Library / Structural dynamics and vibration testing / SDY-25
Structural dynamics and vibration testing
Directional effective modal mass
Part of the total mass that a mode moves in a chosen base direction.
- Ref
- SDY-25
- Rev
- v1
- Field
- Structural dynamics and vibration testing
- Result
- me [kg]
The effective modal mass me = (φᵀMr)²/(φᵀMφ) is the part of the total mass that the mode moves in the chosen base direction. It does not change with the scaling of the shape, and the effective masses of all modes add up to the total mass in that direction. A mode with a small effective mass is hardly excited by base motion.
Part of the total mass that a mode moves in a chosen base direction. It does not change if the shape is rescaled, and the effective masses of all modes add up to the total mass.
Calculator
- Participation factor, φᵀMr/φᵀMφ0.6667
- Total mass in the direction, rᵀMr3 kg
- Share of the total mass0.8889
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.
- This mode carries 2.667 kg of the 3 kg total, 88.9 % of it.
- Scaling φ by any non-zero factor leaves this number unchanged.
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 | 2.017 | 11.52 | 9.507 | +10% | +3.36 % |
| M11 Mass matrix entry 11 | 0.1 to 10 kg | 1.076 | 10.29 | 9.21 | +10% | +6.69 % |
| r1 Influence vector entry 1 | −2 to 2 | 0.002963 | 6 | 5.997 | +0.08 | +8.16 % |
| r2 Influence vector entry 2 | −2 to 2 | 0.002963 | 6 | 5.997 | +0.08 | +8.16 % |
| M12 Mass matrix entry 12 | −1 to 1 kg | 0.5 | 4.9 | 4.4 | +0.04 kg | +3.33 % |
| φ1 Mode shape ordinate 1 | −3 to 3 | 0 | 3 | 3 | +0.12 | +3.58 % |
| φ2 Mode shape ordinate 2 | −3 to 3 | 0 | 3 | 3 | +0.12 | −1.99 % |
Sweep of Mode shape ordinate 1
| Mode shape ordinate 1 | Effective modal mass (kg) |
|---|---|
| −3 | 0.7273 |
| −2.25 | 0.4425 |
| −1.5 | 0.1176 |
| −0.75 | 0.04878 |
| 0 | 1 |
| 0.75 | 2.39 |
| 1.5 | 2.941 |
| 2.25 | 2.991 |
| 3 | 2.909 |
Sweep one variable
Variables and how each one contributes
Effective 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 +3.04 %, from 2.667 to 2.748 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 −1.66 %, from 2.667 to 2.622 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 +6.69 %, from 2.667 to 2.845 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 +8.34 %, from 2.667 to 2.889 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 +3.36 %, from 2.667 to 2.756 kg.
Influence vector entry 1
The influence vector entry at coordinate 1 (dimensionless): how far this coordinate moves per unit base movement. Raises the participation in proportion. Raising it by 0.1 (to 1.1) moves the result by +10.3 %, from 2.667 to 2.94 kg.
Influence vector entry 2
The influence vector entry at coordinate 2 (dimensionless). Raises the participation in proportion. Raising it by 0.1 (to 1.1) moves the result by +10.3 %, from 2.667 to 2.94 kg.
Worked example with the values above
- Numerator: (φᵀMr)² = 4² = 16.
- Denominator: φᵀMφ = 6.
- mₑ = 2.6667 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