# Tuned mass damper

An animation showing the movement of a skyscraper versus the mass damper. Shown in green are the hydraulic cylinders used to damp the motion of the skyscraper.

A tuned mass damper, also known as a harmonic absorber, is a device mounted in structures to reduce the amplitude of mechanical vibrations. Their application can prevent discomfort, damage, or outright structural failure. They are frequently used in power transmission, automobiles, and buildings.

## Principle

A schematic of a simple spring–mass–damper system used to demonstrate the tuned mass damper system.

Tuned mass dampers stabilize against violent motion caused by harmonic vibration. A tuned damper reduces the vibration of a system with a comparatively lightweight component so that the worst-case vibrations are less intense. Roughly speaking, practical systems are tuned to either move the main mode away from a troubling excitation frequency, or to add damping to a resonance that is difficult or expensive to damp directly. An example of the latter is a crankshaft torsional damper. Mass dampers are frequently implemented with a frictional or hydraulic component that turns mechanical kinetic energy into heat, like an automotive shock absorber. An electrical analogue is an LCR circuit.

Given a motor with mass $m_1$ attached via motor mounts to the ground, the motor vibrates as it operates and the soft motor mounts act as a parallel spring and damper, $k_1$ and $c_1$. The force on the motor mounts is $F_0$. In order to reduce the maximum force on the motor mounts as the motor operates over a range of speeds, a smaller mass, $m_2$, is connected to $m_1$ by a spring and a damper, $k_2$ and $c_2$. $F_1$ is the effective force on the motor due to its operation.

Response of the system excited by one unit of force, with (red) and without (blue) the 10% tuned mass. The peak response is reduced from 9 units down to 5.5 units. While the maximum response force is reduced, there are some operating frequencies for which the response force is increased.

The graph shows the effect of a tuned mass damper on a simple spring–mass–damper system, excited by vibrations with an amplitude of one unit of force applied to the main mass, $m_1$. An important measure of performance is the ratio of the force on the motor mounts to the force vibrating the motor, $F_0/F_1$. This assumes that the system is linear, so if the force on the motor were to double, so would the force on the motor mounts. The blue line represents the baseline system, with a maximum response of 9 units of force at around 9 units of frequency. The red line shows the effect of adding a tuned mass of 10% of the baseline mass. It has a maximum response of 5.5, at a frequency of 7. As a side effect, it also has a second normal mode and will vibrate somewhat more than the baseline system at frequencies below about 6 and above about 10.

The heights of the two peaks can be adjusted by changing the stiffness of the spring in the tuned mass damper. Changing the damping also changes the height of the peaks, in a complex fashion. The split between the two peaks can be changed by altering the mass of the damper ($m_2$).

A Bode plot of displacements in the system with (red) and without (blue) the 10% tuned mass.

The Bode plot is more complex, showing the phase and magnitude of the motion of each mass, for the two cases, relative to F1.

In the plots at right, the black line shows the baseline response ($m_2 = 0$). Now considering $m_2 = m_1/10$, the blue line shows the motion of the damping mass and the red line shows the motion of the primary mass. The amplitude plot shows that at low frequencies, the damping mass resonates much more than the primary mass. The phase plot shows that at low frequencies, the two masses are in phase. As the frequency increases $m_2$ moves out of phase with $m_1$ until at around 9.5 Hz it is 180° out of phase with $m_1$, maximizing the damping effect by maximizing the amplitude of $x_2-x_1$, this maximizes the energy dissipated into $c_2$ and simultaneously pulls on the primary mass in the same direction as the motor mounts.

## Mass dampers in automobiles

### Motorsport

The tuned mass damper was introduced as part of the suspension system by Renault, on its 2005 F1 car (the Renault R25), at the 2005 Brazilian Grand Prix. It was deemed to be legal at first, and it was in use up to the 2006 German Grand Prix.

At Hockenheim, the mass damper was deemed illegal by the FIA, because the mass was not rigidly attached to the chassis and, due to the influence it had on the pitch attitude of the car, which in turn significantly affected the gap under the car and hence the ground effects of the car, to be a movable aerodynamic device and hence as a consequence, to be illegally influencing the performance of the aerodynamics.

The Stewards of the meeting deemed it legal, but the FIA appealed against that decision. Two weeks later, the FIA International Court of Appeal deemed the mass damper illegal.[1][2]

### Production cars

Tuned mass dampers are widely used in production cars, typically on the crankshaft pulley to control torsional vibration and, more rarely, the bending modes of the crankshaft. They are also used on the driveline for gearwhine, and elsewhere for other noises or vibrations on the exhaust, body, suspension or anywhere else. Almost all modern cars will have one mass damper, some may have 10 or more.

The usual design of damper on the crankshaft consists of a thin band of rubber between the hub of the pully and the outer rim. This design is often called a harmonic damper. An alternative design is the centrifugal pendulum absorber which is used to reduce the internal combustion engine's torsional vibrations on a few modern cars.

All four wheels of the Citroen 2cv incorporated a tuned mass damper (referred to as a "Batteur" in the original French) of very similar design to that used in the Renault F1 car, from the start of production in 1949 on all four wheels, before being removed from the rear and eventually the front wheels in the mid 1970s.

## Mass dampers in spacecraft

One proposal to reduce vibration on NASA's Ares solid fuel booster was to use 16 tuned mass dampers as part of a design strategy to reduce peak loads from 6g to 0.25 g, the TMDs being responsible for the reduction from 1 g to 0.25 g, the rest being done by conventional vibration isolators between the upper stages and the booster.[3][4]

## Dampers in power transmission lines

Stockbridge dampers on a 400 KV power line near Castle Combe, England

High-tension lines often have small barbell-shaped Stockbridge dampers hanging from the wires to reduce the high-frequency, low-amplitude oscillation termed flutter.[5][6]

## Dampers in wind turbines

A standard tuned mass damper for wind turbines consists of an auxiliary mass which is attached to the main structure by means of springs and dashpot elements. The natural frequency of the tuned mass damper is basically defined by its spring constant and the damping ratio determined by the dashpot. The tuned parameter of the tuned mass damper enables the auxiliary mass to oscillate with a phase shift with respect to the motion of the structure. In a typical configuration an auxiliary mass hung below the nacelle of a wind turbine supported by dampers or friction plates.

## Dampers in buildings and related structures

Location of Taipei 101's largest tuned mass damper
Tuned mass damper atop Taipei 101.

Typically, the dampers are huge concrete blocks or steel bodies mounted in skyscrapers or other structures, and moved in opposition to the resonance frequency oscillations of the structure by means of springs, fluid or pendulums.

### Sources of vibration and resonance

Unwanted vibration may be caused by environmental forces acting on a structure, such as wind or earthquake, or by a seemingly innocuous vibration source causing resonance that may be destructive, unpleasant or simply inconvenient.

#### Earthquakes

The seismic waves caused by an earthquake will make buildings sway and oscillate in various ways depending on the frequency and direction of ground motion, and the height and construction of the building. Seismic activity can cause excessive oscillations of the building which may lead to structural failure. To enhance the building's seismic performance, a proper building design is performed engaging various seismic vibration control technologies. As mentioned above, damping devices had been used in the aeronautics and automobile industries long before they were standard in mitigating seismic damage to buildings. In fact, the first specialized damping devices for earthquakes were not developed until late in 1950.[7]

#### Mechanical human sources

Dampers on the Millennium Bridge in London. The white disk is not part of the damper.

Masses of people walking up and down stairs at once, or great numbers of people stomping in unison, can cause serious problems in large structures like stadiums if those structures lack damping measures.

#### Wind

The force of wind against tall buildings can cause the top of skyscrapers to move more than a meter. This motion can be in the form of swaying or twisting, and can cause the upper floors of such buildings to move. Certain angles of wind and aerodynamic properties of a building can accentuate the movement and cause motion sickness in people. A TMD is usually tuned to a certain building's frequency to work efficiently. However, during their lifetimes, high-rise and slender buildings may experience natural frequency changes under wind speed, ambient temperatures and relative humidity variations, among other factors, which requires a robust TMD design.[8]

#### Examples of buildings and structures with tuned mass dampers

• One Wall Centre in Vancouver — It employs tuned liquid column dampers, at the time of its installation, a unique form of tuned mass damper.
##### Ireland
• Dublin Spire in Dublin, Ireland — This narrow slender structure was designed with a tuned mass damper to ensure aerodynamic stability during a wind storm.
##### Taiwan
• Taipei 101 skyscraper — Contains the world's largest and heaviest tuned mass dampers, at 660 metric tons.[9]

## References

1. ^ Bishop, Matt (2006). "The Long Interview: Flavio Briatore". F1 Racing (October): 66–76.
2. ^ "FIA bans controversial damper system". Pitpass.com. Retrieved 2010-02-07.
3. ^ "Ares I Thrust Oscillation meetings conclude with encouraging data, changes". NASASpaceFlight.com. 2008-12-09. Retrieved 2010-02-07.
4. ^ "Shock Absorber Plan Set for NASA's New Rocket". SPACE.com. 2008-08-19. Retrieved 2010-02-07.
5. ^ "On the hysteresis of wire cables in Stockbridge dampers". Cat.inist.fr. Retrieved 2010-02-07.
6. ^ "Cable clingers - 27 October 2007". New Scientist. Retrieved 2010-02-07.
7. ^ Reitherman, Robert (2012). Earthquakes and Engineers: An International History. Reston, VA: ASCE Press. ISBN 9780784410714.
8. ^ ALY, Aly Mousaad (2012). "Proposed robust tuned mass damper for response mitigation in buildings exposed to multidirectional wind". The Structural Design of Tall and Special Buildings. doi:10.1002/tal.1068.
9. ^ taipei-101.com.tw
10. ^ Petroski, Henry (1996). Invention by Design: How Engineers Get from Thought to Thing. Harvard University Press. pp. 205–208.
11. ^ "Comcast Center". Retrieved 2010-02-07.