Structural Dynamics / Structural control
Tuned mass dampers
A second mass that moves out of phase, the tuning that makes it work, and the mistuning that undoes it.
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.
- Control specialist review needed for modal normalisation, TMD stroke/forces, detuning, pedestrian feedback, robust stability and safe failure states. No regulatory life-safety permission has been verified.
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.
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Read the lesson register and outstanding review notesWhy this matters
A tuned mass damper adds a secondary resonator to the structure. Its motion changes the dynamic forces, while its damper dissipates energy through relative motion. Tuning, damping, placement and available stroke determine how much it helps for the loading being assessed.
By the end of this lesson you should be able to
- Explain how secondary-mass coupling and damping change the response
- 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)
Coupled response
A TMD attaches a secondary mass m₂ through a spring k₂ and damper c₂. Its relative motion changes the dynamic force acting on the structure, and c₂ dissipates energy. Phase varies with forcing frequency; the masses are not always 180° out of phase.
The combined linear system has two modes. A lightly damped response commonly has two peaks separated by a trough whose location depends on tuning, damping and the response measure.
Classical fixed-point tuning
For an undamped primary SDOF system under harmonic force applied to that primary mass, the classical expressions are
f = ω₂/ω₁ = 1/(1+μ),
ζ₂ ≈ √(3μ/[8(1+μ)]).
These are the Den Hartog/Brock fixed-point estimates. The usual damping expression closely approximates the exact minimax solution; it cannot place both peaks exactly on the fixed points simultaneously. The fixed-point magnification R = √(1+2/μ) approximates the controlled peak in this model.
For a structural mode, normalise the attachment ordinate to 1 before using μ = m₂/Mn. More generally μ = m₂φa²/Mn. Generalised modal mass requires its normalisation to be stated.
Primary damping, base excitation, other loading, stroke limits and different objectives change the problem. Use these formulas as initial estimates and assess the coupled system.
Mistuning and commissioning
The teaching system reports peak magnification 8.71 at 1% primary damping, increasing to 12.4 and 16.8 in its 5% and 10% detuning cases. These are model-specific values. State the detuning direction and frequency-sweep range when reproducing them.
Measure relevant modes during commissioning, provide a justified adjustment range and allow inspection and maintenance. Occupancy, amplitude, temperature and structural changes can shift the target. Multiple dampers may cover uncertainty or several modes but require a coupled assessment.
Excitation
TMDs are useful where an important response is concentrated near a target mode. Wind may be broadband, with a narrow resonant structural response. Pedestrian forcing varies and may interact with the bridge. Earthquakes may excite several modes transiently. Assess the actual loading and response rather than inferring suitability from the load’s name.
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: Develop fixed-point tuning estimates for an undamped primary oscillator under harmonic force
The secondary resonator changes dynamic force and dissipates energy through relative motion. In the classical undamped-primary model, response curves for different absorber damping values pass through two fixed points. Equalising their heights gives a tuning estimate, while the standard damping expression approximately controls the nearby peaks.
Worked example
Designing a tuned mass damper for a tall building
Given
- A tall building has first-mode generalised mass 1 200 tonnes with the roof attachment ordinate normalised to 1, and 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-force response is used for this teaching calculation; project wind loading and comfort need a separate assessment
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
The first mode has generalised mass 1 200 tonnes when its TMD attachment ordinate is normalised to 1. At a 2% mass ratio, what physical damper mass is needed, in tonnes?
Practice
For an undamped primary oscillator under harmonic force, what fixed-point magnification √(1+2/μ) is obtained at a 2% TMD mass ratio?
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 couples a secondary resonator to the structure; inertia coupling and damper dissipation shape the response
- One resonant peak becomes two, with the trough at the tuning frequency
- For harmonic force and an undamped primary: f = 1/(1+μ), ζ₂ ≈ √(3μ/[8(1+μ)]), fixed-point R = √(1+2/μ)
- The classical approximation controls two nearly equal peaks; inspect the full curve
- The stated 10% detuning case nearly doubles this example’s peak; test project tuning uncertainty
- Tune to a MEASURED frequency and make the device adjustable
- Assess excitation bandwidth, target modes and transient response for the actual loading