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Technology / Technical guide
Modal Superposition: Coordinates, Coupling & Truncation
Modal superposition represents a linear structural response through mode shapes and their time-dependent coordinates. It can simplify a model by separating spatial patterns from changing amplitudes. However, the word modal does not automatically make every equation independent, and retaining a few modes does not preserve every local quantity equally well.
A mode shape is a basis pattern, not the total motion.
Mass, stiffness and damping assumptions determine whether those coordinates separate into independent response equations.
01 / Linear model
Superposition starts from a defined equation.
A common model is M ü + C u̇ + K u = F(t), with displacement vector u.
A common model is M ü + C u̇ + K u = F(t), with displacement vector u. Assume constant matrices, symmetric positive-definite M and K, and linear damping C. These conditions describe an illustrative model, rather than every rotating or contacting structure.
Linearity allows responses to be combined according to the same model. Changing stiffness with displacement, nonlinear contact or a changing boundary can break that relationship. A physically complex structure can still have a useful local linear approximation, but the approximation’s boundary remains part of its meaning.
02 / Modes
The undamped eigenproblem relates shape and frequency.
For the assumed model, Kφ[j] = ω[j]²Mφ[j].
For the assumed model, Kφ[j] = ω[j]²Mφ[j]. Each φ is an undamped mode shape and ω an angular natural frequency. Its amplitude normalisation is a mathematical choice; doubling φ and halving its modal coordinate leaves the represented displacement unchanged.
COMSOL explains modal expansion and the orthogonality of suitable undamped modes under mass and stiffness weighting. Ordinary geometric perpendicularity is not the general condition. Within a repeated-frequency subspace, an appropriate basis can be chosen; the individual vector label is not unique physical motion.
03 / Expansion
Coordinates combine spatial patterns into displacement.
Write u(t) = Φq(t), or Σφ[j]q[j](t).
Write u(t) = Φq(t), or Σφ[j]q[j](t). With a complete basis, this is a coordinate change within the finite model. Modal coordinates vary over time, whereas each basis shape describes how its contribution is distributed among the represented displacement components.
As a hypothetical arithmetic expansion, use dimensionless vectors (1, 1) and (1, −1) with coordinates 2 mm and 1 mm. Their sum gives displacements (3, 1) mm. The example illustrates vector addition; it does not claim these are modes of an actual turbine.
04 / Damping
Orthogonal mass and stiffness do not guarantee uncoupled damping.
Multiplying the transformed equation by Φᵀ produces modal mass, stiffness and damping matrices.
Multiplying the transformed equation by Φᵀ produces modal mass, stiffness and damping matrices. The appropriate undamped basis can diagonalise mass and stiffness. Damping ΦᵀCΦ can nevertheless retain off-diagonal terms, linking the modal coordinates.
Independent single-mode equations require a compatible damping relationship, commonly called classical damping. A proportional damping model is one sufficient example, rather than a property established by every material or structure. Adding independent mode responses without this condition can omit represented coupling.
05 / Truncation
A reduced basis changes what the model can reproduce.
Retaining only some shapes gives an approximation u ≈ Φr qr.
Retaining only some shapes gives an approximation u ≈ Φr qr. A force component outside their represented span is not fully described. A low-frequency displacement may be represented adequately while another quantity remains sensitive to omitted contributions.
Local gradients, strain and stress can weight the shapes differently from overall movement. A mode that contributes little to a displacement at one location can matter to a local derivative elsewhere. Mode count alone therefore cannot establish equal accuracy for all requested response quantities.
06 / Scope
The actual structure determines whether the simple basis applies.
Gyroscopic terms, unsymmetric relationships and nonlinearity can require a different modal treatment.
Gyroscopic terms, unsymmetric relationships and nonlinearity can require a different modal treatment. A useful undamped basis is not evidence of measured mode frequencies, damping ratios or allowable vibration. Those quantities depend on the declared physical model and represented boundaries.
No MARSTON modal model, measured shape or vibration performance is assigned. This page gives no excitation or field identification procedure. It explains why superposition, damping decoupling and truncation are separate conditions rather than interchangeable labels.
Related technical guides: Two-Inertia Torsional Vibration in Wind Generators · Vibration Displacement, Velocity & Acceleration · Logarithmic Decrement: Free Decay & Damping Assumptions.
General background: COMSOL: mode superposition, modal damping and truncation. These references explain industry concepts; they are separate from the applicable MARSTON product and project documents.
Key concepts and conditions.
Read each quantity with its reference, units and stated conditions. A general relation does not establish a model-specific value.
| Concept / quantity | Meaning / relation | Conditions / limits |
|---|---|---|
| Displacement u | Physical coordinate vector | Specified model boundaries and units |
| Mode shape φ | Spatial basis pattern | Normalisation is arbitrary within a consistent expansion |
| Modal coordinate q | Time-varying shape contribution | Units follow the chosen normalisation |
| Modal damping | ΦᵀCΦ | Diagonal form is needed for independent modal equations |
| Truncation | Retaining a subset of shapes | Accuracy depends on force and requested quantity |
Engineering questions.
Is a mode shape the total displacement?
No. The total response combines shapes with their modal coordinates.
Are mode shapes always geometrically perpendicular?
No. The relevant orthogonality uses the model’s mass and stiffness relationships.
Does diagonal mass make damping diagonal?
No. Compatible damping is a separate requirement.
Do a few modes preserve every stress result?
Not necessarily; local derivatives can depend on omitted contributions.
Are the invented vectors actual turbine modes?
No. They illustrate a coordinate expansion only.
Does the page provide a modal test procedure?
No. It explains model relationships without field excitation or identification.
Technical enquiry
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Modal superposition represents a linear structural response through mode shapes and their time-dependent coordinates. Share the model and applicable operating or design information for an engineering discussion.