Library / Structural dynamics and vibration testing / SDY-26
Structural dynamics and vibration testing
Retained directional mass fraction
Share of the total mass in a direction that the retained modes account for: the sum of their effective masses divided by the total directional mass rᵀMr.
- Ref
- SDY-26
- Rev
- v1
- Field
- Structural dynamics and vibration testing
- Result
- η
The retained mass fraction η = Σmₑ/(rᵀMr) is the share of the total mass in a direction that the retained modes account for. A common rule asks for 90 % or more in every direction, so that the discarded modes contribute little to the base-excited response. Here two retained modes and the total directional mass rᵀMr are entered directly.
Share of the total mass in a direction that the retained modes account for: the sum of their effective masses divided by the total directional mass rᵀMr. It should be close to one.
Calculator
- Sum of the retained effective masses2.4 kg
- Missing mass0.6 kg
- Missing mass fraction, 1 − η0.2
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 modes hold 80 % of the mass in this direction, below the usual 90 % target, so more modes should be kept.
- The 20 % that is missing is carried by the discarded modes, and can be added back with a residual static correction.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| rTMr Total directional mass | 1 to 10 kg | 0.24 | 2.4 | 2.16 | +10% | −9.09 % |
| me,1 Effective mass of retained mode 1 | 0 to 5 kg | 0.3 | 1.967 | 1.667 | +10% | +6.25 % |
| me,2 Effective mass of retained mode 2 | 0 to 5 kg | 0.5 | 2.167 | 1.667 | +10% | +3.75 % |
Sweep of Total directional mass
| Total directional mass (kg) | Retained mass fraction |
|---|---|
| 1 | 2.4 |
| 2.125 | 1.129 |
| 3.25 | 0.7385 |
| 4.375 | 0.5486 |
| 5.5 | 0.4364 |
| 6.625 | 0.3623 |
| 7.75 | 0.3097 |
| 8.875 | 0.2704 |
| 10 | 0.24 |
Sweep one variable
Variables and how each one contributes
Retained mass fraction
What the equation gives you. Everything else feeds it.
Effective mass of retained mode 1
The effective mass of retained mode 1, in kg, in the chosen direction. Adds directly to the retained mass. Raising it by 10 % (to 1.65) moves the result by +6.25 %, from 0.8 to 0.85.
Effective mass of retained mode 2
The effective mass of retained mode 2, in kg, in the chosen direction. Adds directly to the retained mass. Raising it by 10 % (to 0.99) moves the result by +3.75 %, from 0.8 to 0.83.
Total directional mass
The total mass in the chosen direction, rᵀMr, in kg. A larger total lowers the retained fraction. Raising it by 10 % (to 3.3) moves the result by −9.09 %, from 0.8 to 0.7273.
Worked example with the values above
- Retained mass: 1.5 + 0.9 = 2.4 kg.
- Total directional mass: rᵀMr = 3 kg.
- η = 2.4 / 3 = 0.8.
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