Module 7 · Lesson 7.2
Dissipation per cycle and equivalent damping
How much a damper removes each cycle, why frequency appears in the answer, and how to turn a real hysteresis loop — which is not viscous at all — into a ζ that linear analysis can use.
Why this matters
No structure has a dashpot in it. Real dissipation comes from friction at connections, from cracking, from cladding and partitions rubbing against the frame, from radiation into the ground and from material hysteresis. None of it produces a force proportional to velocity.
Yet every method in this course assumes viscous damping, because that is the only model that keeps the equation linear. The bridge between the two is energy: choose c so that the ENERGY removed per cycle is right, and accept that the mechanism is a fiction. This lesson builds that bridge and is honest about what it costs.
By the end of this lesson you should be able to
- Derive the energy dissipated per cycle by a viscous damper
- Explain why frequency appears and what follows from it
- Derive the equivalent viscous damping ratio from a loop area
- State the limitations of equivalent linearisation
From first principles
Energy dissipated per cycle, and equivalent viscous damping
We want to show: Derive ED = πcωX² for steady harmonic motion, then invert it to convert any measured loop area into an equivalent viscous damping ratio.
In steady harmonic motion the structure traces the same path every cycle. The damper force opposes the motion the whole way round, so it takes energy out on every part of the path — there is no part of the cycle where it gives any back. Adding up that work over one complete cycle gives a definite amount, and because the damper force depends on speed, going round faster costs more.
The loop area is the energy
Plot damper force against displacement over one cycle and the curve closes into a loop. The area enclosed is the energy dissipated in that cycle — because area on a force–displacement plot is force times distance, which is work.
For a linear viscous damper the loop is an ellipse, and its area works out to exactly πcωX², as derived. For a friction damper it is a rectangle. For a yielding element it is a parallelogram or something more complicated.
This makes the loop a genuinely useful diagnostic. A wide loop dissipates a lot. A thin one dissipates little. A loop that does not close has an error in it — the structure has not returned to where it started, which means either the cycle is incomplete or the model has drift in it.
Where viscous damping comes from, if not from dashpots
A short and honest answer: from measurements on structures believed to be similar.
Typical values used in design:
| Structure | Damping ratio |
|---|---|
| Bare welded steel frame | 1–2% |
| Bolted steel frame | 2–3% |
| Reinforced concrete, uncracked | 2–3% |
| Reinforced concrete, cracked | 3–5% |
| Completed building with fit-out | 3–7% |
| Assumed for elastic seismic design | commonly 5% |
These are not derived from anything. They come from decades of measurements on real structures, and the scatter within each category is wide — comfortably a factor of two.
That matters because, at resonance, the response is 1/(2ζ). A factor of two in the damping estimate is a factor of two in the answer, and it is the largest single uncertainty in most vibration serviceability assessments. Where the conclusion depends on it, the honest response is to bound it rather than to pick a value.
Worked example
Sizing a damper from an energy target
Given
- A footbridge, first mode: modal mass 22 000 kg, frequency 1.9 Hz
- Inherent damping measured at 0.6% — very low, a bare steel deck
- Target: raise the total damping to 3% to control pedestrian-induced response
Find
The damping coefficient required, and how much energy it removes per cycle at a 15 mm amplitude.
Assumptions
- The added damper is linear and viscous, so damping ratios simply add
- The response is dominated by the first mode
- The damper acts on the modal coordinate directly — a real installation would need the cos²θ correction of Module 18
Practice
A viscous damper with c = 8 kN·s/m operates at 10 rad/s with an amplitude of 20 mm. How much energy does it dissipate per cycle, in J?
Practice
A measured hysteresis loop encloses 450 J at an amplitude of 25 mm. The stiffness is 4.0 MN/m. What is the equivalent viscous damping ratio?
Practice
What fraction of the stored energy does a structure with 4% damping lose per cycle? Give the answer as a percentage, using the small-damping relationship.
Check yourself
Why is the energy dissipated by a viscous damper proportional to frequency?
Check yourself
Over one complete cycle of steady harmonic response, what is the net work done by the inertia force?
Worked example
Turning a hysteresis loop into a damping ratio
Given
- A steel brace tested cyclically, behaving essentially elastic–perfectly plastic
- It is cycled to four times its yield displacement
Find
The equivalent viscous damping
Summary
- ED = πcωX² per cycle — linear in frequency, quadratic in amplitude
- The area of a force–displacement loop IS the energy dissipated in that cycle
- ζeq = ED/(4πEs) converts any loop into an equivalent viscous damping ratio
- The fraction of energy lost per cycle is about 4πζ, independent of amplitude
- No structure has a dashpot; c is calibrated to energy, not derived from a mechanism
- Design damping values come from measurement, with a scatter of about a factor of two
- A damper's ζ goes as 1/ωn, so it delivers differently in different modes
This is educational material. It uses simplified examples to teach principles, and must not be relied on for real design or safety-critical decisions. Module overview and checkpoint