Design the tuned mass damper (TMD) that tames the sway of a tall building or a long-span bridge using Den Hartog optimal-TMD theory. Adjust the main structure mass, natural frequency, mass ratio and TMD damping to see the damper mass, optimal tuning ratio, optimal damping, stiffness and peak magnification update in real time.
Parameters
Main structure mass M
kg
Effective mass of the structure (building, bridge) to be damped
Main natural frequency f
Hz
The frequency at which the structure loves to resonate
Mass ratio µ
TMD mass ÷ structure effective mass. Typically 1-5%
TMD damping ratio ζ_d
The damping ratio actually given to the TMD. Compare with the optimum
Results
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Damper mass m (kg)
—
Optimal tuning ratio
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Optimal TMD frequency (Hz)
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Optimal damping ζ_opt
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TMD stiffness (N/m)
—
Peak magnification (with TMD)
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How a TMD works — out-of-phase motion animation
A TMD (a small mass on a spring and damper) near the top of a tall structure swings out of phase with the structure to reduce its sway, alongside a frequency-response sketch (one tall peak without a TMD versus two small peaks with it).
Den Hartog optimal tuning ratio f_opt and optimal damping ratio ζ_opt. µ is the mass ratio (TMD mass divided by the structure's effective mass). Optimal tuning replaces the single resonant peak with two small, tamed peaks.
$$m=\mu M,\qquad k_d=m\,(2\pi f_d)^{2}$$
TMD mass m and stiffness k_d. M is the structure's effective mass, f_d = f_opt·f is the tuned TMD natural frequency.
$$R_{peak}=\sqrt{1+\frac{2}{\mu}}$$
Peak magnification of the main structure with an optimal TMD. Without a TMD, the resonant magnification of a lightly-damped structure is far larger.
What is the Tuned Mass Damper Simulator?
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I heard there is a giant weight inside the top of some tall buildings to "stop the sway". But wouldn't adding a weight make the building sway more, not less?
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Good question — it is counter-intuitive. But that weight is not just sitting there. It rides on its own spring and damper, so from the structure's point of view it is a secondary oscillator that can move on its own. That is a tuned mass damper, a TMD. A slender building has a natural frequency at which it loves to sway, and when wind, an earthquake or marching footsteps excite that frequency it resonates and the sway keeps growing. A TMD is a device that kills exactly that resonance.
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"Kills exactly that resonance" — how? Is there something special about the weight?
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The special thing is the tuning itself. You adjust the TMD's spring and mass so that the TMD's own natural frequency almost exactly matches the building's troublesome resonant frequency. Then, the instant the building tries to resonate, the TMD starts vibrating too. But thanks to the dynamics of two coupled pendulums, the TMD moves out of phase with the building: as the building leans right, the TMD weight swings left. The weight's inertial force travels back through the spring and pushes the building back, cancelling the sway. In the animation above you can see the body and the weight moving in opposite directions.
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Pushing back out of phase — I see. But where does the energy go? Cancelling doesn't make it disappear.
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Sharp point. That is where the damper inside the TMD comes in. The TMD takes the resonant energy as its own motion, then turns that vibration into heat through its damping and throws it away. So the TMD moves a lot instead of the body, dissipating energy as heat. The amount of damping is critical — too much or too little and it fails. A man named Den Hartog derived the classic optimal damping ratio for an undamped main structure: ζ_opt = √(3µ/(8(1+µ)³)). At the default value (µ=0.02) the optimal damping comes out to about 0.0841.
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In the chart on the right, adding the TMD splits the peak into two. What is happening there?
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That is the heart of the TMD. Without a TMD, the structure's frequency response has one dangerously tall peak at the natural frequency. Add an optimally tuned and damped TMD, and that one sharp peak is replaced by two much lower, well-tamed peaks. The peak splits, and both halves come down. At the default values the peak magnification is √(1+2/0.02) = √101 ≈ 10.05. A lightly-damped structure without a TMD can have a resonant magnification of tens to hundreds, so the difference is dramatic.
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Impressive device. Which buildings actually have one?
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The most famous is the giant pendulum sphere in Taipei 101, weighing about 660 tonnes — it is visible from the observation deck and has become a tourist attraction. London's Millennium Bridge had a serious problem right after opening, when pedestrian footsteps and the lateral sway resonated; it was fixed later by adding many TMDs and dampers. Beyond that, TMDs are hidden in skyscrapers, chimneys, footbridges and tall aerial masts — slender, sway-prone structures everywhere. The mass ratio is usually 1 to 5 percent, meaning just a few percent of the structure's effective mass delivers all that effect.
Frequently Asked Questions
A TMD is a secondary mass-spring-damper attached to a vibrating main structure. Its natural frequency is tuned to almost exactly match the structure's troublesome resonant frequency. When the structure tries to resonate, the TMD resonates too, but because of the dynamics of two coupled oscillators it moves out of phase with the structure: as the building lurches one way the TMD mass swings the other way, and the inertial force fed back through its spring pushes back against the structure's motion, cancelling much of it. The TMD's own damper then bleeds the trapped vibrational energy away as heat.
Den Hartog optimal tuning is the classic recipe, worked out by J.P. Den Hartog for an undamped main structure, for designing a TMD. Once you choose a mass ratio µ (the TMD mass divided by the structure's effective mass), the optimal tuning ratio is f_opt = 1/(1+µ) and the optimal damping ratio is ζ_opt = √(3µ/(8(1+µ)³)). The optimal tuning ratio is slightly below 1, and together with the optimal damping it replaces the structure's single dangerous resonant peak with two much smaller, well-tamed peaks. A larger mass ratio lowers the peaks and makes the TMD more effective.
In practice the TMD mass is typically 1 to 5 percent of the structure's effective mass (µ = 0.01 to 0.05). A larger mass ratio reduces the peak magnification √(1+2/µ) and improves performance, but the TMD mass itself becomes heavier, bigger and more expensive, and the demands on installation space and the supporting structure grow. The damper sphere in Taipei 101 weighs about 660 tonnes, roughly a few percent of the tower's effective mass. Balancing cost against effectiveness, most building TMDs settle at around 1 to 3 percent.
A TMD targets a single resonant mode, so its effectiveness drops sharply when the tuning frequency drifts away from the structure's natural frequency. Ageing, payload changes and temperature shift a building's natural frequency, causing detuning, so engineers use TMDs with adjustable stiffness for re-tuning, or multiple TMDs set to slightly different frequencies (multiple TMDs, MTMD). A TMD is also effective only for narrow-band phenomena where the excitation frequency sits near the natural frequency; for impulsive or broadband input, other devices such as viscous dampers are better suited.
Real-World Applications
Wind sway of skyscrapers: Slender skyscrapers can sway enough in strong wind that occupants feel a sea-sickness-like discomfort. The roughly 660-tonne damper sphere in Taipei 101 is the classic example: hung near the top of the tower, it swings out of phase with the building and greatly reduces the wind-driven sway. Many skyscrapers hide a TMD of about 1 to 3 percent mass ratio in a rooftop machine room, installed not for structural safety but to improve occupant comfort (the felt sway).
Footbridge and long-span bridge damping: London's Millennium Bridge had to be closed right after opening when pedestrian footsteps synchronised with the bridge's lateral sway (synchronous walking) and shook it dangerously. As a fix, many lateral and vertical TMDs and dampers were retrofitted before it could reopen. Long-span bridge decks and suspension cables, prone to vortex-induced vibration in wind, are also major application areas for TMDs.
Vortex-induced vibration of chimneys, towers and masts: Tall cylindrical chimneys and steel towers can resonate across the wind when vortices shed alternately in a crosswind — vortex-induced vibration. A small TMD at the top of the tower effectively suppresses this resonance. Alongside the helical strakes wrapped around a chimney, the TMD is a standard wind countermeasure for slender tower structures.
Machinery, floors and sports facilities: TMDs are also used where the excitation from people or machinery makes a structure resonate near its natural frequency — the foundations of large rotating machines, long-span office floors, stadium grandstands and gymnasium floors. The advantage is that the discomfort of floor vibration from walking or exercise can be suppressed with a small added mass, without rebuilding the floor structure itself.
Common Misconceptions and Pitfalls
The biggest misconception is that "a TMD reduces the sway to zero". A TMD dramatically lowers the resonant peak, but it is not a device that eliminates vibration. Even with optimal tuning, the main structure's peak magnification of √(1+2/µ) remains. At the default value (µ=0.02) that is about 10 — still orders of magnitude smaller than the tens to hundreds without a TMD. Understand that the TMD's job is to replace one dangerously sharp peak with two tamed, lower peaks, not to make the response zero.
Next, assuming the tuning frequency is correct forever once it is set. A TMD targets a specific natural frequency, so its effect drops sharply when the main structure's natural frequency shifts. A building's natural frequency easily shifts a few percent due to ageing, changes in internal payload, temperature and construction tolerances. Because even a few percent of detuning greatly degrades TMD performance, real installations fine-tune the device with on-site measurements, and an increasing number of designs build in robustness with adjustable-stiffness TMDs that can be re-tuned, or with multiple TMDs (MTMD) set at slightly offset frequencies.
Finally, the misconception that "a TMD is a universal device that works against any vibration". A TMD is highly effective against narrow-band resonance near a natural frequency, but a single TMD cannot fully handle broadband input such as a shock, or a response with many closely-spaced modes. Against strong, broadband input such as an earthquake, other families of vibration-control and isolation technology — viscous dampers, oil dampers, base isolation — are often better suited. Position the TMD as a device that shows its true strength for narrow-band, steady-state resonance problems such as improving wind-sway comfort, suppressing bridge or floor resonance from walking, and suppressing the vortex-induced vibration of chimneys.
How to Use
Enter the primary structure mass (kg) and fundamental frequency (Hz)—for example, a 500-tonne office tower with f₁ = 0.5 Hz.
Set the mass ratio (TMD mass / primary mass), typically 1–5% for buildings; the simulator calculates optimal tuning frequency using Den Hartog theory.
Adjust TMD damping ratio ζ; the tool displays peak magnification and TMD stiffness (N/m) needed to achieve the target frequency.
Worked Example
A 2000 tonne suspension bridge with f₁ = 0.3 Hz requires vibration control. Apply mass ratio = 2% (40 tonnes TMD). The simulator yields optimal tuning frequency f_TMD = 0.294 Hz, optimal damping ζ_opt = 0.083, required stiffness k = 5440 N/m, and peak magnification 1.8 (reduced from 3.2 without damper). This matches field deployments on the Millennium Bridge and Tacoma Narrows retrofit designs.
Practical Notes
Mass ratio constraints: 1–3% for tall buildings minimizes cost; 2–5% for long-span bridges and offshore platforms where sway dominates fatigue.
Den Hartog optimum assumes linear single-degree-of-freedom primary structure; verify via FEA if secondary modes couple significantly (mode separation <15%).
Damping tuning is critical—undershooting ζ_opt creates split peaks; overshooting broadens the response plateau. Magnetorheological dampers allow real-time adjustment during commissioning.
Environmental drift: temperature changes shift stiffness by ±8% in hydraulic TMDs; re-tune after seasonal extremes or retrofit with frequency-tracking controllers.