TMD Project

University Of Sharjah

Collage of Higher Education and Scientific Research

M.Sc. in Civil Engineering Program

A Project to

Structural Dynamics & Earthquake Engineering (0401512)

Titled

Tuned Mass Dampers: Theory, Types and Applications

Done By Hussain Osama Qasem

U00028420

Submitted to Prof. Mohammed Maalej

Hussain Osama Qasem U00028420 Project/0401512

A1 | P a g e

Abstract:

Structures are subjected to various types of dynamic loads that cause them to oscillate as

an elastic response to these loads, but, if this response was violent, then the structural health will

be endangered, that is, the probability of having plastic response in the structure will increase as

the severity of oscillations increases. On the other hand, wave scientists states that it is possible

to damp a wave if this wave was subjected to another wave that has same magnitude but reverse

effect and this was the idea of structures damping systems.

In this report, a quick view of history of structures that was damaged due to excessive

oscillation will be done, the theory behind dampers will be shown and some types and

applications will be presented


A2 | P a g e

Table of Contents

Title Page Number

Introduction 1

Theory Behind TMDs 4

Applications of TMDs 7

Case Example 1 9

Case Example 2 12

Conclusions 16

References 17


1 | P a g e

1. Introduction:

Many engineers have encountered the collapse of Tacoma Narrows Bridge or Galloping

Gertie Bridge in their studies. The collapse resulted from violent vibration due to winds that

didn’t exceed 64 km/hr speed that resulted in what so called the Kármán Vortex Street as shown

in (figure 1), many hypotheses was stated under the umbrella of this question “would this have

saved bridge?” (Figure 1) to figure out a solution that if it was present at that time the bridge

was not collapsed and also, damping science would not have acquired this concern.

a b c

The speech said “Necessity is the mother of invention”. The necessity was something that

makes the structure doesn’t oscillate or oscillate without endangering structural health, this will

be showed in details in the next part. Wave scientists state that it is possible to damp a wave if

this wave was subjected to another wave that has same magnitude but reverse effect. This was

the idea that guided earthquake engineers to create more efficient structural damping systems. In

general, damping systems has many types

1. Material Damping:

Fig-1:a) Hypotheses to save bridge. b) Collapse moment of the bridge

c) Kármán Vortex Street


2 | P a g e

This arises mainly due to energy dissipations caused by micro-structural interactions

resulting in defects such as grain boundaries, local thermal effects due to temperature gradients,

dislocations in the grain lamina etc.

2. Colomb Damping:

This arises due to energy dissipations caused by the frictional forces that is caused by

sliding of dry surfaces one on each other. The damping force depends on the nature of the

surfaces and is directly proportional to the normal force being applied to the surface as expressed

in friction equation

Where:

µk = coefficient of kinetic friction (since the system is in motion)

N = the normal force

3. Viscous Damping

This is related to bodies that move in moderate speed within fluids and/or fluids moving

by its inertia in a closed boundary system. The damping force here is directly proportional to the

velocity of the damper.

Tuned mass dampers (or TMDs) are considered as viscous dampers since the mass moves

the fluid or the fluid itself moves within a closed and fixed boundary. In any way, the energy of

the vibration is absorbed by this motion of the damper. TMDs has many types depending on the

nature of the vibration of the damper

1


3 | P a g e

1. Pendulum TMD (PTMD): in which the damping masses (or pendulum masses) moves

hydraulic cylinders to dissipate vibration energy. It is active in two axes

simultaneously.

2. Coupled Pendulum TMD (CPTMD): the same as above but the benefit is that the height

of the damper can be reduced

3. Sloshing Liquid TMD (SLTMD: in which the fluid moves within a specifically shaped

tank to dissipate vibration energy. It is active in two axes simultaneously.

4. Liquid Column TMD (LCTMD): Same principle as the previous one but the liquid

moves in a U shaped tank and that’s why this damper is effective in one axis only

which is the axis of the tank. It is good to mention that SLTMDs’ & LCTMDs’ water

can be used for other purposes for the building but the amount of water must be

maintained within the operating range.


4 | P a g e

2. Theory behind TMDs

In its simplest forms as shown in (Fig. 2) with a load like an

earthquake or wind (a sinusoidal wave). The equation of motion of the

given system is given by the next equation:

The solution of this equation is given as follows

If we call the following

Then the solution of the system will be then

Now, it is obvious if we tuned the second mass, the first mass will be affected by this

tuning. If the luck was our friend, tuning the damper to the ressonnence of the system yeilds in

the best elimination of the effect of the vibration.

Fig-2) TMD basics


5 | P a g e

This system now is acting as if the second mass (m2) is excerting an opposite force on the

first system. the first mass is the building or the structure. To generalize more. One can think of

many masses tied to gether with the TMD on top of it. So it will dessipate the vibration caused

by an earthquake or wind. The force is inverted if the sign was different or if the face difference

between the basic forcing and damper forcing function is half a cycle.

The next figure (Fig-3) illustrate the system with TMD and without TMD. The benefit of

TMD is clearly shown by graphs. The next graph compares between DMF of an SDOF once

without the TMD and once again but with TMD

(a) (b)

Fig-3) Difference between SDOF systems in DMF

a) Without TMD b) with TMD


6 | P a g e

Another way of thinking is using the principle of overlaping waves. Knowing that the

forcing function of the second mass is a sinosodial wave of which the basic system is of the same

type. Then if the seconed mass was tuned to be in the opposite phase of the exciting function.

Then the total response will diminish (if not zero) and this is called the distructive overlapping.

The opposite is correct, if the second mass was tuned to the same phase of the basic exciting

force, then, the response will be amplified and this is called constructive overlapping. These facts

are illustrated in the next figure note that the third line represents the total response (Fig-4).

The fact that wants to show is the damping term in the equation is not there! This means

the system is practically damped but mathematically speaking it is not damped. The damping is

because of difference of phase between degrees of freedom that is resulted from total inertia of

the tunned mass damper that is tuned to diminish the response to an acceptable range.

(a) (b)

Fig-4) Overlapping waves

a) Constructive b) Distructive


7 | P a g e

3. Applications of TMDs

TMDs are widely used where ever a wave is to be damped.

The crankshaft dampers are TMDs (figure 5) (it is widely named

here كراسي احملرك). It is used to absorb the torsional vibration of the

crankshaft and that’s why if these parts are damaged in the car

the noise will raise.

Also TMDs are used in space shuttles to reduce violent vibrations because of rapid

ascending of the shuttle.

TMDs are used in power lines (here named Stockbridge

dampers) (figure 6) to absorb flutters which are electrical vibration

of the wire that has high frequency and low amplitude induced by

the high voltage alternating current. These dampers are found at

the connection point between the power lines and cable towers.

Fig-5) Crankshaft Damper

Fig-6) Stockbridge damper

Fig-7) PTMD of Taipei 101


8 | P a g e

TMD are used in structures also (and that’s why this report is written) the types of the

TMDs previously discussed are clearly understood here.

The PTMD in Taipei 101 (figure 7) is the simplest type of TMD. The ball weighs 730

tons and has the height of 15 m and it is positioned at the 87 floor of the tower (observers can use

level 91 & 89)

Princess

tower in Dubai,

United Arab

Emirates (figure 8)

has another type of

TMDs. The

SLTMD in that skyscraper is not as big as the Taipei’s 101 since the expected

winds and earthquake loads on the princess tower is smaller than which is

expected in T-101 an example on SLTMD is shown on (figure 9).

In the One Wall Center of Vancouver, Canada (figure 9). The LCTMD

is used here. At that time this was a boom. The shape of the tower makes us

predict easily where the LCTMD is placed.

Fig-9) SLTMD

Fig-8) Princess Tower

Fig-10) OWC Tower


9 | P a g e

4. Case Example (1)

Taipei 101 in Taiwan is one of the tallest buildings of the world. For Taiwanese this

building is their sign of glory and pride of their country, so, protecting such architectural state of

the art is a nationwide priority. This building consist of 101 stories over ground and 5 basements

with a height of 508 m it makes it one of the tallest buildings in the world. This, besides being

very slender in terms of story area, makes us predict the need of not only one TMD system but

two systems of which one is principle and the other is secondary. The principle TMD is the

pendulum that is hanged in the 90th

story, where the secondary one is the TMD that is placed in

the pinnacle of the tower. For the pendulum, the frequency of it is given by this equation:

This is the frequency that we need to play with to reach the resonance of the system of

101 degrees of freedom (say) to make this pendulum vibrate with the building but with opposite

phase. Here the equation of motion of the pendulum takes the responsibility of engaging the

mass of the pendulum into the equation. It is well known that the pendulum vibrates regardless

of the mass of it. But when we want this pendulum to vibrate with a certain phase and under

certain excitation force the equation of motion of it comes to solve the problem. This equation of

the motion is given by (for the pendulum degree of freedom only)

Which diminish to the next equation.


10 | P a g e

The next is a part of an article about TMD’s:

“The building TMD, as shown in Figure 10, is

essentially a pendulum that spans 5 floors of the structure.

Having worked out the amplitude requirements under

extreme loading scenarios, as discussed in the following

sections, the architect was able to incorporate this vibration

absorber into the architectural scheme of the uppermost

occupied floors. From the restaurant and bar, through the

center of which the TMD penetrates, patrons will be able to

see the 800 ton steel ball swinging slightly many days of the

year, under light winds. During the strongest wind storm

expected to occur in half of a year, according to the Taipei local meteorological records, the

zbuilding TMD will reduce the peak acceleration of the top occupied floor from 7.9milli-g to

5.0milli-g (where 1milli-g is 1/1000 of Earth’s standard gravity). This performance is shown

graphically in Figure 11, and contrasted against the ISO acceleration criteria, as well as the

Taiwanese criteria of 5.0cm/s2 (5.1milli-g).

However, being a completely passive device means that the building TMD is also in

motion during substantially stronger wind events, e.g. 100 years. The design of the TMD must be

economically justifiable with regards to possible damage to component parts and the surrounding

structure. At such wind levels, the most sensitive devices in this assembly are the Viscous

Damping Devices (VDD). These VDDs must be able to dissipate, as heat, enough of the energy

that they are removing from the structure to avoid overheating and subsequent failure. An

example time history of the power absorbed by a single VDD is shown in Figure 12. This is

Fig-10) Drawing of Taipei 101 TMD


11 | P a g e

achieved without the use of supplemental liquid cooling by a heat-resistant VDD design (e.g.

high temperature seals, a working fluid which is thermally stable, etc.). Extensive testing by the

chosen supplier has demonstrated such a level of capability, far beyond the norm in the hydraulic

damper industry.”

Fig-11) Effectiveness of building TMD under moderate winds

Fig-12) VDD power handling requirements during 100-year wind event


12 | P a g e

5. Example Case (2)

An Example of a frame previously discussed in the course will be used to demonstrate

the power of a TMD.

Using the given value of k = 9634 kN/m, one can find the dynamic characteristics of the frame

That is

And

Using a harmonic forcing function that has the same natural frequency as the structure

One can find the response of the structure under this function that is represented in the given

equation

Total Weight = 500 kN

7 m

4 m I = 199 x 106 mm4

I = 257 x 106 mm4

fy = 300 MPa

E = 200 GPa

k = 9634 kN/m

F(t)

Fig-13) Subject Structure


13 | P a g e

One can plot this expression to see how the response of the structure will be:

Now we introduce to the structure the TMD

-80

-60

-40

-20

0

20

40

60

80

0 20 40 60 80 100 120

Fig-14) Un-damped Response of Subject Structure


14 | P a g e

TMD is now tuned to the natural frequency of the subject structure, we want also to use a TMD

that is 0.25 of the subject structure mass.

It is good to say that regardless of the mass and stiffness of the TMD. It is to be selected to have

the same frequency of the subject structure with keeping the mass of the TMD under

consideration so the TMD will work well and it will not affect the structure by its weight.

So, for any TMD that is tuned to have the same natural frequency of the subject structure, the

most important thing is the mass ratio or (µ) = (m2 / m1)

Implementing this equation into the main equation of the TMD result in the fact that the subject

structure doesn’t move

u1o = 0

Thus

u1(t) = 0

7 m

4 m

F(t) TMD, K, M


15 | P a g e

On the other hand, the TMD will vibrate but with another response motion

The All Equation are represented in the following chart

Now one might ask, if the TMD is supposed to damp the motion, so why in larger structures the

TMD doesn’t really 100 % damp the motion of the structure?

The answer is, methods used in calculating stiffnesses and frequencies of structures are all

approximate and doesn’t give the correct answer 100%. But even though the TMD shows its

power by dramatically damp the motion of the structure.

-80

-60

-40

-20

0

20

40

60

80

0 20 40 60 80 100 120

Motion Without TMD

Motion with TMD

Motion of TMD


16 | P a g e

6. Conclusion

In this report we have encountered the necessity of tuned mass damper to damp various

dynamic loads that a structure might face, the Galloping Gertie was used as an example. Also the

different types of damping available in structures were also discussed.

The basics of tuned mass dampers dynamics was explained in two different ways; the

first was to deal with the subject from differential equations point of view by considering a

system of two DOF that the TMD is one of these DOF. The other way was to explain the subject

through the concepts of waves overlapping by saying that the TMD is always tuned to be in

destructive overlapping with the mother structure.

Some examples from life were discussed to show to which degree we are attached to

TMD’s and how much useful it is in our lives. A case example of Taipei 101 was addressed in a

little detailing.


17 | P a g e

7. References

1. http://en.wikipedia.org/wiki/Tuned_mass_damper

2. http://mechanicalengg4u.blogspot.com/2009/06/types-of-damping.html

3. http://www.structuremag.org/Archives/2007-6/C-PracSolTunedLiqDampersRobinsonpac-5-11-

07.pdf

4. http://www.grk802.tu-braunschweig.de/Mitglieder/TommasoMassai.htm

5. http://en.wikipedia.org/wiki/Tacoma_Narrows_Bridge_(1940)

6. Toni Liedes, Improving the Performance of the Semi-Activated Mass Damper, 2009,

OULU University.

7. H. Paul et. al., Investigation into Tuning of a Vibration Absorber for a Single Degree of

Freedom System, lab report, Dublin Institute of Technology.

8. http://electron9.phys.utk.edu/phys135d/modules/m10/waves.htm

9. Haskett, T. et.al., Tuned Mass Dampers Under Excessive Structural Excitations,

Technical paper

Documents

TMD Project