Library / Structural dynamics and vibration testing / SDY-20
Structural dynamics and vibration testing
Longitudinal rod characteristic impedance
Ratio of axial force to particle velocity for a single travelling compression wave in a rod: density times area times wave speed.
- Ref
- SDY-20
- Rev
- v1
- Field
- Structural dynamics and vibration testing
- Result
- Z [N·s/m]
For a single travelling compression wave in a rod, the force is the characteristic impedance Z = ρAc = A√(ρE) times the particle velocity. When a wave meets a change of section, the reflected share depends on the impedance ratio. A reflection at the free end returns the wave with a change of sign. Inputs are in cm² and GPa.
Ratio of axial force to particle velocity for a single travelling compression wave in a rod: density times area times wave speed. It governs how much of a wave is reflected at a change of section.
Calculator
- Wave speed, √(E/ρ)5,000 m/s
- Force per unit velocity of 1 m/s40,000 N
- Impedance with the area doubled80,000 N·s/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.
- A rod of 10 cm² has an impedance of 40,000 N·s/m: a force of 1 N is carried by a particle velocity of 2.5×10−5 m/s.
- When a wave goes from a rod into one with half the area, the impedance halves, and a reflected wave of the opposite sign carries part of the force back.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (N·s/m) | Local step | Change in result |
|---|---|---|---|---|---|---|
| A Cross-section area | 0.5 to 100 cm² | 2,000 | 400,000 | 398,000 | +10% | +10 % |
| E Young’s modulus | 1 to 400 GPa | 2,828 | 56,570 | 53,740 | +10% | +4.88 % |
| ρ Density | 1,000 to 20,000 kg/m³ | 14,140 | 63,250 | 49,100 | +10% | +4.88 % |
Sweep of Cross-section area
| Cross-section area (cm²) | Characteristic impedance (N·s/m) |
|---|---|
| 0.5 | 2,000 |
| 12.94 | 51,750 |
| 25.38 | 101,500 |
| 37.81 | 151,300 |
| 50.25 | 201,000 |
| 62.69 | 250,800 |
| 75.13 | 300,500 |
| 87.56 | 350,300 |
| 100 | 400,000 |
Sweep one variable
Variables and how each one contributes
Characteristic impedance
What the equation gives you. Everything else feeds it.
Cross-section area
The cross-section area of the rod, in cm². Raises the impedance in direct proportion. Raising it by 10 % (to 11) moves the result by +10 %, from 40,000 to 44,000 N·s/m.
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 +4.88 %, from 40,000 to 41,950 N·s/m.
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 40,000 to 41,950 N·s/m.
Worked example with the values above
- Convert the area: 10 cm² = 0.001 m².
- Product of density and modulus: ρE = 8,000 × 200 × 10⁹ = 1.6×1015, whose root is 4×107.
- Z = 0.001 × 4×107 = 40,000 N·s/m.
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