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PASSIVE VIBRATION CONTROL OF

HIGH-RISE STRUCTURES USING

TUNED MASS DAMPERS

UNDER THE GUIDANCE OF

DR.PRADEEP RAMANCHARLA

BY

BHARATHI THADIGOTLA

MTECH CASE

200711002

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INDEX

TOPIC PGNO

ABSTRACT 3

INTRODUCTION 4

EQUATIONS OF MOTION 7

AND CLASSICAL SOLUTION

PRINCIPLE OF TMD 9

DESIGN PROCEDURE OF 10

TMD

TYPES OF TMD 11

ANALYSIS AND RESULTS 12

STAADPRO MODELS 14

MATLAB CODES 17

SDOF WITHOUT TMD 25

MDOF WITHOUT TMD 27

SDOF WITH TMD 28

MDOFWITH TMD 29

SAP2000 RESULTS 41

CONCLUSIONS 57

REFERENCES 58

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ABSTRACT

This project summarizes the results of reduction in maximum displacement of G+20 structure by

using a roof top tuned mass damper. The El Centro data is considered for Time history analysis.First considering whole structure as a two degree freedom system combined with TMD, response

is known for different percentages of mass. Then as multi degree freedom system and performing

optimization techniques response is known for different percentages of mass. A check in SAP2000

is also performed and the results are compared. This shows how effective Tuned mass damper

works for the reduction of maximum displacement ensuring structural stability when subjected to

earthquake loads.

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INTRODUCTION

In earthquake engineering, vibration control is a set of technical means aimed to mitigateseismic

impacts inbuilding and non-building structures.

All seismic vibration control devices may be classified as passive, active or hybrid where:

passive control devices have no feedbackcapability between them, structural elements and

the ground;

active control devices incorporate real-time recoding instrumentation on the groundintegrated with earthquake input processing equipment and actuators within the structure;

hybrid control devices have combined features of active and passive control systems

When groundseismic waves reach up and start to penetrate a base of a building, their energy flow

density, due to reflections, reduces dramatically: usually, up to 90%. However, the remainingportions of the incident waves during a major earthquake still bear a huge devastating potential.

After the seismic waves enter a superstructure, there is a number of ways to control them in order

to sooth their damaging effect and improve the building's seismic performance, for instance:

to dissipatethe wave energy inside asuperstructure with properly engineered dampers;

to disperse the wave energy between a wider range of frequencies;

to absorb the resonant portions of the whole wave frequencies band with the help of so

called mass dampers

Devices of the last kind, abbreviated correspondingly as TMD for the tuned (passive), as AMD forthe active, and as HMD for the hybrid mass dampers, have been studied and installed in high-rise

buildings, predominantly in Japan, for a quarter of a century

However, there is quite another approach: partial suppression of the seismic energy flow into the

superstructure known as seismic orbase isolation which has been implemented in a number ofhistorical buildings all over the world and remains in the focus ofearthquake engineering research

for years.

For this, some pads are inserted into all major load-carrying elements in the base of the building

which should substantiallydecouple a superstructure from itssubstructure resting on a shakingground. It also requires creating a rigiditydiaphragm and a moat around the building, as well as

making provisions against overturning and P-delta effect.
http://en.wikipedia.org/wiki/Earthquake_engineeringhttp://en.wikipedia.org/wiki/Seismichttp://en.wikipedia.org/wiki/Seismichttp://en.wikipedia.org/wiki/Impacthttp://en.wikipedia.org/wiki/Buildinghttp://en.wikipedia.org/wiki/Buildinghttp://en.wikipedia.org/wiki/Non-buildinghttp://en.wikipedia.org/wiki/Feedbackhttp://en.wikipedia.org/wiki/Actuatorhttp://en.wikipedia.org/wiki/Seismic_waveshttp://en.wikipedia.org/wiki/Seismic_waveshttp://en.wikipedia.org/wiki/Seismic_waveshttp://en.wikipedia.org/wiki/Superstructurehttp://en.wikipedia.org/wiki/Seismic_performancehttp://en.wikipedia.org/wiki/Dissipatehttp://en.wikipedia.org/wiki/Dissipatehttp://en.wikipedia.org/wiki/Superstructurehttp://en.wikipedia.org/wiki/Superstructurehttp://en.wikipedia.org/wiki/Damperhttp://en.wikipedia.org/wiki/Dispersehttp://en.wikipedia.org/wiki/Absorption_(acoustics)http://en.wikipedia.org/wiki/Resonanthttp://en.wikipedia.org/wiki/Mass_damperhttp://en.wikipedia.org/wiki/High-rise_buildinghttp://en.wikipedia.org/wiki/High-rise_buildinghttp://en.wikipedia.org/wiki/Suppressionhttp://en.wikipedia.org/wiki/Superstructurehttp://en.wikipedia.org/wiki/Base_isolationhttp://en.wikipedia.org/wiki/Earthquake_engineering_researchhttp://en.wikipedia.org/wiki/Earthquake_engineering_researchhttp://en.wikipedia.org/wiki/Decouplehttp://en.wikipedia.org/wiki/Decouplehttp://en.wikipedia.org/wiki/Superstructurehttp://en.wikipedia.org/wiki/Substructurehttp://en.wikipedia.org/wiki/Substructurehttp://en.wikipedia.org/wiki/Diaphragmhttp://en.wikipedia.org/wiki/Diaphragmhttp://en.wikipedia.org/wiki/Moathttp://en.wikipedia.org/wiki/Seismichttp://en.wikipedia.org/wiki/Impacthttp://en.wikipedia.org/wiki/Buildinghttp://en.wikipedia.org/wiki/Non-buildinghttp://en.wikipedia.org/wiki/Feedbackhttp://en.wikipedia.org/wiki/Actuatorhttp://en.wikipedia.org/wiki/Seismic_waveshttp://en.wikipedia.org/wiki/Seismic_waveshttp://en.wikipedia.org/wiki/Superstructurehttp://en.wikipedia.org/wiki/Seismic_performancehttp://en.wikipedia.org/wiki/Dissipatehttp://en.wikipedia.org/wiki/Superstructurehttp://en.wikipedia.org/wiki/Damperhttp://en.wikipedia.org/wiki/Dispersehttp://en.wikipedia.org/wiki/Absorption_(acoustics)http://en.wikipedia.org/wiki/Resonanthttp://en.wikipedia.org/wiki/Mass_damperhttp://en.wikipedia.org/wiki/High-rise_buildinghttp://en.wikipedia.org/wiki/High-rise_buildinghttp://en.wikipedia.org/wiki/Suppressionhttp://en.wikipedia.org/wiki/Superstructurehttp://en.wikipedia.org/wiki/Base_isolationhttp://en.wikipedia.org/wiki/Earthquake_engineering_researchhttp://en.wikipedia.org/wiki/Decouplehttp://en.wikipedia.org/wiki/Superstructurehttp://en.wikipedia.org/wiki/Substructurehttp://en.wikipedia.org/wiki/Diaphragmhttp://en.wikipedia.org/wiki/Moathttp://en.wikipedia.org/wiki/Earthquake_engineering

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Tuned mass dampers(TMDs) are amongst the oldest structural vibration control devices in

existence. Frahm invented a vibration control device for the first time in 1909 named dunamic

vibration absorber.Frahm did not have any inherent damping. so it is effective only when

absorbers natural frequency was very close to the excitation frequency .this shortcoming waseliminated when Ormondroyd and Den Hartog showed that if certain amount of damping is

introduced in frahms absorber,performance deterioration under changing excitation frequency will

not be very sharp and response at resonance can also be significantly reduced.

Tuned mass dampers stabilize against violent motion caused by harmonic vibration. A tuned

damper balances the vibration of a system with comparatively lightweight component so that theworst-case vibrations are less intense.

Consider a motor with mass m1 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, k1 and c1. The force on the

motor mounts is F0; suppose we wish to reduce the maximum force on the motor mounts as themotor operates over a range of speeds.

Let F1 be the effective force on the motor due to its operation. We will add a smaller mass, m2,

connected to m1 by a spring and a damper, k2 and c2.
http://en.wikipedia.org/wiki/Normal_modehttp://en.wikipedia.org/wiki/Normal_mode

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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 responseforce is reduced, there are some operating frequencies for which the response force is increased.

Graph is shown above

The graph shows the effect of a tuned mass damper on a simple springmassdamper system,excited by vibrations with an amplitude of one unit of force applied to the main mass, m1. An

important measure of performance is the ratio of the force on the motor mounts to the force

vibrating the motor, F0 / F1. (We are assuming 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 baselinesystem, 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 vibratesomewhat 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 (m2).

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

The Bode plotis more complex, showing the phase and magnitude of the motion of each mass, forthe two cases, relative to F1.Graphs are shown above.

In the plots at right, the black line shows the baseline response (m2 = 0). Now considering m2 = m1 /

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 muchmore than the primary mass. The phase plot shows that at low frequencies, the two masses are in
http://en.wikipedia.org/wiki/Bode_plothttp://en.wikipedia.org/wiki/Bode_plothttp://en.wikipedia.org/wiki/Bode_plothttp://en.wikipedia.org/wiki/Bode_plothttp://en.wikipedia.org/wiki/File:2dof_plots.pnghttp://en.wikipedia.org/wiki/File:Tuned_mass_damper.pnghttp://en.wikipedia.org/wiki/File:Tuned_mass_damper.pnghttp://en.wikipedia.org/wiki/Bode_plothttp://en.wikipedia.org/wiki/Bode_plot

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phase. As the frequency increases m2 moves out of phase with m1 until at around 9.5 Hz it is 180

out of phase with m1, maximizing the damping effect by maximizing the amplitude ofx2 x1, this

maximizes the energy dissipated into c2 and simultaneously pulls on the primary mass in the samedirection as the motor mounts.

Typically, the tuned mass dampers are huge concrete blocks mounted in skyscrapers or other

structures and moved in opposition to the resonance frequency oscillations of the structures by

means of some sort of spring mechanism.

Taipei 101 skyscraper needs to withstand typhoon winds and earthquake tremors common in itsarea of the Asia-Pacific. For this purpose, a steel pendulum weighing 660 metric tons that serves as

a tuned mass damper was designed and installed atop the structure. Suspended from the 92nd to

the 88th floor, the pendulum sways to decrease resonant amplifications of lateral displacements inthe building caused by earthquakes and strong gusts.

Equations of motion and classical solution:

For figure 1 the equations of motion of a SDOF structure-TMD mechanism are given as:

MX(t) + KX(t) - [c{Sc(t)-X(t)} + k{x(t)-X(t)}] = P(t) (1)

mY(t) + c{x(t)-J((t)} + k(x(t)-X(t)} = p(t) (2)

where

M = main mass

M= absorber mass

K = main spring stiffness

k =absorber spring stiffness

C =absorber damping

P(t)= force acting on main mass. In case of base excitation
http://en.wikipedia.org/wiki/Tuned_mass_damperhttp://en.wikipedia.org/wiki/Skyscraperhttp://en.wikipedia.org/wiki/Resonance_frequencyhttp://en.wikipedia.org/wiki/Taipei_101http://en.wikipedia.org/wiki/Typhoonhttp://en.wikipedia.org/wiki/Tremorhttp://en.wikipedia.org/wiki/Pendulumhttp://en.wikipedia.org/wiki/Gusthttp://en.wikipedia.org/wiki/Tuned_mass_damperhttp://en.wikipedia.org/wiki/Skyscraperhttp://en.wikipedia.org/wiki/Resonance_frequencyhttp://en.wikipedia.org/wiki/Taipei_101http://en.wikipedia.org/wiki/Typhoonhttp://en.wikipedia.org/wiki/Tremorhttp://en.wikipedia.org/wiki/Pendulumhttp://en.wikipedia.org/wiki/Gust

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P(t)=with acceleration g(t),P(t) = -MY~(t).

force acting on damper mass. It is given as:

P(t)= (m/M)*P(t) for base excitationP(t)=0for main mass excitation

To facilitate further discussions, additional notations areintroduced here as follows:

= damper mass to main mass ratio = m/M.

frequency of a harmonic excitation.

= natural frequency of main mass, =

a = natural frequency of damper mass, a =

g1 = ratio of excitation frequency to main mass natural

frequency g1 = /.

for MDOF structures, g1 = /1 where 1 is thefirst modal frequency of the structure.

f = frequency ratio, f = a/. d= damping ratio of TMD. = damping ratio of main mass.

Den Hartog developed closed form expressions of optimum damper parameters f and d. whichminimize the steady-state response of the main mass subjected to a harmonic main mass excitation.

These expressions for the calculating optimum damper parameters are given as:

For the case when the structure is subjected to a harmonic base excitation, the corresponding

expressions can be easily found to be:

Using the values of , and , optimum values of damping c and stiffness k of the damper

can be calculated as

Which gives

Similarly

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Which gives

Principle of TMD(Tuned Mass Damper)

Developments in computer-aided structural design and high-strength materials led to more flexible

and lightly damped structures. When subjected to dynamic loads such a traffic loads, wind,

earthquake, wave, vibration lasting for long duration may be easily induced in this type of

structures.

TMD is a vibration system with mT spring Kt and viscous cT usually installed on the top of the

movement of the structure. Hence the kinetic energy of the structure goes into the TMD system to

be absorbed by the viscous damper of the TMD. To achieve the most efficient energy absorbing

capacity of TMD, natural period of TMD by itself is tunes with natural period of the structure by

itself, from which the system is called Tuned Mass Damper. The viscous damper of TMD shall

also be adjusted to the optimum value to maximize the absorbed energy.TMD is a mechanically

simple system which does not need any external energy supply for operation. Because of easy

maintenance and high realiability, TMD is used in many flexible and lightly-damped towers.

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Design Procedure of TMD:

In the design of actual TMD, there are many design constraints arising from structural properties of

the main structure. Design procedure for general flexible and lightly-damped structures is as

follow.

1. Identify dynamic structural properties of a main structure. Determine natural frequency,

vibrations mass, damping ratio of the specific vibration mode of the structure to becontrolled by TMD.

2. Determine design ratio eq to be generated by TMD.

3. Assume appropriate mass ratio (determined mass of mT of TMD).

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4. Calculate opt and opt from (determined stiffness K T and damping coefficient C T of

TMD).

5. Calculate maximum displacement of TMD. If it is too large go back to 3.

6. Design mechanical system of TMD including the methods on how to adjust T and T .

7. Verification tests of TMD with shaking table or with other methods

8. Installation of TMD to the main structure.

9. Monitor behavior of TMD and adjust T and T on site.

10. The following issues shall be examined , if necessary:

If the vibration mode of the structure to be controlled by TMD has anothervibration mode close to it, 3- or higher-DOF modeling of the main structure and the

TMD has to be used to determine the optimum TMD.

Multiple TMD of which are distributed around the vibration period of the structure

has robustness in tuning effect.

If the displacement of TMD is too large, use of larger mass or higher damping

becomes inevitable, which may not always give the optimum TMD.

Types of TMD:

There are many types of TMD for implementation. Innovative challenge is highly expected

in this field.Examples of different TMDS are

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ANALYSIS AND RESULTS:

A G+20 structure having dimensions 75x41m is taken and modeled in StaadPro.

The column dimensions are 600x600 and the beam dimensions are taken as 300x450.

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Mass of first and intermediate floors of building using static check results of Staadpro = 4.57*105

Kg.

Mass of the last floor = (4.57*105)-(0.3*240*25*0.6*0.6/2)

= 4246 KN

= 4.446e5 Kg

Mass of the whole structure = (21*4.57*105)-(0.3*240*25*0.6*0.6/2)

= 95646 KN

= 9.5646*106 Kg.

Stiffness of each floor = 1.2*240*108

= 2.88*108

N/m

Stiffness of whole structure = (2.88*108)/21

= 13.71*108 N/m

PLAN OF PLOT

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PLAN OF ONE HOUSE IN PLOT

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Structure is analyzed in staadpro and the results are checked.

STAADPRO MODEL:

Elevation:

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Plan:

3DModel:

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Matlab code to find maximum response of Single degree of freedom system:

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%%% Numerical evaluation of dynamic response by

%% Newmark's method,Elcentro Record is taken.

%% for ground accelerations.

%%% data input

clear all

clc

format long

m=15.23e6

k=11.40e8%input('please enter the stiffness : ');

z=5/100

wn=sqrt(k/m)

c=z*2*wn*m;

%c=19.16%input('please enter the damping constant c ');

u(1)=0%input('please enter the intial displacement : ');

v(1)=0%input('please enter the initial velocity : ');

dt= 0.02

%%% load input file

load elcentro.txt;

t=elcentro(:,1);

a=elcentro(:,2);

f=m*a; %% change to f=m*a if a =acc

d=size(f);

g=d(1);

Be=1/4; %Average acceleration Be= 1/4 && R =1/2

R=1/2; %Linear acceleration R=1/2 && Be=1/6

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%%% intial calulations :::

A(1)=(f(1)-(c*v(1))-(k*u(1)))/m;

Kcap=(k)+(R*c/(Be*dt))+(m/(Be*dt^2));

aa= ((Be*dt)^-1*m)+((R*c)/Be);

bb= ((2*Be)^-1*m) +(((R*(2*Be) -1)-1)*c*dt);

%%%% iteration :::

for j= 1:g-1

dPcap(j)=(f(j+1)-f(j))+(aa*v(j))+(bb*(A(j)));

du(j)=dPcap(j)/Kcap;

dv(j)=(R*du(j)/(Be*dt))-(R*v(j)/Be)+((1-(R/(2*Be)))*(A(j)*dt));

dA(j)=(du(j)/(Be*dt^2))-(v(j)/(Be*dt))-(A(j)/(2*Be));

u(j+1)=u(j)+du(j);

v(j+1)=v(j)+dv(j);

A(j+1)=A(j)+dA(j);

end

w=1:length(a)

t=1:length(a)

%subplot(3,2,1)

plot(t,u(w))

display('max displacement')

max(abs(u(w)))*1000

Matlab code to find maximum response of Multi degree of freedom system:

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%%% Numerical evaluation of dynamic response (time histroy) by

%% Newmark's method

%% for ground accelerations. for MDOF

%%% data input :::

clear all

clc

%%% mass data input

dof=22%input('Enter the number degrees of freedom');

m1=4.57e5;%input('Enter the mass of first and intermediate floors');

m2=4.246e5%input('Enter the mass of last floor');

mn=4.9014e6;

k1=2.88e8%input('Enter the stiffness of each floor');

pm=input('percentage of mass');

k2=input('Enter the stiffenss of tmd');

c1=input('damping of tmd');

mtmd=pm*mn;

for i=1:dof-2

M(i,i)=m1;

end

M(dof-1,dof-1)=m2;

M(dof,dof)=mtmd;

%% stiffness data input.

for i=1:dof-1

k(i)=k1;

end

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k(dof)=k2;

%%% stiffness matrix generator

for i=1:dof

if i==dof

K(i,i)=k(i);

else

K(i,i)=k(i)+k(i+1);

K(i,i+1)=-k(i+1);

K(i+1,i)=K(i,i+1);

end

I(i)=1;

end

%%% Finding the eigen values

[V,W]=eig(inv(M)*K);

W=sqrt(W);

g=size(W);

%%%% bubble sort

for i=1:g(:,2)

for j=1:(g(:,2)-1)

if W(j,j)>W(j+1,j+1)

temp=W(j,j);

temp1=V(:,j);

V(:,j)=V(:,j+1);

V(:,j+1)=temp1;

W(j,j)=W(j+1,j+1);

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W(j+1,j+1)=temp;

end

end

end

fprintf('The eigen Values are \n ');

W;

fprintf('The eigen Vectors are \n');

V;

for i=1:dof

V(:,i)=V(:,i)/max(abs(V(:,i)));

end

display('Modal mass matrix')

MM=V'*M*V;

%for i=1:dof

% V(:,i)=V(:,i)/sqrt(MM(i,i));

%end

%MM=V'*M*V;

MK=V'*K*V;

display('Modal stiffness matrix')

for i=1:dof

u(i,1)=0

v(i,1)=0

cc(i,i)=(2*0.05*W(i,i)*MM(i,i)); %% damping matix (classical damping)

end

cc(dof,dof)=c1;

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%cc(5,5)=12333.1704;

%%%% newmark's method...

dt= 0.02 %input('please enter the time step :');

%%% load input file

load elcentro.txt;

t=elcentro(:,1);

a=elcentro(:,2);

dd=size(a);

gg=dd(1);

Be=1/4; %% BETA Value here %Average acceleration Be= 1/4 && R =1/2

R=1/2; % GAmma Value here Linear acceleration R=1/2 && Be=1/6

%%% intial calulations :::

for i=1:dof

PF(i)=V(:,i)'*M*I';

f=-1*PF(i)*a;

A(i,1)=(f(1)-(cc(i,i)*v(i,1))-(MK(i,i)*u(i,1)))/MM(i,i);

Kcap(i)=(MK(i,i))+(R*cc(i,i)/(Be*dt))+(MM(i,i)/(Be*dt^2));

aa(i)= ((Be*dt)^-1*MM(i,i))+((R/Be)*cc(i,i));

bb(i)= ((2*Be)^-1*MM(i,i)) +(((R*(2*Be)^-1)-1)*cc(i,i)*dt);

%%%% iteration :::

for j= 1:gg-1

dPcap(i,j)=(f(j+1)-f(j))+(aa(i)*v(i,j))+(bb(i)*(A(i,j)));

du(i,j)=dPcap(i,j)/Kcap(i);

dv(i,j)=(R*du(i,j)/(Be*dt))-(R*v(i,j)/Be)+((1-(R/(2*Be)))*(A(i,j)*dt));

dA(i,j)=(du(i,j)/(Be*dt^2))-(v(i,j)/(Be*dt))-(A(i,j)/(2*Be));

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u(i,j+1)=u(i,j)+du(i,j);

v(i,j+1)=v(i,j)+dv(i,j);

A(i,j+1)=A(i,j)+dA(i,j);

end

end

S=triu(ones(dof,dof),0)

SS=S*K;

%%%%% Mode superposition Method......

X=zeros(dof,gg);

t=1;

for i=1:dof

for j=1:gg-1

for t=1:dof

X(i,j)=X(i,j)+V(i,t)*u(t,j);

end

end

end

Force=zeros(dof,gg);

Shearforce=zeros(dof,gg);

t=1;

for i=1:dof

for j=1:gg-1

for t=1:dof

Force(i,j)=Force(i,j)+K(i,t)*X(t,j);

Shearforce(i,j)=Shearforce(i,j)+SS(i,t)*X(t,j);

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end

end

end

i=input('Storey value');

figure(1);

j=1:gg

h=0:0.02:53.76

Optimization:

clear all

m=4.9107e6

mu=0.1

m2=mu*m

ws=9.1488

fopt=(1/(1+mu))*sqrt(((2-mu)/2))

zetaopt=sqrt(3*mu/(8*(1+mu)))*(2/(2-mu))

wa=fopt*ws

ka=(m2*ws^2)*fopt^2

copt=2*zetaopt*fopt*ws*m2

Response of the SDOF structure without TMD

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Graph showing maximum displacement on y-axis

From the Matlab code of SDOF system we got maximum displacement of 60.473 mm.

MAXIMUM DISPLACEMENT FROM SAP CONSIDERING SDOF SYSTEM

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From SAP2000 maximum displacement without TMD is 60.270 mm. So the code value is nearer

to the value from SAP. Hence it is correct.

Response of the MDOF structure without TMD

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The maximum displacement Of MDOF system is found to be 336.539 mm form matlab code and

the SAP2000 result also satisfied. So there is great difference in maximum displacement if the

whole structure is considered SDOF. So MDOF system is opted as it is giving higher

displacement.

TUNED MASS DAMPER FOR SDOF SYSTEM

In this procedure mass and stiffness both are reduced by equal percentages.

Whole G+20 structure is taken as one degree and TMD is taken as another degree.So with TMD it

becomes a two degree freedom system.

%mass

Maximumdisplacement

PercentageReductionindisplacement

NO TMD 60.723 0

1 52.9507 13.71

248.5278855

20.00

344.9002943

26.05

441.5010257

31.67

5 38.6906863

6.67

636.1115661

40.00

734.4809817

43.33

833.8025945

43.60

933.1998715

44.83

1032.7284226

45.45

11 32.2832446

.20

1231.8623244

46.90

1331.4638538

47.57

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1432.1660868

46.40

1533.4905839

44.18

Graph Between maximum displacement versus percentage of mass reduction

TUNED MASS DAMPER FOR MDOF SYSTEM:

In this procedure mass is reduced by a percentage and respective to the reduction of the mass the

optimized stiffness and damping of TMD is found using some numerical techniques.

Each floor is taken as one degree and TMD is taken as another degree. So combined with TMD it

becomes a twenty two degree freedom system.

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MODAL MASS MATRIX AND MODAL FREQUENCY MATRIX FROM CODE

MM =

1.0e+006 *

Columns 1 through 11

4.9014 0.0000 -0.0000 -0.0000 0.0000 0.0000 0.0000 -0.0000 0.0000 0.0000

-0.0000

0.0000 4.9017 0.0000 -0.0000 0.0000 0.0000 0.0000 0.0000 -0.0000 -0.0000

-0.0000

-0.0000 0.0000 4.9107 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 0.0000 0.0000

0.0000

-0.0000 -0.0000 -0.0000 4.9031 0.0000 -0.0000 -0.0000 -0.0000 0.0000 0.0000

-0.0000

0.0000 0.0000 -0.0000 0.0000 4.9177 0.0000 -0.0000 -0.0000 -0.0000 -0.0000

0.0000

0.0000 0.0000 -0.0000 -0.0000 0.0000 4.9091 -0.0000 -0.0000 -0.0000 0.0000

-0.0000

0.0000 0.0000 -0.0000 -0.0000 -0.0000 -0.0000 4.9148 -0.0000 0.0000 0.0000

0.0000

-0.0000 0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 4.9043 0.0000 -0.0000

0.0000

0.0000 -0.0000 0.0000 0.0000 -0.0000 -0.0000 0.0000 0.0000 4.9544 -0.0000

-0.0000

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0.0000 -0.0000 0.0000 0.0000 -0.0000 0.0000 0.0000 -0.0000 -0.0000 4.9530

0.0000

-0.0000 -0.0000 0.0000 -0.0000 0.0000 -0.0000 0.0000 0.0000 -0.0000 0.0000

4.9162

0.0000 -0.0000 -0.0000 0.0000 -0.0000 0.0000 -0.0000 -0.0000 -0.0000 -0.0000

0.0000

0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 0.0000 0.0000 -0.0000 0.0000

-0.0000

0.0000 0.0000 0.0000 0.0000 0.0000 -0.0000 0.0000 -0.0000 0.0000 -0.0000

-0.0000

0.0000 -0.0000 -0.0000 0.0000 0.0000 0.0000 0.0000 -0.0000 0.0000 -0.0000

-0.0000

-0.0000 0.0000 -0.0000 -0.0000 0.0000 0 0.0000 -0.0000 -0.0000 0.0000

0.0000

-0.0000 -0.0000 -0.0000 -0.0000 -0.0000 0.0000 0.0000 -0.0000 0.0000 0.0000

-0.0000

-0.0000 -0.0000 -0.0000 -0.0000 0.0000 0.0000 0.0000 -0.0000 -0.0000 0.0000

-0.0000

-0.0000 0.0000 -0.0000 -0.0000 0.0000 0.0000 -0.0000 -0.0000 -0.0000 -0.00000.0000

-0.0000 0.0000 0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 0.0000 0.0000

0.0000

-0.0000 0.0000 -0.0000 0.0000 0.0000 0.0000 0.0000 -0.0000 0.0000 -0.0000

-0.0000


0.0000 0.0000 0.0000 0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000

-0.0000 -0.0000 0.0000 -0.0000 0.0000 -0.0000 -0.0000 0.0000 0.0000 0.0000

-0.0000 -0.0000 0.0000 0.0000 -0.0000 -0.0000 -0.0000 -0.0000 0.0000 -0.0000

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0.0000 -0.0000 0.0000 0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 0.0000

-0.0000 -0.0000 0.0000 0.0000 -0.0000 -0.0000 0.0000 0.0000 -0.0000 0.0000

0.0000 -0.0000 -0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0000 0.0000

-0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0000 -0.0000 0.0000

-0.0000 0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000

-0.0000 -0.0000 0.0000 0.0000 -0.0000 0.0000 -0.0000 -0.0000 0.0000 0.0000

-0.0000 0.0000 -0.0000 -0.0000 0.0000 0.0000 0.0000 -0.0000 0.0000 -0.0000

0.0000 -0.0000 -0.0000 -0.0000 0.0000 -0.0000 -0.0000 0.0000 0.0000 -0.0000

4.9142 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0000 -0.0000 -0.0000 0.0000

0.0000 4.9172 -0.0000 -0.0000 0.0000 -0.0000 -0.0000 0.0000 -0.0000 0.0000

0.0000 -0.0000 4.9309 -0.0000 -0.0000 0.0000 -0.0000 -0.0000 0.0000 -0.0000

0.0000 -0.0000 -0.0000 4.9246 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 -0.0000 0.0000 4.9242 0.0000 -0.0000 0.0000 -0.0000 -0.0000

0.0000 -0.0000 0.0000 0.0000 0.0000 4.9530 -0.0000 -0.0000 0.0000 0.0000

-0.0000 -0.0000 -0.0000 0.0000 -0.0000 -0.0000 4.9418 -0.0000 -0.0000 -0.0000

-0.0000 0.0000 -0.0000 0.0000 0.0000 -0.0000 -0.0000 4.9297 -0.0000 0.0000

-0.0000 -0.0000 0.0000 0.0000 -0.0000 0.0000 -0.0000 -0.0000 4.9451 -0.0000

0.0000 0.0000 -0.0000 0.0000 -0.0000 0.0000 -0.0000 0.0000 -0.0000 4.9359

>> W

W =


1.8397 0 0 0 0 0 0 0 0 0 0

0 5.5092 0 0 0 0 0 0 0 0 0

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0 0 9.1488 0 0 0 0 0 0 0 0

0 0 0 12.7388 0 0 0 0 0 0 0

0 0 0 0 16.2597 0 0 0 0 0 0

0 0 0 0 0 19.6925 0 0 0 0 0

0 0 0 0 0 0 23.0187 0 0 0 0

0 0 0 0 0 0 0 26.2204 0 0 0

0 0 0 0 0 0 0 0 29.2805 0 0

0 0 0 0 0 0 0 0 0 32.1827 0

0 0 0 0 0 0 0 0 0 0 34.9115

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0


0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

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0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

37.4525 0 0 0 0 0 0 0 0 0

0 39.7925 0 0 0 0 0 0 0 0

0 0 41.9193 0 0 0 0 0 0 0

0 0 0 43.8220 0 0 0 0 0 0

0 0 0 0 45.4907 0 0 0 0 0

0 0 0 0 0 46.9170 0 0 0 0

0 0 0 0 0 0 48.0937 0 0 0

0 0 0 0 0 0 0 49.0151 0 0

0 0 0 0 0 0 0 0 49.6764 0

0 0 0 0 0 0 0 0 0 50.0745

So here mass of TMD should be reduced by some percentage of modal mass.Here modal mass is

considered in place of total mass of the structure

Modal mass = 4.9014e6.

Modal frequency with respect to 4.9014e6 mass is 1.8397.

For 1%mass we have to take 1% of the mass of 4.9014e6 to get the mass of TMD.Response

graphs for different percentages are shown .

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1%

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2%

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3% mass

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4%

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5%

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6%

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7%

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8%

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9% mass

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10% mass

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11% mass

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12% mass

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13% mass

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14% mass

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15% mass

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%mass

Maximumdisplacement

Stiffness oftmd

Damping oftmd

PercentageReductionindisplacement

NOTMD 336.5394 0 0 0

1 329.00681.62E+05

1.09E+04

2.238

2 322.0323.16E+05

3.05E+04

4.311

3 315.6074.62E+05

5.53E+04

6.250

4 309.67366.01E+05

8.41E+04

8.036

5 304.17177.34E+05

1.16E+05

9.524

6 299.23528.59E+05

1.51E+05

11.012

7 294.79549.79E+05

1.88E+05

12.500

8 290.91291.09E+06

2.27E+05

13.690

9 287.15721.20E+06

2.68E+05

14.583

10 283.99531.30E+06

3.11E+05

15.613

11 280.8887

1.40E+

06

3.54E+

05

16.

667

12 278.36051.49E+06

3.99E+05

17.262

13 275.87171.58E+06

4.46E+05

18.155

14 273.94641.66E+06

4.93E+05

18.750

15 272.08791.74E+06

5.41E+05

19.150

Graph of maximum displacement versus percentage of mass

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SAP2000 MODEL:

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TABLE:BaseReactions Column1

Column2

Column3 Column4

Column5

Column6

Column7

Column8

OutputCase

CaseType StepType

StepNum

GlobalFX Global

GlobalFZ

GlobalMX

GlobalMY

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FY

Text Text TextUnitless KN KN KN KN-m KN-m

DEAD LinStatic 0 0 0 0 0

MODAL LinModal Mode 1 1023.916 0 0 0

3071.74

81

MODAL LinModal Mode 2 -3047.417 0 0 0

-9142.2514


-14995.5613

MODAL LinModal Mode 4 6830.929 0 0 020492.7872

MODAL LinModal Mode 5 8501.246 0 0 025503.7371


-

29910.0815

MODAL LinModal Mode 711202.737 0 0 0

33608.2124

MODAL LinModal Mode 8

-12170.574 0 0 0

-36511.7209

MODAL LinModal Mode 912851.143 0 0 0

38553.4304


-13228.977 0 0 0

-39686.9306


-13295.858 0 0 0

-39887.5741


-13050.971 0 0 0

-39152.9123

ACASE1 LinModHist Max1172218.931 0 0 0

3516656.79

ACASE1 LinModHist Min

-942064.446 0 0 0

-2826193.34

TABLE: Modal LoadParticipation Ratios Column1

Column2

Column3

Column4

OutputCaseItemType Item Static

Dynamic

Text Text Text Percent Percent

MODALAcceleration UX 99.9996 99.8254

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MODALAcceleration UY 0 0

MODALAcceleration UZ 0 0

ABLE: Modalarticipating

Mass RatiosColumn1

Column2

Column3

Column4

Column5 Column6

Colum7

utputCaseStepType

StepNum Period UX SumUX RY SumR

ext Text Unitless Sec Unitless Unitless Unitless Unitle

ODAL Mode 110.296256 0.83008 0.83008 0.98557 0.985

ODAL Mode 2 3.438933 0.0915 0.92159 0.01216 0.997

ODAL Mode 3 2.071611 0.03242 0.95401 0.00157 0.999ODAL Mode 4 1.488622 0.01614 0.97015 0.00041 0.999

ODAL Mode 5 1.167134 0.00945 0.9796 0.00015 0.9998

ODAL Mode 6 0.964572 0.00606 0.985660.00006607 0.9999

ODAL Mode 7 0.826117 0.00412 0.989780.00003349 0.9999

ODAL Mode 8 0.726199 0.0029 0.992680.00001859 0.9999

ODAL Mode 9 0.651295 0.00209 0.994770.00001101 0.9999

ODAL Mode 10 0.593591 0.00153 0.9963 0.000006839 0.9999

ODAL Mode 11 0.548262 0.00113 0.997430.000004389 0.9999

ODAL Mode 12 0.512178 0.00083 0.998250.000002878 0.9999

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BLE:odalarticipatnactors

Column1

Column2

Column3 Column4 Column5 Column6

Column7

tputCas StepType

StepNum Period UX RY

ModalMass

ModalStiff

xt TextUnitless Sec KN-s2 KN-m-s2 KN-m-s2 KN-m

DAL Mode 1

10.2962

56

2749.5602

32

107326.41

45 1 0.37239

DAL Mode 23.438933

-912.892625

11920.15747 1 3.3382

DAL Mode 32.071611

-543.373097

-4287.3001 1 9.19906

DAL Mode 41.488622

383.432417

-2183.90837 1

17.81521

DAL Mode 51.167134

293.335104

1317.782668 1

28.98134

DAL Mode 60.964572

-234.966694

878.762569 1

42.43166

DAL Mode 70.826117

193.663453

625.621958 1

57.84642

DAL Mode 80.726199

-162.578239

466.111805 1

74.85979

DAL Mode 90.651295

138.082138

358.771615 1

93.06883

DAL Mode 100.593591

-118.070241

282.718472 1

112.04328

DAL Mode 110.548262

-101.235444

-226.498119 1 131.336

DAL Mode 120.512178

-86.721025

183.395491 1

150.49373

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TABLE: ModalPeriods AndFrequencies

Column1

Column2

Column3

Column4

Column5

Column6

OutputCaseStepType

StepNum Period

Frequency

CircFreq

Eigenvalue

Text TextUnitless Sec Cyc/sec rad/sec

rad2/sec2

MODAL Mode 110.296256

0.097123

0.61024 0.37239

MODAL Mode 23.438933 0.29079 1.8271 3.3382

MODAL Mode 32.071611 0.48272 3.033 9.1991

MODAL Mode 41.488622 0.67176 4.2208 17.815

MODAL Mode 51.167134 0.8568 5.3834 28.981

MODAL Mode 60.964572 1.0367 6.514 42.432

MODAL Mode 70.826117 1.2105 7.6057 57.846

MODAL Mode 80.726199 1.377 8.6522 74.86

MODAL Mode 90.651295 1.5354 9.6472 93.069

MODAL Mode 100.593591 1.6847 10.585 112.04

MODAL Mode 110.548262 1.8239 11.46 131.34

MODAL Mode 120.512178 1.9524 12.268 150.49

TABLE: TotalEnergyComponents

Column1 Column2 Column3 Column4 Column5

Column6

OutputCaseCaseType Input Kinetic Potential

ModalDamp Error

Text Text KN-m KN-m KN-m KN-m KN-m

DEAD LinStatic 0 0 0 0 0

MODAL LinModal 0 0 0 0 0

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ACASE1LinModHist

770533.9263

630068.9029

123981.9311

547678.3559 0.0017

MAXIMUM DISPLACEMENT FROM SAP CONSIDERING MDOF SYSTEM:

A check for 8 and 10 percent mass reduction is done using SAP2000.

The maximum displacement is 224.8 using SAP2000 . From code the value is 283.995 mm. So a

difference of 60 mm is found. A reduction of 33.33% at 10% mass.So TMD can be designed for

8% mass.

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So mass of TMD = 4.1094*104 Kg

Stiffness of TMD= 1.0900*105 N/m.

So accordingly fix the sizes of beams and columns of the roof top TMD to have a mass of

4.1094*10

4

and stiffness of 1.0900*10

5

N/m.

CONCLUSIONS:

1. The maximum displacement of the structure considering it as SDOF is less compared to

MDOF system. So it is preferred to analyze as a MDOF system.

2. As the percentage of mass is increased the reduction in displacement is also increased.

3. But it is not recommended to go beyond some percentage as it is not economical and is

not an optimized structure.

4. As the stiffness is increased the damping of TMD is also increased while the response of

the structure decreased.

5. When gone beyond certain percentages there is less significant decrease in maximum

displacement.

6. In SAP2000 33.33% reduction in response is observed at percentage 0f 10 of mass.

7. Consider between 8 to 10 percentages of mass where the reduction is around 15% and

design will also be economical. While for an SDOF system the reduction is more(25%)

for a less mass percentage compared to MDOF system.

8. Here only first mode is considered for tuning to required natural frequency.

9. Using SAP2000 the maximum displacement is found to be 224.8 compared to 283.995

by using matlab codes.

10.So finally design TMD for percentages in range of 6 to 8 to get both an optimized TMD

with significant reduction in maximum displacement.

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REFERENCES:

1. Optimal design theories and applications of tuned mass dampers by Chien-Liang Lee,Yung

Chen, Lap-Loi Chung,Ten-Po Wang.

2. Parametric study and simplified design of tuned mass dampers by Rahul Rana and

T.T.Soong.

3. Passive and active structural vibration control in civil engineering by T.T.soong and M.C.

Constantinou, State university of Newyork at buffalo.

4. Structural dynamics and vibration in practice by Douglas Thorby.

5. Modal Analysis He& Fu.

6. Li C,Liu Y.Ground motion dominant frequency effect on design of multiple tuned mass

dampers, Jouranl of Earthquake Engineering 2004;8:89-105.

7. SimiuLi E,Scanlan RH.Wind effects on Strcutres,fundamental and applications todesign.3rd ed.NY: John Wiley and Sons Inc.:1996.

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