parametric study and simplified design of tuned mass dampers

ELSEVIER PIh S0141-0296(97)00078-3

Engineering Structures, Vol. 20, No. 3, pp. 193-204, 1998 © 1997 Elsevier Science Ltd

All rights reserved. Printed in Great Britain

0141-0296/98 $19.00 + 0.00

Parametric study and simplified design of tuned mass dampers Rahul Rana and T. T. Soong

Civil Engineering Department, State University of New York, Buffalo, NY 14260, USA

This paper summarizes the results of a parametric study performed to enhance the understanding of some important characteristics of tuned mass dampers (TMD). The effect of detuning on some of the TMD parameters on the performance is studied using steady-state harmonic excitation analysis and time-history analysis. The El Cen- tro and Mexico excitations are used for time-history analysis. The effects of tuning criteria and significance of numerical tuning are also studied. The correspondence between the design of a TMD for a SDOF structure and a certain mode of a MDOF structure is drawn to simplify TMD design to control a single mode of a multi- modal structure. An example is given to illustrate the design procedure. Investigations are made regarding controlling multiple structural modes using multi-tuned mass dampers (MTMD). © 1997 Elsevier Science Ltd.

Keywords: tuned mass dampers, multi-tuned mass dampers, parametric study, design

1. Introduction

Tuned mass dampers (TMDs) are amongst the oldest structural vibration control devices in existence. The concept of vibration control using a mass damper dates back to the year 1909, when Frahm invented a vibration control device called a dynamic vibration absorber. While TMDs are among the oldest structural vibration control devices, there has been a resurgence of interest in their study in recent years. A number of newer tall buildings, particularly in Japan, are now equipped with various versions of such a system for vibration mitigation under wind and moderate earthquakes. A summary of these recent applications can be found in Soong et al. ~ Since a passive TMD (Figure 1) is the fundamental mechanism, present in all such appli-

A b s o r b e r m a s s

/ / / / / / / / / / / / / / / /

Figure 1 A schematic representation of damped vibration absorber suggested by Den Hartog (1928)

cations, the understanding of TMD behavior and its design become important problems.

The vibration control device invented by Frahm did not have any inherent damping. It was effective only when absorber's natural frequency was very close to the excitation frequency and it suffered a sharp deterioration in its performance if the excitation frequency deviated away from absorber's natural frequency. In addition, if the excitation frequency approached any of the two natural frequencies of the structure-absorber system, a very large response could occur at resonance. Therefore, it was effective only for the case where the frequency of the exictation was known so that the absorber could be designed with a natural frequency equal to the excitation frequency. This short- coming was later eliminated when Ormondroyd and Den Hartog 2 showed that, if a certain amount of damping is introduced in Frahm's absorber, performance deterioration under changing excitation frequency will not be very sharp and response at resonance can also be significantly reduced.

Den Hartog also derived closed form expressions for optimum damper parameters. He assumed no damping to be present in the main mass to facilitate the derivations. Later, damping in the main mass was included in the analysis by Bishop and Welbourn. 3 While Den Hartog considered absorbers with viscous damping only, Showdown 4 extended it to include different types of absorber damping. Falcon et al. 5 devised a procedure for optimizing an absorber incorporating a restricted amount of damping applied to a damped main system. Ioi and Ikeda 6 developed

193

194 Tuned mass dampers: R. Rana and 7-. 7-. Soong

correction factors for the absorber parameters as functions of the main mass damping assuming light main mass damping. Warburton and Ayorinde 7 tabulated numerically searched optimum values of absorber parameters for certain values of absorber to main mass ratio and main mass damping ratio. Thompson ~ presented a frequency locus method to obtain optimum damper parameters. Warburton 9 derived closed form expressions for optimum absorber parameters for undamped single-degree-of-freedom (SDOF) system for harmonic and white noise random excitations. Vickery et al. m considered a damped SDOF structure-damper system with damper to main mass ratio of 5%. They developed graphs to obtain absorber response and added effective damping due to absorber. Tsai and Lin l~ numerically developed plots to obtain optimum damper parameters for harmonic excitations. They also presented empirical expressions which fit the obtained plots.

The objective of this paper is to present a simplified procedure for TMD design and enhance the understanding of TMD behavior with the help of a parametric study. In addition, a study is performed to investigate the possibility of controlling multiple structural modes using multi-tuned mass dampers.

2. Equations of motion and classical solution

Referring to 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 m K k C P(t )

p(t)

p( t ) =

main mass absorber mass main spring stiffness absorber spring stiffness absorber damping force acting on main mass. In case of base excitation with acceleration £g(t), P(t) = -MY~(t). force acting on damper mass. It is given as:

P(t) for base excitation

for main mass excitation

To facilitate further discussions, additional notations are introduced here as follows:

/~ damper mass to main mass ratio, /~ = m/M. oJ frequency of a harmonic excitation. [ l natural frequency of main mass, ~ = ~/K/M. ~o, natural frequency of damper mass, w, = ~/k/m. g, ratio of excitation frequency to main mass natural

frequency, g~ = ¢o/~. for MDOF structures, g, = w/l~,, where [~l is the first modal frequency of the structure.

f frequency ratio, f = WalO. ~'d damping ratio of TMD.

damping ratio of main mass.

Den Hartog ~2 developed closed form expressions of optimum damper parameters f and G which minimize the ste-

ady-state response of the main mass subjected to a harmonic main mass excitation. These expressions for calculating optimum damper parameters are given as:

1 f ' " ' = i+/x (3)

~'d"e' = 8(i+/z) (4)

For the case when the structure is subjected to a harmonic base excitation, the corresponding expressions can be easily found to be:

' ( , / y ) (5)

(6)

Using the values of G.p, and f,pt, optimum values of damping c and stiffness k of the damper can be calculated as

w.,,p,_ \/kopt/m f , pt : f l -- [~

which gives

kopt =,~p, [)~2m

Similarly

(7)

Copt C o p l Y bd°pt C c 2e%m

which gives

Cop t = 2~d,,pt fopt Dm (8)

3. Inherent damping in main mass: numerical optimization

In Den Hartog's derivation of optimal damper parameters, it is assumed that the main mass is undamped. In the presence of damping in the main mass, no closed-form expressions can be derived for the optimum damper parameters. However, they may be obtained by numerical trials with the aim of achieving a system with smallest possible value of its higher response peak. Figure 2 demonstrates

8 8

"~ p-O.02

i ~-2%

/ ! //' '~

/ \

/ ' ~ ! ~-0.02 ~-2~'0

f 1:

/ /r' ",

/

/ \ 2 / " .... "Optlmlzalion method "" Optimization method -

- - Numedc~d: ¢.09~8479, ~ .000~T81 Numgrk~ : f . , 0 . 9 7 0 ~ E . . -o~=' . . . __ Cloelsd 4OTm:f.0 g7~478 ~.a =0.08a~49 ~ - - C I o N d form: f.O.geO~9~,

m~

°o'.s ?.o ~.s % ,'.0 1~ Frequen~'y ratio gl Frequency ratio gl

Main mass excitation Base excitation

Figure2 N u m e r i c a l o p t i m i z a t i o n , n e c e s s a r y in p r e s e n c e o f i n h e r e n t d a m p i n g

Tuned mass dampers: R. Rana and T. T. Soong

the necessity of numerical optimization of damper parameters when damping is present in the main mass. In this case, the equation of motion of the main mass, equation (1), is modified by adding the term C~((t) to the left-hand side. Under a harmonic excitation Psintot, steady-state response of the main mass and damper mass can be expressed in a dimensionless form as:

X S,, Z

X/1 + 4 ( ~ ) 2 ( 1 + A / . z ) 2

(9)

(10) X~t Z

Z = [ ~ _ 14 gl 2 4~'~fg2 + f2 gZ(l+/-t) - _ _ ]j2 ~'dg, _ 2,~g, ( l l ) + 2 + 2 3 ( l + / x ) _ z T

where X~t displacement of main mass under a static load of

magnitude P, Xst = P/K and

A = I 1_ for base excitation L0 for main mass excitation

Quantities X/X~, and x/Xs, are also termed as main mass response ratio and damper mass response ratio, respectively.

In a numerical approach, several combinations of damper parameters ffa and f are investigated in a systematic manner until the best combination is reached. Since real structures have only a small amount of inherent damping, available closed form expressions for zero inherent damping give good starting values in the numerical search. A mimimax approach was used by Randall et al . j3 to develop design graphs for obtaining optimum damper parameters for SDOF structure under main mass excitation. Given the main structure's damping ratio ff and damper to structure mass ratio

optimum damper parameters ffd andfopt can be directly ' . o p t

obtained from these design graphs. Thus, the presence of inherent damping makes numerical

tuning necessary. Realizing the necessity of numerical tuning and the fact that damper's optimum parameters are different for main mass excitation and base excitation, practical design tables TM have been developed to facilitate the design of a TMD in the presence of inherent structural damping. An example of such design tables is given in Table 2 and its use is illustrated in an example presented later.

4. P a r a m e t r i c s tudy

It is well known that a TMD can be designed to control a single structural mode only. Given the properties of the mode which needs to be controlled, the design problem is essentially the same as designing a TMD for a SDOF structure. Parametric studies were performed on a SDOF structure-TMD system to enhance the understanding of TMD behavior. The numerical optimization using minimax approach, as described in a previous section, was used for obtaining optimum TMD parameters.

195

18 16

| 1 0

I 8

4

" 2

0

Main mass damping t;-O.O0 v,-o.oz ~.o.o6

. . . . I . . . . , . . . . . i . . . .

=f~mtion detunh~g of 1; d

. . . . . . . . . 0.20 0.00 (optimum)

~ - - +0.20 . . . . i . . . .

Main mass damping ~,-0.06 ~-0.02 ~0.06

1618 . . . . , . . . . I._'_ ' ' ' ' ' ' ' 1 | 12

1 0

8

| 4 ,

- 0 . . . . I , , , , i i i i I . . . .

.06 0.00 0,05 -0.06 0.00 0.05 Fraction cletuning of f Fraction detuning of f

Figure3 Ef fec t o f d e t u n i n g : h a r m o n i c m a i n m a s s e x c i t a t i o n

4.1. Effect of detuning If the TMD parameters shift away from their respective optimum values, the response control is expected to degrade. A steady-state harmonic analysis, with varying excitation frequencies, is performed to study the effect of detuning. Obviously, the structure's steady-state response varies with changing excitation frequency and Figures 3 and 4 show the obtained peak structural response ratio with detuning in damper parameters. These figures show the effect of detuning of damper parameters optimized for harmonic main mass excitation and base excitation, respectively. The following can be observed from these plots:

(1) The detuning effect of parameter f is more pronounced than that of the parameter ¢d-

(2) With increasing damping ¢ of the main mass, the effect of detuning becomes less severe.

(3) With increasing mass ratio/x also, the effect of detuning becomes less severe.

18 16

12 10

8

~ 4 & 2

0

! @

O .

18 16 14 12 10

4

2 0 . . . . i : , , ,

- 0 . 0 5 0 . ~ 0 . ~

Fraction detuning of f

Figure 4

Main mass damping~-O.O0 p.o.o2 ~-0,06

• Fraction deluning of ~'d . . . . . . . . . 0.20

0.00 (optimum) - - - - +0.20 , , , , I , i i i . . . . i , h , ,

Main mass damping ~-0 .06 ~,o.o2 ~o.o6

. . . . i . . . . I . . . . i . . . .

F : . ..... / t - 0 . 0 5 O . C O O . C ~


E f fec t o f d e t u n i n g : h a r m o n i c base e x c i t a t i o n

196

c

8

0.4

0.2

0.0

-0.2

-0.4

i I i

, / I i , 0 10 20 30 40

Time (s)

0,2

0.1

0.0

-0.1

-0,2 I , i , I /

Time (s)

0.015 ~ 0.015

0.010 0.010

0.005 0.005

0.000 0.000 ~!o 0.0 10.O 20.0 30.0 0.0 2 0 4 0

Frequency (Hz) Frequency (Hz)

Figure 5

8 0

(a) El C, entro excitation (b) Mexico Excitation

El Centro and Mexico exci tat ions

The effects of detuning in TMD parameters are also studied by performing a time-history analysis on a SDOF structure of mass m=10 kg and stiffness K=1000 N/m, resulting in natural frequency ~=1.592 Hz. The El Centro and Mex- ico earthquake time-histories, which significantly differ in frequency contents (Figure 5), are used as excitations. Peak and root-mean-square (RMS) displacements are plotted with increasing detuning of damper parameters. The results are shown in Figures 6-9. The following observations can be made:

(1) The response control effect of TMD is not always found to be significant for the particular SDOF structure and earthquakes considered, particularly at higher values of main mass damping. Also, it is seen that the values of optimum damper parameters f , pt and ~d,,p,, obtained for steady-state harmonic base excitation,

11

8

O .

Main mass damping ~ - 0 . 0 0

1~0.02

. . . . . . . f l '

Fraction detunlng o f 1; (~

. . . . . . . . . 0 . 4 0

o.oo (optimum) - - - - + 0 . 4 0

. . , , , ,

~ 0 . 0 2

11 . . . . .

a' 8

UNCONTROt.U~. 7JUl om

a, .

6 . . . . . : " " - 0 . 7 5 o . ~ 0 . 7 5


Figure 6 quake

~,-o.o~

,,".,,..'*'

IJNCOi~ROU.E~. I0.1S ~

Main mass damping ~=0.06 v...o.o~

" " " * ' ~ . . ~ . : . . . . . . . .

- 0 . 7 5 0 .00 0,75 Fraction detuning of f

Effect of detun ing on peak response: El Centro earth-


7

.e 3

==-2

7

4

3 09 ~2 u , c o ~ r ~ a . ~ ~.~ ~

1 . . . . . . . . -0 .7

Figure 7 quake

Main mass damping ~-0.00 1~.0.02 . . . . . . *

P rac t ion d e t u n i n g o f l ; d . : / /

. . . . . . .

Main mass damping ~-0.06 ~=0.02 ~-0.0~

.....-'" ~

...r'

.:" . f

UNCONTROLLB~ 4.e2 am

UNCONTROU.ED: 1.79 am

0.00 0.75 -0.75 0.00 0 ,75

Fraction d e t u n i n g o f f Fraction detuning of t

Effect of detuning on RMS response: El Centro earth-

6

g4

O.

Main mass damplng~=0.00 1,-0.02

Fraction detunlng o f 1~, 4 ,. , .-"

. . . . . . . o : : , o "

'. - - - + o.~ ...//..~

' UNCONT1ROLLIED R ESPON.gE: 0.(10 ¢¢n'

~-0,06

UNCONTROLLED: ¢6 om

6 . . .

~3 u.com'~ou.~, z4= ~.

-0.175 . . . . i . . . . 0 .00

Figure 8 quake

Main mass'damping ~ ; = 0 . 0 6

v,-o.oa v=o.o~

UNCOh'T ROL.LED: 2-43 om

" ~ , , . ~ . . . . . . ~ . . ~ -0.75 0.00 0,75


Effect of detuning on peak response: Mexico earth-

0.75


(2)

hold reasonably well for the earthquake excitations considered. This fact essentially states the practicality of the use of steady-state harmonic excitation in TMD design. The effect of detuning is not as significant as that seen in the steady-state harmonic analysis. Qualitatively, the effect of detuning on RMS response corresponds well to what is predicted by the steady-state harmonic analysis. The peak response results of the time-history show less correspondence to the steady-state analysis results.

4.2. Significance of tuning criteria The optimum damper parameters ~d and f,,n, are different depending on whether the excitation is acting at the base or at the main mass. The optimum damper parameters are

Tuned mass dampers: R. Rana and T. 7-. Soong 1 9 7

1 . 0

. , 0 . 9

I 0 . 8

0.7 u )

=~ 0 . 6 m

Main mass damping~,-0.00 p,,-O.~

{ " - ; , I

'~ . . . . . . o~ . . ' ~ ~ ~ ( ~ . . " / )

. . . / /

:/7 / ' :(

"...... UNOONTP~.LI~2.04om

0.5

1.0

~ 0 . 9

Z o.e

~ o . 7

0.6

Main mass damping ~-0.06 w-o.o2 ~-o.06

. . . . .

~ 0 J ~

0.5 . . . . ' . . . . -0.75 0.00 0,75

F r a c t i o n d e t u n i n g o f f

Figure 9 q u a k e

uNco NT~DLL~3: 0.M ¢~

, , , , , . . . .

-0.75 O,OO 0.75 F r a c t i o n d e t u n i n g o f f

E f f e c t o f d e t u n i n g o n R M S response: M e x i c o e a r t h -

calculated for the two criteria for varying structural parameters and are plotted in Figure 10. One can make the following observations:

( 1 ) As inherent damping ff in the main mass increases, the difference between the values of parameter fop, obtained by the two criteria also increase.

(2) The same as above is also observed for increasing mass ratio /z.

1.00

0 . 9 5

0.85

0.80 0

0 . 2 0

0.15

0 . 1 0

E f f e c t o n o p t i m u m t u n i n g r a t i o

i i i i

- - B u o excllnUon ~

, I , I , I , I ,

2 4 6 8

E f f e c t o n o p t i m u m d a m p i n g r a t i o

~" , t ' i i r _

i t - 2 %

• T u n i n g o r R e d a ~ , - 1 0 %

- - ~ m a ~ excRatJon - - - - ~ o ~ o n

0 . 0 ~ , I , I , I , l

0 2 4 6 8

~. ( % )

Figure 10 E f f e c t o f t u n i n g criteria o n o p t i m u m p a r a m e t e r s

1 0

1 0

(3) The parameter ~'do-, follows a similar pattern as fop,, though with less c[~ity.

One refers to the results of the previous section, since the effect of detuning of damper parameters becomes less severe with increasing /x and if, at higher values o f /x and if, increasing differences between optimum parameters based on the two criteria will be compensated by decreasing detuning effects.

4.3. Significance of numerical tuning The presence of inherent damping, ~', makes the closed form expressions for calculating optimum damper parameters inapplicable (Figure2). The plots are shown in Figure 11 to investigate how much is the difference between optimum parameters obtained by numerical search and those by closed form expressions. This figure illus- trates that

(1) As the inherent damping increases, the difference between the 'true' f~p, (by numerical search) and 'approximate' fop, (by close form expressions) increases, which is an obvious result since close form expressions are obtained assuming zero inherent damping. This difference also increases with increasing mass ratio /z.

(2) A similar result is obtained for parameter ffdop,, although a smaller difference is observed between the 'true' and 'approximate' values.

Based on detuning characteristics discussed previously, it can be argued that since the effect of detuning on response is less severe at higher values of ~ and /x, and since at small values of ~ and /z the difference between ~dop, and fopt, obtained numerically and by close form expression, is not significant, numerical tuning perhaps can be avoided all together. However, it should be noted that

1 .00

0 . 9 5

~ ' 0 . 9 0

0 . 9 5

,,_ 1~'0.90

Main mass excitation

• v • i , i , i •

I ~ . . . . . 0 .15

0 .10 0 . 8 5 T u n i n g m e t h o d

- - Nun~m~ ~ - - Cio~KSform~lln 0

0 . 8 0 ' ~ ' ' ' ' ' ' ' 0 . 05 , 0 2 4 6 8 10

Base excitation 1 .00 , ,

0 . 1 5

0 .10

0 .85

0 .20 . , . , . , . , •

p.,,2%

p.-6%

Tun ing method ~-10~ - - Numeric4d - ~ CIoI4KI form

I , I , I , L * 2 4 @ 8 10

S i g n i f i c a n c e o f n u m e r i c a l t u n i n g Figure 11

0 . 8 0 0 " ' " I . , = I = 0 . 0 5 , I , I , I , t , = , ( ; , ,o 2 , , ,

% ¢ ( % )

J r - r 'uning method v.-10~

- - Num~'k~d -- ~ C , I ~ form

1 it-6%

it-2%

0 . 2 0 , . , . , . ,

198 T u n e d mass d a m p e r s : R. Rana a n d 7-. T. S o o n g

a difference does exist between the two and, with the help of design aids ~4, numerically searched ~'dop ' and fop, are as easy to use as the close form expressions.

mE(t) + c{Jc(t)-,~j(t)} + k{x( t ) -X j ( t ) } = Ap(t) (15)

where A has same meaning as in equations (9) and (10) and

5. D e s i g n i n g T M D to c o n t r o l a p a r t i c u l a r s t r u c t u r a l m o d e

Equations of motion for a SDOF undamped structure- TMD system were written as equations ( 1 ) and (2). For an N DOF proportionally damped structure-TMD system with a TMD placed at the jth floor of the structure (Figure 12), the equations of motion can be written as follows: For a general rth floor mass

M ~ ( t ) + C ~ ( t ) + K~Xr(t) - ~U[c{±(t) -

J(,(t)} + k{x(t)-X~(t)} ] = P,(t) (12)

where

6ri = {~ r=AJ - r = j

for the damper mass

mY(t) + c{Sc(t)-,~,(t)} + k{x ( t ) -Xr ( t ) } = p( t ) (13)

where

m p( t ) = IM: Pj(t) for base (earthquake) excitation

[0 j for super structure (wind) excitation

Pi(t) = ~ivP(t), P(t) being the load vector acting on the structure; y( t ) generalized displacement of ith mode

Comparing equations (14) and (15) to the equations of motion for the SDOF structure-TMD system, equations ( 1 ) and (2), one can observe that these two pairs of equations differ on two accounts, namely, the presence of the term &u in equation (14) and the presence of term Xi(t) instead of y( t ) in both equations (14) and (15). However, if the structure's ith mode shape vector is normalized with respect to its jth element, which corresponds to TMD location (jth floor), ~b 6 becomes unity and Xs(t) = chuy(t) = y(t), and equations (14) and (15) reduce to the same form as equations (1) and (2). Thus if 4)0 is unity, the expressions for calculating the steady-state ith modal response and damper response in a MDOF structure-TMD system will be exactly same as those for main mass and damper mass responses, respectively, in a SDOF structure-TMD system.

It is stated here that, simple design aids j3,i4, which are developed for designing a TMD for a SDOF structure can be directly used to design a TMD for a certain structural mode of a MDOF structure, provided that the corresponding modal mass is obtained using a mode shape vector, which is normalized with respect to its element corresponding to the TMD location. Design procedure using the above-described approach is simple and is illustrated by the following example.

If the TMD is to be designed for ith structural mode with modal properties Mi, Ks and Ci, the design problem is essentially similar to that of designing a TMD for a SDOF structure. Relevant equations of motion can be written as:

M/y'(t) + Cig(t) + Kiy(t) - d)u[c{(Sc(t)-Xi(t)} (14)

+ k {x ( t ) -X j ( t ) } ] = Pi(t)

6. A d e s i g n e x a m p l e

In this example, the design of a TMD tuned to the first mode of a three DOF structure, described in Table 1, is carried out under harmonic base excitation. A practical problem of controlling a MDOF structure with a dominant first mode under moderate earthquakes will be similar to the discussed design problem.

Referring to Table 1, the first-mode mode-shape vector ~j of this structure is given as:

&l = [0.737 0.59! 0.328] r

It is clear that, in the first mode, the top floor will undergo the largest steady-state deflection under a harmonic excitation. Therefore, the TMD should be placed at the top floor for best control of the first mode. Since the TMD will be placed at the top floor, the mode-shape vector ~b~ should be normalized with respect to its first element to calculate the structure's first modal mass. Therefore, the normalized &j is given as:

[ "" I N ~ : " .......... z:~...

/ / / / /

Figure 12 A general N DOF structure-damper system

&l,, = [1.000 0.802 0.445]r

NOW, the first-mode modal-mass can be calculated as:

M, = 6~,,M6L,, = 18.41 kg (16)

If the damper mass is taken to be 2% of the entire building mass, then

Tuned mass dampers: R. Rana and T. T. Soong 1 9 9

Table 1 Propert ies of the three DOF structure considered

[ 00] [,00o_1O0Oo M(kg) = 10 0 K(N/m) = -1000 2000

-1000 0 10

C(N-s/rn) : 1.519 0.272 0.0971 ['0.737 --0.591 0.272 11343 0.176 / ~ : 01591 01328 0.097 0.176 1.246J L0.328 0.737

~1 = 0.708306 Hz,~2 = 1.98463 Hz, ~3 = 2.86787 Hz

~1 = 2.0%, ~'2 = 0.5%, ~'3 = 0.3%

-1000 2oooj

-0 .328] 0.737 /

-0.591J

m = 0.6 kg (17)

Therefore, the damper mass to structure's first-mode modal mass ratio is:

0.6 - 0.03259

/xj - 18.41

The first-mode modal damping ratio is known to be ff~ = 2%. Using these known values of/~t and ff~, the optimum damper parameters, fopt and ffao-,, can be found from numerical search, or more easily /;rom the appropriate design table (Table 2). One obtains

fopt = 0.952, ~'dov, = 0.11

Using 1 ~ = 2 ~ (0.708) rad/s (Table 1) and equations (7) and (8), the stiffness and damping coefficient of the TMD are found to be

kopt = 10.77 N/m

and

(18)

Cop, = 0.559 N - s / m (19)

Thus, given by equations (17), (18) and (19), all three parameters of the damper are now known. This completes the design of the TMD tuned to the first mode of a three DOF structure.

7. Multi-tuned mass dampers

A single TMD can only control the mode for which it is designed. For controlling an additional mode a separate TMD can be used. Therefore, the concept of multi-tuned mass dampers (MTMD), i.e. having a separate TMD for every structural mode appears to be worth investigating.

The concept of MTMD is relatively new as compared to that of TMD. Much of the research in this area has been done not with the aim of controlling multiple modes, but a single mode only. Igusa and Xu ~5 and Xu and Igusa j6 examined a SDOF structure-MTMD system with dampers' natural frequencies distributed over a range and analyzed it for controlling structures subjected to a wide-band random input. They found MTMD to be more effective and robust than a single TMD of equal mass. Yamaguchi et al. t7 and Kareem et al. ~8 performed detailed parametric studies on SDOF structure-MTMD systems under harmonic and random excitations, respectively. Like Xu and Igusa ~6, their results show that multiple mass dampers with natural frequencies distributed about the natural frequency of the structure are more effective for both robustness and response control. In this paper, the purpose of MTMD design is to control multiple modes. The design method described earlier is used to tune each damper to a particular mode.

7.1. Preliminary harmonic analysis

An analysis is carried out using a single TMD tuned to different modes in turn. Each TMD is placed at its most effective location, namely, the antinode of the mode. For

Table 2 A design table of op t imum damper parameters for harmonic base exci tat ion

/~ = 3 . 0 % /~ = 3 . 5 %

~(°/o) fop, ~'~op, { X ~ ~(°/o) fop, ~ ))(st peak

~aoo t

0.0 0.963384 0.103248 8.411 0.0 0.5 0.961585 0.104848 7.861 0.5 1.0 0.959585 0.105448 7.372 1.0 1.5 0.957585 0.106848 6.938 1.5 2.0 0.955385 0.106648 6.552 2.0 2.5 0.953186 0.107248 6.204 2.5 3.0 0.950786 0.106448 5.893 3.0 3.5 0.948386 0.106648 5.609 3.5 4.0 0.945987 0.107248 5.350 4.0 4.5 0.943387 0.107048 5.114 4.5 5.0 0.940787 0.107248 4.898 5.0

0.957585 0.955585 0.953386 0.951186 0.948786 0.946387 0.943987 0.941387 0.938788 0.936188 0.933588

0.111648 0.113448 0.113048 0.113848 0.113048 0.113448 0.113848 0.114248 0.114448 0.114848 0.115848

7.830 7.349 6.925 6.544 6.204 5.893 5.614 5.355 5.121 4.907 4.708

200 Tuned mass dampers: R. Rana and T. T. Soong

16 14 12

1~ 10

4 2 0

Third floor response 8 ' I '

4

2

0 ' 1 1 6 • •

1 2 ~ 6 1~ 10

4 2 2 0 0

Second floor response I i

16 ' I" ' 14 i ~ 12 s l

4 2 0

0 . 5

I 4

2

0 1.0 1.5 2 .25

Frequency ratio g l

First floor response

2.75 3.25 3 4.0 4.5 Frequency ratio g l Frequency ratio g l

Figure 13a Preliminary harmonic analysis with first mode TMD

these preliminary harmonic analyses, TMDs have been designed with a mass ratio /x = 2%. Results are shown in Figure 13a-c.

It is clear from these analyses that the presence of a first- mode TMD reduces the second and third mode responses as well. The second-mode response is moderately reduced and the third-mode response is slightly reduced. The presence of a second-mode TMD reduces the third-mode

response significantly, but slightly increases the first-mode response. The presence of a third-mode TMD reduces the second-mode response moderately but slightly increases the first-mode response. Since the second- and third-mode TMDs slightly increase the first-mode response, one might expect the overall response to increase in the presence of second- and/or third-mode TMD. The above observations are specific to the structure considered and similar analyses may yield different results for another structure.

60 5O

~;40

2O 10 o

Third floor response

L A 1 t° r ~ , , ' ~'--~ 0.0 , I ' " ~ --">~ 0.0

fl

~t

t~ I

Second floor response

,~80 0.5 0.5 2O 10 0 0.0 ~ ' " ~ ' '0.0

t 6 0 i

5O 1~40

2O , o

0 0 .5 1.0

Frequency ratio gl

First floor response

i l o , ;, , / , °

0.5 0.5

0.0 L. - , ~ , JO.O 1.5 2.25 2.75 Frequency ratio g l

Figure 13b prel iminary harmonic analysis with second mode TMD

• I ' t '

I

3.25 3.5 4.0 4 .5

Frequency ratio g l


6O

1 ~ 4 0 >~ m with TM

3 0 - - w/oTM 2 0 10

0

Third floor response

I ' i i ~ 4 I I ~

2 ~ _ I It ~ ' ~

0 , t ~ 0 . 0 / I =

6O 5O

1 4 0 30 2 0 10

0

Second floor response 6 , • i ' ,0.3

2

0 ~ 1 ' , 7 0 . 0

6O 5 0

1 ~ 4 0 so 2O 10

0o.5

, 6

C 1 .o 1.5 Frequency ratio g l

Rrst floor response

z.25 2.7s 3.25 3.5 4.o 4.~ Frequency ratio g l Frequency ratio g l

Figure 13c pre l im inary harmonic analysis wi th third mode TMD

2 0 1

7.2. Preliminao' time-histo O' analysis

Before actually installing a MTMD in a structture, it is desirable to observe the performance of a single TMD tuned to each structural mode in turn. Time-history analysis is performed on the three DOF structure considered under El Centro and Mexico earthquakes and a summary is presented in Table 3. Following results are obtained from these analyses:

( 1 ) Under E1 Centro earthquake, the presence of the first- mode TMD significantly reduces the RMS response and moderately reduces the peak response. Second- mode TMD results in minute and third-mode TMD results in moderate increase in the RMS and peak response.

(2) Under Mexico earthquake, first-mode TMD results in minute reduction of both RMS and peak responses. Second- and third-mode TMDs result in minute to moderate increase in the RMS and peak responses.

Table3 Structure-TMD system t ime-h is tory analyses results

Tuned to ( locat ion)

TMD descr ipt ion Floor no. Parameters

El Centro Mex ico Peak (cm) RMS (cm) Peak (cm) RMS (cm)

1st mode m = 0.6 kg 3rd (3rd f loor) k = 10.77 N/m 2nd

c = 0.56 N-s/m 1st TMD

2nd mode m = 0.6 kg 3rd (1st f loor) k = 85.72 N/m 2nd


3rd mode m = 0.6 kg 3rd (2nd f loor) k = 179.29 N/m 2rid


w/o TMD

13.42 4.50 -22.34 5.63 9.98 3.54 -18.00 4.54 5.32 2.08 -10.39 2.60

-40.73 14.62 71.41 18.39

14.77 6.49 -23.80 5.72 11.71 5.19 -19.39 4.67 6.15 2.91 -11.14 2.69

-9.82 3.84 -13.38 3.21

16.11 6.59 -24.75 5.84 11.03 5.25 -20.05 4.77 6.58 3.02 -11.30 2.73

11.45 5.67 -21.35 5.10

3rd 15.65 6.45 -23.47 5.66 2nd 11.02 5.13 -19.07 4.61 1st 6.18 2.94 -10.92 2.64

2 0 2 Tuned mass dampers: R. Rana and 7-. T. Soong

The above results indicate that the performance of the s t ruc ture -TMD system depends on the following:

( 1 ) The relative significance of individual structural modes in determining the overall response. A TMD tuned to the most dominant structural mode will be most effective. Also, as could be predicted, based on the harmonic analysis, the structural response observed in t ime-history analysis is found to be slightly deterio- rated by the presence of a second- or third-mode TMD, due to the deterioration in the first-mode response.

(2) The natural frequencies of the structure and the frequency content of the excitation. If the major frequency content of the excitation is away from the structure's natural frequencies, the presence of a TMD may not cause much response reduction. This is the reason of less effectiveness of the first-mode TMD under the Mexico earthquake. Since the Mexico earthquake is characterized by a frquency content highly concen- trated at a frequency of approximately 0 . 5 H z (Figure 5b), while the first mode natural frequency of the structure considered is 0.71 Hz (Table 1).

7.3. Analysis using MTMD As found above, the presence of a second- or third-mode TMD alone does not always result in a response reduction of the structures considered. In this section, an analysis is performed with more than one TMD to determine whether a net response reduction is possible when multiple tuned mass dampers are used simultaneously.

Assuming that 2% of the building-mass is the total available mass for all the dampers, an appropriate mass distri- bution among the various TMDs must first be determined. A response analysis of the structures with harmonic base excitation of frequencies, varying over a range which covers all three natural frequencies, was done. Peak

response ratios were found to be, in the first mode- -50 .0 (at the top floor), in the second mode- -7 .5 (at the first floor) and in the third mode- -2 .5 (at the middle floor). These responses show the relative importance of various modes in determining the overall structural response and TMD masses can be distributed in the ratio of 50.0:7.5:2.5 for the first-, second- and third-mode TMD, respectively.

7.3.1. TMD effect on other modes When a TMD is installed in the structure to control a particular mode, properties of the finally obtained system become different from those of the original structure. Now, if an additional TMD tuned to another mode is also to be installed, it may not perform as expected because of this effective change in structural parameters. Also, the addition of a TMD may affect the performance of TMD(s) already present.

This interaction effect is discussed with the help of a harmonic base excitation analysis. The parameters of the various TMDs used are given in Table 4. Figure 14a-c demonstrates the effect of a TMD on the other modes of the structure. The following observations are made:

(1) To effectively control any particular mode, a separate TMD, specifically tuned to that mode, must be provided.

(2) The structural response of the first controlled mode is marginally increased due to the presence of TMDs tuned to other modes.

(3) The structural response of second and third controlled mode is marginally reduced due to the presence of TMDs tuned to other modes.

7.3.2. Time-history analysis It was observed in the previous section that, the presence of higher-mode TMDs causes some deterioration in the first-mode response although this deterioration is very small. At the same time,

Table4 Struc ture-MTMD system t ime-h is tory analyses results

TMD descr ipt ion Floor No. of TMD(s) Tuned to Parameters no.

( locat ion)

El Centro Mexico Peak RMS (cm) Peak (cm) RMS (cm) (cm)

1 1st mode m = 0.5 kg 3rd (3rd f loor) k = 9.11 N/m 2nd

c = 0.43 N-s/m 1st TMD-1

1st mode: as above 3rd (3rd f loor) 2nd 2nd mode: m = 0.075 kg 1st (1st f loor) k = 11.53 N/m TMD-1

c = 0.07 N-s/m TMD-2

1st mode: as above 3rd (3rd f loor) 2nd 2nd mode: as above 1st (1st f loor) TMD-1 3rd mode: m = 0.025 kg TMD-2 (2rid f loor ) k = 8.09 N/m TMD-3

c = 0.02 N-s/m

w /o TMD 3rd 2nd 1st

13.29 4.58 22.33 5.60 9.87 3.61 -17.76 4.52 5.26 2.12 -10.20 2.59

-43.00 16.06 75.51 19.02

13.62 4.57 -22.22 5.62 9.66 3.61 17.89 4.53 5.14 2.06 -10.20 2.59

-43.08 16.09 75.70 19.06 18.24 6.45 12.82 3.25

13.63 4.57 22.26 5.63 9.69 3.61 17.91 4.54 5.15 2.07 10.21 2.60

-43.13 16.11 75.84 19.09 18.21 6.45 -12.84 3.25 11.41 4.28 19.23 4.83

15.65 6.45 23.47 5.66 11.02 5.13 19.07 4.61 6.18 2.94 10.92 2.64


14.5

14.0

13.5

~ 13.0

12.5

12.0

1 1 . 5

11.5

11.0

"~ 10.5

10.0

9.5

9.0

6.5

6.0

5.5

5.0 0.875

Third floor response I I ' I I 1 I I

TMD (s) present for

,I

, I , I i I i I i I ~ I i I i

Second f loor r e s p o n s e

, I , I ~ I ~ I , I ~ I , I ,

Firs t f l o o r r e s p o n s e

I I I i I I I

0.900 0.925 0.950 0.975 1.000 1.025 1.050 1.075 F requency rat io g l

1.75

1.25

0.75

0.25

1.0

~ 0 . 5

0.0

1.5

0.5 2.65

~ 1.0

203

Thi rd f loor r e s p o n s e

. / I . ;

o,'

S e c o n d f l o o r r e s p o n s e

I . . .

. . . . . . - - ° " "

F i rs t f loor response

' ." I ' i ".

o, . . , , " ' " I i I i

2.95 2.75 2.85 Frequency ratio gl

0.2 t~

0.1

0.4

Thi rd f loor r e s p o n s e

0 . 3 I / ' 1 ' I I " . , ~

/ / " '~_. •

/ " ~ . . . "7 - - - TM-D~;) ; r ~ t for ....... l l1 mode only - - - l i t l a l d 21XI modes

. . '~" - - 3¢d mode only . .~- '~" -- -- l i t lmd3rdmodes - - - NI throe m o d e s

0 . 0 ~ I , I ,

Second f loor r e s p o n s e 0 .6 " , . ' 1 I ' I " ; \ "

"" .N N

0.2 , I , I i I ,"-1%

First llOor r e s p o n s e 0.45 I : i • I ' I . , ~

/ / - - .

0.35

j • . , " / /

0.25

0 . 1 5 ,"=~" I i t i I i 3.95 4.00 4.05 4.10 4.15

F r e q u e n c y ra t io 91

Figure 14a Effect of var ious TMD(s) on first mode; b effect of various TMD(s) on second mode; c effect of various TMD(s) on third mode

to control higher modes, a separate TMD must be provided. Therefore, it is reasonable to continue with the design of MTMD and investigate its performance under some excitation histories. Time-history analyses are done under E! Centro and Mexico earthquakes using one, two and three TMDs, respectively. The parameters of the TMDs and the results of these analyses are summarized in Table 4.

Noting the various response values in Table 3 and comparing them to the corresponding uncontrolled responses, it is clear that a MTMD does not result in appreciable response reduction in addition to what is already possible by a first-mode TMD.

It is concluded that, the effect of controlling the higher modes gets nullified by a marginal increase in the first

2 0 4 Tuned mass dampers: R. Rana and T. T. Soong

mode response. Therefore, the overall response is changed by only a small amount. However, it should be noted that the results obtained correspond to only the structure considered.

8. C o n c l u s i o n s

From the study performed in this work, the fol lowing main conclusions can be drawn:

( l ) The effect of detuning in TMD parameters becomes less detrimental with increasingly structural damping and/or mass ratio.

(2) From the time-history analyses on SDOF structure- TMD system, it was seen that for large damping of structure, TMD was not found to give much response reduction. In the time-history analysis using El Centro and Mexico earthquakes, the TMD designed for harmonic excitation was observed to be performing close to the best possible. Considering the significantly different natures of these two earthquakes, this suggests the practicality of use of harmonic excitation in the TMD design.

(3) Optimal parameter values differ based on whether the TMD is designed for base excitation or main mass excitation. This difference between the two grows with increasing structural damping and/or mass ratio.

(4) In the presence of inherent structural damping, optimum TMD parameters must be obtained by a numerical search. The design of a TMD for a certain mode of a MDOF structure can be easily done fol lowing a simple procedure, as illustrated by a design example.

(5) From the harmonic and time-history analyses performed on a three DOF structure, equipped in turn with a TMD and MTMD, it was observed that a modal con- tamination problem was present. Because of second- and third-mode TMD deteriorating the first mode response, the MTMD was not found to be effective in response control.

R e f e r e n c e s

1 Soong, T. T., Reinhorn, A. M., Aizawa, S. and Higashino, M. 'Recent structural applications of active control technology', J. Struct. Con- trol 1994, 1 (2), 5-21

2 Ormondroyd, J. and Den Hartog, J. P. 'The theory of dynamic vibration absorber', Trans.. ASME 1928, APM-50-7, 9-22

3 Bishop, R. E. D. and Welbourn, D. B. 'The problem of the dynamic vibration absorber', Engineering, London, 1952, p. 174 and 769

4 Snowdown, J. C. 'Steady-state behavior of tile dynamic absorber', J. Acoust. Soc. Am. 1960, 31 (8), 1096 1103

5 Falcon, K. C., Stone, B. J., Simcock, W. D. and Andrew, C. 'Optimiz- ation of vibration absorbers: a graphical method for use on idealized systems with restricted damping', J. Mech. Engng Sci. 1967, 9, 374 381

6 Ioi, T. and lkeda, K. 'On the dynamic vibration damped absorber of the vibration system', Bull. Jpnese Soc. Mech. Engng 1978, 21 ( 151 ), 64-71

7 Warburton, G. B. and Ayorinde, E. O. 'Optimum absorber parameters for simple systems', Earthq. Engng Struct. Dvn 1980, 8, 197 217

8 Thompson, A. G. 'Optimum damping and tuning of a dynamic vibration absorber applied to a force excited and damped primary system', J. Sound Vib. 1981, 77, 403-415

9 Warburton, G. B. 'Optimal absorber parameters for various combinations of response and excitation parameters', Earthq. Engng Struct. Dyn. 1982, 10, 381-401

10 Vickery, B. J., lsyumov, N. and Davenport. A. G., "The role of damping, mass and acceleration', .1. Wind Engng lnd. Aerodynam. 1983, I I , 285 294

1 I Tsai, H. C. and Lin, G. C. 'Optimum tuned mass dampers for minim- izing steady-state response of support excited and damped systems', Earthq. Engng Struct. Dynam. 1993, 22, 957-973

12 Den Hartog, J. P. Mechanical vibrations, 4th edn, McGraw-Hill, New York, 1956

13 Randall, S. E., Halsted, D. M. and Taylor, D. L. 'Optimum vibration absorbers for linear damped systems', J. Mech. Des. ASME 1981, 103, 908-913

14 Rana, R. 'A parametric study of tuned mass dampers and their gen- eralizations', M.S. thesis, State University of New York, Buffalo, 1995

15 lgusa, T. and Xu, K. 'Vibration reduction characteristics of distributed tuned mass dampers'. Proc. 4th hit. Cor!f Struct. Dynam.: Recent Advances, 1991

16 Xu, K. and Igusa, T. "Dynamics characteristics of multiple substruc- tures with closely spaced frequencies', Earthq. Engng Struct. Dynam. 1992, 21, 1059-1070

17 Yamaguchi, H. and Harnpornchai, N. "Fundamental characteristics of multiple tuned mass dampers ['or suppressing harmonically forced oscillations', Earthq. Engng Struct. Dynam. 1993, 22, 51 62

18 Kareem, A. and Kline, S. 'Performance of multiple mass dampers under random loading', J. Struct. Engng 1995; 121 (2), 348-361

parametric study and simplified design of tuned mass dampers

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