Structural dynamics / tutorial
Fitting Rayleigh damping, and checking it
Anchor the damping at two frequencies, then find out what every other mode received.
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 notesThe three-storey building from the previous tutorial is to be run by direct integration, which needs a damping MATRIX rather than modal ratios. Rayleigh damping is to be fitted to give 5% at the first and third modes. Fit it — then check what the second mode actually got.
- ω₁ = 19.903 rad/s, ω₂ = 55.75 rad/s, ω₃ = 80.585 rad/s
- Target damping 5% at modes 1 and 3
- C = a₀M + a₁K
Practice
What is the mass-proportional coefficient a₀?
Practice
What is the stiffness-proportional coefficient a₁?
Practice
What damping ratio does the second mode actually receive, as a percentage?
Practice
By what percentage is the second mode under-damped relative to the 5% target?
Rayleigh damping is exact at two frequencies and wrong everywhere else, and 'wrong' has a direction that depends on where the mode sits. Fit it, then tabulate the ratio every retained mode actually received. That table takes a minute and is the difference between a damping model and a damping assumption.