WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

46 Chapter 5 Advanced optimization procedures
log(error)
0
-10
-20
-30
-40
N = 5
N = 3
N = 7
-50
0 10 20 30 40 50
Significant digits
Figure 5.2 Error of the bi−coefficients in the computation of the Padé approximant
versus the numerical precision (significant digits) used in the computations. From paper
[18].
5.2 Nonlinear transients
Section 2.5 analyzes the performance of a mass-spring structure with attached oscillators.
The results indicate that it could be beneficial to optimize the parameters
of each oscillator, including the nonlinear stiffness parameter, in order to minimize
the transmission of waves through the structure.
The steady-state formulation used in Chapter 3 and Chapter 4 is not directly
applicable to this system due to the presence of nonlinearities that add higher-order
harmonics to the response. Instead, a time-domain optimization formulation can
be applied which is based on transient simulation of the wave propagation through
the system. The method is adapted to the present nonlinear system which turns
out to pose no significant complications with regards to simulation and sensitivity
analysis.
Fig. 5.3 shows the considered system with a one-dimensional mass-spring structure
with a number of oscillators attached to the masses by linear viscous dampers
and nonlinear springs. The design variables in the system are the mass ratio for each
oscillator (relative to the mass it is attached to), the natural frequency, the viscous
damping ratio and the nonlinear stiffness parameter. Thus, we have four design
variables per attached oscillator and all oscillators may possess different parameters.
A propagating steady-state wave pulse is simulated by adding a sinusoidal
time-dependent force at the first mass and absorbing boundaries in the form of
properly tuned viscous dampers at both ends.
Fig. 5.4 shows an example of the optimized distribution of the natural frequency,