | 15.44 | +1 | +8.47 % |
| a Side length a | 0.5 to 5 m | 20.1 | 28.28 | 8.185 | +10% | −0.512 % |
Sweep of Edge tension per unit length
| Edge tension per unit length (N/m) | Natural frequency (Hz) |
|---|---|
| 10 | 6.519 |
| 133.8 | 23.84 |
| 257.5 | 33.08 |
| 381.3 | 40.25 |
| 505 | 46.33 |
| 628.8 | 51.69 |
| 752.5 | 56.55 |
| 876.3 | 61.03 |
| 1,000 | 65.19 |
Sweep one variable
Variables and how each one contributes
Natural 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 20.62 to 21.62 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 20.62 to 19.66 Hz.
Mode index along side a
The mode index along side a (a whole number from 1). More half-waves along a side raise the frequency. Raising it by 1 (to 2) moves the result by +8.47 %, from 20.62 to 22.36 Hz.
Mode index along side b
The mode index along side b (a whole number from 1). More half-waves along a side raise the frequency. Raising it by 1 (to 3) moves the result by +47.5 %, from 20.62 to 30.41 Hz.
Side length a
The side length a, in m. A longer side lowers the frequency. Raising it by 10 % (to 2.2) moves the result by −0.512 %, from 20.62 to 20.51 Hz.
Side length b
The side length b, in m. A longer side lowers the frequency. Raising it by 10 % (to 1.1) moves the result by −8.53 %, from 20.62 to 18.86 Hz.
Worked example with the values above
- Wave speed: √(T/σ) = √(100 / 0.25) = 20 m/s.
- Shape term: √((i/a)² + (j/b)²) = √((1/2)² + (2/1)²) = 2.0616 1/m.
- f = ½ × 20 × 2.0616 = 20.616 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