Library / Structural dynamics and vibration testing / SDY-24
Structural dynamics and vibration testing
Participation factor for base translation
How strongly a mode is driven by a uniform translation of the base, from the shape φ, the mass matrix M and an influence vector r that says how each coordinate moves with the base.
- Ref
- SDY-24
- Rev
- v1
- Field
- Structural dynamics and vibration testing
- Result
- Γr
The participation factor Γr = φᵀMr/(φᵀMφ) says how strongly a uniform base translation drives the mode. The vector r is the influence vector: its entries say how far each coordinate moves when the base moves one unit in the chosen direction. In relative coordinates the base-acceleration forcing on the mode has the sign −Γᵣ.
How strongly a mode is driven by a uniform translation of the base, from the shape φ, the mass matrix M and an influence vector r that says how each coordinate moves with the base.
Calculator
- Numerator, φᵀMr4 kg
- Modal mass, φᵀMφ6 kg
- Effective modal mass, (φᵀMr)²/(φᵀMφ)2.667 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 mode has a participation factor of 0.6667 for this base direction.
- Rescaling the shape rescales Γᵣ in the opposite direction, so only the product Γᵣ × shape is meaningful; the effective mass is scale-free.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| r1 Influence vector entry 1 | −2 to 2 | −0.3333 | 1 | 1.333 | +0.08 | +4 % |
| r2 Influence vector entry 2 | −2 to 2 | −0.3333 | 1 | 1.333 | +0.08 | +4 % |
| φ2 Mode shape ordinate 2 | −3 to 3 | −0.09091 | 1.111 | 1.202 | +0.12 | −4.84 % |
| φ1 Mode shape ordinate 1 | −3 to 3 | −0.1829 | 0.6818 | 0.8647 | +0.12 | −2.29 % |
| M22 Mass matrix entry 22 | 0.1 to 10 kg | 0.5238 | 0.9167 | 0.3929 | +10% | −1.56 % |
| M11 Mass matrix entry 11 | 0.1 to 10 kg | 0.5122 | 0.8571 | 0.3449 | +10% | +1.61 % |
| M12 Mass matrix entry 12 | −1 to 1 kg | 0.5 | 0.7 | 0.2 | +0.04 kg | +0.325 % |
Sweep of Mode shape ordinate 1
| Mode shape ordinate 1 | Participation factor |
|---|---|
| −3 | −0.1818 |
| −2.25 | −0.177 |
| −1.5 | −0.1176 |
| −0.75 | 0.09756 |
| 0 | 0.5 |
| 0.75 | 0.6829 |
| 1.5 | 0.5882 |
| 2.25 | 0.4602 |
| 3 | 0.3636 |
Sweep one variable
Variables and how each one contributes
Participation factor
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 −1.87 %, from 0.6667 to 0.6542.
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 −4.06 %, from 0.6667 to 0.6396.
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 +1.61 %, from 0.6667 to 0.6774.
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 +0.781 %, from 0.6667 to 0.6719.
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 −1.56 %, from 0.6667 to 0.6563.
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 +5 %, from 0.6667 to 0.7.
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 +5 %, from 0.6667 to 0.7.
Worked example with the values above
- Numerator: φᵀMr = 4 kg.
- Denominator: φᵀMφ = 6 kg.
- Γᵣ = 0.66667.
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