Optimal Design and Practical Considerations of Tuned Mass Dampers for Structural Control - Design Optimization of Active and Passive Structural Control Systems - page 127

Den Hartog's Design Formulas
for Undamped Structures

All of these applications show that TMDs can
reduce structural vibrations effectively. However,
a TMD could face some drawbacks in seismic
applications: large stroke and detuning problem,
due to large earthquake forces. To solve the detun-
ing problem, a new device, called multiple tuned
mass dampers (MTMD), with less sensitivity to
frequency change was first proposed by Xu and
Igusa in 1992. Followed by numerous studies
(i.e., Yamaguchi & Harnpornchai, 1993; Abe
& Fujino, 1994, 1995; Kareem & Kline, 1995;
Jangid, 1995; Li, 2000, 2002, 2003, Lin et al.,
1999; Lin et al., 2005; Wang & Lin, 2005; Hoang
& Warnitchai, 2005; Zuo & Nayfeh, 2005; Li &
Ni, 2007), various design theories and control
efficiency of an MTMD were well established.
Still, the large stroke problem was not taken into
account. In 2010, a research paper by the authors
using a brand-new MTMD design theory with the
consideration of stroke limitation was published.
A shaking table test was also conducted to verify
the control effectiveness of a fabricated MTMD
device for a large-scale building. The objective
of this chapter is to provide a quick review on the
development of TMD and MTMD and to introduce
the new findings by Lin et al . (2010).

In Den Hartog's study, an undamped SDOF dy-
namic system subjected to sinusoidal loading was
considered. For a given mass ratio of TMD to the
primary structure, μ, he suggested that the optimal
frequency ratio of the TMD to the primary, ( )

r f opt

and the optimal damping ratio of TMD, ( ξ s opt ,

can be calculated by the following equations

)

1 3

8 1

µ

( ) r f opt and ( )

= ξ =

1 + µ

s opt

( + µ )

(1)

From Equation (1), the physical parameters
(e.g., mass, damping coefficient, and stiffness
coefficient) of a TMD can be obtained if the mass
and natural frequency of the primary structure are
given. Although the design problem is signifi-
cantly simplified, it appears that the damping
ratio of the primary structure is unrelated to the
design of a TMD. As a result, the applicability of
the equation to damped structures is not clear. The
equation also implies that the larger the TMD
mass, the smaller the value of ( ) r f opt . It is a rea-

sonable situation because a larger TMD mass
means larger system mass which will have a
smaller resonant frequency to which a TMD is
tuned.

BACKGROUND

A TMD is a single-degree-of-freedom (SDOF)
dynamic system. The design of TMD is to deter-
mine its mass, damping coefficient, and stiffness
coefficient based on the characteristics of primary
structure to which it is installed and/or the external
excitation. The time-domain equations of mo-
tion of a linear structure-TMD system involves
second-order differential. In the early stage, the
application of TMD mainly focused on the vibra-
tion problem of mechanical systems. Although
the concept of TMD dated back to 1909 (Frahm,
1911), Den Hartog (1956) could be the first one
who provided a detail description and design
formulas for TMD.

Optimally Designed Formulas
for Damped Structures

After Den Hartog's work, numerous studies have
investigated the TMD design for damped struc-
tural systems. Lin et al . (1994) developed theories
for optimal design of TMD's parameters and in-
vestigated its vibration control effectiveness for
building structures. In their study, a linear multi-
story planar building model with viscous damping
was applied. The earthquake excitation was con-