Structural Dynamics / module 12
Modal analysis and modal superposition
Turning one coupled problem into many independent ones — what participation and effective mass actually measure, what truncation costs, and the point at which the whole method stops being valid.
What you will understand
- Derive the modal decoupling of the equations of motion
- Distinguish the participation factor from the effective modal mass
- Explain the 90% effective-mass criterion and what it does and does not catch
- Derive the Rayleigh damping relationship and read what it delivers
- Say what classical damping means and recognise when it fails
- Judge how many modes an analysis needs, and on what evidence
Lessons
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.
Participation, effective mass and truncation
Two quantities that are constantly confused, one criterion that is widely misapplied, and an honest account of what stopping at n modes actually costs.
Module summary
- u = Φq turns n coupled equations into n independent SDOF equations
- Decoupling works because of orthogonality, and for no other reason
- Γₙ = φᵀMι/φᵀMφ depends on scaling; effective mass Lₙ²/Mₙ does not
- Effective modal masses sum to the total mass — every kilogram belongs to some mode
- The 90% rule catches too few modes; it does NOT catch a wrong mass source
- Displacement converges with far fewer modes than acceleration does
- Rayleigh damping is exact at two frequencies and wrong at every other
- Modal superposition requires ΦᵀCΦ diagonal — base isolation and local dampers break it
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 mode has L = 2 400 tonnes and modal mass 2 000 tonnes, on a building of total mass 3 000 tonnes. What percentage of the total mass does this mode mobilise?
Practice
A mode with modal mass 600 tonnes and frequency 25 rad/s is given 3% damping. What is its modal damping coefficient, in kN·s/m?
Practice
Rayleigh damping is anchored at 4% on ω = 5 rad/s and ω = 30 rad/s. What damping does it deliver at ω = 60 rad/s?
Check yourself
Why do design codes accumulate effective modal mass rather than participation factors?
Check yourself
Which situation makes modal superposition invalid rather than merely inaccurate?