Structural Dynamics / Supplemental damping devices
How a viscous damper works
Velocity-dependent force, brace geometry and how the exponent affects device and connection demands.
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 notesWhy this matters
Damping has appeared in every module as a number, ζ, taken from a table. For an ordinary building 2% to 5% is a reasonable estimate of what friction in connections, cracking and non-structural elements happen to provide. It is not designed; it is observed.
A supplemental damper changes that. It is a manufactured component with a stated force law, placed where the engineer chooses, and it can raise the effective damping to 20% or 30%. Damping stops being a property you estimate and becomes one you specify.
The consequences are more interesting than simply 'less response', and they turn on one fact: a viscous damper's force depends on velocity, and velocity is ninety degrees out of phase with displacement.
By the end of this lesson you should be able to
- State the damper force law and its phase relationship to displacement
- Explain why combined damper, column and connection forces require assessment
- Say what the velocity exponent does and why it is set below 1
- Compute energy dissipated per cycle
What you should already know
- Viscous damping in the equation of motion (Module 3)
- Energy dissipated per cycle (Module 7)
- Harmonic response and phase (Module 6)
Force and phase
An ideal fluid viscous damper develops an axial resisting force with magnitude C|v|^α, where v is the relative axial velocity of its ends. With the resisting term on the left of the equation of motion, F = C|v|^α sign(v). The units of C depend on α.
For harmonic axial motion u = u₀ sin ωt, velocity is u₀ω cos ωt. Force is zero at the axial displacement extrema and peaks at its zero crossings. The columns are not necessarily unloaded then: gravity, other modes and the damper’s vertical and horizontal components still contribute. Trace combined forces through connections, members and foundations.
Velocity exponent
For α = 1 the device is linear. For 0 < α < 1, doubling speed multiplies force by 2^α; at α = 0.4 this is 1.32. This law imposes no finite force cap.
Using C = 900 in each device’s respective kN·(s/m)^α units, axial velocities of 0.35 and 0.70 m/s give linear forces of 315 and 630 kN, and α = 0.4 forces of 591 and 780 kN. Equal numerical coefficients have different units here. Compare force–velocity curves over the operating range.
Brace geometry
With horizontal storey velocity vh, a rigid diagonal at θ has axial velocity vh cos θ. Its horizontal force magnitude is C|vh|^α cos^(1+α)θ. The familiar cos²θ coefficient applies only when α = 1. Vertical relative motion and brace flexibility require the actual axial kinematics.
Energy and analysis
For a linear damper under harmonic axial motion, ED = πCωu₀² per cycle. The relevant amplitude is relative displacement across the device.
A concentrated linear layout can produce coupled modal equations. A nonlinear device has no fixed exact equivalent modal damping ratio. Use a suitable nonlinear response method, or justify the limits and accuracy of an equivalent-linear approximation.
Try it
A manufactured viscous damper
Force from velocity, energy from the loop, and what the velocity exponent is really for.
1 is linear. 0.3–0.5 is typical for seismic dampers, and it caps the force in a rare event.
- α = 0.50
- α = 1 (linear)
- Displacement (mm)
- Damper force (kN ÷ 10)
- Peak relative velocity
- 0.188 m/s
- Peak damper force (axial)
- 868 kN
- Brace efficiency cos²θ
- 0.750
- Effective horizontal force
- 752 kN
- Damper stroke required
- 52 mm
- Energy dissipated per cycle
- 91.1 kJ
- Equivalent modal damping (linear damper)
- 2.7%
- Peak response without dampers
- 90 mm
- Peak response with dampers
- 46.2 mm
- Reduction
- 49%
Shown for α = 1; a nonlinear damper has no exact modal damping ratio.
With α = 0.50, doubling the velocity raises the force by only 41%. That is the point of a nonlinear damper: it delivers nearly full force at moderate velocity and refuses to deliver an enormous force in a rare, fast event — which protects the braces, the connections and the frame behind them.
Sizing and placement
- Added damping goes with the SQUARE of the relative modal displacement across the damper. A damper in a storey where the mode barely drifts does almost nothing, however large it is.
- Dampers add damping without adding stiffness, so they do not change the period and do not attract more force to the structure — unlike a brace, which does both.
- Damper force and structural force peak at different instants, because one depends on velocity and the other on displacement. Adding their peaks arithmetically over-designs the connection.
- The stroke is a hard limit. A damper that bottoms out becomes a rigid strut, and the force that follows is not a damping force at all.
What this shows: A viscous damper's force depends on VELOCITY, not displacement, so it peaks where the displacement is zero — which is why it adds damping without adding stiffness, and why it does not change the structure's period.
Worked example
Sizing a damper and checking what it does to the frame
Given
- Peak relative velocity along the damper axis is 0.35 m/s in the illustrative design event
- Peak relative velocity along the damper axis reaches 0.70 m/s in the illustrative larger event
- Two candidate dampers, both with C = 900 kN·(s/m)^α: one with α = 1.0, one with α = 0.4
- Damper braced at 38° to the horizontal
Find
The force each delivers in both events, and what the brace and connection must be designed for.
Assumptions
- The given velocities are axial device velocities, already projected from the structural motion. Resolve axial force horizontally using cos θ. The separate cos² comparison applies only to a linear device under equal horizontal storey velocity.
- Rate effects and damper stiffness neglected, as is usual at this stage of sizing
Predict first
An ideal viscous damper follows horizontal storey drift through a rigid diagonal with no vertical relative motion. At peak drift, what is its force?
Practice
A linear damper has C = 900 kN·s/m. What force does it deliver at a relative velocity of 0.35 m/s, in kN?
Practice
A damper with C = 900 kN·(s/m)0.4 and α = 0.4 is driven at 0.35 m/s. What force does it deliver, in kN?
Practice
The same α = 0.4 damper is driven at 0.70 m/s. What force does it deliver, in kN?
Practice
A linear damper is braced at38° to the horizontal. What fraction of its horizontal damping coefficient is retained?
Practice
A linear damper with C = 2.0 MN·s/m is cycled at 30 mm amplitude at ω = 4.19 rad/s. How much energy does it dissipate per cycle, in kJ?
Check yourself
Why are seismic viscous dampers usually made with a velocity exponent below 1?
Worked example
Why a viscous damper adds no stiffness
Given
- A linear viscous damper with coefficient c, in a structure oscillating harmonically
Find
The phase of the damper force relative to the displacement, and why it matters
Summary
- Damper force depends on velocity: F = C|v|^α sign(v)
- Under harmonic axial motion, ideal viscous force is zero at displacement extrema
- Assess combined column and connection forces rather than assuming the damper has little effect
- An exponent below1 slows force growth; nonlinear response requires a suitable nonlinear method
- For a linear damper the horizontal coefficient scales as cos²θ; for exponent α it scales as cos^(1+α)θ
- ED = πCωu₀² for a linear damper — amplitude matters most, and it enters squared