Library / Structural dynamics and vibration testing / SDY-22
Structural dynamics and vibration testing
Generalised modal stiffness
Modal (generalised) stiffness of a shape φ for a two-coordinate model: the shape multiplied through the stiffness matrix K on both sides.
- Ref
- SDY-22
- Rev
- v1
- Field
- Structural dynamics and vibration testing
- Result
- kr [N/m]
The modal stiffness kr = φᵀKφ is the stiffness that the mode behaves as. Divided by the modal mass of the same shape it gives the square of the natural angular frequency: ω² = kᵣ/mᵣ. For an assumed shape this is the Rayleigh quotient, an upper bound on the lowest frequency.
Modal (generalised) stiffness of a shape φ for a two-coordinate model: the shape multiplied through the stiffness matrix K on both sides. Divided by the modal mass it gives the squared natural frequency.
Calculator
- Sum of the diagonal stiffnesses10 N/m
- Modal stiffness if the shape is doubled56 N/m
- Strain energy for a unit modal coordinate, ½kᵣ7 J
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 shape (1, 2) gives a modal stiffness of 14 N/m.
- With the modal mass of the same shape, ω = √(kᵣ/mᵣ) gives a frequency estimate that is never below the true fundamental.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (N/m) | Local step | Change in result |
|---|---|---|---|---|---|---|
| K22 Stiffness matrix entry 22 | 0.1 to 50 N/m | −1.6 | 198 | 199.6 | +10% | +11.4 % |
| φ1 Mode shape ordinate 1 | −3 to 3 | 13.36 | 94 | 80.64 | +0.12 | +4.05 % |
| K12 Stiffness matrix entry 12 | −10 to 10 N/m | −18 | 62 | 80 | +0.4 N/m | +11.4 % |
| K11 Stiffness matrix entry 11 | 0.1 to 50 N/m | 8.1 | 58 | 49.9 | +10% | +4.29 % |
| φ2 Mode shape ordinate 2 | −3 to 3 | 5.04 | 54 | 48.96 | +0.12 | +10.7 % |
Sweep of Mode shape ordinate 1
| Mode shape ordinate 1 | Modal stiffness (N/m) |
|---|---|
| −3 | 94 |
| −2.25 | 64.38 |
| −1.5 | 41.5 |
| −0.75 | 25.38 |
| 0 | 16 |
| 0.75 | 13.38 |
| 1.5 | 17.5 |
| 2.25 | 28.38 |
| 3 | 46 |
Sweep one variable
Variables and how each one contributes
Modal stiffness
What the equation gives you. Everything else feeds it.
Mode shape ordinate 1
The mode shape ordinate at coordinate 1 (dimensionless). Only the ratio v₁ : v₂ defines the shape; the scale is arbitrary. Enters squared in the quadratic forms. Raising it by 0.1 (to 1.1) moves the result by +3.29 %, from 14 to 14.46 N/m.
Mode shape ordinate 2
The mode shape ordinate at coordinate 2 (dimensionless). Enters squared in the quadratic forms. Raising it by 0.1 (to 2.1) moves the result by +8.86 %, from 14 to 15.24 N/m.
Stiffness matrix entry 11
The stiffness matrix entry for coordinate 1, in N/m. Raises the modal stiffness. Raising it by 10 % (to 6.6) moves the result by +4.29 %, from 14 to 14.6 N/m.
Stiffness matrix entry 12
The off-diagonal stiffness entry, in N/m: negative for a spring linking the two coordinates. Couples the two coordinates. Raising it by 0.5 (to −1.5) moves the result by +14.3 %, from 14 to 16 N/m.
Stiffness matrix entry 22
The stiffness matrix entry for coordinate 2, in N/m. Raises the modal stiffness. Raising it by 10 % (to 4.4) moves the result by +11.4 %, from 14 to 15.6 N/m.
Worked example with the values above
- Expand: φ₁²K₁₁ + 2φ₁φ₂K₁₂ + φ₂²K₂₂.
- Substitute: 1 × 6 + 4 × −2 + 4 × 4.
- kᵣ = 14 N/m.
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