Library / Structural dynamics and vibration testing / SDY-27
Structural dynamics and vibration testing
Modal coordinate from an initial displacement
Starting value of a modal coordinate from a physical initial displacement, using the mass-weighted projection onto the shape.
- Ref
- SDY-27
- Rev
- v1
- Field
- Structural dynamics and vibration testing
- Result
- qr(0) [mm]
The initial modal coordinate qr(0) = φᵀMu₀/(φᵀMφ) projects a physical initial displacement onto the shape using the mass matrix as a weight. For mass-orthogonal modes it is exact. For a single assumed shape it is the mass-weighted least-squares fit. Output is in mm for displacements in mm.
Starting value of a modal coordinate from a physical initial displacement, using the mass-weighted projection onto the shape. For mass-orthogonal modes it is exact.
Calculator
- Numerator, φᵀMu₀90 kg·mm
- Modal mass, φᵀMφ6 kg
- Displacement the mode accounts for at coordinate 230 mm
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 starting shape of 30 mm and 15 mm corresponds to a modal coordinate of 15 mm.
- If the starting shape is a pure multiple of φ, the projection recovers that multiple exactly, here 30 mm.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (mm) | Local step | Change in result |
|---|---|---|---|---|---|---|
| u1 Initial displacement 1 | −100 to 100 mm | −28.33 | 38.33 | 66.67 | +4 mm | +8.89 % |
| u2 Initial displacement 2 | −100 to 100 mm | −23.33 | 43.33 | 66.67 | +4 mm | +8.89 % |
| φ2 Mode shape ordinate 2 | −3 to 3 | 1.364 | 30.88 | 29.52 | +0.12 | −5.76 % |
| φ1 Mode shape ordinate 1 | −3 to 3 | −7.5 | 15 | 22.5 | +0.12 | −0.442 % |
| M22 Mass matrix entry 22 | 0.1 to 10 kg | 8.571 | 26.25 | 17.68 | +10% | −3.13 % |
| M11 Mass matrix entry 11 | 0.1 to 10 kg | 8.049 | 23.57 | 15.52 | +10% | +3.23 % |
| M12 Mass matrix entry 12 | −1 to 1 kg | 7.5 | 16.5 | 9 | +0.04 kg | +0.649 % |
Sweep of Initial displacement 1
| Initial displacement 1 (mm) | Initial modal coordinate (mm) |
|---|---|
| −100 | −28.33 |
| −75 | −20 |
| −50 | −11.67 |
| −25 | −3.333 |
| 0 | 5 |
| 25 | 13.33 |
| 50 | 21.67 |
| 75 | 30 |
| 100 | 38.33 |
Sweep one variable
Variables and how each one contributes
Initial modal coordinate
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 −0.312 %, from 15 to 14.95 mm.
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 −4.84 %, from 15 to 14.27 mm.
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 +3.23 %, from 15 to 15.48 mm.
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 +1.56 %, from 15 to 15.23 mm.
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 −3.13 %, from 15 to 14.53 mm.
Initial displacement 1
The initial displacement at coordinate 1, in mm. Raises the modal coordinate in proportion. Raising it by 10 % (to 33) moves the result by +6.67 %, from 15 to 16 mm.
Initial displacement 2
The initial displacement at coordinate 2, in mm. Raises the modal coordinate in proportion. Raising it by 10 % (to 16.5) moves the result by +3.33 %, from 15 to 15.5 mm.
Worked example with the values above
- Numerator: φᵀMu₀ = 90 kg·mm.
- Denominator: φᵀMφ = 6 kg.
- q(0) = 15 mm.
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