WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

y
2.5 Nonlinearities 13
Response in B (log-scale)
200
150
100
50
-50
0
a) b)
0
Mx = My = 3
Mx = My = 7
Mx = My = 15
Mx = My = 21
2 4 6 8 10
Frequency (kHz)
Figure2.6 a) A6×6mass-springunitcell thatmodelsaheavy stiff resonator(center 2×2
masses and springs) in soft suspension (surrounding springs) connected to a surrounding
matrix material, b) The acceleration response in point B (cf. Fig. 2.2) with different
numbers of unit cells included in the structure. From paper [1].
Figure 2.7 Linear mass-spring chain with local oscillators attached to the masses via
linear or nonlinear spring (as well as linear viscous dampers). From paper [3].
structurewithidentical oscillatorsattachedtothemassesinthemainstructure. The
structure behaves in a similar manner as the two-dimensional structure described in
Section 2.4. Consequently, the corresponding periodic material has a bandgap near
the eigenfrequency of the local oscillators.
If the local oscillators are attached to the main masses via nonlinear springs, the
bandgap behavior is significantly modified. Cubic hardening springs make the attached
system stiffer for high vibration amplitudes and therefore move the bandgap
up in the frequency range. As a result, the bandgap becomes a local property as its
location in the frequency spectrum depends on the local vibration amplitude of the
attached oscillator. The resulting complex behavior is illustrated in the transmission
of waves through a finite chain 2 . Fig. 2.8 shows the transmission of a harmonic
wave through a chain with 2000 attached oscillators (each having a normalized linear
eigenfrequency of 1). For linear oscillators (Fig. 2.8a) the bandgap behavior is
noticed with a low wave transmission in a well defined frequency range. With cubic
hardening nonlinear oscillators the low transmission region is moved up in the fre-
2 The chain is finite, but absorbing boundaries conditions are added to both ends in order to
simulate wave transmission.