Optimal Design and Practical Considerations of Tuned Mass Dampers for Structural Control - Design Optimization of Active and Passive Structural Control Systems - page 128

sidered as a random process of white noise or
filtered white noise. Because of structural damp-
ing, analytical solutions for the optimal frequen-
cy ratio, ( )

ture by tuning its vibration frequency to that of
the structure. Figure 1(a) gives plots of typical
transfer functions of a building with fundamental
frequency ω p = 1 . Hz and damping ratio ξ p =2
%, without and with TMD µ =

ξ s opt
of TMD with a given mass ratio, µ , are inacces-
sible. Instead, by conventional curve fitting, the
optimal design parameters of TMD were obtained
as

r f opt , and optimal damping ratio, (

)

( )

2 8. % . Here,
two different TMD design parameter sets were
utilized: (a) optimal TMD: r f = 0.953, ξ s = 8.3%
obtained from Equation (2); (b) Den Hartog's
TMD: r f = 0.972, ξ s = 10.2% obtained from Equa-
tion (1). It is seen that the amplitude of transfer
function is significantly reduced near the resonant
frequency (1.5Hz) when the structure is equipped
with TMD. The frequency bandwidth where TMD
is effective is called the operating range. Notice
that insignificant amplifications are observed
outside the operating range. Figure 1(a) also shows
that the optimal TMD can suppress the transfer
function amplitude more than the Den Hartog's
TMD near the resonant frequency of the structure.
This simple case shows a TMD, merely requir-
ing a small room compared to the primary struc-
tures, is able to reduce structural resonant response
by means of increasing additional damping. This
is economically advantageous for gigantic struc-
tures with enough stiffness but limited inherent
damping, such as a high-rise building or a long-

a b

( ) r = ( ) ;

f opt

1 + µ

ξ

3 2 . ξ

p

a = 1 0 . − , b = 1 35 . e and (2)

p

4

0 . 48

( ξ ) = 0 46 . µ

s opt

where ξ p is the structural damping ratio. Compar-
ing with Equation (1), it is seen that the influence
of structural damping on ( ) r f opt is included in

Equation (2).

Philosophy and Advantages of TMD

The vibration control effectiveness of TMD is
achieved by changing the dynamic characteristics,
specifically the damping properties, of the struc-

Figure 1. Transfer functions of (a) structure without and with TMD based on two different design equa-
tions (“optimal TMD”: based on Eq. (2); “Den Hartog's TMD”: based on Eq. (1)); (b) TMD's stroke