Library / Structural dynamics and vibration testing / SDY-37
Structural dynamics and vibration testing
Least-squares modal scale factor
Factor that, multiplied into shape a, best fits shape b in the least-squares sense.
- Ref
- SDY-37
- Rev
- v1
- Field
- Structural dynamics and vibration testing
- Result
- s
The least-squares scale factor s = aᴴb/(aᴴa) is the number that, multiplied into shape a, gives the best fit to shape b over the common coordinates. It is the projection of b on a. It can be complex for complex modes. Use it to scale a computed mode to a measured one before comparing them point by point. This entry uses three real coordinates.
Factor that, multiplied into shape a, best fits shape b in the least-squares sense. It is the projection of b onto a. This entry uses three real coordinates.
Calculator
- Dot product a·b4
- Length squared of a, a·a5
- Residual length after scaling, |b − s a|1.342
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.
- Multiplying shape a by 0.8 gives the closest match to shape b in the least-squares sense.
- The residual after scaling shows how much of b is not a multiple of a: here 1.34.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| b2 Shape b, coordinate 2 | −3 to 3 | −0.8 | 1.6 | 2.4 | +0.12 | +6 % |
| a2 Shape a, coordinate 2 | −3 to 3 | −0.1 | 2.115 | 2.215 | +0.12 | −6.27 % |
| b1 Shape b, coordinate 1 | −3 to 3 | −0.2 | 1 | 1.2 | +0.12 | +3 % |
| a1 Shape a, coordinate 1 | −3 to 3 | −0.3077 | 0.8088 | 1.117 | +0.12 | +0.868 % |
| a3 Shape a, coordinate 3 | −3 to 3 | 0.2857 | 0.8 | 0.5143 | +0.12 | −0.287 % |
| b3 Shape b, coordinate 3 | −3 to 3 | 0.8 | 0.8 | 0 | +0.12 | 0 % |
Sweep of Shape b, coordinate 1
| Shape b, coordinate 1 | Scale factor |
|---|---|
| −3 | −0.2 |
| −2.25 | −0.05 |
| −1.5 | 0.1 |
| −0.75 | 0.25 |
| 0 | 0.4 |
| 0.75 | 0.55 |
| 1.5 | 0.7 |
| 2.25 | 0.85 |
| 3 | 1 |
Sweep one variable
Variables and how each one contributes
Scale factor
What the equation gives you. Everything else feeds it.
Shape a, coordinate 1
The ordinate of shape a at coordinate 1 (dimensionless). Changes how well shape a lines up with shape b. Raising it by 0.1 (to 1.1) moves the result by +0.768 %, from 0.8 to 0.8061.
Shape a, coordinate 2
The ordinate of shape a at coordinate 2 (dimensionless). Changes how well shape a lines up with shape b. Raising it by 0.1 (to 2.1) moves the result by −5.27 %, from 0.8 to 0.7579.
Shape a, coordinate 3
The ordinate of shape a at coordinate 3 (dimensionless). Zero if only two coordinates are used. Changes how well shape a lines up with shape b. Raising it by 0.1 (to 0.1) moves the result by −0.2 %, from 0.8 to 0.7984.
Shape b, coordinate 1
The ordinate of shape b at coordinate 1 (dimensionless). Changes how well shape b lines up with shape a. Raising it by 0.1 (to 2.1) moves the result by +2.5 %, from 0.8 to 0.82.
Shape b, coordinate 2
The ordinate of shape b at coordinate 2 (dimensionless). Changes how well shape b lines up with shape a. Raising it by 0.1 (to 1.1) moves the result by +5 %, from 0.8 to 0.84.
Shape b, coordinate 3
The ordinate of shape b at coordinate 3 (dimensionless). Zero if only two coordinates are used. Changes how well shape b lines up with shape a. Raising it by 0.1 (to 0.1) moves the result by +0 %, from 0.8 to 0.8.
Worked example with the values above
- Dot product: a·b = 4.
- Length squared of a: a·a = 5.
- s = 4 / 5 = 0.8.
Assumptions and limits
- Both shapes are measured or computed at the same coordinates and directions, in the same order.
- Real shapes are used here; for complex modes use the conjugate transpose.
Common mistakes
- Comparing shapes that do not share the same coordinates or sign convention.
- Trusting a high MAC alone when two modes have close frequencies, since closely spaced modes can have similar shapes.
- Using too few coordinates, so that different shapes look alike.
Statistics and uncertainty
A correlation measure from a handful of noisy coordinates is itself uncertain. Measurement noise and too few sensors both lower the apparent agreement, so a threshold such as 0.9 should be judged against the test quality.
Where it is used
- Matching test modes to finite-element modes and tracking modes between test runs.
- Scaling a computed shape to a measured one before updating a model.
Maths you need
- Dot product: the sum of products of two vectors
- Vector norm: the length of a vector
- Least squares: the best fit in the sum-of-squares sense