Library / Structural dynamics and vibration testing / SDY-23
Structural dynamics and vibration testing
Generalised force from nodal loads
Force felt by a mode: the dot product of the mode shape with the simultaneous nodal forces.
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- SDY-23
- Rev
- v1
- Field
- Structural dynamics and vibration testing
- Result
- Qr [N]
The generalised force Qr = φᵀF is the dot product of the mode shape with the nodal forces. It measures how well a load pattern matches the shape. Loads that push opposite ways on a mode can cancel, and a load that is orthogonal to a mode gives zero generalised force and cannot excite it.
Force felt by a mode: the dot product of the mode shape with the simultaneous nodal forces. Loads that act in opposite senses on the shape cancel in the projection.
Calculator
- Contribution of coordinate 13 N
- Contribution of coordinate 210 N
- Sum of the nodal forces8 N
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 nodal loads 3 N and 5 N drive this mode with a generalised force of 13 N.
- Changing the sign of one load changes the result to −7 N, so the direction of each load matters as much as its size.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (N) | Local step | Change in result |
|---|---|---|---|---|---|---|
| F2 Nodal force 2 | −20 to 20 N | −37 | 43 | 80 | +0.8 N | +12.3 % |
| F1 Nodal force 1 | −20 to 20 N | −10 | 30 | 40 | +0.8 N | +6.15 % |
| φ2 Mode shape ordinate 2 | −3 to 3 | −12 | 18 | 30 | +0.12 | +4.62 % |
| φ1 Mode shape ordinate 1 | −3 to 3 | 1 | 19 | 18 | +0.12 | +2.77 % |
Sweep of Nodal force 1
| Nodal force 1 (N) | Generalised force (N) |
|---|---|
| −20 | −10 |
| −15 | −5 |
| −10 | 0 |
| −5 | 5 |
| 0 | 10 |
| 5 | 15 |
| 10 | 20 |
| 15 | 25 |
| 20 | 30 |
Sweep one variable
Variables and how each one contributes
Generalised force
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 +2.31 %, from 13 to 13.3 N.
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 +3.85 %, from 13 to 13.5 N.
Nodal force 1
The force at coordinate 1, in N, signed. Contributes in proportion to the shape ordinate at 1. Raising it by 10 % (to 3.3) moves the result by +2.31 %, from 13 to 13.3 N.
Nodal force 2
The force at coordinate 2, in N, signed. Contributes in proportion to the shape ordinate at 2. Raising it by 10 % (to 5.5) moves the result by +7.69 %, from 13 to 14 N.
Worked example with the values above
- Coordinate 1: φ₁F₁ = 1 × 3 = 3.
- Coordinate 2: φ₂F₂ = 2 × 5 = 10.
- Qᵣ = 13 N.
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