Module 19 · Lesson 19.1
Tuned mass dampers
A second mass that moves out of phase, the tuning that makes it work, and the mistuning that undoes it.
Why this matters
Every device so far has acted between the structure and the ground, or between two parts of the structure. A tuned mass damper does something different: it adds a mass, hangs it off the structure on a spring, and lets it move.
The idea is counter-intuitive enough to be worth stating plainly. Adding mass usually makes a vibration problem worse. Added in the right place, tuned to the right frequency, and given the right damping, it makes it dramatically better — because the added mass moves OUT OF PHASE with the structure, and its inertia force opposes the structure's motion.
This is the most widely used control device in tall buildings and long footbridges, and it is the one where the theory is precise enough to be genuinely useful.
By the end of this lesson you should be able to
- Explain the mechanism in terms of phase, not energy
- Apply Den Hartog's optimum tuning and say what it optimises
- Read the two-peak response curve and know where the trough sits
- Quantify the effect of mistuning
What you should already know
- Two-degree-of-freedom systems (Module 9)
- Harmonic response and resonance (Module 6)
- Damping and its effect on resonant amplitude (Module 5)
The mechanism
Attach a secondary mass m₂ to the structure through a spring k₂ and a damper c₂. Tune it so its own natural frequency ω₂ = √(k₂/m₂) is close to the structure's ω₁.
Drive the structure at ω₁. The secondary mass, being at its own resonance, responds strongly and — this is the whole trick — with a phase lag approaching 90° relative to the force, which puts it close to 180° out of phase with the structure's motion.
When the structure moves right, the damper mass moves left. Its inertia force pulls back through the spring, and that force opposes the motion. The structure is being held.
A TMD does not primarily absorb energy. It applies a force out of phase with the motion. The damper c₂ is there to stop the secondary mass running away and to control the width of the effect — not to be the mechanism.
One peak becomes two
The combined system has two degrees of freedom, so it has two natural frequencies: one below the original and one above. The frequency response therefore has TWO peaks, with a trough between them at the tuning frequency.
The original single tall peak is gone. At the frequency that used to be catastrophic, the response is now small.
The height of the two new peaks depends on the damper's own damping ζ₂:
- ζ₂ too small — the two peaks are very tall and very narrow. The damper mass swings enormously, and the system is exquisitely sensitive to tuning.
- ζ₂ too large — the secondary mass is locked to the structure and moves with it. It is now just added mass, and the single peak returns.
- ζ₂ optimum — in between, and it is a genuine optimum rather than a compromise.
What the optimum actually optimises
Den Hartog's result comes from a remarkable observation. Plot the response curve for several values of ζ₂ and they all pass through two FIXED POINTS, whose height does not depend on ζ₂ at all.
Since no choice of ζ₂ can lower those two points, the best available response is to make them equal in height, and then choose the damping that puts the two peaks exactly on them. That gives
fopt = ω₂/ω₁ = 1/(1 + μ), ζ₂,opt = √(3μ/(8(1 + μ)))
where μ = m₂/m₁ is the mass ratio.
The optimum deliberately produces two EQUAL peaks. It does not produce a deep trough. Anyone judging a TMD design by how deep the trough is has misunderstood what is being optimised — a deeper trough always comes with taller peaks, and the peaks are what governs.
The achievable peak, for an undamped main structure, is
Rpeak = √(1 + 2/μ)
At μ = 0.02 that is √101 = 10.0. Against an uncontrolled structure with 1% damping, whose peak magnification is 1/(2ζ) = 50, that is a fivefold reduction from a damper weighing 2% of the structure.
Mistuning: the practical limit
The optimum is sharp, and that is the problem.
On the course's teaching system — μ = 0.02, main damping 1%, optimally tuned — the peak magnification is 8.71. Detune the damper by 5% and it rises to 12.4. Detune by 10% and it rises to 16.8, which is nearly double the optimum.
The structure's own frequency is not known to 5%. It changes with occupancy, with cladding, with temperature, with concrete maturity, with cracking, and it is rarely predicted to better than 10% before the building exists.
The consequences are practical and they are the reason TMDs are engineering projects rather than catalogue purchases:
Measure the real frequency. Every serious installation is commissioned by measuring the completed structure's actual frequency, not the predicted one.
Make the tuning adjustable. Movable mass, adjustable spring, or both, so it can be re-tuned after measurement and again later in life.
Use a larger mass ratio than the minimum. A larger μ gives a broader effective band. Doubling μ makes the device markedly more forgiving, which is often worth more than the extra performance at perfect tuning.
Consider multiple dampers. Several units at slightly different frequencies cover a wider band and degrade gracefully.
Why a TMD is better for wind than for earthquake
A TMD is a NARROW-BAND device. It works at the frequency it is tuned to and does progressively less either side.
Wind and footfall excitation is narrow-band by nature: vortex shedding at a particular frequency, pedestrians at a particular pace, machinery at a particular speed. The excitation sits where the damper is tuned, and the match is excellent. This is why TMDs in tall buildings are usually comfort devices — they exist to reduce sway that occupants can feel, and they do it very well.
Earthquake excitation is broad-band and transient. It contains everything, so a device tuned to one frequency addresses only part of it. Worse, a TMD needs several cycles to build up its own response before it starts helping, and the strong-motion phase of an earthquake may be over before it has. TMDs do help in earthquakes and are installed for that purpose, but the benefit is smaller and less reliable than the wind case, and it should not be claimed to be otherwise.
Try it
A tuned mass damper
Add a small secondary mass on its own spring, tuned near the structure's frequency, and one resonance becomes two.
3.0% of the structure's mass
- Without the damper
- With the damper
- Auxiliary mass
The structure and its auxiliary mass, driven at ω/ω₁ = 1.00. Watch the phase between them.
- Optimum f (Den Hartog)
- 0.9709
- Optimum ζ₂
- 0.1045
- In use: f
- 0.9709
- In use: ζ₂
- 0.1045
- Peak without the damper
- 50.0
- Peak with the damper
- 7.34
- Reduction
- 85%
- Auxiliary mass amplitude at ω₁
- 29.1× static
- Peak after a 10% tuning error
- 13.02
The damper moves far more than the structure does. Its stroke is a real design constraint.
77% worse than the tuned case
A mass ratio of 3.0% has taken the peak from 50 to 7.3 — a reduction of 85%. Notice what happened to the SHAPE: the single tall peak has split into two shorter ones, and the frequency the structure used to resonate at now sits in the trough between them. The auxiliary mass is moving 4.6 times as far as the structure, and that motion is where the energy is going.
Why the optimum tuning looks flat
- Den Hartog's optimum makes the two peaks equal in height rather than digging one deep notch. A deep narrow notch would give a lower minimum and would miss it entirely as soon as the structure's real frequency differed from the model's.
- A tuned mass damper is commissioned by measurement, not by calculation. The structure's real frequency is measured after construction and the damper is retuned to it.
- Turn off the optimum and set f well away from 1. The device stops working, and the structure's original resonance comes back.
- The optimum here is for harmonic forcing of an undamped structure. Under wind or earthquake, and with the structure's own damping present, the best tuning shifts — this is a starting point, not a final answer.
- The auxiliary mass has to travel. On a real building that means a stroke of the order of a metre, plus a stopper for the case where the design event is exceeded.
What this shows: A tuned mass damper does not absorb energy by being large. It works by moving out of phase with the structure so that its inertia opposes the motion — which is why a 3% mass ratio can halve the response, and why it stops working if the tuning is wrong.
From first principles
The tuned mass damper principle and Den Hartog's optimum
We want to show: Find the tuning ratio and damper damping that minimise a structure's peak resonant response for a given secondary mass.
Hold a heavy weight on a spring and shake the top of the spring at just the right rate: the weight lags behind and ends up moving the opposite way to your hand. Now imagine your hand is the top of a building. The weight's inertia is pulling back through the spring every time the building tries to move, and the building is being held by something that is only moving because the building moved first. The remarkable part, which Den Hartog noticed, is that when you draw the response curves for every possible damper damping, they all cross at two points that no amount of damping can lower. That turns an open-ended design problem into a closed one: you cannot do better than those two points, so make them level and put your peaks on them.
Worked example
Designing a tuned mass damper for a tall building
Given
- A tall building, first-mode generalised mass 1 200 tonnes, first period 2.5 s
- Inherent damping 1% — typical for a tall steel building in wind
- A mass ratio of 2% is proposed
- The complaint is wind-induced sway felt by occupants on the upper floors
Find
The damper mass, its tuning, its damping, and the improvement — including what happens if the tuning is 10% out.
Assumptions
- First mode dominant; the damper is at roof level where the mode shape is greatest
- Harmonic forcing assumed for the tuning; wind excitation is narrow-band enough to justify it
Predict first
A TMD is designed using Den Hartog's optimum. Looking at the frequency response, the trough between the two peaks is fairly shallow. What does that indicate?
Practice
A tuned mass damper has a mass ratio of 2%. What is Den Hartog's optimum tuning ratio f = ω₂/ω₁?
Practice
For the same 2% mass ratio, what is the optimum damping ratio of the damper itself?
Practice
A building with a first-mode generalised mass of 1 200 tonnes is fitted with a TMD at a mass ratio of 2%. What is the damper mass, in tonnes?
Practice
An undamped structure has a TMD with a 2% mass ratio at Den Hartog's optimum. What peak magnification does the theory predict?
Practice
A structure with 1% damping has a peak magnification of 8.71 with an optimally tuned TMD, and 16.8 when the damper is mistuned by 10%. By what factor does the peak response increase?
Check yourself
How does a tuned mass damper reduce a structure's resonant response?
Worked example
Tuning a mass damper by Den Hartog
Given
- A footbridge with a problematic mode at 2.0 Hz
- A tuned mass damper of 3 % of the modal mass is proposed
Find
The optimum tuning ratio and damper damping
Worked example
Why a tuned mass damper needs to be tuned
Given
- The same footbridge and 3 % damper, tuned to 1.942 Hz
- The bridge's actual frequency turns out to be 2.15 Hz rather than 2.0 Hz
Find
What the mistuning costs
Summary
- A TMD works by phase, not by absorption: its inertia force opposes the motion
- One resonant peak becomes two, with the trough at the tuning frequency
- fopt = 1/(1 + μ), ζ₂,opt = √(3μ/(8(1 + μ))), Rpeak = √(1 + 2/μ)
- The optimum deliberately gives two EQUAL peaks, not a deep trough
- Mistuning by 10% nearly doubles the peak — the practical limit on the device
- Tune to a MEASURED frequency and make the device adjustable
- Narrow-band: excellent for wind and footfall, weaker and slower for earthquake
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