Library / Structural dynamics and vibration testing / SDY-16
Structural dynamics and vibration testing
Transverse string harmonics
Natural frequencies of a uniform flexible string fixed at both ends, from the tension T, the mass per metre μ, the span L and the harmonic number n.
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- SDY-16
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- v1
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- Structural dynamics and vibration testing
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- fn [Hz]
A flexible string fixed at both ends vibrates at fₙ = n√(T/μ)/(2L), where n is the harmonic number. The frequencies are whole multiples of the fundamental, which is why a string gives a harmonic series. The wave speed is √(T/μ). Stiffness of the string is neglected.
Natural frequencies of a uniform flexible string fixed at both ends, from the tension T, the mass per metre μ, the span L and the harmonic number n.
Calculator
- Wave speed, √(T/μ)100 m/s
- Fundamental frequency (n = 1)25 Hz
- Wavelength of this harmonic, 2L/n1.333 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.
- Harmonic 3 of this string vibrates at 75 Hz, 3 times the fundamental of 25 Hz.
- Quadrupling the tension doubles every frequency, which is how a string is tuned.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (Hz) | Local step | Change in result |
|---|---|---|---|---|---|---|
| L Span | 0.5 to 5 m | 30 | 300 | 270 | +10% | −9.09 % |
| n Harmonic number | 1 to 10 | 25 | 250 | 225 | +1 | +33.3 % |
| T Tension | 10 to 1,000 N | 23.72 | 237.2 | 213.5 | +10% | +4.88 % |
| μ Mass per metre | 0.001 to 0.1 kg/m | 23.72 | 237.2 | 213.5 | +10% | −4.65 % |
Sweep of Tension
| Tension (N) | Natural frequency (Hz) |
|---|---|
| 10 | 23.72 |
| 133.8 | 86.74 |
| 257.5 | 120.4 |
| 381.3 | 146.4 |
| 505 | 168.5 |
| 628.8 | 188.1 |
| 752.5 | 205.7 |
| 876.3 | 222 |
| 1,000 | 237.2 |
Sweep one variable
Variables and how each one contributes
Natural frequency
What the equation gives you. Everything else feeds it.
Harmonic 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 4) moves the result by +33.3 %, from 75 to 100 Hz.
Tension
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 75 to 78.66 Hz.
Mass per metre
The mass per metre of the string, in kg/m. A heavier string lowers the frequency. Raising it by 10 % (to 0.011) moves the result by −4.65 %, from 75 to 71.51 Hz.
Span
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 75 to 68.18 Hz.
Worked example with the values above
- Wave speed: √(T/μ) = √(100 / 0.01) = 100 m/s.
- Half the span: 2L = 4 m.
- fₙ = 3 × 100 / 4 = 75 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