# Tuned Mass Dampers

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A Tuned Mass Damper (TMD) is a mechanical device designed to add damping to a structure for a certain range of exciting frequencies. The extra damping will reduce the movement of the structure to an acceptable level.

A tuned mass damper contains a mass that is able to oscillate in the same direction as the structure. The oscillation frequency of the mass can be tuned using springs, suspension bars, or ball transfers. When the structure starts to oscillate, the mass of the TMD will initially remain stationary due to inertia. A frictional or hydraulic component connected between the structure and the TMD mass then turns the kinetic energy of the structure into thermal energy, which results in a lower vibration amplitude of the structure.

The design of an TMD depends on the oscillation frequency and mass of the structure, direction of the movements (one horizontal direction, two horizontal directions, or vertical) and the available space.

**Guarantee**

Flow Engineering has more than 20 years of experience in the calculation, designing, fabrication, and installation of Tuned Mass Damper (TMD) systems. Our TMD systems are applied to bridges, flagpoles, chimneys, distillation columns and other slender structures.

Our TMD systems have proven themselves in practice. We guarantee that the solutions we offer ensure the reduction of unwanted vibrations to an acceptable level, thereby preventing fatigue damage.

**Tuning**

By cleverly designing a Tuned Mass Damper (TMD), a single device can add sufficient damping for two or more of the structures natural frequencies, reducing the number of necessary TMDs. In order to perform such optimizations, we model the structure with a finite element package to calculate the modes of vibration of the structure. The TMD is then tuned to add at least the necessary damping for each natural frequency.

**Design Considerations**

In simple situations a structure with a connected Tuned Mass Damper (TMD) can be modelled as in the following figure.

Here \(k\) is the spring constant, \(c\) is the damper constant, and \(m\) is the mass. Subscript \(1\) pertains to the structure and subscript \(2\) to the TMD.

A TMD can significantly reduce the response of a structure, as can be seen from the following graph.

The effects of varying several design parameters are given below.

**Mass Ratio \(\mu\)**

Increasing the mass ratio \(\mu\) (increasing the damper mass) will decrease the structural displacement. The normalized structural displacement amplitude can be computed with the formula given by J.P. Den Hartog in “Mechanical Vibrations”:

$$ \frac{\left| z_{1} \right|}{x_{st}} = \sqrt{1+\frac{2}{\mu}} $$

As can be seen from the figure, Den Hartog’s approach, calculating with \(\zeta_{1}=0\), is slightly conservative for steel structures (\(\zeta_{1}=0.2\%\)) at the lower mass ratios.

**Damper Frequency \(f\)**

The eigenfrequencies of a structure may not be known to a sufficient level of accuracy at the time that the TMDs are designed. It is then useful to define a range in which the frequency of the eigenmode to be damped is sure to reside. By designing an appropriate TMD for the entire range, the need for measuring a structures eigenfrequencies before a TMD can be produced is negated.

In the case that the structure has multiple eigenfrequencies relatively near to each other, a wide range TMD may be used to add damping to several eigenmodes. Reducing the cost of the vibration damping system.

**Internal Damping Ratio \(\zeta_{2}\)**

The increase in amplitude from mis-tuned internal damping can be significant. It is because of this effect that we advise changing the internal dampers at set intervals of 15 to 25 years, depending on the damper used.

Our ongoing research into maintenance free tuned mass dampers has solved this issue for linear tuned mass dampers. See our solution for linear tuned mass dampers: Magnovisco Linear Dampers