Library / Structural dynamics and vibration testing / SDY-21
Structural dynamics and vibration testing
Generalised modal mass
Modal (generalised) mass of a shape φ for a two-coordinate model: the shape multiplied through the mass matrix M on both sides.
- Ref
- SDY-21
- Rev
- v1
- Field
- Structural dynamics and vibration testing
- Result
- mr [kg]
The modal mass mr = φᵀMφ is the mass that the mode behaves as when it is described by a single coordinate. It depends on how the shape is scaled: doubling the shape quadruples the modal mass. In a mass-normalised mode it equals one. Here M is a symmetric two-coordinate mass matrix.
Modal (generalised) mass of a shape φ for a two-coordinate model: the shape multiplied through the mass matrix M on both sides. Its value depends on how the shape is scaled.
Calculator
- Mass-normalising scale factor, 1/√mᵣ0.4082
- Sum of the diagonal masses3 kg
- Modal mass if the shape is doubled24 kg
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 mass of 6 kg for these masses.
- Dividing the shape by √6 would make the modal mass one, the usual mass normalisation.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (kg) | Local step | Change in result |
|---|---|---|---|---|---|---|
| M22 Mass matrix entry 22 | 0.1 to 10 kg | 2.4 | 42 | 39.6 | +10% | +6.67 % |
| φ1 Mode shape ordinate 1 | −3 to 3 | 4 | 22 | 18 | +0.12 | +8.48 % |
| M11 Mass matrix entry 11 | 0.1 to 10 kg | 4.1 | 14 | 9.9 | +10% | +3.33 % |
| φ2 Mode shape ordinate 2 | −3 to 3 | 2 | 11 | 9 | +0.12 | +8.24 % |
| M12 Mass matrix entry 12 | −1 to 1 kg | 2 | 10 | 8 | +0.04 kg | +2.67 % |
Sweep of Mode shape ordinate 1
| Mode shape ordinate 1 | Modal mass (kg) |
|---|---|
| −3 | 22 |
| −2.25 | 14.13 |
| −1.5 | 8.5 |
| −0.75 | 5.125 |
| 0 | 4 |
| 0.75 | 5.125 |
| 1.5 | 8.5 |
| 2.25 | 14.13 |
| 3 | 22 |
Sweep one variable
Variables and how each one contributes
Modal mass
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 +7 %, from 6 to 6.42 kg.
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 +6.83 %, from 6 to 6.41 kg.
Mass matrix entry 11
The mass matrix entry for coordinate 1, in kg. More mass at coordinate 1 raises the modal mass. Raising it by 10 % (to 2.2) moves the result by +3.33 %, from 6 to 6.2 kg.
Mass matrix entry 12
The off-diagonal mass matrix entry, in kg: zero for a lumped mass model. Couples the two coordinates. Raising it by 0.1 (to 0.1) moves the result by +6.67 %, from 6 to 6.4 kg.
Mass matrix entry 22
The mass matrix entry for coordinate 2, in kg. More mass at coordinate 2 raises the modal mass. Raising it by 10 % (to 1.1) moves the result by +6.67 %, from 6 to 6.4 kg.
Worked example with the values above
- Expand: φ₁²M₁₁ + 2φ₁φ₂M₁₂ + φ₂²M₂₂.
- Substitute: 1 × 2 + 4 × 0 + 4 × 1.
- mᵣ = 6 kg.
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