Evolutionary Optimization of Passive Compensators to Improve Earthquake Resistance - Design Optimization of Active and Passive Structural Control Systems - page 263

some selected mass-points is not representative
of the system's energy intake. Instead we take the
elastic energy stored in the system as indication
of the potential damage:

the stiffness. After some searching of the space
of possible values for the m i , c i and k i , we find
that the total energy stored in the system may be
reduced to about 55% as indicated in Figure 4c for
a 16-mass system with additional 16 compensators.
The plot shows the fitness function, the elastic
energy of the 3 best parents and the worst parent
vs. the generation of the optimization process.
We used 20 parents, 60 kids and a large muta-
tion radius for 100 generations. In Figure 4c we
observe an additional process. Having found an
energy level that is close to the minimum of the
search, we want to minimise the summed mass of
the compensators. Figure 4c indicates that there
might be some designs with the same compen-
sation efficiency but essentially smaller masses.
We have found compensating systems which are
nearly as efficient as the best found up to now but
have a significantly smaller mass - about 20% of
the mass found before (cf. Figure 4c). So we are
able to propose efficient and relatively lightweight
designs for compensators of the 16 masses prob-
lem. There should be no doubt that this approach
may be transferred to other dynamic models with
an arbitrary number of masses.
The two step optimisation mentioned above
could be easily reduced to a single step process
by introducing a combined fitness function by the
weighted sum of energy intake (step 1) and mass
(step 2). The procedure following Equation (33) is
outlined in section 3.5.2. Here we demonstrate the
stepped approach to give an idea how to proceed
if necessary.
To use the method of modal contributions
mentioned above would be a rather tricky task in
the case of many compensators applied at many
mass points. The interaction of original masses
and dampers has to be taken into account, so a
multidimensional optimisation problem has to
be solved simultaneously. Here evolutionary ap-
proaches are superior to conventional gradient
methods.

= 1

T

W el ( ω )

2 u

( ω ) Ku ( ω ) (31)

k k k

(cf. Equation (7) and Equation (8)) either at a
given frequency ω or summed over the range of
frequencies we are interested in:

k max

∑ ω
0

W = W ( ) (32)

el tot , el k

k =

In Equation (31) u stands for the complex
displacement derived from Equation (17). The
frequency range [0, ω max ] should again cover all
relevant Eigen frequencies of the problem. Figure
4b compares the amplitudes for dynamic systems
(2 − 32 mass oscillators). We see the peaks at the
Eigen frequencies and the decreasing amplitudes
vs. the increasing number of masses. If we sum the
energy (Equation (32)) over the frequency range,
we learn that the total elastic energy stored in the
chain is nearly constant for the different oscillat-
ing systems. This is not a very surprising fact as
the external energy is delivered from the same
source, the oscillating force or the corresponding
ground motion.
To reduce the dynamic response, we may add
some compensators here as indicated in Figure
4a. We assume that each mass point has its own
compensator. Again we use the summed elastic
energy stored in the system (Equation (32)) without
the compensators as indication of the internally
acting destructive load.
The optimization problem to minimize the
energy in the chain is multidimensional now.
We use evolutionary approaches to find efficient
designs for the compensators' mass and stiffness.
We assume the damping to be a small fraction of