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Structural Dynamics / Supplemental damping devices

Sizing and placing dampers

Where a damper earns its keep, how much damping it actually adds, and what analysis a damped structure needs.

Lesson review: expert verification remains open

This lesson has received a full-context editorial review and a check of its technical explanations. Selected arithmetic has been checked separately. This is not independent engineering sign-off.

Checks needed for project use

  • A dynamics reviewer must confirm excitation models, damping, modal assumptions and any comfort or serviceability interpretation. Small displacement alone is not an acceptance criterion.
  • Independent validation of the associated interactive implementation, input domain and project applicability remains required; this review checks lesson context and selected calculations, not design certification.
  • Damper specialist review needed for force–velocity test data, exponent-dependent geometry, brace compliance, stroke, temperature, tolerances, non-classical coupling and combined member/foundation actions.

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 notes

Why this matters

A damper only works if the two ends of it move relative to one another. That obvious statement decides almost everything about where dampers go, and it is the single point most often got wrong in a retrofit.

Dampers are frequently placed where there is room for them — a plant floor, a stair core, a storey with a convenient bay. If that storey happens to have little inter-storey drift, the damper is an expensive ornament.

This lesson is about placing them where the structure is actually moving, quantifying what they add, and knowing what analysis is then required.

By the end of this lesson you should be able to

  • Compute the damping ratio a linear damper adds to a mode
  • Use modal drift to decide where dampers go
  • Explain why added damping is mode-specific
  • State the analysis a damped structure needs and why

What you should already know

  • Damper force and energy per cycle (Lesson 18.1)
  • Mode shapes and modal mass (Module 11)
  • Classical and non-classical damping (Module 12)

Added modal damping for linear devices

The diagonal contribution associated with mode n is

ζadded = Σj Cj cos²θj φ²_rel,j /(2ωn Mn).

Here φrel,j is the horizontal modal displacement difference across the storey and Mn = φnᵀMφ_n uses the same normalisation. The cos² term already projects horizontal motion onto the damper axis. Do not apply that projection again if φrel is measured along the axis.

This is an exact uncoupled modal ratio only when the full modal damping matrix is diagonal. Otherwise it is a useful layout indicator, with coupling requiring separate analysis.

Placement

For fixed mode properties, doubling C doubles the contribution; doubling relative modal drift multiplies it by four. Check all modes relevant to displacement, acceleration and torsion, alongside space, stroke, brace flexibility and force paths. A large roof displacement does not by itself imply large drift across a storey damper.

Stiffness and forces

An ideal viscous element adds no elastic stiffness. Its supporting assembly and any real device stiffness can still affect the system. Bracing changes period, but the direction of the change in spectral demand depends on the spectrum. Neither intervention guarantees lower foundation demand.

Analysis and targets

Examine the modal damping matrix with a consistent normalisation. Retain significant coupling through direct integration or a suitable complex-mode method. Nonlinear devices require an appropriate nonlinear method or a validated equivalent-linear approximation.

For an SDOF oscillator under fixed-amplitude harmonic force at its unchanged undamped natural frequency, response is proportional to 1/ζ. Raising damping from 2% to 10% gives a fivefold reduction in that comparison. This is not a general earthquake-response reduction or a coupled TMD prediction. Establish targets through response and device-demand checks.

Worked example

Placing dampers by modal drift

Given

  • A five-storey shear building, first mode ω₁ = 4.19 rad/s, modal mass M₁ = 900 tonnes
  • First-mode shape, normalised to unity at roof: 0.20, 0.44, 0.66, 0.85, 1.00
  • Linear dampers available with C = 2.0 MN·s/m, braced at 30°
  • Target: at least 5% added damping in the first mode

Find

Which storeys to damp, and the damping added.

Assumptions

  • Linear devices; modal mass uses the roof-unity normalisation. Diagonal first-mode contributions are preliminary layout indicators; coupling and other modes remain to be checked.

    Predict first

    Keeping a linear damper, brace angle and modal normalisation unchanged, move it from horizontal modal drift 0.24 to 0.12. What happens to its diagonal modal damping contribution?

    Practice

    A linear damper with C = 2.0 MN·s/m braced at 30° spans a storey with first-mode relative displacement 0.35. The first mode has ω₁ = 4.19 rad/s and modal mass 900 tonnes. What damping ratio does it add?

    Practice

    For equal linear coefficients and identical horizontal storey motion, what fraction of a horizontal damper’s horizontal coefficient is retained by a rigid diagonal at 60°?

    Practice

    A structure has 2% inherent damping. Six dampers add 5.81%. By what factor does the resonant response fall, taking response at resonance as proportional to 1/ζ?

    Check yourself

    Why might viscous dampers be preferred to added bracing in the seismic retrofit of an existing building?

    Check yourself

    Two dampers are quoted with numerically equal C in their respective N·(s/m)^α units but different exponents. How should their performance be compared?

    Worked example

    Where to put a damper in a three-storey frame

    Given

    • A three-storey frame, first mode shape roughly 0.33, 0.67, 1.00 from bottom to top
    • One damper is available, and it acts on the drift of the storey it is in

    Find

    Which storey it should go in

      Summary

      • ζadded = Σ C cos²θ φ²_rel / (2ωM), and the squared drift term dominates
      • Place dampers by modal drift, not by mode-shape ordinate or by available space
      • Added damping is mode-specific; check the modes that matter, not just the first
      • Ideal viscous dampers add no elastic stiffness; assess their force paths and combined foundation demands
      • Concentrated dampers give non-classical damping — test it, do not assume it
      • Nonlinear dampers require nonlinear time-history analysis
      • The 1/ζ comparison applies to fixed-force SDOF resonance; establish project targets by analysis