Nonlinear Structural Control Using Magnetorheological Damper - Design Optimization of Active and Passive Structural Control Systems - page 310

the vector of base accelerations (at isolation
level) relative to the ground, and  U g is the vector
of ground accelerations. The equation of motion
for the base is given as follows

that the difference between analytical results and
experimental data is minimized.
For optimization using PSO the building is
idealized to be a three storey shear building model.
PSO is used to minimize the cost function as given
in Equation (8).

( ) ( )

   

R M Ü R U

T

U m U U

+ +

+

+

a g b b g b



+ cU + kU + = f 0

ψ = W J W J W J + + (8)

(7)

b b b b c

ω ω ζ ζ φ φ

where f c is the force across the MR damper. Sub-
script (b) represents parameters from the base of
the building.
Impulse hammer tests (IHT) are conducted to
determine the building parameters without isolator
attached and sinusoidal base excitation is used to
determine the damping characteristics of the slid-
ing bearings. The stiffness at the base provided
by the linear springs is determined experimentally
using a servo hydraulic closed loop universal test
rig. As noted earlier a simple Bouc-Wen model is
used to describe the MR damper hysteretic char-
acteristic. Details of experiments on MR damper
are discussed in Ali and Ramaswamy (2009b).
For IHT, impulsive force is given at the top
floor and the acceleration responses are measured
at all the floors along the direction of impulse.
The frequency response functions (FRFs) char-
acteristic of the building is obtained in IOtech
DaqBoard-2000 device and with DaisyLab
Software (Ver. 7.02). From the FRFs, natural
frequencies, damping coefficients and the mass
normalized mode shapes of the fixed base build-
ing are determined using 'MEscope' software.
Finally, the building parameters for analytical
simulations (mass, damping and stiffness) are
updated to match experimentally obtained results
using particle swarm optimization (PSO).
Model updating aims at introducing correction
to an initial model so that it predicts accurate and
reliable dynamic behavior of the structure. The
process of updating is performed by adjusting
parameters of the initial model in such a way

where J ω , J ς and J φ are the cost function compo-
nents related to the natural frequencies, damping
coefficients and mode shapes, respectively, and
W ω , W ς and W φ are the relative weight factors.
The cost functions J ω and J ς are given as:

2 2

m a m a

ω − ω ζ − ζ

3 3

∑ ∑

J =

k k

J =

k k

ω m ζ m

ω ζ

k = k = 1 1

k k

(9)

whereas the cost function related to mode shapes
(J φ ) of the building is given by:

{ }

3

( )

T

( )

∑ 1

J MSF

m a

MSF

m a

= φ − φ φ − φ

φ k k k k

k =

(10)

ω k , ς k , and φ k are the k th natural frequency,
coefficient of damping and mode shape vector,
respectively. The superscripts (m) and (a) represent
measured and analytical data, respectively and
(T) is the transpose operator. MSF is the modal
scale factor that is used to keep the experimental
and analytical mode shapes at equal scale (Ewins,
2000). Equal weight factors for J ω , J ς and J φ , i.e.,
W ω = 1/3, W ς = 1/3 and W φ = 1/3 are assumed.
The obtained updated mass ( M a ), damping
( C a ) and stiffness ( K a ) matrices are as given in
Equation (11).