Sizing and placing dampers

The added damping ratio

For linear dampers, the damping ratio added to mode n is

ζ_added = Σ_j C_j cos²θ_j φ²_rel,j / (2 ω_n M_n)

where φ_rel,j is the relative modal displacement ACROSS damper j — the difference in the mode shape between its two ends — and M_n is the modal mass.

Every term in it is worth reading as a design instruction.

C_j, linear. Doubling the damper doubles its contribution. Predictable and unexciting.

cos²θ_j. Brace geometry, from the previous lesson. A damper at 60° retains 25% of the effectiveness of a horizontal one — and 60° is not an unusual angle in a tall narrow bay.

φ²_rel,j, SQUARED. This is the dominant term and the one that decides placement. A damper in a storey with twice the modal drift contributes four times as much.

1/ω_n. Higher modes receive less added damping from the same damper — the opposite of what many people expect, and it follows directly from the modal mass and frequency in the denominator.

Mode-specific. There is no single 'added damping' for the structure. Each mode gets its own, and a damper layout tuned to the first mode may do very little for the second.

The squared modal-drift term is why damper placement is a modal question, not an architectural one. Plot the mode shape, look at the drift in each storey, and put the dampers where the drift is largest.

Placement in practice

Compute the modal drift in every storey for the modes that matter, and rank them. A first-mode-dominated shear building has its largest drift in the lower storeys, so that is usually where the dampers go. A building with a soft storey has almost all of its drift there, and a damper in that storey does an enormous amount.

Two qualifications that stop this being purely mechanical:

Do not tune only to the first mode. If the second mode matters — for floor accelerations, for a tall building, for torsion — check the added damping in that mode too. A layout that is excellent for mode 1 can be nearly useless for mode 2 if the dampers sit near mode 2's node.

Damper force has to go somewhere. A damper delivers a large force into the beam and column it connects to. Those members and their connections need checking for it, and a retrofit that concentrates all the dampers in one storey concentrates all that force there too.

Dampers add damping without adding stiffness

This is the property that distinguishes damping from bracing as a retrofit strategy.

Add a brace, and the storey stiffens. The period shortens, the structure moves UP the acceleration branch of the spectrum, and it attracts more force. It also attracts more force to the foundation, which on an existing building is frequently the binding constraint.

Add a viscous damper, and the period is essentially unchanged. The structure attracts the same spectral acceleration and responds less because it is dissipating energy. Nothing new is demanded of the foundation.

For a retrofit where the foundations cannot be strengthened economically, that difference decides the strategy.

What analysis is required

Linear dampers, and classical damping. If the damper layout happens to produce a damping matrix that the modes diagonalise, modal superposition survives and a spectrum analysis with modified modal damping is defensible. This is rare, and it must be TESTED rather than assumed — the orthogonality check from Module 12 answers it directly.

Linear dampers, non-classical damping. The usual case with concentrated dampers. The modes no longer decouple. Either use complex modal analysis, or run a direct integration.

Nonlinear dampers. No modal damping ratio exists at all. A nonlinear time-history analysis is required, and it is not optional.

The commonest error in damped-structure analysis is to compute an added damping ratio, put it into a response-spectrum analysis, and report the result. For concentrated dampers the damping is non-classical, the spectrum method's central assumption is violated, and the answer is not conservative in a knowable direction.

How much damping is worth adding

Response falls roughly as 1/ζ at resonance, so the returns diminish sharply.

Going from 2% to 10% cuts the resonant response by a factor of five. Going from 10% to 20% cuts it by a further two. Going from 20% to 30% gains a third. Beyond about 30% the dampers are large, the forces they impose on the frame are large, and the additional benefit is small.

Most supplemental damping schemes target 15% to 25% total, and that is a considered engineering target rather than a limit of the technology.