Library / Structural dynamics and vibration testing / SDY-29
Structural dynamics and vibration testing
Modal sum for transfer receptance
Magnitude of the displacement-per-force frequency response between two points, as a sum of two modal terms with classical damping.
- Ref
- SDY-29
- Rev
- v1
- Field
- Structural dynamics and vibration testing
- Result
- |H| [mm/N]
For real modes and classical damping, the frequency response between a response point j and a force point k is a sum of modal terms: Hjk = Σφjrφkr/[mr(ωr² − ω² + 2iζrωrω)]. This entry uses two modes, takes the shape products as inputs and returns the magnitude in mm/N; the derived values give the real and imaginary parts and the phase under an exp(+iωt) convention.
Magnitude of the displacement-per-force frequency response between two points, as a sum of two modal terms with classical damping. The shape ordinates enter as the product of the values at the response and force points.
Calculator
- Real part of the receptance−1.659 mm/N
- Imaginary part of the receptance−49.89 mm/N
- Phase angle−91.9 °
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.
- At 10 rad/s the receptance is 49.92 mm/N; at a mode’s natural frequency it is dominated by that mode’s damping term.
- With the two shape products of opposite sign there is an anti-resonance between the modes where the response dips sharply.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (mm/N) | Local step | Change in result |
|---|---|---|---|---|---|---|
| m1 Modal mass, mode 1 | 0.1 to 20 kg | 5.163 | 999.9 | 994.7 | +10% | −9.1 % |
| ζ1 Damping ratio, mode 1 | 0.005 to 0.2 | 12.5 | 499.9 | 487.4 | +10% | −9.1 % |
| φj1φk1 Shape product, mode 1 | −2 to 2 | 1.663 | 100.1 | 98.46 | +0.08 | +8.01 % |
| ω Excitation angular frequency | 1 to 60 rad/s | 0.01365 | 49.92 | 49.9 | +10% | −54.7 % |
| ω1 Natural angular frequency, mode 1 | 2 to 60 rad/s | 0.1419 | 49.92 | 49.78 | +10% | −60.8 % |
| m2 Modal mass, mode 2 | 0.1 to 20 kg | 49.9 | 58.18 | 8.281 | +10% | +0.0105 % |
| ω2 Natural angular frequency, mode 2 | 5 to 100 rad/s | 46.5 | 50 | 3.5 | +10% | +0.0512 % |
| φj2φk2 Shape product, mode 2 | −2 to 2 | 49.89 | 50.33 | 0.4415 | +0.08 | +0.00923 % |
| ζ2 Damping ratio, mode 2 | 0.005 to 0.2 | 49.61 | 50.02 | 0.4072 | +10% | −0.022 % |
Sweep of Excitation angular frequency
| Excitation angular frequency (rad/s) | Receptance magnitude (mm/N) |
|---|---|
| 1 | 3.797 |
| 8.375 | 14.65 |
| 15.75 | 6.501 |
| 23.13 | 2.425 |
| 30.5 | 0.34 |
| 37.88 | 0.1092 |
| 45.25 | 0.04719 |
| 52.63 | 0.02405 |
| 60 | 0.01365 |
Sweep one variable
Variables and how each one contributes
Receptance magnitude
What the equation gives you. Everything else feeds it.
Shape product, mode 1
The product of the ordinates of mode 1 at the response point j and the force point k (dimensionless). Scales the contribution of mode 1. Raising it by 0.1 (to 1.1) moves the result by +10 %, from 49.92 to 54.91 mm/N.
Shape product, mode 2
The product of the ordinates of mode 2 at the response point and the force point (dimensionless). A negative value means the two points move in opposite senses in that mode. Scales the contribution of mode 2. Raising it by 0.1 (to −0.9) moves the result by +0.0117 %, from 49.92 to 49.92 mm/N.
Modal mass, mode 1
The modal mass of mode 1, in kg. A heavier mode responds less. Raising it by 10 % (to 2.2) moves the result by −9.1 %, from 49.92 to 45.37 mm/N.
Modal mass, mode 2
The modal mass of mode 2, in kg. A heavier mode responds less. Raising it by 10 % (to 2.2) moves the result by +0.0105 %, from 49.92 to 49.92 mm/N.
Natural angular frequency, mode 1
The natural angular frequency of mode 1, in rad/s. Moves the first resonance. Raising it by 10 % (to 11) moves the result by −60.8 %, from 49.92 to 19.58 mm/N.
Natural angular frequency, mode 2
The natural angular frequency of mode 2, in rad/s. Moves the second resonance. Raising it by 10 % (to 22) moves the result by +0.0512 %, from 49.92 to 49.94 mm/N.
Damping ratio, mode 1
The damping ratio of mode 1 (a plain number). More damping lowers the peak at that mode. Raising it by 10 % (to 0.055) moves the result by −9.1 %, from 49.92 to 45.37 mm/N.
Damping ratio, mode 2
The damping ratio of mode 2 (a plain number). More damping lowers the peak at that mode. Raising it by 10 % (to 0.055) moves the result by −0.022 %, from 49.92 to 49.91 mm/N.
Excitation angular frequency
The excitation angular frequency, in rad/s, at which the response is evaluated. Moves the frequency at which the response is read; near a natural frequency the effect is large. Raising it by 10 % (to 11) moves the result by −54.7 %, from 49.92 to 22.62 mm/N.
Worked example with the values above
- Evaluate each modal term at ω = 10 rad/s for modes at 10 and 20 rad/s.
- Add the real and imaginary parts: H = −1.6593 − 49.889i mm/N.
- |H| = 49.917 mm/N.
Assumptions and limits
- Real mode shapes, classical (proportional) damping and an exp(+iωt) time convention.
- Only the two modes listed are kept, so the result is valid only in the band between and around them.
Common mistakes
- Dropping the higher modes and expecting accuracy at high frequency: truncation leaves a residual flexibility.
- Using the wrong sign for the shape product when the two points are on opposite sides of a node.
- Mixing Hz and rad/s.
Statistics and uncertainty
Near a resonance the response depends strongly on damping and on the exact frequency, so small errors in either are amplified. Away from resonances the result is driven by the mass and stiffness terms and is much less sensitive.
Where it is used
- Reconstructing and checking frequency response functions from modal parameters.
- Understanding resonance and anti-resonance between modes.
Maths you need
- Modal superposition: the response as a sum over modes
- Complex numbers: real and imaginary parts, magnitude and phase
- Receptance: displacement per unit force