Structural Dynamics / module 10
Multi-degree-of-freedom systems
Assembling the matrices of a real model — lumped against consistent mass, rotational inertia, rigid diaphragms — and the entries that get left at zero without anyone noticing.
What you will understand
- Assemble mass and stiffness matrices for a multi-storey shear building
- Explain the difference between lumped and consistent mass and when each is right
- Derive the consistent mass matrix of an axial element from its shape functions
- Compute the mass moment of inertia of a floor diaphragm and say why it matters
- Explain how mass eccentricity couples translation and rotation
- Recognise the modelling faults that leave a matrix entry silently at zero
Lessons
Assembling the matrices
The two-storey pattern, extended — and the reason every entry of a shear-building stiffness matrix can still be checked by inspection at fifty storeys.
Lumped and consistent mass
Two ways to describe where the mass is, both approximations, missing the exact answer from opposite sides — which makes the gap between them a mesh check that needs no exact solution.
Diaphragms, rotational inertia and eccentricity
A floor has three degrees of freedom, not two — and the third one is the entry most often left at zero, which does not make the torsional mode wrong but removes it.
Module summary
- The shear-building pattern extends unchanged to any number of storeys — tridiagonal K
- Lumped mass gives a diagonal M; consistent mass does not, and couples the element ends
- The two bracket the exact frequency from opposite sides, so their gap measures the mesh
- A rigid floor has THREE degrees of freedom: two translations and a rotation
- I₀ = m(a² + b²)/12 for a rectangular floor about its centre of mass
- Omitting I₀ does not make the torsional mode wrong — it deletes it
- Mass eccentricity puts −me terms in M and couples translation with rotation
- A model with N rigid-diaphragm levels has exactly 3N modes — a free check
Teaching basis and review status
These are educational models and exercises. Check their assumptions before using the results. The full technical content and every diagram have not yet been independently verified.
Concrete and steel calculations retain their stated first-generation Eurocode teaching basis. Use the standards and amendments specified for your project; do not mix generations without an appropriate design basis.
Read the lesson register and outstanding review notesCheck your understanding
Practice
A six-storey shear building has all storey stiffnesses of 550 MN/m. What is the sum of the diagonal entries of its stiffness matrix, in MN/m?
Practice
A bar element has ρAL = 1 200 kg. What is the off-diagonal entry of its consistent mass matrix, in kg?
Practice
A square floor is 20 m by 20 m with a mass of 600 tonnes. What is its mass moment of inertia, in kg·m²?
Check yourself
A modal analysis of an eight-storey rigid-diaphragm model returns 26 modes. What does this indicate?
Check yourself
Why do lumped and consistent mass models bracket the exact natural frequency?