Library / Structural dynamics and vibration testing / SDY-30
Structural dynamics and vibration testing
Residual static flexibility after mode truncation
Static flexibility that the discarded modes still contribute: the exact static displacement per unit force minus the part supplied by the retained mode.
- Ref
- SDY-30
- Rev
- v1
- Field
- Structural dynamics and vibration testing
- Result
- R [mm/N]
When higher modes are dropped, their combined static contribution is lost. The residual flexibility is R = s₀ − Σφjφk/(mrωr²), the exact static flexibility minus the part the retained modes supply. Adding R to the modal sum restores accuracy at low frequency. For several retained modes subtract each term. Output is in mm/N.
Static flexibility that the discarded modes still contribute: the exact static displacement per unit force minus the part supplied by the retained mode. For several retained modes subtract each term.
Calculator
- Retained mode static flexibility5 mm/N
- Share of the exact flexibility that the retained mode supplies0.8
- Modal stiffness of the retained mode, mᵣωᵣ²200 N/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.
- The retained mode supplies 80 % of the exact static flexibility; the other 20 % comes from the discarded modes.
- Leaving the residual out underestimates the static deflection and the low-frequency response by that amount.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (mm/N) | Local step | Change in result |
|---|---|---|---|---|---|---|
| ωr Natural angular frequency of the retained mode | 2 to 100 rad/s | −118.8 | 6.2 | 125 | +10% | +69.4 % |
| mr Modal mass of the retained mode | 0.1 to 20 kg | −93.75 | 5.75 | 99.5 | +10% | +36.4 % |
| φjφk Shape product of the retained mode | −2 to 2 | −3.75 | 16.25 | 20 | +0.08 | −32 % |
| s0 Exact static flexibility at the coordinate pair | 1 to 20 mm/N | −4 | 15 | 19 | +10% | +50 % |
Sweep of Exact static flexibility at the coordinate pair
| Exact static flexibility at the coordinate pair (mm/N) | Residual flexibility (mm/N) |
|---|---|
| 1 | −4 |
| 3.375 | −1.625 |
| 5.75 | 0.75 |
| 8.125 | 3.125 |
| 10.5 | 5.5 |
| 12.88 | 7.875 |
| 15.25 | 10.25 |
| 17.63 | 12.63 |
| 20 | 15 |
Sweep one variable
Variables and how each one contributes
Residual flexibility
What the equation gives you. Everything else feeds it.
Exact static flexibility at the coordinate pair
The exact static flexibility between the two points, in mm/N: the displacement per unit force. Raises the residual in direct proportion. Raising it by 10 % (to 6.875) moves the result by +50 %, from 1.25 to 1.875 mm/N.
Shape product of the retained mode
The product of the retained mode’s ordinates at the response and force points (dimensionless). Raises the part the retained mode supplies. Raising it by 0.1 (to 1.1) moves the result by −40 %, from 1.25 to 0.75 mm/N.
Modal mass of the retained mode
The modal mass of the retained mode, in kg. A heavier mode contributes less static flexibility. Raising it by 10 % (to 2.2) moves the result by +36.4 %, from 1.25 to 1.705 mm/N.
Natural angular frequency of the retained mode
The natural angular frequency of the retained mode, in rad/s. A higher frequency contributes less static flexibility. Raising it by 10 % (to 11) moves the result by +69.4 %, from 1.25 to 2.118 mm/N.
Worked example with the values above
- Retained mode: φⱼφₖ/(mᵣωᵣ²) = 1 / (2 × 10²) = 0.005 m/N = 5 mm/N.
- Exact static flexibility: s₀ = 6.25 mm/N.
- R = 6.25 − 5 = 1.25 mm/N.
Assumptions and limits
- A small, linear model with symmetric mass and stiffness matrices; this entry shows the arithmetic with two (or three) coordinates.
- Shape, mass and stiffness are in consistent units and degree-of-freedom order.
Common mistakes
- Using a shape scaled in one program with a modal mass from another that uses a different normalisation.
- Mixing the order of the degrees of freedom in the shape and in the matrix.
- Forgetting that modal mass depends on how the shape is scaled, while the participation and effective masses do not.
Statistics and uncertainty
These are exact algebraic forms, so their uncertainty comes from the inputs: mass and stiffness matrices from a model, and mode shapes from a modal solution or test. Shape errors at a few points can shift modal quantities noticeably.
Where it is used
- Modal analysis, model reduction and checks that a reduced model has kept enough modes.
- Base-shock and seismic analysis through participation factors and effective modal masses.
Maths you need
- Mass and stiffness matrices: the matrices of M ü + K u = F
- Mode shape: the pattern of motion at one natural frequency
- Quadratic form: vᵀMv, a scalar from a vector and a matrix