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Queensferry

Module 12 · Lesson 12.1

Decoupling the equations

The transformation that separates a fifty-degree-of-freedom building into fifty independent oscillators — and the exact conditions under which it is allowed.

Why this matters

A fifty-storey building has fifty coupled equations. Solving them directly means factorising a fifty-by-fifty matrix at every time step of a forty-second record — perfectly possible, and it tells you nothing about WHY the building responds as it does.

Modal analysis does something different. It changes coordinates so that the fifty equations become fifty separate single-degree-of-freedom equations, each of which is the problem of Module 3. The building's response is then a sum of contributions you can name, rank and discard.

That interpretability is worth more than the computational saving, and it is what makes the response-spectrum method of Module 14 possible at all.

By the end of this lesson you should be able to

  • Perform the modal transformation and see the equations separate
  • Identify the generalised force and what it means physically
  • State the conditions under which decoupling is valid
  • Recognise a non-classically damped system

What you should already know

  • Orthogonality (Module 11)
  • Assembly of M, C and K (Module 10)
  • The SDOF equation of motion (Module 3)

From first principles

Modal decoupling

We want to show: Transform Mü + Cu̇ + Ku = p(t) into n independent single-degree-of-freedom equations, and identify exactly what is required for that to work.

Any deformed shape the structure can take is a mixture of its mode shapes — that is what completeness means. So instead of tracking where each floor is, track how much of each mode is present. Substituting that description into the equations of motion and using orthogonality, every cross term between different modes vanishes, and what is left is a set of equations each involving only one mode.

What the generalised force means

The right-hand side, φₙᵀp(t), is the generalised force for mode n. It measures how well the applied load pattern MATCHES the mode shape.

A load distributed like mode 1 excites mode 1 strongly and everything else weakly. A load applied at a point where a mode has a node excites that mode not at all, whatever its magnitude.

This is the formal version of an idea that should already be familiar: pushing a swing in time and in the right place works, and pushing it at the pivot does nothing.

For base excitation the load vector is p = −Mι üg, so the generalised force is

φₙᵀp = −(φₙᵀMι) üg = −Lₙ üg

and the quantity Lₙ = φₙᵀMι decides how strongly the ground motion drives each mode. That single number is the subject of the next lesson.

When classical damping fails

Three cases matter in practice, and all three are common.

Base isolation. The isolation layer has 15–30% damping; the superstructure has 2–5%. The damping is wildly non-uniform, so ΦᵀCΦ is far from diagonal.

Supplemental dampers. Dampers installed in a few storeys, or a few bays, concentrate damping where the structure does not have it uniformly.

Soil–structure interaction. Radiation damping into the ground affects the modes that move the foundation and not those that do not.

In all three the correct treatments are direct integration of the coupled equations, or a complex-mode analysis in state space. Running an ordinary modal superposition on them produces an answer that is smooth, plausible and wrong — and the error is not small when the damping contrast is large.

The diagnostic is simple and worth applying: compute ΦᵀCΦ and look at the ratio of the largest off-diagonal term to the largest diagonal term. Below about 1% the classical assumption is safe. Above about 10% it is not.

Worked example

Decoupling a three-storey building by hand

Given

  • The uniform three-storey building of Module 11: m = 400 t per floor, k = 600 MN/m per storey
  • Mode 1: ω₁ = 17.24 rad/s, shape {0.445, 0.802, 1.000} normalised to unit roof
  • 5% damping assumed in every mode
  • A lateral force of 200 kN applied at roof level only

Find

The modal equation for mode 1 and the roof displacement it produces at steady state.

Assumptions

  • Classical damping; the force is applied slowly enough to be quasi-static for this illustration

    Predict first

    A base-isolated building is analysed by modal superposition, using 20% damping for the isolated mode and 3% for the superstructure modes. What is wrong?

    Practice

    A mode has shape {0.4, 0.7, 1.0} on a building with 300 tonnes per floor. A horizontal force of 150 kN is applied at the roof only. What is the generalised force φᵀp, in kN?

    Practice

    A mode has modal mass 500 tonnes and natural frequency 20 rad/s. What is its modal stiffness, in MN/m?

    Practice

    A mode with modal mass 800 tonnes and frequency 15 rad/s is assigned 4% damping. What is its modal damping coefficient Cₙ, in kN·s/m?

    Check yourself

    Which matrix is NOT automatically diagonalised by the mode shapes?

    Summary

    • u = Φq, then pre-multiply by φₙᵀ — orthogonality kills every cross term
    • The result is n independent SDOF equations, each solvable by Stage A methods
    • The generalised force φₙᵀp measures how well the load pattern matches the mode
    • Under base excitation φₙᵀp = −Lₙüg with Lₙ = φₙᵀMι
    • M and K are diagonalised as a consequence; C only if it is classical
    • Base isolation, local dampers and SSI all produce non-classical damping
    • Neither non-classical damping nor yielding produces a warning — check before choosing
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    This is educational material. It uses simplified examples to teach principles, and must not be relied on for real design or safety-critical decisions. Module overview and checkpoint