Library / Structural dynamics and vibration testing / SDY-28
Structural dynamics and vibration testing
Physical motion reconstructed from modal coordinates
Displacement at a sensor point as the sum, over the retained modes, of the mode shape ordinate at that point times the modal coordinate.
- Ref
- SDY-28
- Rev
- v1
- Field
- Structural dynamics and vibration testing
- Result
- uj [mm]
The displacement at a sensor point j is the sum of the contributions of the retained modes: uj = Σφjrqr. Each term is the mode shape ordinate at the sensor times that mode’s coordinate. Signs and normalisation must match those used to find the modal coordinates. Modes that are not retained are lost.
Displacement at a sensor point as the sum, over the retained modes, of the mode shape ordinate at that point times the modal coordinate.
Calculator
- Contribution of mode 13 mm
- Contribution of mode 2−1 mm
- Contribution of mode 31 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 sensor reads 3 mm from three modes; the largest single contribution is 3 mm in size.
- Terms of opposite sign partly cancel, so a sensor can read almost nothing even when the modes are strongly excited.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (mm) | Local step | Change in result |
|---|---|---|---|---|---|---|
| φj3 Sensor ordinate of mode 3 | −2 to 2 | −8 | 12 | 20 | +0.08 | +13.3 % |
| q1 Modal coordinate 1 | −10 to 10 mm | −10 | 10 | 20 | +0.4 mm | +13.3 % |
| φj1 Sensor ordinate of mode 1 | −2 to 2 | −6 | 6 | 12 | +0.08 | +8 % |
| q2 Modal coordinate 2 | −10 to 10 mm | −1 | 9 | 10 | +0.4 mm | −6.67 % |
| φj2 Sensor ordinate of mode 2 | −2 to 2 | 0 | 8 | 8 | +0.08 | +5.33 % |
| q3 Modal coordinate 3 | −10 to 10 mm | 0 | 4 | 4 | +0.4 mm | +2.67 % |
Sweep of Modal coordinate 1
| Modal coordinate 1 (mm) | Displacement at the sensor (mm) |
|---|---|
| −10 | −10 |
| −7.5 | −7.5 |
| −5 | −5 |
| −2.5 | −2.5 |
| 0 | 0 |
| 2.5 | 2.5 |
| 5 | 5 |
| 7.5 | 7.5 |
| 10 | 10 |
Sweep one variable
Variables and how each one contributes
Displacement at the sensor
What the equation gives you. Everything else feeds it.
Sensor ordinate of mode 1
The mode shape ordinate at the sensor for mode 1 (dimensionless). Scales the contribution of mode 1. Raising it by 0.1 (to 1.1) moves the result by +10 %, from 3 to 3.3 mm.
Sensor ordinate of mode 2
The mode shape ordinate at the sensor for mode 2 (dimensionless). Scales the contribution of mode 2. Raising it by 0.1 (to −0.4) moves the result by +6.67 %, from 3 to 3.2 mm.
Sensor ordinate of mode 3
The mode shape ordinate at the sensor for mode 3 (dimensionless). Scales the contribution of mode 3. Raising it by 0.1 (to 0.3) moves the result by +16.7 %, from 3 to 3.5 mm.
Modal coordinate 1
The modal coordinate of mode 1, in mm. Contributes in proportion to its ordinate. Raising it by 10 % (to 3.3) moves the result by +10 %, from 3 to 3.3 mm.
Modal coordinate 2
The modal coordinate of mode 2, in mm. Contributes in proportion to its ordinate. Raising it by 10 % (to 2.2) moves the result by −3.33 %, from 3 to 2.9 mm.
Modal coordinate 3
The modal coordinate of mode 3, in mm. Contributes in proportion to its ordinate. Raising it by 10 % (to 5.5) moves the result by +3.33 %, from 3 to 3.1 mm.
Worked example with the values above
- Mode 1: 1 × 3 = 3 mm.
- Modes 2 and 3: −1 mm and 1 mm.
- uⱼ = 3 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