Structural Dynamics / Energy in vibrating structures
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.
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Read the lesson register and outstanding review notesWhy this matters
Structures with installed fluid dampers can contain physical dashpots. For inherent structural damping, an equivalent viscous model represents the combined effects of several dissipation mechanisms.
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. If the path is not closed, separate stored-energy change from dissipated work. An incomplete cycle, transient response, plastic residual or ratcheting can produce an open path without a coding error.
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 can be a major 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
A linear viscous oscillator with 4% damping is driven steadily at its undamped natural frequency. What is the energy dissipated per cycle divided by its peak stored elastic energy, as a percentage?
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
- At steady resonance, ED/Es,max=4πζ; free-cycle energy loss is 1−exp[−4πζ/√(1−ζ²)]
- For inherent damping,c is often calibrated from response; installed viscous devices have their own measured force–velocity behaviour
- 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