tmd project
TRANSCRIPT
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
Hussain Osama Qasem U00028420 Project/0401512
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
Hussain Osama Qasem U00028420 Project/0401512
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
Hussain Osama Qasem U00028420 Project/0401512
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
Hussain Osama Qasem U00028420 Project/0401512
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.
Hussain Osama Qasem U00028420 Project/0401512
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
Hussain Osama Qasem U00028420 Project/0401512
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
Hussain Osama Qasem U00028420 Project/0401512
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
Hussain Osama Qasem U00028420 Project/0401512
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
Hussain Osama Qasem U00028420 Project/0401512
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
Hussain Osama Qasem U00028420 Project/0401512
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.
Hussain Osama Qasem U00028420 Project/0401512
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
Hussain Osama Qasem U00028420 Project/0401512
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
Hussain Osama Qasem U00028420 Project/0401512
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
Hussain Osama Qasem U00028420 Project/0401512
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
Hussain Osama Qasem U00028420 Project/0401512
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
Hussain Osama Qasem U00028420 Project/0401512
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
Hussain Osama Qasem U00028420 Project/0401512
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.
Hussain Osama Qasem U00028420 Project/0401512
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